{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Model mieszany + estymacja parametrów wariancji + dodanie markerów genetycznych"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Rozważmy model: "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\mathbf{y} = \\mathbf{X\\beta+Za+\\epsilon}$, gdzie"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\mathbf{y}$ - wektor z wartościami fenotypowmi ($n\\times 1$ w naszym przypadku n = ilość zwierząt) "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\mathbf{\\beta}$ - wektor efektów stałych ($p\\times 1$)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\mathbf{a}$ - wektor efektów losowych ($q\\times 1$)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\mathbf{\\epsilon}$ - wektor błędów ($n\\times 1$)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\mathbf{X}$ - macierz wystąpien dla efektów stałych ($n\\times p$)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\mathbf{Z}$ - macierz wystąpien dla efektów losowych ($n\\times q$)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Zakładamy, że macierze $\\mathbf{X}$ i $\\mathbf{Z}$ są niezależne. Dodatkowo zakładamy, że:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$E(\\mathbf{y}) = \\mathbf{X\\beta}$, $E(\\mathbf{a})=E(\\mathbf{\\epsilon})=0$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$V=Var(\\mathbf{y})=Var(\\mathbf{Za})+ Var(\\mathbf{\\epsilon})=\\mathbf{Z}Var(\\mathbf{a})\\mathbf{Z}^{T}+\\mathbf{I}Var(\\mathbf{\\epsilon})\\mathbf{I}^{T}=\\mathbf{ZGZ}^{T}+\\mathbf{R}$, gdzie"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\mathbf{G}=\\mathbf{A}\\cdot\\sigma^{2}_{a}$, $\\mathbf{R}=\\mathbf{I}\\cdot\\sigma^{2}_{\\epsilon}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$ \\left[ \\begin{array}{cc}\n",
    "         \\mathbf{X}^{T}\\mathbf{X} & \\mathbf{X}^{T}\\mathbf{Z} \\\\\n",
    "         \\mathbf{Z}^{T}\\mathbf{X} & \\mathbf{Z}^{T}\\mathbf{Z}+\\mathbf{A}^{-1}\\alpha \\end{array}\\right]\\cdot\n",
    "  \\left[ \\begin{array}{c}\n",
    "         \\widehat{\\mathbf{\\beta}} \\\\\n",
    "         \\widehat{\\mathbf{a}} \\end{array}\\right]=\n",
    "  \\left[ \\begin{array}{c}\n",
    "         \\mathbf{X}^{T}\\mathbf{y} \\\\\n",
    "         \\mathbf{Z}^{T}\\mathbf{y} \\end{array}\\right]$, gdzie"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\alpha=\\frac{\\sigma^{2}_{\\epsilon}}{\\sigma^{2}_{a}}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$ \\left[ \\begin{array}{c}\n",
    "         \\widehat{\\mathbf{\\beta}} \\\\\n",
    "         \\widehat{\\mathbf{a}} \\end{array}\\right]=\n",
    "  \\left[ \\begin{array}{cc}\n",
    "         \\mathbf{X}^{T}\\mathbf{X} & \\mathbf{X}^{T}\\mathbf{Z} \\\\\n",
    "         \\mathbf{Z}^{T}\\mathbf{X} & \\mathbf{Z}^{T}\\mathbf{Z}+\\mathbf{A}^{-1}\\alpha \\end{array}\\right]^{-1}\\cdot\n",
    "  \\left[ \\begin{array}{c}\n",
    "         \\mathbf{X}^{T}\\mathbf{y} \\\\\n",
    "         \\mathbf{Z}^{T}\\mathbf{y} \\end{array}\\right]$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$ C = \\left[ \\begin{array}{cc}\n",
    "         \\mathbf{X}^{T}\\mathbf{X} & \\mathbf{X}^{T}\\mathbf{Z} \\\\\n",
    "         \\mathbf{Z}^{T}\\mathbf{X} & \\mathbf{Z}^{T}\\mathbf{Z}+\\mathbf{A}^{-1}\\alpha \\end{array}\\right] = \n",
    "         \\left[ \\begin{array}{cc}\n",
    "         \\mathbf{C}_{11} & \\mathbf{C}_{12} \\\\ \\mathbf{C}_{21} & \\mathbf{C}_{22} \\end{array}\\right]$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$ C^{-1} = \\left[ \\begin{array}{cc}\n",
    "         \\mathbf{C}^{11} & \\mathbf{C}^{12} \\\\ \\mathbf{C}^{21} & \\mathbf{C}^{22} \\end{array}\\right]$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Dokładność oszacowania wynosi: $r^2 = diag(1-\\mathbf{C}^{22}\\cdot\\alpha$)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "1. Liczymy macierz A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<thead><tr><th scope=col>id</th><th scope=col>sire</th><th scope=col>dam</th></tr></thead>\n",
       "<tbody>\n",
       "\t<tr><td>1</td><td>0</td><td>0</td></tr>\n",
       "\t<tr><td>2</td><td>0</td><td>0</td></tr>\n",
       "\t<tr><td>3</td><td>0</td><td>0</td></tr>\n",
       "\t<tr><td>4</td><td>1</td><td>0</td></tr>\n",
       "\t<tr><td>5</td><td>3</td><td>2</td></tr>\n",
       "\t<tr><td>6</td><td>1</td><td>2</td></tr>\n",
       "\t<tr><td>7</td><td>4</td><td>5</td></tr>\n",
       "\t<tr><td>8</td><td>3</td><td>6</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{lll}\n",
       " id & sire & dam\\\\\n",
       "\\hline\n",
       "\t 1 & 0 & 0\\\\\n",
       "\t 2 & 0 & 0\\\\\n",
       "\t 3 & 0 & 0\\\\\n",
       "\t 4 & 1 & 0\\\\\n",
       "\t 5 & 3 & 2\\\\\n",
       "\t 6 & 1 & 2\\\\\n",
       "\t 7 & 4 & 5\\\\\n",
       "\t 8 & 3 & 6\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "| id | sire | dam |\n",
       "|---|---|---|\n",
       "| 1 | 0 | 0 |\n",
       "| 2 | 0 | 0 |\n",
       "| 3 | 0 | 0 |\n",
       "| 4 | 1 | 0 |\n",
       "| 5 | 3 | 2 |\n",
       "| 6 | 1 | 2 |\n",
       "| 7 | 4 | 5 |\n",
       "| 8 | 3 | 6 |\n",
       "\n"
      ],
      "text/plain": [
       "     id sire dam\n",
       "[1,] 1  0    0  \n",
       "[2,] 2  0    0  \n",
       "[3,] 3  0    0  \n",
       "[4,] 4  1    0  \n",
       "[5,] 5  3    2  \n",
       "[6,] 6  1    2  \n",
       "[7,] 7  4    5  \n",
       "[8,] 8  3    6  "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "library(AGHmatrix)\n",
    "\n",
    "id = 1:8\n",
    "sire = c(0, 0, 0, 1, 3, 1, 4, 3)\n",
    "dam = c(0, 0, 0, 0, 2, 2, 5, 6)\n",
    "\n",
    "(ped = cbind(id, sire, dam))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Verifying conflicting data... \n",
      "Organizing data... \n",
      "Your data was chronologically organized with success. \n",
      "Constructing matrix A using ploidy = 2 \n",
      "Completed! Time = 0  minutes \n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<thead><tr><th scope=col>1</th><th scope=col>2</th><th scope=col>3</th><th scope=col>4</th><th scope=col>5</th><th scope=col>6</th><th scope=col>7</th><th scope=col>8</th></tr></thead>\n",
       "<tbody>\n",
       "\t<tr><td>1.00 </td><td>0.00 </td><td>0.00 </td><td>0.500</td><td>0.000</td><td>0.50 </td><td>0.25 </td><td>0.250</td></tr>\n",
       "\t<tr><td>0.00 </td><td>1.00 </td><td>0.00 </td><td>0.000</td><td>0.500</td><td>0.50 </td><td>0.25 </td><td>0.250</td></tr>\n",
       "\t<tr><td>0.00 </td><td>0.00 </td><td>1.00 </td><td>0.000</td><td>0.500</td><td>0.00 </td><td>0.25 </td><td>0.500</td></tr>\n",
       "\t<tr><td>0.50 </td><td>0.00 </td><td>0.00 </td><td>1.000</td><td>0.000</td><td>0.25 </td><td>0.50 </td><td>0.125</td></tr>\n",
       "\t<tr><td>0.00 </td><td>0.50 </td><td>0.50 </td><td>0.000</td><td>1.000</td><td>0.25 </td><td>0.50 </td><td>0.375</td></tr>\n",
       "\t<tr><td>0.50 </td><td>0.50 </td><td>0.00 </td><td>0.250</td><td>0.250</td><td>1.00 </td><td>0.25 </td><td>0.500</td></tr>\n",
       "\t<tr><td>0.25 </td><td>0.25 </td><td>0.25 </td><td>0.500</td><td>0.500</td><td>0.25 </td><td>1.00 </td><td>0.250</td></tr>\n",
       "\t<tr><td>0.25 </td><td>0.25 </td><td>0.50 </td><td>0.125</td><td>0.375</td><td>0.50 </td><td>0.25 </td><td>1.000</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{r|llllllll}\n",
       " 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8\\\\\n",
       "\\hline\n",
       "\t 1.00  & 0.00  & 0.00  & 0.500 & 0.000 & 0.50  & 0.25  & 0.250\\\\\n",
       "\t 0.00  & 1.00  & 0.00  & 0.000 & 0.500 & 0.50  & 0.25  & 0.250\\\\\n",
       "\t 0.00  & 0.00  & 1.00  & 0.000 & 0.500 & 0.00  & 0.25  & 0.500\\\\\n",
       "\t 0.50  & 0.00  & 0.00  & 1.000 & 0.000 & 0.25  & 0.50  & 0.125\\\\\n",
       "\t 0.00  & 0.50  & 0.50  & 0.000 & 1.000 & 0.25  & 0.50  & 0.375\\\\\n",
       "\t 0.50  & 0.50  & 0.00  & 0.250 & 0.250 & 1.00  & 0.25  & 0.500\\\\\n",
       "\t 0.25  & 0.25  & 0.25  & 0.500 & 0.500 & 0.25  & 1.00  & 0.250\\\\\n",
       "\t 0.25  & 0.25  & 0.50  & 0.125 & 0.375 & 0.50  & 0.25  & 1.000\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |\n",
       "|---|---|---|---|---|---|---|---|\n",
       "| 1.00  | 0.00  | 0.00  | 0.500 | 0.000 | 0.50  | 0.25  | 0.250 |\n",
       "| 0.00  | 1.00  | 0.00  | 0.000 | 0.500 | 0.50  | 0.25  | 0.250 |\n",
       "| 0.00  | 0.00  | 1.00  | 0.000 | 0.500 | 0.00  | 0.25  | 0.500 |\n",
       "| 0.50  | 0.00  | 0.00  | 1.000 | 0.000 | 0.25  | 0.50  | 0.125 |\n",
       "| 0.00  | 0.50  | 0.50  | 0.000 | 1.000 | 0.25  | 0.50  | 0.375 |\n",
       "| 0.50  | 0.50  | 0.00  | 0.250 | 0.250 | 1.00  | 0.25  | 0.500 |\n",
       "| 0.25  | 0.25  | 0.25  | 0.500 | 0.500 | 0.25  | 1.00  | 0.250 |\n",
       "| 0.25  | 0.25  | 0.50  | 0.125 | 0.375 | 0.50  | 0.25  | 1.000 |\n",
       "\n"
      ],
      "text/plain": [
       "  1    2    3    4     5     6    7    8    \n",
       "1 1.00 0.00 0.00 0.500 0.000 0.50 0.25 0.250\n",
       "2 0.00 1.00 0.00 0.000 0.500 0.50 0.25 0.250\n",
       "3 0.00 0.00 1.00 0.000 0.500 0.00 0.25 0.500\n",
       "4 0.50 0.00 0.00 1.000 0.000 0.25 0.50 0.125\n",
       "5 0.00 0.50 0.50 0.000 1.000 0.25 0.50 0.375\n",
       "6 0.50 0.50 0.00 0.250 0.250 1.00 0.25 0.500\n",
       "7 0.25 0.25 0.25 0.500 0.500 0.25 1.00 0.250\n",
       "8 0.25 0.25 0.50 0.125 0.375 0.50 0.25 1.000"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "(A = as.matrix(Amatrix(ped)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "2. Definiujemy kolejne źródła informacji"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<tbody>\n",
       "\t<tr><td>4.5</td><td>2.9</td><td>3.9</td><td>3.5</td><td>5  </td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{lllll}\n",
       "\t 4.5 & 2.9 & 3.9 & 3.5 & 5  \\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "| 4.5 | 2.9 | 3.9 | 3.5 | 5   |\n",
       "\n"
      ],
      "text/plain": [
       "     [,1] [,2] [,3] [,4] [,5]\n",
       "[1,] 4.5  2.9  3.9  3.5  5   "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<tbody>\n",
       "\t<tr><td>1</td><td>0</td></tr>\n",
       "\t<tr><td>0</td><td>1</td></tr>\n",
       "\t<tr><td>0</td><td>1</td></tr>\n",
       "\t<tr><td>1</td><td>0</td></tr>\n",
       "\t<tr><td>1</td><td>0</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{ll}\n",
       "\t 1 & 0\\\\\n",
       "\t 0 & 1\\\\\n",
       "\t 0 & 1\\\\\n",
       "\t 1 & 0\\\\\n",
       "\t 1 & 0\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "| 1 | 0 |\n",
       "| 0 | 1 |\n",
       "| 0 | 1 |\n",
       "| 1 | 0 |\n",
       "| 1 | 0 |\n",
       "\n"
      ],
      "text/plain": [
       "     [,1] [,2]\n",
       "[1,] 1    0   \n",
       "[2,] 0    1   \n",
       "[3,] 0    1   \n",
       "[4,] 1    0   \n",
       "[5,] 1    0   "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<thead><tr><th scope=col>factor(sex)0</th><th scope=col>factor(sex)1</th></tr></thead>\n",
       "<tbody>\n",
       "\t<tr><td>0</td><td>1</td></tr>\n",
       "\t<tr><td>1</td><td>0</td></tr>\n",
       "\t<tr><td>1</td><td>0</td></tr>\n",
       "\t<tr><td>0</td><td>1</td></tr>\n",
       "\t<tr><td>0</td><td>1</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{r|ll}\n",
       " factor(sex)0 & factor(sex)1\\\\\n",
       "\\hline\n",
       "\t 0 & 1\\\\\n",
       "\t 1 & 0\\\\\n",
       "\t 1 & 0\\\\\n",
       "\t 0 & 1\\\\\n",
       "\t 0 & 1\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "| factor(sex)0 | factor(sex)1 |\n",
       "|---|---|\n",
       "| 0 | 1 |\n",
       "| 1 | 0 |\n",
       "| 1 | 0 |\n",
       "| 0 | 1 |\n",
       "| 0 | 1 |\n",
       "\n"
      ],
      "text/plain": [
       "  factor(sex)0 factor(sex)1\n",
       "1 0            1           \n",
       "2 1            0           \n",
       "3 1            0           \n",
       "4 0            1           \n",
       "5 0            1           "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "y = as.matrix(c(4.5, 2.9, 3.9, 3.5, 5.0))\n",
    "t(y)\n",
    "\n",
    "sex = c(1, 0, 0, 1, 1)\n",
    "X = matrix(0, 5, 2)\n",
    "X[,1] = sex\n",
    "X[,2] = 1-sex\n",
    "X\n",
    "\n",
    "model.matrix(~factor(sex) - 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<tbody>\n",
       "\t<tr><td>0</td><td>0</td><td>0</td><td>1</td><td>0</td><td>0</td><td>0</td><td>0</td></tr>\n",
       "\t<tr><td>0</td><td>0</td><td>0</td><td>0</td><td>1</td><td>0</td><td>0</td><td>0</td></tr>\n",
       "\t<tr><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1</td><td>0</td><td>0</td></tr>\n",
       "\t<tr><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1</td><td>0</td></tr>\n",
       "\t<tr><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{llllllll}\n",
       "\t 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\\\\n",
       "\t 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\\\\n",
       "\t 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\\\\n",
       "\t 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\\\\n",
       "\t 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |\n",
       "| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |\n",
       "| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |\n",
       "| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |\n",
       "| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |\n",
       "\n"
      ],
      "text/plain": [
       "     [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8]\n",
       "[1,] 0    0    0    1    0    0    0    0   \n",
       "[2,] 0    0    0    0    1    0    0    0   \n",
       "[3,] 0    0    0    0    0    1    0    0   \n",
       "[4,] 0    0    0    0    0    0    1    0   \n",
       "[5,] 0    0    0    0    0    0    0    1   "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "I = diag(5)\n",
    "Z = matrix(0, 5, 8)\n",
    "Z[1:5, 4:8] = I\n",
    "Z"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "3. Rozwiązujemy układ równan mieszanych"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<dl>\n",
       "\t<dt>$C</dt>\n",
       "\t\t<dd><table>\n",
       "<tbody>\n",
       "\t<tr><td>3            </td><td>0            </td><td> 0.000000e+00</td><td> 0.000000e+00</td><td> 0.000000e+00</td><td> 1.000000e+00</td><td> 0.000000e+00</td><td> 0.000000e+00</td><td> 1.000000e+00</td><td> 1.000000e+00</td></tr>\n",
       "\t<tr><td>0            </td><td>2            </td><td> 0.000000e+00</td><td> 0.000000e+00</td><td> 0.000000e+00</td><td> 0.000000e+00</td><td> 1.000000e+00</td><td> 1.000000e+00</td><td> 0.000000e+00</td><td> 0.000000e+00</td></tr>\n",
       "\t<tr><td>0            </td><td>0            </td><td> 3.666667e+00</td><td> 1.000000e+00</td><td> 5.440093e-15</td><td>-1.333333e+00</td><td> 1.554312e-15</td><td>-2.000000e+00</td><td>-9.880985e-15</td><td>-4.996004e-15</td></tr>\n",
       "\t<tr><td>0            </td><td>0            </td><td> 1.000000e+00</td><td> 4.000000e+00</td><td> 1.000000e+00</td><td>-4.440892e-16</td><td>-2.000000e+00</td><td>-2.000000e+00</td><td>-1.110223e-15</td><td>-6.938894e-17</td></tr>\n",
       "\t<tr><td>0            </td><td>0            </td><td> 2.220446e-15</td><td> 1.000000e+00</td><td> 4.000000e+00</td><td> 5.551115e-16</td><td>-2.000000e+00</td><td> 1.000000e+00</td><td>-5.551115e-16</td><td>-2.000000e+00</td></tr>\n",
       "\t<tr><td>1            </td><td>0            </td><td>-1.333333e+00</td><td>-2.220446e-16</td><td>-3.108624e-15</td><td> 4.666667e+00</td><td> 1.000000e+00</td><td>-4.440892e-16</td><td>-2.000000e+00</td><td> 1.727785e-15</td></tr>\n",
       "\t<tr><td>0            </td><td>1            </td><td> 2.220446e-16</td><td>-2.000000e+00</td><td>-2.000000e+00</td><td> 1.000000e+00</td><td> 6.000000e+00</td><td>-1.776357e-15</td><td>-2.000000e+00</td><td>-2.636780e-15</td></tr>\n",
       "\t<tr><td>0            </td><td>1            </td><td>-2.000000e+00</td><td>-2.000000e+00</td><td> 1.000000e+00</td><td>-2.109424e-15</td><td>-3.330669e-15</td><td> 6.000000e+00</td><td> 2.664535e-15</td><td>-2.000000e+00</td></tr>\n",
       "\t<tr><td>1            </td><td>0            </td><td> 7.771561e-16</td><td>-3.330669e-15</td><td>-8.881784e-16</td><td>-2.000000e+00</td><td>-2.000000e+00</td><td> 1.332268e-15</td><td> 5.000000e+00</td><td> 1.346145e-15</td></tr>\n",
       "\t<tr><td>1            </td><td>0            </td><td>-3.497203e-15</td><td>-8.590351e-15</td><td>-2.000000e+00</td><td>-5.280498e-15</td><td> 2.303713e-15</td><td>-2.000000e+00</td><td> 6.231127e-15</td><td> 5.000000e+00</td></tr>\n",
       "</tbody>\n",
       "</table>\n",
       "</dd>\n",
       "\t<dt>$est</dt>\n",
       "\t\t<dd><table>\n",
       "<tbody>\n",
       "\t<tr><td> 4.358502330</td></tr>\n",
       "\t<tr><td> 3.404430006</td></tr>\n",
       "\t<tr><td> 0.098444576</td></tr>\n",
       "\t<tr><td>-0.018770099</td></tr>\n",
       "\t<tr><td>-0.041084203</td></tr>\n",
       "\t<tr><td>-0.008663123</td></tr>\n",
       "\t<tr><td>-0.185732099</td></tr>\n",
       "\t<tr><td> 0.176872088</td></tr>\n",
       "\t<tr><td>-0.249458555</td></tr>\n",
       "\t<tr><td> 0.182614688</td></tr>\n",
       "</tbody>\n",
       "</table>\n",
       "</dd>\n",
       "</dl>\n"
      ],
      "text/latex": [
       "\\begin{description}\n",
       "\\item[\\$C] \\begin{tabular}{llllllllll}\n",
       "\t 3             & 0             &  0.000000e+00 &  0.000000e+00 &  0.000000e+00 &  1.000000e+00 &  0.000000e+00 &  0.000000e+00 &  1.000000e+00 &  1.000000e+00\\\\\n",
       "\t 0             & 2             &  0.000000e+00 &  0.000000e+00 &  0.000000e+00 &  0.000000e+00 &  1.000000e+00 &  1.000000e+00 &  0.000000e+00 &  0.000000e+00\\\\\n",
       "\t 0             & 0             &  3.666667e+00 &  1.000000e+00 &  5.440093e-15 & -1.333333e+00 &  1.554312e-15 & -2.000000e+00 & -9.880985e-15 & -4.996004e-15\\\\\n",
       "\t 0             & 0             &  1.000000e+00 &  4.000000e+00 &  1.000000e+00 & -4.440892e-16 & -2.000000e+00 & -2.000000e+00 & -1.110223e-15 & -6.938894e-17\\\\\n",
       "\t 0             & 0             &  2.220446e-15 &  1.000000e+00 &  4.000000e+00 &  5.551115e-16 & -2.000000e+00 &  1.000000e+00 & -5.551115e-16 & -2.000000e+00\\\\\n",
       "\t 1             & 0             & -1.333333e+00 & -2.220446e-16 & -3.108624e-15 &  4.666667e+00 &  1.000000e+00 & -4.440892e-16 & -2.000000e+00 &  1.727785e-15\\\\\n",
       "\t 0             & 1             &  2.220446e-16 & -2.000000e+00 & -2.000000e+00 &  1.000000e+00 &  6.000000e+00 & -1.776357e-15 & -2.000000e+00 & -2.636780e-15\\\\\n",
       "\t 0             & 1             & -2.000000e+00 & -2.000000e+00 &  1.000000e+00 & -2.109424e-15 & -3.330669e-15 &  6.000000e+00 &  2.664535e-15 & -2.000000e+00\\\\\n",
       "\t 1             & 0             &  7.771561e-16 & -3.330669e-15 & -8.881784e-16 & -2.000000e+00 & -2.000000e+00 &  1.332268e-15 &  5.000000e+00 &  1.346145e-15\\\\\n",
       "\t 1             & 0             & -3.497203e-15 & -8.590351e-15 & -2.000000e+00 & -5.280498e-15 &  2.303713e-15 & -2.000000e+00 &  6.231127e-15 &  5.000000e+00\\\\\n",
       "\\end{tabular}\n",
       "\n",
       "\\item[\\$est] \\begin{tabular}{l}\n",
       "\t  4.358502330\\\\\n",
       "\t  3.404430006\\\\\n",
       "\t  0.098444576\\\\\n",
       "\t -0.018770099\\\\\n",
       "\t -0.041084203\\\\\n",
       "\t -0.008663123\\\\\n",
       "\t -0.185732099\\\\\n",
       "\t  0.176872088\\\\\n",
       "\t -0.249458555\\\\\n",
       "\t  0.182614688\\\\\n",
       "\\end{tabular}\n",
       "\n",
       "\\end{description}\n"
      ],
      "text/markdown": [
       "$C\n",
       ":   \n",
       "| 3             | 0             |  0.000000e+00 |  0.000000e+00 |  0.000000e+00 |  1.000000e+00 |  0.000000e+00 |  0.000000e+00 |  1.000000e+00 |  1.000000e+00 |\n",
       "| 0             | 2             |  0.000000e+00 |  0.000000e+00 |  0.000000e+00 |  0.000000e+00 |  1.000000e+00 |  1.000000e+00 |  0.000000e+00 |  0.000000e+00 |\n",
       "| 0             | 0             |  3.666667e+00 |  1.000000e+00 |  5.440093e-15 | -1.333333e+00 |  1.554312e-15 | -2.000000e+00 | -9.880985e-15 | -4.996004e-15 |\n",
       "| 0             | 0             |  1.000000e+00 |  4.000000e+00 |  1.000000e+00 | -4.440892e-16 | -2.000000e+00 | -2.000000e+00 | -1.110223e-15 | -6.938894e-17 |\n",
       "| 0             | 0             |  2.220446e-15 |  1.000000e+00 |  4.000000e+00 |  5.551115e-16 | -2.000000e+00 |  1.000000e+00 | -5.551115e-16 | -2.000000e+00 |\n",
       "| 1             | 0             | -1.333333e+00 | -2.220446e-16 | -3.108624e-15 |  4.666667e+00 |  1.000000e+00 | -4.440892e-16 | -2.000000e+00 |  1.727785e-15 |\n",
       "| 0             | 1             |  2.220446e-16 | -2.000000e+00 | -2.000000e+00 |  1.000000e+00 |  6.000000e+00 | -1.776357e-15 | -2.000000e+00 | -2.636780e-15 |\n",
       "| 0             | 1             | -2.000000e+00 | -2.000000e+00 |  1.000000e+00 | -2.109424e-15 | -3.330669e-15 |  6.000000e+00 |  2.664535e-15 | -2.000000e+00 |\n",
       "| 1             | 0             |  7.771561e-16 | -3.330669e-15 | -8.881784e-16 | -2.000000e+00 | -2.000000e+00 |  1.332268e-15 |  5.000000e+00 |  1.346145e-15 |\n",
       "| 1             | 0             | -3.497203e-15 | -8.590351e-15 | -2.000000e+00 | -5.280498e-15 |  2.303713e-15 | -2.000000e+00 |  6.231127e-15 |  5.000000e+00 |\n",
       "\n",
       "\n",
       "$est\n",
       ":   \n",
       "|  4.358502330 |\n",
       "|  3.404430006 |\n",
       "|  0.098444576 |\n",
       "| -0.018770099 |\n",
       "| -0.041084203 |\n",
       "| -0.008663123 |\n",
       "| -0.185732099 |\n",
       "|  0.176872088 |\n",
       "| -0.249458555 |\n",
       "|  0.182614688 |\n",
       "\n",
       "\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "$C\n",
       "      [,1] [,2]          [,3]          [,4]          [,5]          [,6]\n",
       " [1,]    3    0  0.000000e+00  0.000000e+00  0.000000e+00  1.000000e+00\n",
       " [2,]    0    2  0.000000e+00  0.000000e+00  0.000000e+00  0.000000e+00\n",
       " [3,]    0    0  3.666667e+00  1.000000e+00  5.440093e-15 -1.333333e+00\n",
       " [4,]    0    0  1.000000e+00  4.000000e+00  1.000000e+00 -4.440892e-16\n",
       " [5,]    0    0  2.220446e-15  1.000000e+00  4.000000e+00  5.551115e-16\n",
       " [6,]    1    0 -1.333333e+00 -2.220446e-16 -3.108624e-15  4.666667e+00\n",
       " [7,]    0    1  2.220446e-16 -2.000000e+00 -2.000000e+00  1.000000e+00\n",
       " [8,]    0    1 -2.000000e+00 -2.000000e+00  1.000000e+00 -2.109424e-15\n",
       " [9,]    1    0  7.771561e-16 -3.330669e-15 -8.881784e-16 -2.000000e+00\n",
       "[10,]    1    0 -3.497203e-15 -8.590351e-15 -2.000000e+00 -5.280498e-15\n",
       "               [,7]          [,8]          [,9]         [,10]\n",
       " [1,]  0.000000e+00  0.000000e+00  1.000000e+00  1.000000e+00\n",
       " [2,]  1.000000e+00  1.000000e+00  0.000000e+00  0.000000e+00\n",
       " [3,]  1.554312e-15 -2.000000e+00 -9.880985e-15 -4.996004e-15\n",
       " [4,] -2.000000e+00 -2.000000e+00 -1.110223e-15 -6.938894e-17\n",
       " [5,] -2.000000e+00  1.000000e+00 -5.551115e-16 -2.000000e+00\n",
       " [6,]  1.000000e+00 -4.440892e-16 -2.000000e+00  1.727785e-15\n",
       " [7,]  6.000000e+00 -1.776357e-15 -2.000000e+00 -2.636780e-15\n",
       " [8,] -3.330669e-15  6.000000e+00  2.664535e-15 -2.000000e+00\n",
       " [9,] -2.000000e+00  1.332268e-15  5.000000e+00  1.346145e-15\n",
       "[10,]  2.303713e-15 -2.000000e+00  6.231127e-15  5.000000e+00\n",
       "\n",
       "$est\n",
       "              [,1]\n",
       " [1,]  4.358502330\n",
       " [2,]  3.404430006\n",
       " [3,]  0.098444576\n",
       " [4,] -0.018770099\n",
       " [5,] -0.041084203\n",
       " [6,] -0.008663123\n",
       " [7,] -0.185732099\n",
       " [8,]  0.176872088\n",
       " [9,] -0.249458555\n",
       "[10,]  0.182614688\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "library(MASS)\n",
    "\n",
    "mme = function(y, X, Z, A, sigma_a, sigma_e) {\n",
    "    alpha = sigma_e / sigma_a\n",
    "    invA = ginv(A)\n",
    "    C = rbind(cbind(t(X)%*%X, t(X)%*%Z),\n",
    "              cbind(t(Z)%*%X, t(Z)%*%Z+invA*c(alpha)))\n",
    "    rhs = rbind(t(X)%*%y, t(Z)%*%y)\n",
    "    invC = ginv(C)\n",
    "    estimators = invC%*%rhs\n",
    "    list(C = C, est = estimators)\n",
    "}\n",
    "\n",
    "mme(y, X, Z, A, 20, 40)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "-0.00572209096278567"
      ],
      "text/latex": [
       "-0.00572209096278567"
      ],
      "text/markdown": [
       "-0.00572209096278567"
      ],
      "text/plain": [
       "[1] -0.005722091"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "0.157163480373342"
      ],
      "text/latex": [
       "0.157163480373342"
      ],
      "text/markdown": [
       "0.157163480373342"
      ],
      "text/plain": [
       "[1] 0.1571635"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "est = mme(y, X, Z, A, 20, 40)$est[3:10]\n",
    "(m = mean(est))\n",
    "(s = sd(est))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "100"
      ],
      "text/latex": [
       "100"
      ],
      "text/markdown": [
       "100"
      ],
      "text/plain": [
       "[1] 100"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "10"
      ],
      "text/latex": [
       "10"
      ],
      "text/markdown": [
       "10"
      ],
      "text/plain": [
       "[1] 10"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>106.627918039179</li>\n",
       "\t<li>99.1697811662677</li>\n",
       "\t<li>97.7499790740002</li>\n",
       "\t<li>99.8128679962949</li>\n",
       "\t<li>88.5463208053009</li>\n",
       "\t<li>111.618104804649</li>\n",
       "\t<li>84.4915330653251</li>\n",
       "\t<li>111.983495048983</li>\n",
       "</ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item 106.627918039179\n",
       "\\item 99.1697811662677\n",
       "\\item 97.7499790740002\n",
       "\\item 99.8128679962949\n",
       "\\item 88.5463208053009\n",
       "\\item 111.618104804649\n",
       "\\item 84.4915330653251\n",
       "\\item 111.983495048983\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 106.627918039179\n",
       "2. 99.1697811662677\n",
       "3. 97.7499790740002\n",
       "4. 99.8128679962949\n",
       "5. 88.5463208053009\n",
       "6. 111.618104804649\n",
       "7. 84.4915330653251\n",
       "8. 111.983495048983\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "[1] 106.62792  99.16978  97.74998  99.81287  88.54632 111.61810  84.49153\n",
       "[8] 111.98350"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "est = 10*(est - m) / s + 100\n",
    "mean(est)\n",
    "sd(est)\n",
    "est"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "4. Dokładność wartości hodowlanych"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<tbody>\n",
       "\t<tr><td> 0.59556097 </td><td> 0.15730213 </td><td>-0.164119413</td><td>-0.083624565</td><td>-0.13059083 </td><td>-0.26455749 </td><td>-0.14827804 </td><td>-0.16632621 </td><td>-0.2842464  </td><td>-0.2378790  </td></tr>\n",
       "\t<tr><td> 0.15730213 </td><td> 0.80245865 </td><td>-0.132863260</td><td>-0.241250738</td><td>-0.11196840 </td><td>-0.08730803 </td><td>-0.29891465 </td><td>-0.30600266 </td><td>-0.1859495  </td><td>-0.1986488  </td></tr>\n",
       "\t<tr><td>-0.16411941 </td><td>-0.13286326 </td><td> 0.471094211</td><td> 0.006928037</td><td> 0.03264668 </td><td> 0.21954371 </td><td> 0.04495225 </td><td> 0.22077427 </td><td> 0.1386223  </td><td> 0.1341923  </td></tr>\n",
       "\t<tr><td>-0.08362457 </td><td>-0.24125074 </td><td> 0.006928037</td><td> 0.492095721</td><td>-0.01030797 </td><td> 0.02039033 </td><td> 0.23734577 </td><td> 0.24515571 </td><td> 0.1198194  </td><td> 0.1106640  </td></tr>\n",
       "\t<tr><td>-0.13059083 </td><td>-0.11196840 </td><td> 0.032646682</td><td>-0.010307967</td><td> 0.45645878 </td><td> 0.04812709 </td><td> 0.20132326 </td><td> 0.02261354 </td><td> 0.1258983  </td><td> 0.2177471  </td></tr>\n",
       "\t<tr><td>-0.26455749 </td><td>-0.08730803 </td><td> 0.219543709</td><td> 0.020390333</td><td> 0.04812709 </td><td> 0.42768015 </td><td> 0.04704420 </td><td> 0.12757186 </td><td> 0.2428012  </td><td> 0.1231911  </td></tr>\n",
       "\t<tr><td>-0.14827804 </td><td>-0.29891465 </td><td> 0.044952254</td><td> 0.237345770</td><td> 0.20132326 </td><td> 0.04704420 </td><td> 0.42810675 </td><td> 0.16972255 </td><td> 0.2197160  </td><td> 0.1780739  </td></tr>\n",
       "\t<tr><td>-0.16632621 </td><td>-0.30600266 </td><td> 0.220774267</td><td> 0.245155707</td><td> 0.02261354 </td><td> 0.12757186 </td><td> 0.16972255 </td><td> 0.44228277 </td><td> 0.1521830  </td><td> 0.2192238  </td></tr>\n",
       "\t<tr><td>-0.28424641 </td><td>-0.18594950 </td><td> 0.138622268</td><td> 0.119819354</td><td> 0.12589831 </td><td> 0.24280124 </td><td> 0.21971599 </td><td> 0.15218301 </td><td> 0.4418562  </td><td> 0.1680818  </td></tr>\n",
       "\t<tr><td>-0.23787901 </td><td>-0.19864885 </td><td> 0.134192262</td><td> 0.110664009</td><td> 0.21774710 </td><td> 0.12319108 </td><td> 0.17807393 </td><td> 0.21922376 </td><td> 0.1680818  </td><td> 0.4223641  </td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{llllllllll}\n",
       "\t  0.59556097  &  0.15730213  & -0.164119413 & -0.083624565 & -0.13059083  & -0.26455749  & -0.14827804  & -0.16632621  & -0.2842464   & -0.2378790  \\\\\n",
       "\t  0.15730213  &  0.80245865  & -0.132863260 & -0.241250738 & -0.11196840  & -0.08730803  & -0.29891465  & -0.30600266  & -0.1859495   & -0.1986488  \\\\\n",
       "\t -0.16411941  & -0.13286326  &  0.471094211 &  0.006928037 &  0.03264668  &  0.21954371  &  0.04495225  &  0.22077427  &  0.1386223   &  0.1341923  \\\\\n",
       "\t -0.08362457  & -0.24125074  &  0.006928037 &  0.492095721 & -0.01030797  &  0.02039033  &  0.23734577  &  0.24515571  &  0.1198194   &  0.1106640  \\\\\n",
       "\t -0.13059083  & -0.11196840  &  0.032646682 & -0.010307967 &  0.45645878  &  0.04812709  &  0.20132326  &  0.02261354  &  0.1258983   &  0.2177471  \\\\\n",
       "\t -0.26455749  & -0.08730803  &  0.219543709 &  0.020390333 &  0.04812709  &  0.42768015  &  0.04704420  &  0.12757186  &  0.2428012   &  0.1231911  \\\\\n",
       "\t -0.14827804  & -0.29891465  &  0.044952254 &  0.237345770 &  0.20132326  &  0.04704420  &  0.42810675  &  0.16972255  &  0.2197160   &  0.1780739  \\\\\n",
       "\t -0.16632621  & -0.30600266  &  0.220774267 &  0.245155707 &  0.02261354  &  0.12757186  &  0.16972255  &  0.44228277  &  0.1521830   &  0.2192238  \\\\\n",
       "\t -0.28424641  & -0.18594950  &  0.138622268 &  0.119819354 &  0.12589831  &  0.24280124  &  0.21971599  &  0.15218301  &  0.4418562   &  0.1680818  \\\\\n",
       "\t -0.23787901  & -0.19864885  &  0.134192262 &  0.110664009 &  0.21774710  &  0.12319108  &  0.17807393  &  0.21922376  &  0.1680818   &  0.4223641  \\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "|  0.59556097  |  0.15730213  | -0.164119413 | -0.083624565 | -0.13059083  | -0.26455749  | -0.14827804  | -0.16632621  | -0.2842464   | -0.2378790   |\n",
       "|  0.15730213  |  0.80245865  | -0.132863260 | -0.241250738 | -0.11196840  | -0.08730803  | -0.29891465  | -0.30600266  | -0.1859495   | -0.1986488   |\n",
       "| -0.16411941  | -0.13286326  |  0.471094211 |  0.006928037 |  0.03264668  |  0.21954371  |  0.04495225  |  0.22077427  |  0.1386223   |  0.1341923   |\n",
       "| -0.08362457  | -0.24125074  |  0.006928037 |  0.492095721 | -0.01030797  |  0.02039033  |  0.23734577  |  0.24515571  |  0.1198194   |  0.1106640   |\n",
       "| -0.13059083  | -0.11196840  |  0.032646682 | -0.010307967 |  0.45645878  |  0.04812709  |  0.20132326  |  0.02261354  |  0.1258983   |  0.2177471   |\n",
       "| -0.26455749  | -0.08730803  |  0.219543709 |  0.020390333 |  0.04812709  |  0.42768015  |  0.04704420  |  0.12757186  |  0.2428012   |  0.1231911   |\n",
       "| -0.14827804  | -0.29891465  |  0.044952254 |  0.237345770 |  0.20132326  |  0.04704420  |  0.42810675  |  0.16972255  |  0.2197160   |  0.1780739   |\n",
       "| -0.16632621  | -0.30600266  |  0.220774267 |  0.245155707 |  0.02261354  |  0.12757186  |  0.16972255  |  0.44228277  |  0.1521830   |  0.2192238   |\n",
       "| -0.28424641  | -0.18594950  |  0.138622268 |  0.119819354 |  0.12589831  |  0.24280124  |  0.21971599  |  0.15218301  |  0.4418562   |  0.1680818   |\n",
       "| -0.23787901  | -0.19864885  |  0.134192262 |  0.110664009 |  0.21774710  |  0.12319108  |  0.17807393  |  0.21922376  |  0.1680818   |  0.4223641   |\n",
       "\n"
      ],
      "text/plain": [
       "      [,1]        [,2]        [,3]         [,4]         [,5]        [,6]       \n",
       " [1,]  0.59556097  0.15730213 -0.164119413 -0.083624565 -0.13059083 -0.26455749\n",
       " [2,]  0.15730213  0.80245865 -0.132863260 -0.241250738 -0.11196840 -0.08730803\n",
       " [3,] -0.16411941 -0.13286326  0.471094211  0.006928037  0.03264668  0.21954371\n",
       " [4,] -0.08362457 -0.24125074  0.006928037  0.492095721 -0.01030797  0.02039033\n",
       " [5,] -0.13059083 -0.11196840  0.032646682 -0.010307967  0.45645878  0.04812709\n",
       " [6,] -0.26455749 -0.08730803  0.219543709  0.020390333  0.04812709  0.42768015\n",
       " [7,] -0.14827804 -0.29891465  0.044952254  0.237345770  0.20132326  0.04704420\n",
       " [8,] -0.16632621 -0.30600266  0.220774267  0.245155707  0.02261354  0.12757186\n",
       " [9,] -0.28424641 -0.18594950  0.138622268  0.119819354  0.12589831  0.24280124\n",
       "[10,] -0.23787901 -0.19864885  0.134192262  0.110664009  0.21774710  0.12319108\n",
       "      [,7]        [,8]        [,9]       [,10]     \n",
       " [1,] -0.14827804 -0.16632621 -0.2842464 -0.2378790\n",
       " [2,] -0.29891465 -0.30600266 -0.1859495 -0.1986488\n",
       " [3,]  0.04495225  0.22077427  0.1386223  0.1341923\n",
       " [4,]  0.23734577  0.24515571  0.1198194  0.1106640\n",
       " [5,]  0.20132326  0.02261354  0.1258983  0.2177471\n",
       " [6,]  0.04704420  0.12757186  0.2428012  0.1231911\n",
       " [7,]  0.42810675  0.16972255  0.2197160  0.1780739\n",
       " [8,]  0.16972255  0.44228277  0.1521830  0.2192238\n",
       " [9,]  0.21971599  0.15218301  0.4418562  0.1680818\n",
       "[10,]  0.17807393  0.21922376  0.1680818  0.4223641"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>0.0578115770821044</li>\n",
       "\t<li>0.0158085581151144</li>\n",
       "\t<li>0.0870824309247233</li>\n",
       "\t<li>0.144639692852924</li>\n",
       "\t<li>0.143786506530158</li>\n",
       "\t<li>0.11543446872744</li>\n",
       "\t<li>0.116287655050207</li>\n",
       "\t<li>0.155271707028943</li>\n",
       "</ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item 0.0578115770821044\n",
       "\\item 0.0158085581151144\n",
       "\\item 0.0870824309247233\n",
       "\\item 0.144639692852924\n",
       "\\item 0.143786506530158\n",
       "\\item 0.11543446872744\n",
       "\\item 0.116287655050207\n",
       "\\item 0.155271707028943\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 0.0578115770821044\n",
       "2. 0.0158085581151144\n",
       "3. 0.0870824309247233\n",
       "4. 0.144639692852924\n",
       "5. 0.143786506530158\n",
       "6. 0.11543446872744\n",
       "7. 0.116287655050207\n",
       "8. 0.155271707028943\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "[1] 0.05781158 0.01580856 0.08708243 0.14463969 0.14378651 0.11543447 0.11628766\n",
       "[8] 0.15527171"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
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    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>0.240440381554564</li>\n",
       "\t<li>0.125732088645319</li>\n",
       "\t<li>0.295097324496044</li>\n",
       "\t<li>0.38031525456248</li>\n",
       "\t<li>0.379191912532635</li>\n",
       "\t<li>0.33975648445238</li>\n",
       "\t<li>0.341009757998517</li>\n",
       "\t<li>0.394045310883079</li>\n",
       "</ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item 0.240440381554564\n",
       "\\item 0.125732088645319\n",
       "\\item 0.295097324496044\n",
       "\\item 0.38031525456248\n",
       "\\item 0.379191912532635\n",
       "\\item 0.33975648445238\n",
       "\\item 0.341009757998517\n",
       "\\item 0.394045310883079\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 0.240440381554564\n",
       "2. 0.125732088645319\n",
       "3. 0.295097324496044\n",
       "4. 0.38031525456248\n",
       "5. 0.379191912532635\n",
       "6. 0.33975648445238\n",
       "7. 0.341009757998517\n",
       "8. 0.394045310883079\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "[1] 0.2404404 0.1257321 0.2950973 0.3803153 0.3791919 0.3397565 0.3410098\n",
       "[8] 0.3940453"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "C = as.matrix(mme(y, X, Z, A, 20, 40)$C)\n",
    "(invC = ginv(C))\n",
    "\n",
    "invC22 = invC[3:10, 3:10]\n",
    "(r2 = diag(1 - invC22*2))\n",
    "(r = sqrt(r2))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "5. Istotność efektów stałych"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Test Walda:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$W = \\frac{\\widehat{\\beta}}{se(\\widehat{\\beta})}\\sim\\mathcal{N}(0,1)$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$H_{0}: \\beta=0$ vs. $H_{1}: \\beta\\neq 0$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$var(\\widehat{\\beta}) = \\left(\\mathbf{X}^{T}\\mathbf{V}^{-1}\\mathbf{X}\\right)^{-1}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<tbody>\n",
       "\t<tr><td>60.0</td><td> 0.0</td><td> 5  </td><td>10  </td><td> 2.5</td></tr>\n",
       "\t<tr><td> 0.0</td><td>60.0</td><td> 5  </td><td>10  </td><td> 7.5</td></tr>\n",
       "\t<tr><td> 5.0</td><td> 5.0</td><td>60  </td><td> 5  </td><td>10.0</td></tr>\n",
       "\t<tr><td>10.0</td><td>10.0</td><td> 5  </td><td>60  </td><td> 5.0</td></tr>\n",
       "\t<tr><td> 2.5</td><td> 7.5</td><td>10  </td><td> 5  </td><td>60.0</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{lllll}\n",
       "\t 60.0 &  0.0 &  5   & 10   &  2.5\\\\\n",
       "\t  0.0 & 60.0 &  5   & 10   &  7.5\\\\\n",
       "\t  5.0 &  5.0 & 60   &  5   & 10.0\\\\\n",
       "\t 10.0 & 10.0 &  5   & 60   &  5.0\\\\\n",
       "\t  2.5 &  7.5 & 10   &  5   & 60.0\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "| 60.0 |  0.0 |  5   | 10   |  2.5 |\n",
       "|  0.0 | 60.0 |  5   | 10   |  7.5 |\n",
       "|  5.0 |  5.0 | 60   |  5   | 10.0 |\n",
       "| 10.0 | 10.0 |  5   | 60   |  5.0 |\n",
       "|  2.5 |  7.5 | 10   |  5   | 60.0 |\n",
       "\n"
      ],
      "text/plain": [
       "     [,1] [,2] [,3] [,4] [,5]\n",
       "[1,] 60.0  0.0  5   10    2.5\n",
       "[2,]  0.0 60.0  5   10    7.5\n",
       "[3,]  5.0  5.0 60    5   10.0\n",
       "[4,] 10.0 10.0  5   60    5.0\n",
       "[5,]  2.5  7.5 10    5   60.0"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<tbody>\n",
       "\t<tr><td>23.822439</td><td> 6.292085</td></tr>\n",
       "\t<tr><td> 6.292085</td><td>32.098346</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{ll}\n",
       "\t 23.822439 &  6.292085\\\\\n",
       "\t  6.292085 & 32.098346\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "| 23.822439 |  6.292085 |\n",
       "|  6.292085 | 32.098346 |\n",
       "\n"
      ],
      "text/plain": [
       "     [,1]      [,2]     \n",
       "[1,] 23.822439  6.292085\n",
       "[2,]  6.292085 32.098346"
      ]
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     "metadata": {},
     "output_type": "display_data"
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    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>4.88082357807458</li>\n",
       "\t<li>5.66554023294585</li>\n",
       "</ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item 4.88082357807458\n",
       "\\item 5.66554023294585\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 4.88082357807458\n",
       "2. 5.66554023294585\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "[1] 4.880824 5.665540"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>0.892985017822407</li>\n",
       "\t<li>0.600901214346598</li>\n",
       "</ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item 0.892985017822407\n",
       "\\item 0.600901214346598\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 0.892985017822407\n",
       "2. 0.600901214346598\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "[1] 0.8929850 0.6009012"
      ]
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     "metadata": {},
     "output_type": "display_data"
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    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>0.371865195985295</li>\n",
       "\t<li>0.547905784351094</li>\n",
       "</ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item 0.371865195985295\n",
       "\\item 0.547905784351094\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 0.371865195985295\n",
       "2. 0.547905784351094\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "[1] 0.3718652 0.5479058"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "G = A*20\n",
    "R = diag(5)*40\n",
    "V = Z%*%G%*%t(Z) + R\n",
    "V\n",
    "\n",
    "(varB = ginv(t(X)%*%ginv(V)%*%X))\n",
    "(seB = sqrt(diag(varB)))\n",
    "\n",
    "(testWalda = mme(y, X, Z, A, 20, 40)$est[1:2] / seB)\n",
    "(p_value = 2*pnorm(abs(testWalda), lower.tail = FALSE))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "6. Estymacja parametrów wariancji"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<tbody>\n",
       "\t<tr><td>0.678</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{l}\n",
       "\t 0.678\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "| 0.678 |\n",
       "\n"
      ],
      "text/plain": [
       "     [,1] \n",
       "[1,] 0.678"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "var(y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "sigma_a = 100.01  #starting value for random effect\n",
    "sigma_e = 9.01 #starting value for error variance"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "EM = function(y, X, Z, A, sigma_a, sigma_e) {\n",
    "  n = nrow(X)\n",
    "  p = ncol(X) \n",
    "  q = nrow(A) \n",
    "  \n",
    "  t = 1 #iteration number 1\n",
    "  tmp = 0.1 #test for convergance\n",
    "  \n",
    "  while (tmp > 0.00001) {\n",
    "    mme_new = mme(y, X, Z, A, sigma_a, sigma_e)\n",
    "    C_new = ginv(mme_new$C)\n",
    "    Ck = C_new[(p+1):(p+q), (p+1):(p+q)]\n",
    "    mme2 = mme_new$est\n",
    "    \n",
    "    a = as.matrix(mme2[(p+1):(p+q)])\n",
    "    sigma_a_new = (t(a)%*%ginv(A)%*%a + sum(diag(ginv(A)%*%Ck))*c(sigma_e))/q\n",
    "    \n",
    "    res = as.matrix(y-X%*%as.matrix(mme2[1:p]) - Z%*%as.matrix(mme2[(p+1):(p+q)]))\n",
    "    X.tmp1 = cbind(X,Z) %*% C_new\n",
    "    X.tmp2 = t(cbind(X,Z))\n",
    "    sigma_e_new = (t(res)%*%res + sum(diag(X.tmp1%*%X.tmp2))*c(sigma_e))/n\n",
    "    \n",
    "    tmp = max(abs(sigma_a - sigma_a_new), abs(sigma_e - sigma_e_new))\n",
    "    sigma_a = sigma_a_new\n",
    "    sigma_e = sigma_e_new\n",
    "    \n",
    "    t = t + 1\n",
    "  }\n",
    "  list(t = t, sigma_a = sigma_a, sigma_e = sigma_e)\n",
    "}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### $\\sigma^{2}_{\\epsilon[t+1]} = \\frac{\\widehat{\\epsilon}^{'}_{[t]}\\widehat{\\epsilon}_{[t]} + tr([X, Z]C^{22}_{[t]}[X, Z]^{'})\\cdot\\sigma^{2}_{\\epsilon[t]}}{n}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### $\\sigma^{2}_{a[t+1]} = \\frac{\\widehat{a}^{'}_{[t]}A^{-1}\\widehat{a}_{[t]} + tr(A^{-1}C^{22}_{[t]})\\cdot\\sigma^{2}_{\\epsilon[t]}}{q}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<dl>\n",
       "\t<dt>$t</dt>\n",
       "\t\t<dd>657</dd>\n",
       "\t<dt>$sigma_a</dt>\n",
       "\t\t<dd><table>\n",
       "<tbody>\n",
       "\t<tr><td>0.6694952</td></tr>\n",
       "</tbody>\n",
       "</table>\n",
       "</dd>\n",
       "\t<dt>$sigma_e</dt>\n",
       "\t\t<dd><table>\n",
       "<tbody>\n",
       "\t<tr><td>0.005739873</td></tr>\n",
       "</tbody>\n",
       "</table>\n",
       "</dd>\n",
       "</dl>\n"
      ],
      "text/latex": [
       "\\begin{description}\n",
       "\\item[\\$t] 657\n",
       "\\item[\\$sigma\\_a] \\begin{tabular}{l}\n",
       "\t 0.6694952\\\\\n",
       "\\end{tabular}\n",
       "\n",
       "\\item[\\$sigma\\_e] \\begin{tabular}{l}\n",
       "\t 0.005739873\\\\\n",
       "\\end{tabular}\n",
       "\n",
       "\\end{description}\n"
      ],
      "text/markdown": [
       "$t\n",
       ":   657\n",
       "$sigma_a\n",
       ":   \n",
       "| 0.6694952 |\n",
       "\n",
       "\n",
       "$sigma_e\n",
       ":   \n",
       "| 0.005739873 |\n",
       "\n",
       "\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "$t\n",
       "[1] 657\n",
       "\n",
       "$sigma_a\n",
       "          [,1]\n",
       "[1,] 0.6694952\n",
       "\n",
       "$sigma_e\n",
       "            [,1]\n",
       "[1,] 0.005739873\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<tbody>\n",
       "\t<tr><td>0.675235</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{l}\n",
       "\t 0.675235\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "| 0.675235 |\n",
       "\n"
      ],
      "text/plain": [
       "     [,1]    \n",
       "[1,] 0.675235"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<tbody>\n",
       "\t<tr><td>0.678</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{l}\n",
       "\t 0.678\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "| 0.678 |\n",
       "\n"
      ],
      "text/plain": [
       "     [,1] \n",
       "[1,] 0.678"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "(wyniki = EM(y, X, Z, A, sigma_a, sigma_e))\n",
    "wyniki$sigma_a+wyniki$sigma_e\n",
    "var(y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "7. Odziedziczalność"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "0.970839380554511"
      ],
      "text/latex": [
       "0.970839380554511"
      ],
      "text/markdown": [
       "0.970839380554511"
      ],
      "text/plain": [
       "[1] 0.9708394"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "(h2 = 0.6555448 / (0.6694952 + 0.005739873))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<tbody>\n",
       "\t<tr><td> 4.44171842</td></tr>\n",
       "\t<tr><td> 3.48850299</td></tr>\n",
       "\t<tr><td> 0.28002080</td></tr>\n",
       "\t<tr><td>-0.13721162</td></tr>\n",
       "\t<tr><td>-0.08518631</td></tr>\n",
       "\t<tr><td> 0.05357608</td></tr>\n",
       "\t<tr><td>-0.58605426</td></tr>\n",
       "\t<tr><td> 0.40904828</td></tr>\n",
       "\t<tr><td>-0.93033132</td></tr>\n",
       "\t<tr><td> 0.55159998</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{l}\n",
       "\t  4.44171842\\\\\n",
       "\t  3.48850299\\\\\n",
       "\t  0.28002080\\\\\n",
       "\t -0.13721162\\\\\n",
       "\t -0.08518631\\\\\n",
       "\t  0.05357608\\\\\n",
       "\t -0.58605426\\\\\n",
       "\t  0.40904828\\\\\n",
       "\t -0.93033132\\\\\n",
       "\t  0.55159998\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "|  4.44171842 |\n",
       "|  3.48850299 |\n",
       "|  0.28002080 |\n",
       "| -0.13721162 |\n",
       "| -0.08518631 |\n",
       "|  0.05357608 |\n",
       "| -0.58605426 |\n",
       "|  0.40904828 |\n",
       "| -0.93033132 |\n",
       "|  0.55159998 |\n",
       "\n"
      ],
      "text/plain": [
       "      [,1]       \n",
       " [1,]  4.44171842\n",
       " [2,]  3.48850299\n",
       " [3,]  0.28002080\n",
       " [4,] -0.13721162\n",
       " [5,] -0.08518631\n",
       " [6,]  0.05357608\n",
       " [7,] -0.58605426\n",
       " [8,]  0.40904828\n",
       " [9,] -0.93033132\n",
       "[10,]  0.55159998"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "mme(y, X, Z, A, wyniki$sigma_a, wyniki$sigma_e)$est"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "-0.0555672951330806"
      ],
      "text/latex": [
       "-0.0555672951330806"
      ],
      "text/markdown": [
       "-0.0555672951330806"
      ],
      "text/plain": [
       "[1] -0.0555673"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "0.501885441017205"
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      "text/latex": [
       "0.501885441017205"
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      "text/markdown": [
       "0.501885441017205"
      ],
      "text/plain": [
       "[1] 0.5018854"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "100"
      ],
      "text/latex": [
       "100"
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      "text/markdown": [
       "100"
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      "text/plain": [
       "[1] 100"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "10"
      ],
      "text/latex": [
       "10"
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      "text/markdown": [
       "10"
      ],
      "text/plain": [
       "[1] 10"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<thead><tr><th scope=col>est</th><th scope=col>est2</th></tr></thead>\n",
       "<tbody>\n",
       "\t<tr><td>106.62792</td><td>106.68655</td></tr>\n",
       "\t<tr><td> 99.16978</td><td> 98.37325</td></tr>\n",
       "\t<tr><td> 97.74998</td><td> 99.40985</td></tr>\n",
       "\t<tr><td> 99.81287</td><td>102.17467</td></tr>\n",
       "\t<tr><td> 88.54632</td><td> 89.43012</td></tr>\n",
       "\t<tr><td>111.61810</td><td>109.25740</td></tr>\n",
       "\t<tr><td> 84.49153</td><td> 82.57044</td></tr>\n",
       "\t<tr><td>111.98350</td><td>112.09773</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{ll}\n",
       " est & est2\\\\\n",
       "\\hline\n",
       "\t 106.62792 & 106.68655\\\\\n",
       "\t  99.16978 &  98.37325\\\\\n",
       "\t  97.74998 &  99.40985\\\\\n",
       "\t  99.81287 & 102.17467\\\\\n",
       "\t  88.54632 &  89.43012\\\\\n",
       "\t 111.61810 & 109.25740\\\\\n",
       "\t  84.49153 &  82.57044\\\\\n",
       "\t 111.98350 & 112.09773\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "| est | est2 |\n",
       "|---|---|\n",
       "| 106.62792 | 106.68655 |\n",
       "|  99.16978 |  98.37325 |\n",
       "|  97.74998 |  99.40985 |\n",
       "|  99.81287 | 102.17467 |\n",
       "|  88.54632 |  89.43012 |\n",
       "| 111.61810 | 109.25740 |\n",
       "|  84.49153 |  82.57044 |\n",
       "| 111.98350 | 112.09773 |\n",
       "\n"
      ],
      "text/plain": [
       "     est       est2     \n",
       "[1,] 106.62792 106.68655\n",
       "[2,]  99.16978  98.37325\n",
       "[3,]  97.74998  99.40985\n",
       "[4,]  99.81287 102.17467\n",
       "[5,]  88.54632  89.43012\n",
       "[6,] 111.61810 109.25740\n",
       "[7,]  84.49153  82.57044\n",
       "[8,] 111.98350 112.09773"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "est2 = mme(y, X, Z, A, wyniki$sigma_a, wyniki$sigma_e)$est[3:10]\n",
    "(m = mean(est2))\n",
    "(s = sd(est2))\n",
    "\n",
    "est2 = 10*(est2 - m) / s + 100\n",
    "mean(est2)\n",
    "sd(est2)\n",
    "cbind(est, est2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "8. Test Walda z nowymi $\\sigma^{2}_{a}$ i $\\sigma^{2}_{\\epsilon}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<tbody>\n",
       "\t<tr><td>0.6752350</td><td>0.0000000</td><td>0.1673738</td><td>0.3347476</td><td>0.0836869</td></tr>\n",
       "\t<tr><td>0.0000000</td><td>0.6752350</td><td>0.1673738</td><td>0.3347476</td><td>0.2510607</td></tr>\n",
       "\t<tr><td>0.1673738</td><td>0.1673738</td><td>0.6752350</td><td>0.1673738</td><td>0.3347476</td></tr>\n",
       "\t<tr><td>0.3347476</td><td>0.3347476</td><td>0.1673738</td><td>0.6752350</td><td>0.1673738</td></tr>\n",
       "\t<tr><td>0.0836869</td><td>0.2510607</td><td>0.3347476</td><td>0.1673738</td><td>0.6752350</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{lllll}\n",
       "\t 0.6752350 & 0.0000000 & 0.1673738 & 0.3347476 & 0.0836869\\\\\n",
       "\t 0.0000000 & 0.6752350 & 0.1673738 & 0.3347476 & 0.2510607\\\\\n",
       "\t 0.1673738 & 0.1673738 & 0.6752350 & 0.1673738 & 0.3347476\\\\\n",
       "\t 0.3347476 & 0.3347476 & 0.1673738 & 0.6752350 & 0.1673738\\\\\n",
       "\t 0.0836869 & 0.2510607 & 0.3347476 & 0.1673738 & 0.6752350\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "| 0.6752350 | 0.0000000 | 0.1673738 | 0.3347476 | 0.0836869 |\n",
       "| 0.0000000 | 0.6752350 | 0.1673738 | 0.3347476 | 0.2510607 |\n",
       "| 0.1673738 | 0.1673738 | 0.6752350 | 0.1673738 | 0.3347476 |\n",
       "| 0.3347476 | 0.3347476 | 0.1673738 | 0.6752350 | 0.1673738 |\n",
       "| 0.0836869 | 0.2510607 | 0.3347476 | 0.1673738 | 0.6752350 |\n",
       "\n"
      ],
      "text/plain": [
       "     [,1]      [,2]      [,3]      [,4]      [,5]     \n",
       "[1,] 0.6752350 0.0000000 0.1673738 0.3347476 0.0836869\n",
       "[2,] 0.0000000 0.6752350 0.1673738 0.3347476 0.2510607\n",
       "[3,] 0.1673738 0.1673738 0.6752350 0.1673738 0.3347476\n",
       "[4,] 0.3347476 0.3347476 0.1673738 0.6752350 0.1673738\n",
       "[5,] 0.0836869 0.2510607 0.3347476 0.1673738 0.6752350"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<tbody>\n",
       "\t<tr><td>0.3451040</td><td>0.2065156</td></tr>\n",
       "\t<tr><td>0.2065156</td><td>0.3626311</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{ll}\n",
       "\t 0.3451040 & 0.2065156\\\\\n",
       "\t 0.2065156 & 0.3626311\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "| 0.3451040 | 0.2065156 |\n",
       "| 0.2065156 | 0.3626311 |\n",
       "\n"
      ],
      "text/plain": [
       "     [,1]      [,2]     \n",
       "[1,] 0.3451040 0.2065156\n",
       "[2,] 0.2065156 0.3626311"
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     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>0.587455495770437</li>\n",
       "\t<li>0.602188575392395</li>\n",
       "</ol>\n"
      ],
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       "\\begin{enumerate*}\n",
       "\\item 0.587455495770437\n",
       "\\item 0.602188575392395\n",
       "\\end{enumerate*}\n"
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      "text/markdown": [
       "1. 0.587455495770437\n",
       "2. 0.602188575392395\n",
       "\n",
       "\n"
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      "text/plain": [
       "[1] 0.5874555 0.6021886"
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       "<ol class=list-inline>\n",
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       "\\begin{enumerate*}\n",
       "\\item 7.5609445323216\n",
       "\\item 5.79304080974346\n",
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       "1. 7.5609445323216\n",
       "2. 5.79304080974346\n",
       "\n",
       "\n"
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       "[1] 7.560945 5.793041"
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     "metadata": {},
     "output_type": "display_data"
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    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>4.00152869669226e-14</li>\n",
       "\t<li>6.91233023736263e-09</li>\n",
       "</ol>\n"
      ],
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       "\\begin{enumerate*}\n",
       "\\item 4.00152869669226e-14\n",
       "\\item 6.91233023736263e-09\n",
       "\\end{enumerate*}\n"
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      "text/markdown": [
       "1. 4.00152869669226e-14\n",
       "2. 6.91233023736263e-09\n",
       "\n",
       "\n"
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      "text/plain": [
       "[1] 4.001529e-14 6.912330e-09"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "G = A*c(wyniki$sigma_a)\n",
    "R = diag(5)*c(wyniki$sigma_e)\n",
    "V = Z%*%G%*%t(Z) + R\n",
    "V\n",
    "\n",
    "(varB = ginv(t(X)%*%ginv(V)%*%X))\n",
    "(seB = sqrt(diag(varB)))\n",
    "\n",
    "(testWalda = mme(y, X, Z, A, wyniki$sigma_a, wyniki$sigma_e)$est[1:2] / seB)\n",
    "(p_value = 2*pnorm(abs(testWalda), lower.tail = FALSE))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "9. Dokładność oceny"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>0.189935890368132</li>\n",
       "\t<li>0.0776482185901066</li>\n",
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       "\\item 0.483237179540816\n",
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      "text/markdown": [
       "1. 0.189935890368132\n",
       "2. 0.0776482185901066\n",
       "3. 0.291998383422087\n",
       "4. 0.482418666354055\n",
       "5. 0.459808525687841\n",
       "6. 0.457012381382101\n",
       "7. 0.479622522048315\n",
       "8. 0.483237179540816\n",
       "\n",
       "\n"
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      "text/plain": [
       "[1] 0.18993589 0.07764822 0.29199838 0.48241867 0.45980853 0.45701238 0.47962252\n",
       "[8] 0.48323718"
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     "data": {
      "text/html": [
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       "\t<li>0.694563651765665</li>\n",
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       "\\begin{enumerate*}\n",
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       "\\item 0.540368747636359\n",
       "\\item 0.694563651765665\n",
       "\\item 0.678091826884708\n",
       "\\item 0.676026908770724\n",
       "\\item 0.692547848201346\n",
       "\\item 0.695152630391928\n",
       "\\end{enumerate*}\n"
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      "text/markdown": [
       "1. 0.435816349358457\n",
       "2. 0.278654299428713\n",
       "3. 0.540368747636359\n",
       "4. 0.694563651765665\n",
       "5. 0.678091826884708\n",
       "6. 0.676026908770724\n",
       "7. 0.692547848201346\n",
       "8. 0.695152630391928\n",
       "\n",
       "\n"
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      "text/plain": [
       "[1] 0.4358163 0.2786543 0.5403687 0.6945637 0.6780918 0.6760269 0.6925478\n",
       "[8] 0.6951526"
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     "data": {
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       "<table>\n",
       "<thead><tr><th scope=col>r</th><th scope=col>r_2</th></tr></thead>\n",
       "<tbody>\n",
       "\t<tr><td>0.2404404</td><td>0.4358163</td></tr>\n",
       "\t<tr><td>0.1257321</td><td>0.2786543</td></tr>\n",
       "\t<tr><td>0.2950973</td><td>0.5403687</td></tr>\n",
       "\t<tr><td>0.3803153</td><td>0.6945637</td></tr>\n",
       "\t<tr><td>0.3791919</td><td>0.6780918</td></tr>\n",
       "\t<tr><td>0.3397565</td><td>0.6760269</td></tr>\n",
       "\t<tr><td>0.3410098</td><td>0.6925478</td></tr>\n",
       "\t<tr><td>0.3940453</td><td>0.6951526</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{ll}\n",
       " r & r\\_2\\\\\n",
       "\\hline\n",
       "\t 0.2404404 & 0.4358163\\\\\n",
       "\t 0.1257321 & 0.2786543\\\\\n",
       "\t 0.2950973 & 0.5403687\\\\\n",
       "\t 0.3803153 & 0.6945637\\\\\n",
       "\t 0.3791919 & 0.6780918\\\\\n",
       "\t 0.3397565 & 0.6760269\\\\\n",
       "\t 0.3410098 & 0.6925478\\\\\n",
       "\t 0.3940453 & 0.6951526\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "| r | r_2 |\n",
       "|---|---|\n",
       "| 0.2404404 | 0.4358163 |\n",
       "| 0.1257321 | 0.2786543 |\n",
       "| 0.2950973 | 0.5403687 |\n",
       "| 0.3803153 | 0.6945637 |\n",
       "| 0.3791919 | 0.6780918 |\n",
       "| 0.3397565 | 0.6760269 |\n",
       "| 0.3410098 | 0.6925478 |\n",
       "| 0.3940453 | 0.6951526 |\n",
       "\n"
      ],
      "text/plain": [
       "     r         r_2      \n",
       "[1,] 0.2404404 0.4358163\n",
       "[2,] 0.1257321 0.2786543\n",
       "[3,] 0.2950973 0.5403687\n",
       "[4,] 0.3803153 0.6945637\n",
       "[5,] 0.3791919 0.6780918\n",
       "[6,] 0.3397565 0.6760269\n",
       "[7,] 0.3410098 0.6925478\n",
       "[8,] 0.3940453 0.6951526"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "C = as.matrix(mme(y, X, Z, A, wyniki$sigma_a, wyniki$sigma_e)$C)\n",
    "invC = ginv(C)\n",
    "\n",
    "invC22 = invC[3:10, 3:10]\n",
    "alpha = wyniki$sigma_e / wyniki$sigma_a\n",
    "(r_22 = diag(1 - invC22*c(alpha)))\n",
    "(r_2 = sqrt(r_22))\n",
    "\n",
    "cbind(r, r_2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "10. Korelacje oceny"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "0.986408003048259"
      ],
      "text/latex": [
       "0.986408003048259"
      ],
      "text/markdown": [
       "0.986408003048259"
      ],
      "text/plain": [
       "[1] 0.986408"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "0.977062095530124"
      ],
      "text/latex": [
       "0.977062095530124"
      ],
      "text/markdown": [
       "0.977062095530124"
      ],
      "text/plain": [
       "[1] 0.9770621"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "cor(est, est2)\n",
    "cor(r, r_2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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d3f7wsJtgjxUEZ4RTp83HpF4qGMcI50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\nEBIECAkChAQBQoKA/wB3/WKFfAkbVwAAAABJRU5ErkJggg==",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot(est, est2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "Call:\n",
       "lm(formula = est ~ est2)\n",
       "\n",
       "Residuals:\n",
       "     Min       1Q   Median       3Q      Max \n",
       "-2.33224 -1.18757  0.04123  1.00186  2.48653 \n",
       "\n",
       "Coefficients:\n",
       "            Estimate Std. Error t value Pr(>|t|)    \n",
       "(Intercept)  1.35920    6.73740   0.202    0.847    \n",
       "est2         0.98641    0.06708  14.705 6.21e-06 ***\n",
       "---\n",
       "Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1\n",
       "\n",
       "Residual standard error: 1.775 on 6 degrees of freedom\n",
       "Multiple R-squared:  0.973,\tAdjusted R-squared:  0.9685 \n",
       "F-statistic: 216.2 on 1 and 6 DF,  p-value: 6.214e-06\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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R3pqBoZQ1ocdwdtu+e3nK6/JAnpwPKo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\nD57NCuHFe9kMAAAAAElFTkSuQmCC",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot(est, est2)\n",
    "model = lm(est~est2)\n",
    "summary(model)\n",
    "abline(model, lwd= 2, col = \"red\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "11. Macierz z genotypami"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 211,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>'BB'</li>\n",
       "\t<li>'AB'</li>\n",
       "\t<li>'BB'</li>\n",
       "\t<li>'AA'</li>\n",
       "\t<li>'AA'</li>\n",
       "\t<li>'AA'</li>\n",
       "</ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item 'BB'\n",
       "\\item 'AB'\n",
       "\\item 'BB'\n",
       "\\item 'AA'\n",
       "\\item 'AA'\n",
       "\\item 'AA'\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 'BB'\n",
       "2. 'AB'\n",
       "3. 'BB'\n",
       "4. 'AA'\n",
       "5. 'AA'\n",
       "6. 'AA'\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "[1] \"BB\" \"AB\" \"BB\" \"AA\" \"AA\" \"AA\""
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "Genotypes = sample(c(\"AA\", \"AB\", \"BB\"), 5000, replace = TRUE)\n",
    "head(Genotypes)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 212,
   "metadata": {},
   "outputs": [],
   "source": [
    "Genotypes = as.data.frame(matrix(Genotypes, 5, 1000))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 213,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<thead><tr><th scope=col>V1</th><th scope=col>V2</th><th scope=col>V3</th><th scope=col>V4</th><th scope=col>V5</th><th scope=col>V6</th><th scope=col>V7</th><th scope=col>V8</th><th scope=col>V9</th><th scope=col>V10</th><th scope=col>...</th><th scope=col>V991</th><th scope=col>V992</th><th scope=col>V993</th><th scope=col>V994</th><th scope=col>V995</th><th scope=col>V996</th><th scope=col>V997</th><th scope=col>V998</th><th scope=col>V999</th><th scope=col>V1000</th></tr></thead>\n",
       "<tbody>\n",
       "\t<tr><td>BB </td><td>AA </td><td>BB </td><td>AA </td><td>BB </td><td>AA </td><td>AA </td><td>AB </td><td>AB </td><td>BB </td><td>...</td><td>BB </td><td>AB </td><td>BB </td><td>AA </td><td>AA </td><td>AB </td><td>AB </td><td>AB </td><td>AB </td><td>AB </td></tr>\n",
       "\t<tr><td>AB </td><td>AA </td><td>BB </td><td>BB </td><td>AB </td><td>BB </td><td>AA </td><td>AB </td><td>AA </td><td>BB </td><td>...</td><td>AA </td><td>BB </td><td>BB </td><td>AA </td><td>AA </td><td>AA </td><td>BB </td><td>BB </td><td>BB </td><td>AA </td></tr>\n",
       "\t<tr><td>BB </td><td>AA </td><td>AA </td><td>AA </td><td>BB </td><td>AA </td><td>BB </td><td>AA </td><td>BB </td><td>AA </td><td>...</td><td>AB </td><td>AA </td><td>AA </td><td>BB </td><td>BB </td><td>AB </td><td>AA </td><td>AB </td><td>AB </td><td>AB </td></tr>\n",
       "\t<tr><td>AA </td><td>AB </td><td>BB </td><td>BB </td><td>BB </td><td>BB </td><td>AB </td><td>BB </td><td>AB </td><td>BB </td><td>...</td><td>AA </td><td>AA </td><td>BB </td><td>BB </td><td>AB </td><td>AB </td><td>AA </td><td>AB </td><td>BB </td><td>AA </td></tr>\n",
       "\t<tr><td>AA </td><td>AB </td><td>AA </td><td>BB </td><td>AA </td><td>AA </td><td>AA </td><td>BB </td><td>AA </td><td>BB </td><td>...</td><td>BB </td><td>AB </td><td>AA </td><td>BB </td><td>BB </td><td>AB </td><td>AA </td><td>BB </td><td>BB </td><td>AA </td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{r|llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll}\n",
       " V1 & V2 & V3 & V4 & V5 & V6 & V7 & V8 & V9 & V10 & ... & V991 & V992 & V993 & V994 & V995 & V996 & V997 & V998 & V999 & V1000\\\\\n",
       "\\hline\n",
       "\t BB  & AA  & BB  & AA  & BB  & AA  & AA  & AB  & AB  & BB  & ... & BB  & AB  & BB  & AA  & AA  & AB  & AB  & AB  & AB  & AB \\\\\n",
       "\t AB  & AA  & BB  & BB  & AB  & BB  & AA  & AB  & AA  & BB  & ... & AA  & BB  & BB  & AA  & AA  & AA  & BB  & BB  & BB  & AA \\\\\n",
       "\t BB  & AA  & AA  & AA  & BB  & AA  & BB  & AA  & BB  & AA  & ... & AB  & AA  & AA  & BB  & BB  & AB  & AA  & AB  & AB  & AB \\\\\n",
       "\t AA  & AB  & BB  & BB  & BB  & BB  & AB  & BB  & AB  & BB  & ... & AA  & AA  & BB  & BB  & AB  & AB  & AA  & AB  & BB  & AA \\\\\n",
       "\t AA  & AB  & AA  & BB  & AA  & AA  & AA  & BB  & AA  & BB  & ... & BB  & AB  & AA  & BB  & BB  & AB  & AA  & BB  & BB  & AA \\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "| V1 | V2 | V3 | V4 | V5 | V6 | V7 | V8 | V9 | V10 | ... | V991 | V992 | V993 | V994 | V995 | V996 | V997 | V998 | V999 | V1000 |\n",
       "|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|\n",
       "| BB  | AA  | BB  | AA  | BB  | AA  | AA  | AB  | AB  | BB  | ... | BB  | AB  | BB  | AA  | AA  | AB  | AB  | AB  | AB  | AB  |\n",
       "| AB  | AA  | BB  | BB  | AB  | BB  | AA  | AB  | AA  | BB  | ... | AA  | BB  | BB  | AA  | AA  | AA  | BB  | BB  | BB  | AA  |\n",
       "| BB  | AA  | AA  | AA  | BB  | AA  | BB  | AA  | BB  | AA  | ... | AB  | AA  | AA  | BB  | BB  | AB  | AA  | AB  | AB  | AB  |\n",
       "| AA  | AB  | BB  | BB  | BB  | BB  | AB  | BB  | AB  | BB  | ... | AA  | AA  | BB  | BB  | AB  | AB  | AA  | AB  | BB  | AA  |\n",
       "| AA  | AB  | AA  | BB  | AA  | AA  | AA  | BB  | AA  | BB  | ... | BB  | AB  | AA  | BB  | BB  | AB  | AA  | BB  | BB  | AA  |\n",
       "\n"
      ],
      "text/plain": [
       "  V1 V2 V3 V4 V5 V6 V7 V8 V9 V10 ... V991 V992 V993 V994 V995 V996 V997 V998\n",
       "1 BB AA BB AA BB AA AA AB AB BB  ... BB   AB   BB   AA   AA   AB   AB   AB  \n",
       "2 AB AA BB BB AB BB AA AB AA BB  ... AA   BB   BB   AA   AA   AA   BB   BB  \n",
       "3 BB AA AA AA BB AA BB AA BB AA  ... AB   AA   AA   BB   BB   AB   AA   AB  \n",
       "4 AA AB BB BB BB BB AB BB AB BB  ... AA   AA   BB   BB   AB   AB   AA   AB  \n",
       "5 AA AB AA BB AA AA AA BB AA BB  ... BB   AB   AA   BB   BB   AB   AA   BB  \n",
       "  V999 V1000\n",
       "1 AB   AB   \n",
       "2 BB   AA   \n",
       "3 AB   AB   \n",
       "4 BB   AA   \n",
       "5 BB   AA   "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "head(Genotypes)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "11. Macierz G"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\mathbf{G} = \\frac{\\textbf{MM}^{T}}{2\\sum_{i=1}^{N_{SNP}}p_{i}(1-p_{i})}$,  gdzie $\\textbf{M}_{ij}\\in\\{2-2p_{i}, 1-2p_{i}, -2p_{i}\\}$ dla genotypu $SNP_{i} = \\{AA, AB, BB\\}$ i osobnika $j$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 214,
   "metadata": {},
   "outputs": [],
   "source": [
    "tmp = data.frame(matrix(0, 3, 1))\n",
    "colnames(tmp) = c(\"Var1\")\n",
    "tmp[, 1] = c(\"AA\", \"AB\", \"BB\")\n",
    "tmp = merge(tmp, as.data.frame(table(Genotypes[, 1])), by = \"Var1\", all.x = TRUE)\n",
    "tmp[is.na(tmp)] = 0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 215,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<thead><tr><th scope=col>Var1</th><th scope=col>Freq</th></tr></thead>\n",
       "<tbody>\n",
       "\t<tr><td>AA</td><td>2 </td></tr>\n",
       "\t<tr><td>AB</td><td>1 </td></tr>\n",
       "\t<tr><td>BB</td><td>2 </td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{r|ll}\n",
       " Var1 & Freq\\\\\n",
       "\\hline\n",
       "\t AA & 2 \\\\\n",
       "\t AB & 1 \\\\\n",
       "\t BB & 2 \\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "| Var1 | Freq |\n",
       "|---|---|\n",
       "| AA | 2  |\n",
       "| AB | 1  |\n",
       "| BB | 2  |\n",
       "\n"
      ],
      "text/plain": [
       "  Var1 Freq\n",
       "1 AA   2   \n",
       "2 AB   1   \n",
       "3 BB   2   "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "tmp"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 216,
   "metadata": {},
   "outputs": [],
   "source": [
    "p = matrix(0, 3, 1000)\n",
    "for (i in 1:1000) {\n",
    "    tmp = data.frame(matrix(0, 3, 1))\n",
    "    colnames(tmp) = c(\"Var1\")\n",
    "    tmp[, 1] = c(\"AA\", \"AB\", \"BB\")\n",
    "    tmp = merge(tmp, as.data.frame(table(Genotypes[, i])), by = \"Var1\", all.x = TRUE)\n",
    "    tmp[is.na(tmp)] = 0\n",
    "    p[, i] = tmp$Freq\n",
    "}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 217,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<tbody>\n",
       "\t<tr><td>2  </td><td>3  </td><td>2  </td><td>2  </td><td>1  </td><td>3  </td><td>3  </td><td>1  </td><td>2  </td><td>1  </td><td>...</td><td>2  </td><td>2  </td><td>2  </td><td>2  </td><td>2  </td><td>1  </td><td>3  </td><td>0  </td><td>0  </td><td>3  </td></tr>\n",
       "\t<tr><td>1  </td><td>2  </td><td>0  </td><td>0  </td><td>1  </td><td>0  </td><td>1  </td><td>2  </td><td>2  </td><td>0  </td><td>...</td><td>1  </td><td>2  </td><td>0  </td><td>0  </td><td>1  </td><td>4  </td><td>1  </td><td>3  </td><td>2  </td><td>2  </td></tr>\n",
       "\t<tr><td>2  </td><td>0  </td><td>3  </td><td>3  </td><td>3  </td><td>2  </td><td>1  </td><td>2  </td><td>1  </td><td>4  </td><td>...</td><td>2  </td><td>1  </td><td>3  </td><td>3  </td><td>2  </td><td>0  </td><td>1  </td><td>2  </td><td>3  </td><td>0  </td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll}\n",
       "\t 2   & 3   & 2   & 2   & 1   & 3   & 3   & 1   & 2   & 1   & ... & 2   & 2   & 2   & 2   & 2   & 1   & 3   & 0   & 0   & 3  \\\\\n",
       "\t 1   & 2   & 0   & 0   & 1   & 0   & 1   & 2   & 2   & 0   & ... & 1   & 2   & 0   & 0   & 1   & 4   & 1   & 3   & 2   & 2  \\\\\n",
       "\t 2   & 0   & 3   & 3   & 3   & 2   & 1   & 2   & 1   & 4   & ... & 2   & 1   & 3   & 3   & 2   & 0   & 1   & 2   & 3   & 0  \\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "| 2   | 3   | 2   | 2   | 1   | 3   | 3   | 1   | 2   | 1   | ... | 2   | 2   | 2   | 2   | 2   | 1   | 3   | 0   | 0   | 3   |\n",
       "| 1   | 2   | 0   | 0   | 1   | 0   | 1   | 2   | 2   | 0   | ... | 1   | 2   | 0   | 0   | 1   | 4   | 1   | 3   | 2   | 2   |\n",
       "| 2   | 0   | 3   | 3   | 3   | 2   | 1   | 2   | 1   | 4   | ... | 2   | 1   | 3   | 3   | 2   | 0   | 1   | 2   | 3   | 0   |\n",
       "\n"
      ],
      "text/plain": [
       "     [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [,13] [,14]\n",
       "[1,] 2    3    2    2    1    3    3    1    2    1     ...   2     2     2    \n",
       "[2,] 1    2    0    0    1    0    1    2    2    0     ...   1     2     0    \n",
       "[3,] 2    0    3    3    3    2    1    2    1    4     ...   2     1     3    \n",
       "     [,15] [,16] [,17] [,18] [,19] [,20] [,21]\n",
       "[1,] 2     2     1     3     0     0     3    \n",
       "[2,] 0     1     4     1     3     2     2    \n",
       "[3,] 3     2     0     1     2     3     0    "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "p"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 218,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>3</li>\n",
       "\t<li>1000</li>\n",
       "</ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item 3\n",
       "\\item 1000\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 3\n",
       "2. 1000\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "[1]    3 1000"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "p = as.data.frame(p)\n",
    "dim(p)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 220,
   "metadata": {},
   "outputs": [],
   "source": [
    "colMax <- function(data) sapply(data, max, na.rm = TRUE)\n",
    "k_max = which(colMax(p) == 5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 221,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<dl class=dl-horizontal>\n",
       "\t<dt>V36</dt>\n",
       "\t\t<dd>36</dd>\n",
       "\t<dt>V58</dt>\n",
       "\t\t<dd>58</dd>\n",
       "\t<dt>V108</dt>\n",
       "\t\t<dd>108</dd>\n",
       "\t<dt>V131</dt>\n",
       "\t\t<dd>131</dd>\n",
       "\t<dt>V186</dt>\n",
       "\t\t<dd>186</dd>\n",
       "\t<dt>V278</dt>\n",
       "\t\t<dd>278</dd>\n",
       "\t<dt>V510</dt>\n",
       "\t\t<dd>510</dd>\n",
       "\t<dt>V573</dt>\n",
       "\t\t<dd>573</dd>\n",
       "\t<dt>V751</dt>\n",
       "\t\t<dd>751</dd>\n",
       "\t<dt>V850</dt>\n",
       "\t\t<dd>850</dd>\n",
       "</dl>\n"
      ],
      "text/latex": [
       "\\begin{description*}\n",
       "\\item[V36] 36\n",
       "\\item[V58] 58\n",
       "\\item[V108] 108\n",
       "\\item[V131] 131\n",
       "\\item[V186] 186\n",
       "\\item[V278] 278\n",
       "\\item[V510] 510\n",
       "\\item[V573] 573\n",
       "\\item[V751] 751\n",
       "\\item[V850] 850\n",
       "\\end{description*}\n"
      ],
      "text/markdown": [
       "V36\n",
       ":   36V58\n",
       ":   58V108\n",
       ":   108V131\n",
       ":   131V186\n",
       ":   186V278\n",
       ":   278V510\n",
       ":   510V573\n",
       ":   573V751\n",
       ":   751V850\n",
       ":   850\n",
       "\n"
      ],
      "text/plain": [
       " V36  V58 V108 V131 V186 V278 V510 V573 V751 V850 \n",
       "  36   58  108  131  186  278  510  573  751  850 "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "k_max"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 222,
   "metadata": {},
   "outputs": [],
   "source": [
    "Genotypes = Genotypes[,-k_max]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 223,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>5</li>\n",
       "\t<li>990</li>\n",
       "</ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item 5\n",
       "\\item 990\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 5\n",
       "2. 990\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "[1]   5 990"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "dim(Genotypes)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 224,
   "metadata": {},
   "outputs": [],
   "source": [
    "p2 = numeric(ncol(Genotypes))\n",
    "for (i in 1:ncol(Genotypes)) {\n",
    "    p2[i] = (2*p[1, i] + p[2, i]) / 10\n",
    "}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 225,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. \n",
       " 0.0000  0.4000  0.5000  0.4937  0.6000  1.0000 "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "990"
      ],
      "text/latex": [
       "990"
      ],
      "text/markdown": [
       "990"
      ],
      "text/plain": [
       "[1] 990"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "summary(p2)\n",
    "length(p2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 226,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>58</li>\n",
       "\t<li>510</li>\n",
       "</ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item 58\n",
       "\\item 510\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 58\n",
       "2. 510\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "[1]  58 510"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "(k = which(p2 == 0))\n",
    "p2 = p2[-k]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 227,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>107</li>\n",
       "\t<li>130</li>\n",
       "\t<li>571</li>\n",
       "\t<li>749</li>\n",
       "</ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item 107\n",
       "\\item 130\n",
       "\\item 571\n",
       "\\item 749\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 107\n",
       "2. 130\n",
       "3. 571\n",
       "4. 749\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "[1] 107 130 571 749"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "(k2 = which(p2 == 1))\n",
    "p2 = p2[-k2]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 228,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>5</li>\n",
       "\t<li>984</li>\n",
       "</ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item 5\n",
       "\\item 984\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 5\n",
       "2. 984\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "[1]   5 984"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "Genotypes2 = Genotypes[, -c(k, k2)]\n",
    "dim(Genotypes2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 229,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<thead><tr><th scope=col>V1</th><th scope=col>V2</th><th scope=col>V3</th><th scope=col>V4</th><th scope=col>V5</th><th scope=col>V6</th><th scope=col>V7</th><th scope=col>V8</th><th scope=col>V9</th><th scope=col>V10</th><th scope=col>...</th><th scope=col>V991</th><th scope=col>V992</th><th scope=col>V993</th><th scope=col>V994</th><th scope=col>V995</th><th scope=col>V996</th><th scope=col>V997</th><th scope=col>V998</th><th scope=col>V999</th><th scope=col>V1000</th></tr></thead>\n",
       "<tbody>\n",
       "\t<tr><td>BB </td><td>AA </td><td>BB </td><td>AA </td><td>BB </td><td>AA </td><td>AA </td><td>AB </td><td>AB </td><td>BB </td><td>...</td><td>BB </td><td>AB </td><td>BB </td><td>AA </td><td>AA </td><td>AB </td><td>AB </td><td>AB </td><td>AB </td><td>AB </td></tr>\n",
       "\t<tr><td>AB </td><td>AA </td><td>BB </td><td>BB </td><td>AB </td><td>BB </td><td>AA </td><td>AB </td><td>AA </td><td>BB </td><td>...</td><td>AA </td><td>BB </td><td>BB </td><td>AA </td><td>AA </td><td>AA </td><td>BB </td><td>BB </td><td>BB </td><td>AA </td></tr>\n",
       "\t<tr><td>BB </td><td>AA </td><td>AA </td><td>AA </td><td>BB </td><td>AA </td><td>BB </td><td>AA </td><td>BB </td><td>AA </td><td>...</td><td>AB </td><td>AA </td><td>AA </td><td>BB </td><td>BB </td><td>AB </td><td>AA </td><td>AB </td><td>AB </td><td>AB </td></tr>\n",
       "\t<tr><td>AA </td><td>AB </td><td>BB </td><td>BB </td><td>BB </td><td>BB </td><td>AB </td><td>BB </td><td>AB </td><td>BB </td><td>...</td><td>AA </td><td>AA </td><td>BB </td><td>BB </td><td>AB </td><td>AB </td><td>AA </td><td>AB </td><td>BB </td><td>AA </td></tr>\n",
       "\t<tr><td>AA </td><td>AB </td><td>AA </td><td>BB </td><td>AA </td><td>AA </td><td>AA </td><td>BB </td><td>AA </td><td>BB </td><td>...</td><td>BB </td><td>AB </td><td>AA </td><td>BB </td><td>BB </td><td>AB </td><td>AA </td><td>BB </td><td>BB </td><td>AA </td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{r|llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll}\n",
       " V1 & V2 & V3 & V4 & V5 & V6 & V7 & V8 & V9 & V10 & ... & V991 & V992 & V993 & V994 & V995 & V996 & V997 & V998 & V999 & V1000\\\\\n",
       "\\hline\n",
       "\t BB  & AA  & BB  & AA  & BB  & AA  & AA  & AB  & AB  & BB  & ... & BB  & AB  & BB  & AA  & AA  & AB  & AB  & AB  & AB  & AB \\\\\n",
       "\t AB  & AA  & BB  & BB  & AB  & BB  & AA  & AB  & AA  & BB  & ... & AA  & BB  & BB  & AA  & AA  & AA  & BB  & BB  & BB  & AA \\\\\n",
       "\t BB  & AA  & AA  & AA  & BB  & AA  & BB  & AA  & BB  & AA  & ... & AB  & AA  & AA  & BB  & BB  & AB  & AA  & AB  & AB  & AB \\\\\n",
       "\t AA  & AB  & BB  & BB  & BB  & BB  & AB  & BB  & AB  & BB  & ... & AA  & AA  & BB  & BB  & AB  & AB  & AA  & AB  & BB  & AA \\\\\n",
       "\t AA  & AB  & AA  & BB  & AA  & AA  & AA  & BB  & AA  & BB  & ... & BB  & AB  & AA  & BB  & BB  & AB  & AA  & BB  & BB  & AA \\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "| V1 | V2 | V3 | V4 | V5 | V6 | V7 | V8 | V9 | V10 | ... | V991 | V992 | V993 | V994 | V995 | V996 | V997 | V998 | V999 | V1000 |\n",
       "|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|\n",
       "| BB  | AA  | BB  | AA  | BB  | AA  | AA  | AB  | AB  | BB  | ... | BB  | AB  | BB  | AA  | AA  | AB  | AB  | AB  | AB  | AB  |\n",
       "| AB  | AA  | BB  | BB  | AB  | BB  | AA  | AB  | AA  | BB  | ... | AA  | BB  | BB  | AA  | AA  | AA  | BB  | BB  | BB  | AA  |\n",
       "| BB  | AA  | AA  | AA  | BB  | AA  | BB  | AA  | BB  | AA  | ... | AB  | AA  | AA  | BB  | BB  | AB  | AA  | AB  | AB  | AB  |\n",
       "| AA  | AB  | BB  | BB  | BB  | BB  | AB  | BB  | AB  | BB  | ... | AA  | AA  | BB  | BB  | AB  | AB  | AA  | AB  | BB  | AA  |\n",
       "| AA  | AB  | AA  | BB  | AA  | AA  | AA  | BB  | AA  | BB  | ... | BB  | AB  | AA  | BB  | BB  | AB  | AA  | BB  | BB  | AA  |\n",
       "\n"
      ],
      "text/plain": [
       "  V1 V2 V3 V4 V5 V6 V7 V8 V9 V10 ... V991 V992 V993 V994 V995 V996 V997 V998\n",
       "1 BB AA BB AA BB AA AA AB AB BB  ... BB   AB   BB   AA   AA   AB   AB   AB  \n",
       "2 AB AA BB BB AB BB AA AB AA BB  ... AA   BB   BB   AA   AA   AA   BB   BB  \n",
       "3 BB AA AA AA BB AA BB AA BB AA  ... AB   AA   AA   BB   BB   AB   AA   AB  \n",
       "4 AA AB BB BB BB BB AB BB AB BB  ... AA   AA   BB   BB   AB   AB   AA   AB  \n",
       "5 AA AB AA BB AA AA AA BB AA BB  ... BB   AB   AA   BB   BB   AB   AA   BB  \n",
       "  V999 V1000\n",
       "1 AB   AB   \n",
       "2 BB   AA   \n",
       "3 AB   AB   \n",
       "4 BB   AA   \n",
       "5 BB   AA   "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "Genotypes2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 238,
   "metadata": {},
   "outputs": [],
   "source": [
    "p2 = ifelse(p2 > 0.5, 1-p2, p2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 239,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. \n",
       " 0.1000  0.3000  0.4000  0.3535  0.4000  0.5000 "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "summary(p2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 240,
   "metadata": {},
   "outputs": [],
   "source": [
    "Genotypes2_2 = matrix(0, 5, ncol(Genotypes2))\n",
    "for (i in 1:ncol(Genotypes2)) {\n",
    "    Genotypes2_2[Genotypes2[,i] == \"AB\", i] = 1\n",
    "    Genotypes2_2[Genotypes2[,i] == \"BB\", i] = 2\n",
    "}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 241,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<tbody>\n",
       "\t<tr><td>2  </td><td>0  </td><td>2  </td><td>0  </td><td>2  </td><td>0  </td><td>0  </td><td>1  </td><td>1  </td><td>2  </td><td>...</td><td>2  </td><td>1  </td><td>2  </td><td>0  </td><td>0  </td><td>1  </td><td>1  </td><td>1  </td><td>1  </td><td>1  </td></tr>\n",
       "\t<tr><td>1  </td><td>0  </td><td>2  </td><td>2  </td><td>1  </td><td>2  </td><td>0  </td><td>1  </td><td>0  </td><td>2  </td><td>...</td><td>0  </td><td>2  </td><td>2  </td><td>0  </td><td>0  </td><td>0  </td><td>2  </td><td>2  </td><td>2  </td><td>0  </td></tr>\n",
       "\t<tr><td>2  </td><td>0  </td><td>0  </td><td>0  </td><td>2  </td><td>0  </td><td>2  </td><td>0  </td><td>2  </td><td>0  </td><td>...</td><td>1  </td><td>0  </td><td>0  </td><td>2  </td><td>2  </td><td>1  </td><td>0  </td><td>1  </td><td>1  </td><td>1  </td></tr>\n",
       "\t<tr><td>0  </td><td>1  </td><td>2  </td><td>2  </td><td>2  </td><td>2  </td><td>1  </td><td>2  </td><td>1  </td><td>2  </td><td>...</td><td>0  </td><td>0  </td><td>2  </td><td>2  </td><td>1  </td><td>1  </td><td>0  </td><td>1  </td><td>2  </td><td>0  </td></tr>\n",
       "\t<tr><td>0  </td><td>1  </td><td>0  </td><td>2  </td><td>0  </td><td>0  </td><td>0  </td><td>2  </td><td>0  </td><td>2  </td><td>...</td><td>2  </td><td>1  </td><td>0  </td><td>2  </td><td>2  </td><td>1  </td><td>0  </td><td>2  </td><td>2  </td><td>0  </td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll}\n",
       "\t 2   & 0   & 2   & 0   & 2   & 0   & 0   & 1   & 1   & 2   & ... & 2   & 1   & 2   & 0   & 0   & 1   & 1   & 1   & 1   & 1  \\\\\n",
       "\t 1   & 0   & 2   & 2   & 1   & 2   & 0   & 1   & 0   & 2   & ... & 0   & 2   & 2   & 0   & 0   & 0   & 2   & 2   & 2   & 0  \\\\\n",
       "\t 2   & 0   & 0   & 0   & 2   & 0   & 2   & 0   & 2   & 0   & ... & 1   & 0   & 0   & 2   & 2   & 1   & 0   & 1   & 1   & 1  \\\\\n",
       "\t 0   & 1   & 2   & 2   & 2   & 2   & 1   & 2   & 1   & 2   & ... & 0   & 0   & 2   & 2   & 1   & 1   & 0   & 1   & 2   & 0  \\\\\n",
       "\t 0   & 1   & 0   & 2   & 0   & 0   & 0   & 2   & 0   & 2   & ... & 2   & 1   & 0   & 2   & 2   & 1   & 0   & 2   & 2   & 0  \\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "| 2   | 0   | 2   | 0   | 2   | 0   | 0   | 1   | 1   | 2   | ... | 2   | 1   | 2   | 0   | 0   | 1   | 1   | 1   | 1   | 1   |\n",
       "| 1   | 0   | 2   | 2   | 1   | 2   | 0   | 1   | 0   | 2   | ... | 0   | 2   | 2   | 0   | 0   | 0   | 2   | 2   | 2   | 0   |\n",
       "| 2   | 0   | 0   | 0   | 2   | 0   | 2   | 0   | 2   | 0   | ... | 1   | 0   | 0   | 2   | 2   | 1   | 0   | 1   | 1   | 1   |\n",
       "| 0   | 1   | 2   | 2   | 2   | 2   | 1   | 2   | 1   | 2   | ... | 0   | 0   | 2   | 2   | 1   | 1   | 0   | 1   | 2   | 0   |\n",
       "| 0   | 1   | 0   | 2   | 0   | 0   | 0   | 2   | 0   | 2   | ... | 2   | 1   | 0   | 2   | 2   | 1   | 0   | 2   | 2   | 0   |\n",
       "\n"
      ],
      "text/plain": [
       "     [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [,13] [,14]\n",
       "[1,] 2    0    2    0    2    0    0    1    1    2     ...   2     1     2    \n",
       "[2,] 1    0    2    2    1    2    0    1    0    2     ...   0     2     2    \n",
       "[3,] 2    0    0    0    2    0    2    0    2    0     ...   1     0     0    \n",
       "[4,] 0    1    2    2    2    2    1    2    1    2     ...   0     0     2    \n",
       "[5,] 0    1    0    2    0    0    0    2    0    2     ...   2     1     0    \n",
       "     [,15] [,16] [,17] [,18] [,19] [,20] [,21]\n",
       "[1,] 0     0     1     1     1     1     1    \n",
       "[2,] 0     0     0     2     2     2     0    \n",
       "[3,] 2     2     1     0     1     1     1    \n",
       "[4,] 2     1     1     0     1     2     0    \n",
       "[5,] 2     2     1     0     2     2     0    "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "Genotypes2_2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 242,
   "metadata": {},
   "outputs": [],
   "source": [
    "M = matrix(0, 5, length(p2))\n",
    "for (i in 1:5) {\n",
    "    for (j in 1:length(p2)) {\n",
    "       if (Genotypes2_2[i, j] == 0) {\n",
    "           M[i, j] = - 2*p2[j]\n",
    "       } else if (Genotypes2_2[i, j] == 1) {\n",
    "           M[i, j] = 1 - 2*p2[j]\n",
    "       } else {\n",
    "           M[i, j] = 2 - 2*p2[j]\n",
    "       }\n",
    "    }\n",
    "}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 243,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<tbody>\n",
       "\t<tr><td> 1  </td><td>-0.4</td><td> 1.2</td><td>-0.8</td><td> 1.4</td><td>-0.8</td><td>-0.6</td><td> 0.2</td><td> 0.2</td><td> 1.6</td><td>... </td><td> 1  </td><td> 0.4</td><td> 1.4</td><td>-0.6</td><td>-0.4</td><td> 0.8</td><td> 0.8</td><td>0   </td><td>0   </td><td> 0  </td></tr>\n",
       "\t<tr><td> 0  </td><td>-0.4</td><td> 1.2</td><td> 1.2</td><td> 0.4</td><td> 1.2</td><td>-0.6</td><td> 0.2</td><td>-0.8</td><td> 1.6</td><td>... </td><td>-1  </td><td> 1.4</td><td> 1.4</td><td>-0.6</td><td>-0.4</td><td>-0.2</td><td> 1.8</td><td>1   </td><td>1   </td><td>-1  </td></tr>\n",
       "\t<tr><td> 1  </td><td>-0.4</td><td>-0.8</td><td>-0.8</td><td> 1.4</td><td>-0.8</td><td> 1.4</td><td>-0.8</td><td> 1.2</td><td>-0.4</td><td>... </td><td> 0  </td><td>-0.6</td><td>-0.6</td><td> 1.4</td><td> 1.6</td><td> 0.8</td><td>-0.2</td><td>0   </td><td>0   </td><td> 0  </td></tr>\n",
       "\t<tr><td>-1  </td><td> 0.6</td><td> 1.2</td><td> 1.2</td><td> 1.4</td><td> 1.2</td><td> 0.4</td><td> 1.2</td><td> 0.2</td><td> 1.6</td><td>... </td><td>-1  </td><td>-0.6</td><td> 1.4</td><td> 1.4</td><td> 0.6</td><td> 0.8</td><td>-0.2</td><td>0   </td><td>1   </td><td>-1  </td></tr>\n",
       "\t<tr><td>-1  </td><td> 0.6</td><td>-0.8</td><td> 1.2</td><td>-0.6</td><td>-0.8</td><td>-0.6</td><td> 1.2</td><td>-0.8</td><td> 1.6</td><td>... </td><td> 1  </td><td> 0.4</td><td>-0.6</td><td> 1.4</td><td> 1.6</td><td> 0.8</td><td>-0.2</td><td>1   </td><td>1   </td><td>-1  </td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll}\n",
       "\t  1   & -0.4 &  1.2 & -0.8 &  1.4 & -0.8 & -0.6 &  0.2 &  0.2 &  1.6 & ...  &  1   &  0.4 &  1.4 & -0.6 & -0.4 &  0.8 &  0.8 & 0    & 0    &  0  \\\\\n",
       "\t  0   & -0.4 &  1.2 &  1.2 &  0.4 &  1.2 & -0.6 &  0.2 & -0.8 &  1.6 & ...  & -1   &  1.4 &  1.4 & -0.6 & -0.4 & -0.2 &  1.8 & 1    & 1    & -1  \\\\\n",
       "\t  1   & -0.4 & -0.8 & -0.8 &  1.4 & -0.8 &  1.4 & -0.8 &  1.2 & -0.4 & ...  &  0   & -0.6 & -0.6 &  1.4 &  1.6 &  0.8 & -0.2 & 0    & 0    &  0  \\\\\n",
       "\t -1   &  0.6 &  1.2 &  1.2 &  1.4 &  1.2 &  0.4 &  1.2 &  0.2 &  1.6 & ...  & -1   & -0.6 &  1.4 &  1.4 &  0.6 &  0.8 & -0.2 & 0    & 1    & -1  \\\\\n",
       "\t -1   &  0.6 & -0.8 &  1.2 & -0.6 & -0.8 & -0.6 &  1.2 & -0.8 &  1.6 & ...  &  1   &  0.4 & -0.6 &  1.4 &  1.6 &  0.8 & -0.2 & 1    & 1    & -1  \\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "|  1   | -0.4 |  1.2 | -0.8 |  1.4 | -0.8 | -0.6 |  0.2 |  0.2 |  1.6 | ...  |  1   |  0.4 |  1.4 | -0.6 | -0.4 |  0.8 |  0.8 | 0    | 0    |  0   |\n",
       "|  0   | -0.4 |  1.2 |  1.2 |  0.4 |  1.2 | -0.6 |  0.2 | -0.8 |  1.6 | ...  | -1   |  1.4 |  1.4 | -0.6 | -0.4 | -0.2 |  1.8 | 1    | 1    | -1   |\n",
       "|  1   | -0.4 | -0.8 | -0.8 |  1.4 | -0.8 |  1.4 | -0.8 |  1.2 | -0.4 | ...  |  0   | -0.6 | -0.6 |  1.4 |  1.6 |  0.8 | -0.2 | 0    | 0    |  0   |\n",
       "| -1   |  0.6 |  1.2 |  1.2 |  1.4 |  1.2 |  0.4 |  1.2 |  0.2 |  1.6 | ...  | -1   | -0.6 |  1.4 |  1.4 |  0.6 |  0.8 | -0.2 | 0    | 1    | -1   |\n",
       "| -1   |  0.6 | -0.8 |  1.2 | -0.6 | -0.8 | -0.6 |  1.2 | -0.8 |  1.6 | ...  |  1   |  0.4 | -0.6 |  1.4 |  1.6 |  0.8 | -0.2 | 1    | 1    | -1   |\n",
       "\n"
      ],
      "text/plain": [
       "     [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [,13] [,14]\n",
       "[1,]  1   -0.4  1.2 -0.8  1.4 -0.8 -0.6  0.2  0.2  1.6  ...    1     0.4   1.4 \n",
       "[2,]  0   -0.4  1.2  1.2  0.4  1.2 -0.6  0.2 -0.8  1.6  ...   -1     1.4   1.4 \n",
       "[3,]  1   -0.4 -0.8 -0.8  1.4 -0.8  1.4 -0.8  1.2 -0.4  ...    0    -0.6  -0.6 \n",
       "[4,] -1    0.6  1.2  1.2  1.4  1.2  0.4  1.2  0.2  1.6  ...   -1    -0.6   1.4 \n",
       "[5,] -1    0.6 -0.8  1.2 -0.6 -0.8 -0.6  1.2 -0.8  1.6  ...    1     0.4  -0.6 \n",
       "     [,15] [,16] [,17] [,18] [,19] [,20] [,21]\n",
       "[1,] -0.6  -0.4   0.8   0.8  0     0      0   \n",
       "[2,] -0.6  -0.4  -0.2   1.8  1     1     -1   \n",
       "[3,]  1.4   1.6   0.8  -0.2  0     0      0   \n",
       "[4,]  1.4   0.6   0.8  -0.2  0     1     -1   \n",
       "[5,]  1.4   1.6   0.8  -0.2  1     1     -1   "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "M"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 246,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Initial data: \n",
      "\tNumber of Individuals: 5 \n",
      "\tNumber of Markers: 984 \n",
      "\n",
      "Missing data check: \n",
      "\tTotal SNPs: 984 \n",
      "\t 0 SNPs dropped due to missing data threshold of 1 \n",
      "\tTotal of: 984  SNPs \n",
      "MAF check: \n",
      "\tNo SNPs with MAF below 0 \n",
      "Monomorphic check: \n",
      "\tNo monomorphic SNPs \n",
      "Summary check: \n",
      "\tInitial:  984 SNPs \n",
      "\tFinal:  984  SNPs ( 0  SNPs removed) \n",
      " \n",
      "Completed! Time = 0.04  seconds \n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<tbody>\n",
       "\t<tr><td> 1.2288695</td><td>-0.2286356</td><td>-0.3235886</td><td>-0.3399598</td><td>-0.3366855</td></tr>\n",
       "\t<tr><td>-0.2286356</td><td> 1.2536601</td><td>-0.2702652</td><td>-0.3591375</td><td>-0.3956219</td></tr>\n",
       "\t<tr><td>-0.3235886</td><td>-0.2702652</td><td> 1.2227887</td><td>-0.2927172</td><td>-0.3362178</td></tr>\n",
       "\t<tr><td>-0.3399598</td><td>-0.3591375</td><td>-0.2927172</td><td> 1.2017400</td><td>-0.2099256</td></tr>\n",
       "\t<tr><td>-0.3366855</td><td>-0.3956219</td><td>-0.3362178</td><td>-0.2099256</td><td> 1.2784508</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{lllll}\n",
       "\t  1.2288695 & -0.2286356 & -0.3235886 & -0.3399598 & -0.3366855\\\\\n",
       "\t -0.2286356 &  1.2536601 & -0.2702652 & -0.3591375 & -0.3956219\\\\\n",
       "\t -0.3235886 & -0.2702652 &  1.2227887 & -0.2927172 & -0.3362178\\\\\n",
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       "\t -0.3366855 & -0.3956219 & -0.3362178 & -0.2099256 &  1.2784508\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "|  1.2288695 | -0.2286356 | -0.3235886 | -0.3399598 | -0.3366855 |\n",
       "| -0.2286356 |  1.2536601 | -0.2702652 | -0.3591375 | -0.3956219 |\n",
       "| -0.3235886 | -0.2702652 |  1.2227887 | -0.2927172 | -0.3362178 |\n",
       "| -0.3399598 | -0.3591375 | -0.2927172 |  1.2017400 | -0.2099256 |\n",
       "| -0.3366855 | -0.3956219 | -0.3362178 | -0.2099256 |  1.2784508 |\n",
       "\n"
      ],
      "text/plain": [
       "     [,1]       [,2]       [,3]       [,4]       [,5]      \n",
       "[1,]  1.2288695 -0.2286356 -0.3235886 -0.3399598 -0.3366855\n",
       "[2,] -0.2286356  1.2536601 -0.2702652 -0.3591375 -0.3956219\n",
       "[3,] -0.3235886 -0.2702652  1.2227887 -0.2927172 -0.3362178\n",
       "[4,] -0.3399598 -0.3591375 -0.2927172  1.2017400 -0.2099256\n",
       "[5,] -0.3366855 -0.3956219 -0.3362178 -0.2099256  1.2784508"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "Gmatrix(SNPmatrix = Genotypes2_2,\n",
    "  method = \"VanRaden\",\n",
    "  missingValue = -9,\n",
    "  maf = 0.0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "12. Wrzucenie macierzy G do modelu"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\mathbf{y} = \\mathbf{X\\beta+Z_{1}a+Z_{2}g+\\epsilon}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<tbody>\n",
       "\t<tr><td> 4.341304960</td></tr>\n",
       "\t<tr><td> 3.401214252</td></tr>\n",
       "\t<tr><td>-0.003452949</td></tr>\n",
       "\t<tr><td>-0.055133673</td></tr>\n",
       "\t<tr><td> 0.054219329</td></tr>\n",
       "\t<tr><td>-0.072283409</td></tr>\n",
       "\t<tr><td> 0.054390036</td></tr>\n",
       "\t<tr><td> 0.087512029</td></tr>\n",
       "\t<tr><td>-0.246672434</td></tr>\n",
       "\t<tr><td> 0.245158275</td></tr>\n",
       "\t<tr><td>-0.425100763</td></tr>\n",
       "\t<tr><td> 0.335020177</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
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      "text/latex": [
       "\\begin{tabular}{l}\n",
       "\t  4.341304960\\\\\n",
       "\t  3.401214252\\\\\n",
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       "\t -0.055133673\\\\\n",
       "\t  0.054219329\\\\\n",
       "\t -0.072283409\\\\\n",
       "\t  0.054390036\\\\\n",
       "\t  0.087512029\\\\\n",
       "\t -0.246672434\\\\\n",
       "\t  0.245158275\\\\\n",
       "\t -0.425100763\\\\\n",
       "\t  0.335020177\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "|  4.341304960 |\n",
       "|  3.401214252 |\n",
       "| -0.003452949 |\n",
       "| -0.055133673 |\n",
       "|  0.054219329 |\n",
       "| -0.072283409 |\n",
       "|  0.054390036 |\n",
       "|  0.087512029 |\n",
       "| -0.246672434 |\n",
       "|  0.245158275 |\n",
       "| -0.425100763 |\n",
       "|  0.335020177 |\n",
       "\n"
      ],
      "text/plain": [
       "      [,1]        \n",
       " [1,]  4.341304960\n",
       " [2,]  3.401214252\n",
       " [3,] -0.003452949\n",
       " [4,] -0.055133673\n",
       " [5,]  0.054219329\n",
       " [6,] -0.072283409\n",
       " [7,]  0.054390036\n",
       " [8,]  0.087512029\n",
       " [9,] -0.246672434\n",
       "[10,]  0.245158275\n",
       "[11,] -0.425100763\n",
       "[12,]  0.335020177"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "mme2 = function(y, X, Z1, Z2, A, G, sigma_a, sigma_g, sigma_e) {\n",
    "    alpha1 = sigma_e / sigma_a\n",
    "    alpha2 = sigma_e / sigma_g\n",
    "    invA = ginv(A)\n",
    "    invG = ginv(G)\n",
    "    C = rbind(cbind(t(X)%*%X, t(X)%*%Z1, t(X)%*%Z2),\n",
    "              cbind(t(Z1)%*%X, t(Z1)%*%Z1+invA*c(alpha1), t(Z1)%*%Z2),\n",
    "              cbind(t(Z2)%*%X, t(Z2)%*%Z1, t(Z2)%*%Z2 + invG*c(alpha2)))\n",
    "    rhs = rbind(t(X)%*%y, t(Z1)%*%y, t(Z2)%*%y)\n",
    "    invC = ginv(C)\n",
    "    estimators = invC%*%rhs\n",
    "    list(C = C, est = estimators)\n",
    "}\n",
    "\n",
    "(est = mme2(y, X, diag(5), diag(5), A[4:8, 4:8], G, 10, 30, 40)$est)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "13. Efekty markerów SNP"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\widehat{SNP} = \\left(\\left(\\frac{1-k}{N_{SNP}}\\right)^{-1} + \\frac{1}{k}\\textbf{MA}^{-1}\\textbf{M}^{T} \\right)^{-1}\\frac{1}{k}\\textbf{MA}^{-1}\\widehat{g}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [],
   "source": [
    "k = 0.2\n",
    "snp = ginv((t(M)%*%ginv(A)[4:8, 4:8]%*%M)/k + 1 / ((1-k)/length(p2)) ) / k "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [],
   "source": [
    "snp = snp%*%t(M)%*%ginv(A)[4:8, 4:8]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [],
   "source": [
    "snp = snp %*% est[8:12]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
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       "<tbody>\n",
       "\t<tr><td>-0.0013010329</td></tr>\n",
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       "\t  0.0011944542\\\\\n",
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       "\t -0.0002694417\\\\\n",
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       "\\end{tabular}\n"
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      "text/markdown": [
       "\n",
       "| -0.0013010329 |\n",
       "| -0.0006039244 |\n",
       "|  0.0011944542 |\n",
       "| -0.0003928861 |\n",
       "| -0.0002694417 |\n",
       "| -0.0002621532 |\n",
       "\n"
      ],
      "text/plain": [
       "     [,1]         \n",
       "[1,] -0.0013010329\n",
       "[2,] -0.0006039244\n",
       "[3,]  0.0011944542\n",
       "[4,] -0.0003928861\n",
       "[5,] -0.0002694417\n",
       "[6,] -0.0002621532"
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     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "head(snp)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "14. Czy efekty SNP są statystycznie istotne?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "0.000776102778820242"
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       "0.000776102778820242"
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       "0.000776102778820242"
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      "text/plain": [
       "[1] 0.0007761028"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "-5.11112732482814e-20"
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      "text/latex": [
       "-5.11112732482814e-20"
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       "-5.11112732482814e-20"
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      "text/plain": [
       "[1] -5.111127e-20"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "(s = sd(snp))\n",
    "(m= mean(snp))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "1"
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      "text/latex": [
       "1"
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      "text/markdown": [
       "1"
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       "[1] 1"
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     },
     "metadata": {},
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   ],
   "source": [
    "snp2 = snp / s\n",
    "sd(snp2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [],
   "source": [
    "W = snp2\n",
    "p.value = 2*pnorm(abs(W), lower.tail = FALSE)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "40"
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       "40"
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       "[1] 40"
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     "metadata": {},
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   "source": [
    "length(which(p.value < 0.05))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "15. Estymacja efektów markerów - metoda GBLUP"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\mathbf{y} = \\mathbf{X\\beta+Z_{1}a+Z_{2}g+\\epsilon}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$Z_{2} = \\{-1, 0, 1\\}$ odpowiednio dla genotypów $\\{AA, AB, BB\\}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Efekt $\\mathbf{g}\\sim\\mathcal{N}(0, \\mathbf{I}\\cdot\\frac{\\sigma^{2}_{a}}{N_{SNP}})$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [],
   "source": [
    "Z2 = matrix(0, 5, length(p2))\n",
    "for (i in 1:5) {\n",
    "    for (j in 1:length(p2)) {\n",
    "       if (Genotypes2[i, j] == \"AA\") {\n",
    "           Z2[i, j] = -1\n",
    "       } else if (Genotypes2[i, j] == \"AB\") {\n",
    "           Z2[i, j] = 0\n",
    "       } else {\n",
    "           Z2[i, j] = 1\n",
    "       }\n",
    "    }\n",
    "}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
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       "\t -1  &  1  &  1  &  1  & -1  & 1   &  0  & 1   &  1  & 1   & ... &  1  &  0  &  0  &  1  & -1  & 1   &  1  &  0  &  1  &  0 \\\\\n",
       "\t  0  &  0  &  1  &  0  & -1  & 0   & -1  & 1   &  0  & 1   & ... & -1  & -1  &  1  & -1  & -1  & 1   &  1  &  1  &  1  &  1 \\\\\n",
       "\t -1  & -1  &  1  & -1  & -1  & 0   & -1  & 0   &  0  & 1   & ... &  0  &  1  &  1  & -1  &  0  & 0   & -1  & -1  &  0  &  0 \\\\\n",
       "\t  1  &  1  & -1  &  0  & -1  & 1   & -1  & 0   & -1  & 0   & ... &  1  &  1  &  0  &  0  &  1  & 0   &  1  &  0  & -1  & -1 \\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "| -1  | -1  | -1  |  1  |  1  | 1   |  0  | 1   |  1  | 0   | ... |  0  |  0  | -1  | -1  |  1  | 1   |  0  |  1  |  0  |  1  |\n",
       "| -1  |  1  |  1  |  1  | -1  | 1   |  0  | 1   |  1  | 1   | ... |  1  |  0  |  0  |  1  | -1  | 1   |  1  |  0  |  1  |  0  |\n",
       "|  0  |  0  |  1  |  0  | -1  | 0   | -1  | 1   |  0  | 1   | ... | -1  | -1  |  1  | -1  | -1  | 1   |  1  |  1  |  1  |  1  |\n",
       "| -1  | -1  |  1  | -1  | -1  | 0   | -1  | 0   |  0  | 1   | ... |  0  |  1  |  1  | -1  |  0  | 0   | -1  | -1  |  0  |  0  |\n",
       "|  1  |  1  | -1  |  0  | -1  | 1   | -1  | 0   | -1  | 0   | ... |  1  |  1  |  0  |  0  |  1  | 0   |  1  |  0  | -1  | -1  |\n",
       "\n"
      ],
      "text/plain": [
       "     [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [,13] [,14]\n",
       "[1,] -1   -1   -1    1    1   1     0   1     1   0     ...    0     0    -1   \n",
       "[2,] -1    1    1    1   -1   1     0   1     1   1     ...    1     0     0   \n",
       "[3,]  0    0    1    0   -1   0    -1   1     0   1     ...   -1    -1     1   \n",
       "[4,] -1   -1    1   -1   -1   0    -1   0     0   1     ...    0     1     1   \n",
       "[5,]  1    1   -1    0   -1   1    -1   0    -1   0     ...    1     1     0   \n",
       "     [,15] [,16] [,17] [,18] [,19] [,20] [,21]\n",
       "[1,] -1     1    1      0     1     0     1   \n",
       "[2,]  1    -1    1      1     0     1     0   \n",
       "[3,] -1    -1    1      1     1     1     1   \n",
       "[4,] -1     0    0     -1    -1     0     0   \n",
       "[5,]  0     1    0      1     0    -1    -1   "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "head(Z2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [],
   "source": [
    "sigma_a = 0.6555448 \n",
    "sigma_e = 0.01799635\n",
    "sigma_g = sigma_a / length(p2)\n",
    "G = diag(length(p2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [],
   "source": [
    "res2 = mme2(y, X, diag(5), Z2, A[4:8, 4:8], G, sigma_a, sigma_g, sigma_e)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>999</li>\n",
       "\t<li>999</li>\n",
       "</ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item 999\n",
       "\\item 999\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 999\n",
       "2. 999\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "[1] 999 999"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "dim(res2$C)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>0.00107632007119809</li>\n",
       "\t<li>0.00056523214301077</li>\n",
       "\t<li>-0.00117104593778478</li>\n",
       "\t<li>0.000505113920098791</li>\n",
       "\t<li>0.000350269830614274</li>\n",
       "\t<li>0.000329979004801596</li>\n",
       "</ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item 0.00107632007119809\n",
       "\\item 0.00056523214301077\n",
       "\\item -0.00117104593778478\n",
       "\\item 0.000505113920098791\n",
       "\\item 0.000350269830614274\n",
       "\\item 0.000329979004801596\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 0.00107632007119809\n",
       "2. 0.00056523214301077\n",
       "3. -0.00117104593778478\n",
       "4. 0.000505113920098791\n",
       "5. 0.000350269830614274\n",
       "6. 0.000329979004801596\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "[1]  0.0010763201  0.0005652321 -0.0011710459  0.0005051139  0.0003502698\n",
       "[6]  0.0003299790"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "est_snp2 = res2$est[8:999]\n",
    "head(est_snp2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "0.000691430096493191"
      ],
      "text/latex": [
       "0.000691430096493191"
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      "text/markdown": [
       "0.000691430096493191"
      ],
      "text/plain": [
       "[1] 0.0006914301"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "-8.63480392554709e-06"
      ],
      "text/latex": [
       "-8.63480392554709e-06"
      ],
      "text/markdown": [
       "-8.63480392554709e-06"
      ],
      "text/plain": [
       "[1] -8.634804e-06"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "(s = sd(est_snp2))\n",
    "(m= mean(est_snp2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "1"
      ],
      "text/latex": [
       "1"
      ],
      "text/markdown": [
       "1"
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      "text/plain": [
       "[1] 1"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "est_snp2 = est_snp2 / s\n",
    "sd(est_snp2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {},
   "outputs": [],
   "source": [
    "W2 = est_snp2\n",
    "p.value2 = 2*pnorm(abs(W2), lower.tail = FALSE)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "34"
      ],
      "text/latex": [
       "34"
      ],
      "text/markdown": [
       "34"
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      "text/plain": [
       "[1] 34"
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     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "length(which(p.value2 < 0.05))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>38</li>\n",
       "\t<li>42</li>\n",
       "\t<li>69</li>\n",
       "\t<li>75</li>\n",
       "\t<li>93</li>\n",
       "\t<li>107</li>\n",
       "\t<li>136</li>\n",
       "\t<li>138</li>\n",
       "\t<li>145</li>\n",
       "\t<li>204</li>\n",
       "\t<li>223</li>\n",
       "\t<li>230</li>\n",
       "\t<li>257</li>\n",
       "\t<li>268</li>\n",
       "\t<li>302</li>\n",
       "\t<li>312</li>\n",
       "\t<li>346</li>\n",
       "\t<li>349</li>\n",
       "\t<li>376</li>\n",
       "\t<li>434</li>\n",
       "\t<li>436</li>\n",
       "\t<li>452</li>\n",
       "\t<li>462</li>\n",
       "\t<li>463</li>\n",
       "\t<li>507</li>\n",
       "\t<li>543</li>\n",
       "\t<li>561</li>\n",
       "\t<li>576</li>\n",
       "\t<li>604</li>\n",
       "\t<li>626</li>\n",
       "\t<li>695</li>\n",
       "\t<li>715</li>\n",
       "\t<li>733</li>\n",
       "\t<li>739</li>\n",
       "\t<li>768</li>\n",
       "\t<li>787</li>\n",
       "\t<li>791</li>\n",
       "\t<li>799</li>\n",
       "\t<li>811</li>\n",
       "\t<li>972</li>\n",
       "</ol>\n"
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      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item 38\n",
       "\\item 42\n",
       "\\item 69\n",
       "\\item 75\n",
       "\\item 93\n",
       "\\item 107\n",
       "\\item 136\n",
       "\\item 138\n",
       "\\item 145\n",
       "\\item 204\n",
       "\\item 223\n",
       "\\item 230\n",
       "\\item 257\n",
       "\\item 268\n",
       "\\item 302\n",
       "\\item 312\n",
       "\\item 346\n",
       "\\item 349\n",
       "\\item 376\n",
       "\\item 434\n",
       "\\item 436\n",
       "\\item 452\n",
       "\\item 462\n",
       "\\item 463\n",
       "\\item 507\n",
       "\\item 543\n",
       "\\item 561\n",
       "\\item 576\n",
       "\\item 604\n",
       "\\item 626\n",
       "\\item 695\n",
       "\\item 715\n",
       "\\item 733\n",
       "\\item 739\n",
       "\\item 768\n",
       "\\item 787\n",
       "\\item 791\n",
       "\\item 799\n",
       "\\item 811\n",
       "\\item 972\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 38\n",
       "2. 42\n",
       "3. 69\n",
       "4. 75\n",
       "5. 93\n",
       "6. 107\n",
       "7. 136\n",
       "8. 138\n",
       "9. 145\n",
       "10. 204\n",
       "11. 223\n",
       "12. 230\n",
       "13. 257\n",
       "14. 268\n",
       "15. 302\n",
       "16. 312\n",
       "17. 346\n",
       "18. 349\n",
       "19. 376\n",
       "20. 434\n",
       "21. 436\n",
       "22. 452\n",
       "23. 462\n",
       "24. 463\n",
       "25. 507\n",
       "26. 543\n",
       "27. 561\n",
       "28. 576\n",
       "29. 604\n",
       "30. 626\n",
       "31. 695\n",
       "32. 715\n",
       "33. 733\n",
       "34. 739\n",
       "35. 768\n",
       "36. 787\n",
       "37. 791\n",
       "38. 799\n",
       "39. 811\n",
       "40. 972\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       " [1]  38  42  69  75  93 107 136 138 145 204 223 230 257 268 302 312 346 349 376\n",
       "[20] 434 436 452 462 463 507 543 561 576 604 626 695 715 733 739 768 787 791 799\n",
       "[39] 811 972"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
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     "data": {
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       "<ol class=list-inline>\n",
       "\t<li>38</li>\n",
       "\t<li>42</li>\n",
       "\t<li>69</li>\n",
       "\t<li>75</li>\n",
       "\t<li>93</li>\n",
       "\t<li>107</li>\n",
       "\t<li>136</li>\n",
       "\t<li>138</li>\n",
       "\t<li>145</li>\n",
       "\t<li>204</li>\n",
       "\t<li>223</li>\n",
       "\t<li>230</li>\n",
       "\t<li>257</li>\n",
       "\t<li>268</li>\n",
       "\t<li>312</li>\n",
       "\t<li>346</li>\n",
       "\t<li>349</li>\n",
       "\t<li>376</li>\n",
       "\t<li>436</li>\n",
       "\t<li>452</li>\n",
       "\t<li>462</li>\n",
       "\t<li>463</li>\n",
       "\t<li>507</li>\n",
       "\t<li>561</li>\n",
       "\t<li>576</li>\n",
       "\t<li>604</li>\n",
       "\t<li>626</li>\n",
       "\t<li>695</li>\n",
       "\t<li>739</li>\n",
       "\t<li>768</li>\n",
       "\t<li>787</li>\n",
       "\t<li>791</li>\n",
       "\t<li>811</li>\n",
       "\t<li>972</li>\n",
       "</ol>\n"
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      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item 38\n",
       "\\item 42\n",
       "\\item 69\n",
       "\\item 75\n",
       "\\item 93\n",
       "\\item 107\n",
       "\\item 136\n",
       "\\item 138\n",
       "\\item 145\n",
       "\\item 204\n",
       "\\item 223\n",
       "\\item 230\n",
       "\\item 257\n",
       "\\item 268\n",
       "\\item 312\n",
       "\\item 346\n",
       "\\item 349\n",
       "\\item 376\n",
       "\\item 436\n",
       "\\item 452\n",
       "\\item 462\n",
       "\\item 463\n",
       "\\item 507\n",
       "\\item 561\n",
       "\\item 576\n",
       "\\item 604\n",
       "\\item 626\n",
       "\\item 695\n",
       "\\item 739\n",
       "\\item 768\n",
       "\\item 787\n",
       "\\item 791\n",
       "\\item 811\n",
       "\\item 972\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 38\n",
       "2. 42\n",
       "3. 69\n",
       "4. 75\n",
       "5. 93\n",
       "6. 107\n",
       "7. 136\n",
       "8. 138\n",
       "9. 145\n",
       "10. 204\n",
       "11. 223\n",
       "12. 230\n",
       "13. 257\n",
       "14. 268\n",
       "15. 312\n",
       "16. 346\n",
       "17. 349\n",
       "18. 376\n",
       "19. 436\n",
       "20. 452\n",
       "21. 462\n",
       "22. 463\n",
       "23. 507\n",
       "24. 561\n",
       "25. 576\n",
       "26. 604\n",
       "27. 626\n",
       "28. 695\n",
       "29. 739\n",
       "30. 768\n",
       "31. 787\n",
       "32. 791\n",
       "33. 811\n",
       "34. 972\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       " [1]  38  42  69  75  93 107 136 138 145 204 223 230 257 268 312 346 349 376 436\n",
       "[20] 452 462 463 507 561 576 604 626 695 739 768 787 791 811 972"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "which(p.value < 0.05)\n",
    "which(p.value2 < 0.05)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "16. Model wielocechowy"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Mamy $\\textbf{y}_{1} = \\textbf{X}_{1}\\beta_{1}+\\textbf{Z}_{1}\\textbf{a}_{1}+\\epsilon_{1}$ oraz $\\textbf{y}_{2} = \\textbf{X}_{2}\\beta_{2}+\\textbf{Z}_{2}\\textbf{a}_{2}+\\epsilon_{2}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$ \\left[ \\begin{array}{c}\n",
    "         \\mathbf{y}_{1} \\\\\n",
    "         \\mathbf{y}_{2}\\end{array}\\right] = \n",
    "  \\left[ \\begin{array}{cc}\n",
    "         \\mathbf{X}_{1} & \\mathbf{0} \\\\\n",
    "         \\mathbf{0} & \\mathbf{X}_{2} \\end{array}\\right]\\cdot\n",
    "  \\left[ \\begin{array}{c}\n",
    "         \\mathbf{\\beta}_{1} \\\\\n",
    "         \\mathbf{\\beta}_{2} \\end{array}\\right] +\n",
    "  \\left[ \\begin{array}{cc}\n",
    "         \\mathbf{Z}_{1} & \\mathbf{0} \\\\\n",
    "         \\mathbf{0} & \\mathbf{Z}_{2} \\end{array}\\right]\\cdot\n",
    "  \\left[ \\begin{array}{c}\n",
    "         \\mathbf{a}_{1} \\\\\n",
    "         \\mathbf{a}_{2} \\end{array}\\right] +\n",
    "  \\left[ \\begin{array}{c}\n",
    "         \\mathbf{\\epsilon}_{1} \\\\\n",
    "         \\mathbf{\\epsilon}_{2}\\end{array}\\right]$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Korelacja pomiedzy efektami"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$var\\left[ \\begin{array}{c}\n",
    "         \\mathbf{a}_{1} \\\\\n",
    "         \\mathbf{a}_{2} \\\\ \n",
    "         \\mathbf{\\epsilon}_{1} \\\\\n",
    "         \\mathbf{\\epsilon}_{2}\n",
    "         \\end{array}\\right] = \n",
    "    \\left[ \\begin{array}{cccc}\n",
    "         \\mathbf{g}_{11}\\mathbf{A} & \\mathbf{g}_{12}\\mathbf{A} & \\mathbf{0} & \\mathbf{0} \\\\\n",
    "         \\mathbf{g}_{21}\\mathbf{A} & \\mathbf{g}_{22}\\mathbf{A} & \\mathbf{0} & \\mathbf{0} \\\\ \n",
    "         \\mathbf{0} & \\mathbf{0} & \\mathbf{r}_{11}\\mathbf{I} & \\mathbf{r}_{12}\\mathbf{I} \\\\\n",
    "         \\mathbf{0} & \\mathbf{0} & \\mathbf{r}_{21}\\mathbf{I} & \\mathbf{r}_{22}\\mathbf{I}\n",
    "         \\end{array}\\right] $"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Układ równań mieszanych"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$ \\left[ \\begin{array}{cc}\n",
    "         \\mathbf{X}^{T}\\mathbf{R}^{-1}\\mathbf{X} & \\mathbf{X}^{T}\\mathbf{R}^{-1}\\mathbf{Z} \\\\\n",
    "         \\mathbf{Z}^{T}\\mathbf{R}^{-1}\\mathbf{X} & \\mathbf{Z}^{T}\\mathbf{R}^{-1}\\mathbf{Z}+\\mathbf{A}^{-1}\\otimes \\mathbf{G}^{-1} \\end{array}\\right]\\cdot\n",
    "  \\left[ \\begin{array}{c}\n",
    "         \\widehat{\\mathbf{\\beta}} \\\\\n",
    "         \\widehat{\\mathbf{a}} \\end{array}\\right]=\n",
    "  \\left[ \\begin{array}{c}\n",
    "         \\mathbf{X}^{T}\\mathbf{R}^{-1}\\mathbf{y} \\\\\n",
    "         \\mathbf{Z}^{T}\\mathbf{R}^{-1}\\mathbf{y} \\end{array}\\right]$, gdzie"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$ \\mathbf{y} = \\left[ \\begin{array}{c}\n",
    "         \\mathbf{y}_{1} \\\\\n",
    "         \\mathbf{y}_{2}\\end{array}\\right]$, $ \\mathbf{X} = \\left[ \\begin{array}{cc}\n",
    "         \\mathbf{X}_{1} & \\mathbf{0} \\\\\n",
    "         \\mathbf{0} & \\mathbf{X}_{2} \\end{array}\\right]$, \n",
    "$ \\mathbf{\\beta} = \\left[ \\begin{array}{c}\n",
    "         \\mathbf{\\beta}_{1} \\\\\n",
    "         \\mathbf{\\beta}_{2} \\end{array}\\right]$, \n",
    "$ \\mathbf{Z} =  \\left[ \\begin{array}{cc}\n",
    "         \\mathbf{Z}_{1} & \\mathbf{0} \\\\\n",
    "         \\mathbf{0} & \\mathbf{Z}_{2} \\end{array}\\right]$,\n",
    "$ \\mathbf{a} = \\left[ \\begin{array}{c}\n",
    "         \\mathbf{a}_{1} \\\\\n",
    "         \\mathbf{a}_{2} \\end{array}\\right]$,\n",
    "$ \\mathbf{\\epsilon} = \\left[ \\begin{array}{c}\n",
    "         \\mathbf{\\epsilon}_{1} \\\\\n",
    "         \\mathbf{\\epsilon}_{2}\\end{array}\\right]$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Macierz A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "scrolled": true
   },
   "outputs": [],
   "source": [
    "library(AGHmatrix)\n",
    "\n",
    "id = 1:8\n",
    "sire = c(0, 0, 0, 1, 3, 1, 4, 3)\n",
    "dam = c(0, 0, 0, 0, 2, 2, 5, 6)\n",
    "\n",
    "ped = cbind(id, sire, dam)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Verifying conflicting data... \n",
      "Organizing data... \n",
      "Your data was chronologically organized with success. \n",
      "Constructing matrix A using ploidy = 2 \n",
      "Completed! Time = 0  minutes \n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<thead><tr><th scope=col>1</th><th scope=col>2</th><th scope=col>3</th><th scope=col>4</th><th scope=col>5</th><th scope=col>6</th><th scope=col>7</th><th scope=col>8</th></tr></thead>\n",
       "<tbody>\n",
       "\t<tr><td>1.00 </td><td>0.00 </td><td>0.00 </td><td>0.500</td><td>0.000</td><td>0.50 </td><td>0.25 </td><td>0.250</td></tr>\n",
       "\t<tr><td>0.00 </td><td>1.00 </td><td>0.00 </td><td>0.000</td><td>0.500</td><td>0.50 </td><td>0.25 </td><td>0.250</td></tr>\n",
       "\t<tr><td>0.00 </td><td>0.00 </td><td>1.00 </td><td>0.000</td><td>0.500</td><td>0.00 </td><td>0.25 </td><td>0.500</td></tr>\n",
       "\t<tr><td>0.50 </td><td>0.00 </td><td>0.00 </td><td>1.000</td><td>0.000</td><td>0.25 </td><td>0.50 </td><td>0.125</td></tr>\n",
       "\t<tr><td>0.00 </td><td>0.50 </td><td>0.50 </td><td>0.000</td><td>1.000</td><td>0.25 </td><td>0.50 </td><td>0.375</td></tr>\n",
       "\t<tr><td>0.50 </td><td>0.50 </td><td>0.00 </td><td>0.250</td><td>0.250</td><td>1.00 </td><td>0.25 </td><td>0.500</td></tr>\n",
       "\t<tr><td>0.25 </td><td>0.25 </td><td>0.25 </td><td>0.500</td><td>0.500</td><td>0.25 </td><td>1.00 </td><td>0.250</td></tr>\n",
       "\t<tr><td>0.25 </td><td>0.25 </td><td>0.50 </td><td>0.125</td><td>0.375</td><td>0.50 </td><td>0.25 </td><td>1.000</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{r|llllllll}\n",
       " 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8\\\\\n",
       "\\hline\n",
       "\t 1.00  & 0.00  & 0.00  & 0.500 & 0.000 & 0.50  & 0.25  & 0.250\\\\\n",
       "\t 0.00  & 1.00  & 0.00  & 0.000 & 0.500 & 0.50  & 0.25  & 0.250\\\\\n",
       "\t 0.00  & 0.00  & 1.00  & 0.000 & 0.500 & 0.00  & 0.25  & 0.500\\\\\n",
       "\t 0.50  & 0.00  & 0.00  & 1.000 & 0.000 & 0.25  & 0.50  & 0.125\\\\\n",
       "\t 0.00  & 0.50  & 0.50  & 0.000 & 1.000 & 0.25  & 0.50  & 0.375\\\\\n",
       "\t 0.50  & 0.50  & 0.00  & 0.250 & 0.250 & 1.00  & 0.25  & 0.500\\\\\n",
       "\t 0.25  & 0.25  & 0.25  & 0.500 & 0.500 & 0.25  & 1.00  & 0.250\\\\\n",
       "\t 0.25  & 0.25  & 0.50  & 0.125 & 0.375 & 0.50  & 0.25  & 1.000\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |\n",
       "|---|---|---|---|---|---|---|---|\n",
       "| 1.00  | 0.00  | 0.00  | 0.500 | 0.000 | 0.50  | 0.25  | 0.250 |\n",
       "| 0.00  | 1.00  | 0.00  | 0.000 | 0.500 | 0.50  | 0.25  | 0.250 |\n",
       "| 0.00  | 0.00  | 1.00  | 0.000 | 0.500 | 0.00  | 0.25  | 0.500 |\n",
       "| 0.50  | 0.00  | 0.00  | 1.000 | 0.000 | 0.25  | 0.50  | 0.125 |\n",
       "| 0.00  | 0.50  | 0.50  | 0.000 | 1.000 | 0.25  | 0.50  | 0.375 |\n",
       "| 0.50  | 0.50  | 0.00  | 0.250 | 0.250 | 1.00  | 0.25  | 0.500 |\n",
       "| 0.25  | 0.25  | 0.25  | 0.500 | 0.500 | 0.25  | 1.00  | 0.250 |\n",
       "| 0.25  | 0.25  | 0.50  | 0.125 | 0.375 | 0.50  | 0.25  | 1.000 |\n",
       "\n"
      ],
      "text/plain": [
       "  1    2    3    4     5     6    7    8    \n",
       "1 1.00 0.00 0.00 0.500 0.000 0.50 0.25 0.250\n",
       "2 0.00 1.00 0.00 0.000 0.500 0.50 0.25 0.250\n",
       "3 0.00 0.00 1.00 0.000 0.500 0.00 0.25 0.500\n",
       "4 0.50 0.00 0.00 1.000 0.000 0.25 0.50 0.125\n",
       "5 0.00 0.50 0.50 0.000 1.000 0.25 0.50 0.375\n",
       "6 0.50 0.50 0.00 0.250 0.250 1.00 0.25 0.500\n",
       "7 0.25 0.25 0.25 0.500 0.500 0.25 1.00 0.250\n",
       "8 0.25 0.25 0.50 0.125 0.375 0.50 0.25 1.000"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "(A = as.matrix(Amatrix(ped)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Pozostałe źródła informacji"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<tbody>\n",
       "\t<tr><td>20</td><td>18</td></tr>\n",
       "\t<tr><td>18</td><td>40</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{ll}\n",
       "\t 20 & 18\\\\\n",
       "\t 18 & 40\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "| 20 | 18 |\n",
       "| 18 | 40 |\n",
       "\n"
      ],
      "text/plain": [
       "     [,1] [,2]\n",
       "[1,] 20   18  \n",
       "[2,] 18   40  "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<tbody>\n",
       "\t<tr><td>40</td><td>11</td></tr>\n",
       "\t<tr><td>11</td><td>30</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{ll}\n",
       "\t 40 & 11\\\\\n",
       "\t 11 & 30\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "| 40 | 11 |\n",
       "| 11 | 30 |\n",
       "\n"
      ],
      "text/plain": [
       "     [,1] [,2]\n",
       "[1,] 40   11  \n",
       "[2,] 11   30  "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "y1 = as.matrix(c(4.5, 2.9, 3.9, 3.5, 5.0))\n",
    "y2 = as.matrix(c(6.8, 5.0, 6.8, 6.0, 7.5))\n",
    "\n",
    "(G = matrix(c(20, 18, 18, 40), 2, 2))\n",
    "(R = matrix(c(40, 11, 11, 30), 2, 2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "0.333333333333333"
      ],
      "text/latex": [
       "0.333333333333333"
      ],
      "text/markdown": [
       "0.333333333333333"
      ],
      "text/plain": [
       "[1] 0.3333333"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "0.571428571428571"
      ],
      "text/latex": [
       "0.571428571428571"
      ],
      "text/markdown": [
       "0.571428571428571"
      ],
      "text/plain": [
       "[1] 0.5714286"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "20/60\n",
    "40/70"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "sex = c(1, 0, 0, 1, 1)\n",
    "X1 = matrix(0, 5, 2)\n",
    "X1[, 1] = sex\n",
    "X1[, 2] = 1-sex\n",
    "X2 = X1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "I = diag(5)\n",
    "Z1 = matrix(0, 5, 8)\n",
    "Z1[1:5, 4:8] = I\n",
    "Z2 = Z1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "library(MASS)\n",
    "\n",
    "mme3 = function(y, X, Z, A, G, R) {\n",
    "    invA = ginv(A)\n",
    "    invG = ginv(G)\n",
    "    R = kronecker(R, diag(5))\n",
    "    invR = ginv(R)\n",
    "    C = rbind(cbind(t(X)%*%invR%*%X, t(X)%*%invR%*%Z),\n",
    "              cbind(t(Z)%*%invR%*%X, t(Z)%*%invR%*%Z+kronecker(invG, invA)))\n",
    "    rhs = rbind(t(X)%*%invR%*%y, t(Z)%*%invR%*%y)\n",
    "    invC = ginv(C)\n",
    "    estimators = invC%*%rhs\n",
    "    list(C = C, est = estimators)\n",
    "}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<tbody>\n",
       "\t<tr><td>0</td><td>0</td><td>0</td><td>1</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td></tr>\n",
       "\t<tr><td>0</td><td>0</td><td>0</td><td>0</td><td>1</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td></tr>\n",
       "\t<tr><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td></tr>\n",
       "\t<tr><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td></tr>\n",
       "\t<tr><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td></tr>\n",
       "\t<tr><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1</td><td>0</td><td>0</td><td>0</td><td>0</td></tr>\n",
       "\t<tr><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1</td><td>0</td><td>0</td><td>0</td></tr>\n",
       "\t<tr><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1</td><td>0</td><td>0</td></tr>\n",
       "\t<tr><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1</td><td>0</td></tr>\n",
       "\t<tr><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>1</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{llllllllllllllll}\n",
       "\t 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\\n",
       "\t 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\\n",
       "\t 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\\n",
       "\t 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\\n",
       "\t 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\\n",
       "\t 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\\\\n",
       "\t 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\\\\n",
       "\t 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\\\\n",
       "\t 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\\\\n",
       "\t 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |\n",
       "| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |\n",
       "| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |\n",
       "| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |\n",
       "| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |\n",
       "| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |\n",
       "| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |\n",
       "| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |\n",
       "| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |\n",
       "| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |\n",
       "\n"
      ],
      "text/plain": [
       "      [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [,13]\n",
       " [1,] 0    0    0    1    0    0    0    0    0    0     0     0     0    \n",
       " [2,] 0    0    0    0    1    0    0    0    0    0     0     0     0    \n",
       " [3,] 0    0    0    0    0    1    0    0    0    0     0     0     0    \n",
       " [4,] 0    0    0    0    0    0    1    0    0    0     0     0     0    \n",
       " [5,] 0    0    0    0    0    0    0    1    0    0     0     0     0    \n",
       " [6,] 0    0    0    0    0    0    0    0    0    0     0     1     0    \n",
       " [7,] 0    0    0    0    0    0    0    0    0    0     0     0     1    \n",
       " [8,] 0    0    0    0    0    0    0    0    0    0     0     0     0    \n",
       " [9,] 0    0    0    0    0    0    0    0    0    0     0     0     0    \n",
       "[10,] 0    0    0    0    0    0    0    0    0    0     0     0     0    \n",
       "      [,14] [,15] [,16]\n",
       " [1,] 0     0     0    \n",
       " [2,] 0     0     0    \n",
       " [3,] 0     0     0    \n",
       " [4,] 0     0     0    \n",
       " [5,] 0     0     0    \n",
       " [6,] 0     0     0    \n",
       " [7,] 0     0     0    \n",
       " [8,] 1     0     0    \n",
       " [9,] 0     1     0    \n",
       "[10,] 0     0     1    "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "y = as.matrix(c(y1, y2))\n",
    "\n",
    "X = matrix(0, 10, 4)\n",
    "X[1:5, 1:2] = X1\n",
    "X[6:10, 3:4] = X2\n",
    "\n",
    "Z = matrix(0, 10, 16)\n",
    "Z[1:5, 1:8] = Z1\n",
    "Z[6:10, 9:16] = Z2\n",
    "Z"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "scrolled": true
   },
   "outputs": [],
   "source": [
    "results = mme3(y, X, Z, A, G, R)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<dl>\n",
       "\t<dt>$C</dt>\n",
       "\t\t<dd><table>\n",
       "<tbody>\n",
       "\t<tr><td> 0.08341057  </td><td> 0.00000000  </td><td>-0.03058387  </td><td> 0.00000000  </td><td> 0.000000e+00</td><td> 0.000000e+00</td><td> 0.000000e+00</td><td> 2.780352e-02</td><td> 0.000000e+00</td><td> 0.000000e+00</td><td> 2.780352e-02</td><td> 2.780352e-02</td><td> 0.000000e+00</td><td> 0.000000e+00</td><td> 0.000000e+00</td><td>-1.019462e-02</td><td> 0.000000e+00</td><td> 0.000000e+00</td><td>-1.019462e-02</td><td>-1.019462e-02</td></tr>\n",
       "\t<tr><td> 0.00000000  </td><td> 0.05560704  </td><td> 0.00000000  </td><td>-0.02038925  </td><td> 0.000000e+00</td><td> 0.000000e+00</td><td> 0.000000e+00</td><td> 0.000000e+00</td><td> 2.780352e-02</td><td> 2.780352e-02</td><td> 0.000000e+00</td><td> 0.000000e+00</td><td> 0.000000e+00</td><td> 0.000000e+00</td><td> 0.000000e+00</td><td> 0.000000e+00</td><td>-1.019462e-02</td><td>-1.019462e-02</td><td> 0.000000e+00</td><td> 0.000000e+00</td></tr>\n",
       "\t<tr><td>-0.03058387  </td><td> 0.00000000  </td><td> 0.11121409  </td><td> 0.00000000  </td><td> 0.000000e+00</td><td> 0.000000e+00</td><td> 0.000000e+00</td><td>-1.019462e-02</td><td> 0.000000e+00</td><td> 0.000000e+00</td><td>-1.019462e-02</td><td>-1.019462e-02</td><td> 0.000000e+00</td><td> 0.000000e+00</td><td> 0.000000e+00</td><td> 3.707136e-02</td><td> 0.000000e+00</td><td> 0.000000e+00</td><td> 3.707136e-02</td><td> 3.707136e-02</td></tr>\n",
       "\t<tr><td> 0.00000000  </td><td>-0.02038925  </td><td> 0.00000000  </td><td> 0.07414272  </td><td> 0.000000e+00</td><td> 0.000000e+00</td><td> 0.000000e+00</td><td> 0.000000e+00</td><td>-1.019462e-02</td><td>-1.019462e-02</td><td> 0.000000e+00</td><td> 0.000000e+00</td><td> 0.000000e+00</td><td> 0.000000e+00</td><td> 0.000000e+00</td><td> 0.000000e+00</td><td> 3.707136e-02</td><td> 3.707136e-02</td><td> 0.000000e+00</td><td> 0.000000e+00</td></tr>\n",
       "\t<tr><td> 0.00000000  </td><td> 0.00000000  </td><td> 0.00000000  </td><td> 0.00000000  </td><td> 1.540616e-01</td><td> 4.201681e-02</td><td> 2.332401e-16</td><td>-5.602241e-02</td><td> 3.731842e-17</td><td>-8.403361e-02</td><td>-4.478211e-16</td><td>-1.559793e-16</td><td>-6.932773e-02</td><td>-1.890756e-02</td><td>-1.049581e-16</td><td> 2.521008e-02</td><td>-1.679329e-17</td><td> 3.781513e-02</td><td> 2.015195e-16</td><td> 7.019070e-17</td></tr>\n",
       "\t<tr><td> 0.00000000  </td><td> 0.00000000  </td><td> 0.00000000  </td><td> 0.00000000  </td><td> 4.201681e-02</td><td> 1.680672e-01</td><td> 4.201681e-02</td><td>-6.530724e-17</td><td>-8.403361e-02</td><td>-8.403361e-02</td><td> 7.463684e-17</td><td>-9.621155e-18</td><td>-1.890756e-02</td><td>-7.563025e-02</td><td>-1.890756e-02</td><td> 2.938826e-17</td><td> 3.781513e-02</td><td> 3.781513e-02</td><td>-3.358658e-17</td><td> 4.329520e-18</td></tr>\n",
       "\t<tr><td> 0.00000000  </td><td> 0.00000000  </td><td> 0.00000000  </td><td> 0.00000000  </td><td> 9.329605e-17</td><td> 4.201681e-02</td><td> 1.680672e-01</td><td> 1.865921e-17</td><td>-8.403361e-02</td><td> 4.201681e-02</td><td>-3.265362e-17</td><td>-8.403361e-02</td><td>-4.198322e-17</td><td>-1.890756e-02</td><td>-7.563025e-02</td><td>-8.396645e-18</td><td> 3.781513e-02</td><td>-1.890756e-02</td><td> 1.469413e-17</td><td> 3.781513e-02</td></tr>\n",
       "\t<tr><td> 0.02780352  </td><td> 0.00000000  </td><td>-0.01019462  </td><td> 0.00000000  </td><td>-5.602241e-02</td><td>-1.399441e-16</td><td>-1.072905e-16</td><td> 1.818651e-01</td><td> 4.201681e-02</td><td> 5.597763e-17</td><td>-8.403361e-02</td><td> 2.128316e-17</td><td> 2.521008e-02</td><td> 6.297484e-17</td><td> 4.828071e-17</td><td>-7.952236e-02</td><td>-1.890756e-02</td><td>-2.518993e-17</td><td> 3.781513e-02</td><td>-9.577423e-18</td></tr>\n",
       "\t<tr><td> 0.00000000  </td><td> 0.02780352  </td><td> 0.00000000  </td><td>-0.01019462  </td><td> 9.329605e-18</td><td>-8.403361e-02</td><td>-8.403361e-02</td><td> 4.201681e-02</td><td> 2.378876e-01</td><td>-7.463684e-17</td><td>-8.403361e-02</td><td>-8.163405e-17</td><td>-4.198322e-18</td><td> 3.781513e-02</td><td> 3.781513e-02</td><td>-1.890756e-02</td><td>-1.047324e-01</td><td> 3.358658e-17</td><td> 3.781513e-02</td><td> 3.673532e-17</td></tr>\n",
       "\t<tr><td> 0.00000000  </td><td> 0.02780352  </td><td> 0.00000000  </td><td>-0.01019462  </td><td>-8.403361e-02</td><td>-8.403361e-02</td><td> 4.201681e-02</td><td>-9.796086e-17</td><td>-6.530724e-17</td><td> 2.378876e-01</td><td> 9.796086e-17</td><td>-8.403361e-02</td><td> 3.781513e-02</td><td> 3.781513e-02</td><td>-1.890756e-02</td><td> 4.408238e-17</td><td> 2.938826e-17</td><td>-1.047324e-01</td><td>-4.408238e-17</td><td> 3.781513e-02</td></tr>\n",
       "\t<tr><td> 0.02780352  </td><td> 0.00000000  </td><td>-0.01019462  </td><td> 0.00000000  </td><td> 4.664803e-18</td><td>-3.731842e-17</td><td>-9.329605e-18</td><td>-8.403361e-02</td><td>-8.403361e-02</td><td>-4.664803e-18</td><td> 1.958707e-01</td><td> 1.137046e-17</td><td>-2.099161e-18</td><td> 1.679329e-17</td><td> 4.198322e-18</td><td> 3.781513e-02</td><td> 3.781513e-02</td><td> 2.099161e-18</td><td>-8.582488e-02</td><td>-5.116705e-18</td></tr>\n",
       "\t<tr><td> 0.02780352  </td><td> 0.00000000  </td><td>-0.01019462  </td><td> 0.00000000  </td><td>-1.020426e-16</td><td>-3.221629e-16</td><td>-8.403361e-02</td><td>-2.183711e-16</td><td> 9.737775e-17</td><td>-8.403361e-02</td><td> 2.040851e-16</td><td> 1.958707e-01</td><td> 4.591915e-17</td><td> 1.449733e-16</td><td> 3.781513e-02</td><td> 9.826698e-17</td><td>-4.381999e-17</td><td> 3.781513e-02</td><td>-9.183830e-17</td><td>-8.582488e-02</td></tr>\n",
       "\t<tr><td> 0.00000000  </td><td> 0.00000000  </td><td> 0.00000000  </td><td> 0.00000000  </td><td>-6.932773e-02</td><td>-1.890756e-02</td><td>-1.049581e-16</td><td> 2.521008e-02</td><td>-1.679329e-17</td><td> 3.781513e-02</td><td> 2.015195e-16</td><td> 7.019070e-17</td><td> 7.703081e-02</td><td> 2.100840e-02</td><td> 1.166201e-16</td><td>-2.801120e-02</td><td> 1.865921e-17</td><td>-4.201681e-02</td><td>-2.239105e-16</td><td>-7.798967e-17</td></tr>\n",
       "\t<tr><td> 0.00000000  </td><td> 0.00000000  </td><td> 0.00000000  </td><td> 0.00000000  </td><td>-1.890756e-02</td><td>-7.563025e-02</td><td>-1.890756e-02</td><td> 2.938826e-17</td><td> 3.781513e-02</td><td> 3.781513e-02</td><td>-3.358658e-17</td><td> 4.329520e-18</td><td> 2.100840e-02</td><td> 8.403361e-02</td><td> 2.100840e-02</td><td>-3.265362e-17</td><td>-4.201681e-02</td><td>-4.201681e-02</td><td> 3.731842e-17</td><td>-4.810578e-18</td></tr>\n",
       "\t<tr><td> 0.00000000  </td><td> 0.00000000  </td><td> 0.00000000  </td><td> 0.00000000  </td><td>-4.198322e-17</td><td>-1.890756e-02</td><td>-7.563025e-02</td><td>-8.396645e-18</td><td> 3.781513e-02</td><td>-1.890756e-02</td><td> 1.469413e-17</td><td> 3.781513e-02</td><td> 4.664803e-17</td><td> 2.100840e-02</td><td> 8.403361e-02</td><td> 9.329605e-18</td><td>-4.201681e-02</td><td> 2.100840e-02</td><td>-1.632681e-17</td><td>-4.201681e-02</td></tr>\n",
       "\t<tr><td>-0.01019462  </td><td> 0.00000000  </td><td> 0.03707136  </td><td> 0.00000000  </td><td> 2.521008e-02</td><td> 6.297484e-17</td><td> 4.828071e-17</td><td>-7.952236e-02</td><td>-1.890756e-02</td><td>-2.518993e-17</td><td> 3.781513e-02</td><td>-9.577423e-18</td><td>-2.801120e-02</td><td>-6.997204e-17</td><td>-5.364523e-17</td><td> 1.141022e-01</td><td> 2.100840e-02</td><td> 2.798882e-17</td><td>-4.201681e-02</td><td> 1.064158e-17</td></tr>\n",
       "\t<tr><td> 0.00000000  </td><td>-0.01019462  </td><td> 0.00000000  </td><td> 0.03707136  </td><td>-4.198322e-18</td><td> 3.781513e-02</td><td> 3.781513e-02</td><td>-1.890756e-02</td><td>-1.047324e-01</td><td> 3.358658e-17</td><td> 3.781513e-02</td><td> 3.673532e-17</td><td> 4.664803e-18</td><td>-4.201681e-02</td><td>-4.201681e-02</td><td> 2.100840e-02</td><td> 1.421134e-01</td><td>-3.731842e-17</td><td>-4.201681e-02</td><td>-4.081702e-17</td></tr>\n",
       "\t<tr><td> 0.00000000  </td><td>-0.01019462  </td><td> 0.00000000  </td><td> 0.03707136  </td><td> 3.781513e-02</td><td> 3.781513e-02</td><td>-1.890756e-02</td><td> 4.408238e-17</td><td> 2.938826e-17</td><td>-1.047324e-01</td><td>-4.408238e-17</td><td> 3.781513e-02</td><td>-4.201681e-02</td><td>-4.201681e-02</td><td> 2.100840e-02</td><td>-4.898043e-17</td><td>-3.265362e-17</td><td> 1.421134e-01</td><td> 4.898043e-17</td><td>-4.201681e-02</td></tr>\n",
       "\t<tr><td>-0.01019462  </td><td> 0.00000000  </td><td> 0.03707136  </td><td> 0.00000000  </td><td>-2.099161e-18</td><td> 1.679329e-17</td><td> 4.198322e-18</td><td> 3.781513e-02</td><td> 3.781513e-02</td><td> 2.099161e-18</td><td>-8.582488e-02</td><td>-5.116705e-18</td><td> 2.332401e-18</td><td>-1.865921e-17</td><td>-4.664803e-18</td><td>-4.201681e-02</td><td>-4.201681e-02</td><td>-2.332401e-18</td><td> 1.211050e-01</td><td> 5.685228e-18</td></tr>\n",
       "\t<tr><td>-0.01019462  </td><td> 0.00000000  </td><td> 0.03707136  </td><td> 0.00000000  </td><td> 4.591915e-17</td><td> 1.449733e-16</td><td> 3.781513e-02</td><td> 9.826698e-17</td><td>-4.381999e-17</td><td> 3.781513e-02</td><td>-9.183830e-17</td><td>-8.582488e-02</td><td>-5.102128e-17</td><td>-1.610815e-16</td><td>-4.201681e-02</td><td>-1.091855e-16</td><td> 4.868888e-17</td><td>-4.201681e-02</td><td> 1.020426e-16</td><td> 1.211050e-01</td></tr>\n",
       "</tbody>\n",
       "</table>\n",
       "</dd>\n",
       "\t<dt>$est</dt>\n",
       "\t\t<dd><table>\n",
       "<tbody>\n",
       "\t<tr><td> 4.360866999</td></tr>\n",
       "\t<tr><td> 3.397261592</td></tr>\n",
       "\t<tr><td> 6.799897620</td></tr>\n",
       "\t<tr><td> 5.880295937</td></tr>\n",
       "\t<tr><td> 0.150915567</td></tr>\n",
       "\t<tr><td>-0.015392510</td></tr>\n",
       "\t<tr><td>-0.078391896</td></tr>\n",
       "\t<tr><td>-0.010238959</td></tr>\n",
       "\t<tr><td>-0.270331441</td></tr>\n",
       "\t<tr><td> 0.275808258</td></tr>\n",
       "\t<tr><td>-0.316117562</td></tr>\n",
       "\t<tr><td> 0.243755523</td></tr>\n",
       "\t<tr><td> 0.279597973</td></tr>\n",
       "\t<tr><td>-0.007610071</td></tr>\n",
       "\t<tr><td>-0.170341439</td></tr>\n",
       "\t<tr><td>-0.012670709</td></tr>\n",
       "\t<tr><td>-0.477830262</td></tr>\n",
       "\t<tr><td> 0.517238387</td></tr>\n",
       "\t<tr><td>-0.478983695</td></tr>\n",
       "\t<tr><td> 0.391961543</td></tr>\n",
       "</tbody>\n",
       "</table>\n",
       "</dd>\n",
       "</dl>\n"
      ],
      "text/latex": [
       "\\begin{description}\n",
       "\\item[\\$C] \\begin{tabular}{llllllllllllllllllll}\n",
       "\t  0.08341057   &  0.00000000   & -0.03058387   &  0.00000000   &  0.000000e+00 &  0.000000e+00 &  0.000000e+00 &  2.780352e-02 &  0.000000e+00 &  0.000000e+00 &  2.780352e-02 &  2.780352e-02 &  0.000000e+00 &  0.000000e+00 &  0.000000e+00 & -1.019462e-02 &  0.000000e+00 &  0.000000e+00 & -1.019462e-02 & -1.019462e-02\\\\\n",
       "\t  0.00000000   &  0.05560704   &  0.00000000   & -0.02038925   &  0.000000e+00 &  0.000000e+00 &  0.000000e+00 &  0.000000e+00 &  2.780352e-02 &  2.780352e-02 &  0.000000e+00 &  0.000000e+00 &  0.000000e+00 &  0.000000e+00 &  0.000000e+00 &  0.000000e+00 & -1.019462e-02 & -1.019462e-02 &  0.000000e+00 &  0.000000e+00\\\\\n",
       "\t -0.03058387   &  0.00000000   &  0.11121409   &  0.00000000   &  0.000000e+00 &  0.000000e+00 &  0.000000e+00 & -1.019462e-02 &  0.000000e+00 &  0.000000e+00 & -1.019462e-02 & -1.019462e-02 &  0.000000e+00 &  0.000000e+00 &  0.000000e+00 &  3.707136e-02 &  0.000000e+00 &  0.000000e+00 &  3.707136e-02 &  3.707136e-02\\\\\n",
       "\t  0.00000000   & -0.02038925   &  0.00000000   &  0.07414272   &  0.000000e+00 &  0.000000e+00 &  0.000000e+00 &  0.000000e+00 & -1.019462e-02 & -1.019462e-02 &  0.000000e+00 &  0.000000e+00 &  0.000000e+00 &  0.000000e+00 &  0.000000e+00 &  0.000000e+00 &  3.707136e-02 &  3.707136e-02 &  0.000000e+00 &  0.000000e+00\\\\\n",
       "\t  0.00000000   &  0.00000000   &  0.00000000   &  0.00000000   &  1.540616e-01 &  4.201681e-02 &  2.332401e-16 & -5.602241e-02 &  3.731842e-17 & -8.403361e-02 & -4.478211e-16 & -1.559793e-16 & -6.932773e-02 & -1.890756e-02 & -1.049581e-16 &  2.521008e-02 & -1.679329e-17 &  3.781513e-02 &  2.015195e-16 &  7.019070e-17\\\\\n",
       "\t  0.00000000   &  0.00000000   &  0.00000000   &  0.00000000   &  4.201681e-02 &  1.680672e-01 &  4.201681e-02 & -6.530724e-17 & -8.403361e-02 & -8.403361e-02 &  7.463684e-17 & -9.621155e-18 & -1.890756e-02 & -7.563025e-02 & -1.890756e-02 &  2.938826e-17 &  3.781513e-02 &  3.781513e-02 & -3.358658e-17 &  4.329520e-18\\\\\n",
       "\t  0.00000000   &  0.00000000   &  0.00000000   &  0.00000000   &  9.329605e-17 &  4.201681e-02 &  1.680672e-01 &  1.865921e-17 & -8.403361e-02 &  4.201681e-02 & -3.265362e-17 & -8.403361e-02 & -4.198322e-17 & -1.890756e-02 & -7.563025e-02 & -8.396645e-18 &  3.781513e-02 & -1.890756e-02 &  1.469413e-17 &  3.781513e-02\\\\\n",
       "\t  0.02780352   &  0.00000000   & -0.01019462   &  0.00000000   & -5.602241e-02 & -1.399441e-16 & -1.072905e-16 &  1.818651e-01 &  4.201681e-02 &  5.597763e-17 & -8.403361e-02 &  2.128316e-17 &  2.521008e-02 &  6.297484e-17 &  4.828071e-17 & -7.952236e-02 & -1.890756e-02 & -2.518993e-17 &  3.781513e-02 & -9.577423e-18\\\\\n",
       "\t  0.00000000   &  0.02780352   &  0.00000000   & -0.01019462   &  9.329605e-18 & -8.403361e-02 & -8.403361e-02 &  4.201681e-02 &  2.378876e-01 & -7.463684e-17 & -8.403361e-02 & -8.163405e-17 & -4.198322e-18 &  3.781513e-02 &  3.781513e-02 & -1.890756e-02 & -1.047324e-01 &  3.358658e-17 &  3.781513e-02 &  3.673532e-17\\\\\n",
       "\t  0.00000000   &  0.02780352   &  0.00000000   & -0.01019462   & -8.403361e-02 & -8.403361e-02 &  4.201681e-02 & -9.796086e-17 & -6.530724e-17 &  2.378876e-01 &  9.796086e-17 & -8.403361e-02 &  3.781513e-02 &  3.781513e-02 & -1.890756e-02 &  4.408238e-17 &  2.938826e-17 & -1.047324e-01 & -4.408238e-17 &  3.781513e-02\\\\\n",
       "\t  0.02780352   &  0.00000000   & -0.01019462   &  0.00000000   &  4.664803e-18 & -3.731842e-17 & -9.329605e-18 & -8.403361e-02 & -8.403361e-02 & -4.664803e-18 &  1.958707e-01 &  1.137046e-17 & -2.099161e-18 &  1.679329e-17 &  4.198322e-18 &  3.781513e-02 &  3.781513e-02 &  2.099161e-18 & -8.582488e-02 & -5.116705e-18\\\\\n",
       "\t  0.02780352   &  0.00000000   & -0.01019462   &  0.00000000   & -1.020426e-16 & -3.221629e-16 & -8.403361e-02 & -2.183711e-16 &  9.737775e-17 & -8.403361e-02 &  2.040851e-16 &  1.958707e-01 &  4.591915e-17 &  1.449733e-16 &  3.781513e-02 &  9.826698e-17 & -4.381999e-17 &  3.781513e-02 & -9.183830e-17 & -8.582488e-02\\\\\n",
       "\t  0.00000000   &  0.00000000   &  0.00000000   &  0.00000000   & -6.932773e-02 & -1.890756e-02 & -1.049581e-16 &  2.521008e-02 & -1.679329e-17 &  3.781513e-02 &  2.015195e-16 &  7.019070e-17 &  7.703081e-02 &  2.100840e-02 &  1.166201e-16 & -2.801120e-02 &  1.865921e-17 & -4.201681e-02 & -2.239105e-16 & -7.798967e-17\\\\\n",
       "\t  0.00000000   &  0.00000000   &  0.00000000   &  0.00000000   & -1.890756e-02 & -7.563025e-02 & -1.890756e-02 &  2.938826e-17 &  3.781513e-02 &  3.781513e-02 & -3.358658e-17 &  4.329520e-18 &  2.100840e-02 &  8.403361e-02 &  2.100840e-02 & -3.265362e-17 & -4.201681e-02 & -4.201681e-02 &  3.731842e-17 & -4.810578e-18\\\\\n",
       "\t  0.00000000   &  0.00000000   &  0.00000000   &  0.00000000   & -4.198322e-17 & -1.890756e-02 & -7.563025e-02 & -8.396645e-18 &  3.781513e-02 & -1.890756e-02 &  1.469413e-17 &  3.781513e-02 &  4.664803e-17 &  2.100840e-02 &  8.403361e-02 &  9.329605e-18 & -4.201681e-02 &  2.100840e-02 & -1.632681e-17 & -4.201681e-02\\\\\n",
       "\t -0.01019462   &  0.00000000   &  0.03707136   &  0.00000000   &  2.521008e-02 &  6.297484e-17 &  4.828071e-17 & -7.952236e-02 & -1.890756e-02 & -2.518993e-17 &  3.781513e-02 & -9.577423e-18 & -2.801120e-02 & -6.997204e-17 & -5.364523e-17 &  1.141022e-01 &  2.100840e-02 &  2.798882e-17 & -4.201681e-02 &  1.064158e-17\\\\\n",
       "\t  0.00000000   & -0.01019462   &  0.00000000   &  0.03707136   & -4.198322e-18 &  3.781513e-02 &  3.781513e-02 & -1.890756e-02 & -1.047324e-01 &  3.358658e-17 &  3.781513e-02 &  3.673532e-17 &  4.664803e-18 & -4.201681e-02 & -4.201681e-02 &  2.100840e-02 &  1.421134e-01 & -3.731842e-17 & -4.201681e-02 & -4.081702e-17\\\\\n",
       "\t  0.00000000   & -0.01019462   &  0.00000000   &  0.03707136   &  3.781513e-02 &  3.781513e-02 & -1.890756e-02 &  4.408238e-17 &  2.938826e-17 & -1.047324e-01 & -4.408238e-17 &  3.781513e-02 & -4.201681e-02 & -4.201681e-02 &  2.100840e-02 & -4.898043e-17 & -3.265362e-17 &  1.421134e-01 &  4.898043e-17 & -4.201681e-02\\\\\n",
       "\t -0.01019462   &  0.00000000   &  0.03707136   &  0.00000000   & -2.099161e-18 &  1.679329e-17 &  4.198322e-18 &  3.781513e-02 &  3.781513e-02 &  2.099161e-18 & -8.582488e-02 & -5.116705e-18 &  2.332401e-18 & -1.865921e-17 & -4.664803e-18 & -4.201681e-02 & -4.201681e-02 & -2.332401e-18 &  1.211050e-01 &  5.685228e-18\\\\\n",
       "\t -0.01019462   &  0.00000000   &  0.03707136   &  0.00000000   &  4.591915e-17 &  1.449733e-16 &  3.781513e-02 &  9.826698e-17 & -4.381999e-17 &  3.781513e-02 & -9.183830e-17 & -8.582488e-02 & -5.102128e-17 & -1.610815e-16 & -4.201681e-02 & -1.091855e-16 &  4.868888e-17 & -4.201681e-02 &  1.020426e-16 &  1.211050e-01\\\\\n",
       "\\end{tabular}\n",
       "\n",
       "\\item[\\$est] \\begin{tabular}{l}\n",
       "\t  4.360866999\\\\\n",
       "\t  3.397261592\\\\\n",
       "\t  6.799897620\\\\\n",
       "\t  5.880295937\\\\\n",
       "\t  0.150915567\\\\\n",
       "\t -0.015392510\\\\\n",
       "\t -0.078391896\\\\\n",
       "\t -0.010238959\\\\\n",
       "\t -0.270331441\\\\\n",
       "\t  0.275808258\\\\\n",
       "\t -0.316117562\\\\\n",
       "\t  0.243755523\\\\\n",
       "\t  0.279597973\\\\\n",
       "\t -0.007610071\\\\\n",
       "\t -0.170341439\\\\\n",
       "\t -0.012670709\\\\\n",
       "\t -0.477830262\\\\\n",
       "\t  0.517238387\\\\\n",
       "\t -0.478983695\\\\\n",
       "\t  0.391961543\\\\\n",
       "\\end{tabular}\n",
       "\n",
       "\\end{description}\n"
      ],
      "text/markdown": [
       "$C\n",
       ":   \n",
       "|  0.08341057   |  0.00000000   | -0.03058387   |  0.00000000   |  0.000000e+00 |  0.000000e+00 |  0.000000e+00 |  2.780352e-02 |  0.000000e+00 |  0.000000e+00 |  2.780352e-02 |  2.780352e-02 |  0.000000e+00 |  0.000000e+00 |  0.000000e+00 | -1.019462e-02 |  0.000000e+00 |  0.000000e+00 | -1.019462e-02 | -1.019462e-02 |\n",
       "|  0.00000000   |  0.05560704   |  0.00000000   | -0.02038925   |  0.000000e+00 |  0.000000e+00 |  0.000000e+00 |  0.000000e+00 |  2.780352e-02 |  2.780352e-02 |  0.000000e+00 |  0.000000e+00 |  0.000000e+00 |  0.000000e+00 |  0.000000e+00 |  0.000000e+00 | -1.019462e-02 | -1.019462e-02 |  0.000000e+00 |  0.000000e+00 |\n",
       "| -0.03058387   |  0.00000000   |  0.11121409   |  0.00000000   |  0.000000e+00 |  0.000000e+00 |  0.000000e+00 | -1.019462e-02 |  0.000000e+00 |  0.000000e+00 | -1.019462e-02 | -1.019462e-02 |  0.000000e+00 |  0.000000e+00 |  0.000000e+00 |  3.707136e-02 |  0.000000e+00 |  0.000000e+00 |  3.707136e-02 |  3.707136e-02 |\n",
       "|  0.00000000   | -0.02038925   |  0.00000000   |  0.07414272   |  0.000000e+00 |  0.000000e+00 |  0.000000e+00 |  0.000000e+00 | -1.019462e-02 | -1.019462e-02 |  0.000000e+00 |  0.000000e+00 |  0.000000e+00 |  0.000000e+00 |  0.000000e+00 |  0.000000e+00 |  3.707136e-02 |  3.707136e-02 |  0.000000e+00 |  0.000000e+00 |\n",
       "|  0.00000000   |  0.00000000   |  0.00000000   |  0.00000000   |  1.540616e-01 |  4.201681e-02 |  2.332401e-16 | -5.602241e-02 |  3.731842e-17 | -8.403361e-02 | -4.478211e-16 | -1.559793e-16 | -6.932773e-02 | -1.890756e-02 | -1.049581e-16 |  2.521008e-02 | -1.679329e-17 |  3.781513e-02 |  2.015195e-16 |  7.019070e-17 |\n",
       "|  0.00000000   |  0.00000000   |  0.00000000   |  0.00000000   |  4.201681e-02 |  1.680672e-01 |  4.201681e-02 | -6.530724e-17 | -8.403361e-02 | -8.403361e-02 |  7.463684e-17 | -9.621155e-18 | -1.890756e-02 | -7.563025e-02 | -1.890756e-02 |  2.938826e-17 |  3.781513e-02 |  3.781513e-02 | -3.358658e-17 |  4.329520e-18 |\n",
       "|  0.00000000   |  0.00000000   |  0.00000000   |  0.00000000   |  9.329605e-17 |  4.201681e-02 |  1.680672e-01 |  1.865921e-17 | -8.403361e-02 |  4.201681e-02 | -3.265362e-17 | -8.403361e-02 | -4.198322e-17 | -1.890756e-02 | -7.563025e-02 | -8.396645e-18 |  3.781513e-02 | -1.890756e-02 |  1.469413e-17 |  3.781513e-02 |\n",
       "|  0.02780352   |  0.00000000   | -0.01019462   |  0.00000000   | -5.602241e-02 | -1.399441e-16 | -1.072905e-16 |  1.818651e-01 |  4.201681e-02 |  5.597763e-17 | -8.403361e-02 |  2.128316e-17 |  2.521008e-02 |  6.297484e-17 |  4.828071e-17 | -7.952236e-02 | -1.890756e-02 | -2.518993e-17 |  3.781513e-02 | -9.577423e-18 |\n",
       "|  0.00000000   |  0.02780352   |  0.00000000   | -0.01019462   |  9.329605e-18 | -8.403361e-02 | -8.403361e-02 |  4.201681e-02 |  2.378876e-01 | -7.463684e-17 | -8.403361e-02 | -8.163405e-17 | -4.198322e-18 |  3.781513e-02 |  3.781513e-02 | -1.890756e-02 | -1.047324e-01 |  3.358658e-17 |  3.781513e-02 |  3.673532e-17 |\n",
       "|  0.00000000   |  0.02780352   |  0.00000000   | -0.01019462   | -8.403361e-02 | -8.403361e-02 |  4.201681e-02 | -9.796086e-17 | -6.530724e-17 |  2.378876e-01 |  9.796086e-17 | -8.403361e-02 |  3.781513e-02 |  3.781513e-02 | -1.890756e-02 |  4.408238e-17 |  2.938826e-17 | -1.047324e-01 | -4.408238e-17 |  3.781513e-02 |\n",
       "|  0.02780352   |  0.00000000   | -0.01019462   |  0.00000000   |  4.664803e-18 | -3.731842e-17 | -9.329605e-18 | -8.403361e-02 | -8.403361e-02 | -4.664803e-18 |  1.958707e-01 |  1.137046e-17 | -2.099161e-18 |  1.679329e-17 |  4.198322e-18 |  3.781513e-02 |  3.781513e-02 |  2.099161e-18 | -8.582488e-02 | -5.116705e-18 |\n",
       "|  0.02780352   |  0.00000000   | -0.01019462   |  0.00000000   | -1.020426e-16 | -3.221629e-16 | -8.403361e-02 | -2.183711e-16 |  9.737775e-17 | -8.403361e-02 |  2.040851e-16 |  1.958707e-01 |  4.591915e-17 |  1.449733e-16 |  3.781513e-02 |  9.826698e-17 | -4.381999e-17 |  3.781513e-02 | -9.183830e-17 | -8.582488e-02 |\n",
       "|  0.00000000   |  0.00000000   |  0.00000000   |  0.00000000   | -6.932773e-02 | -1.890756e-02 | -1.049581e-16 |  2.521008e-02 | -1.679329e-17 |  3.781513e-02 |  2.015195e-16 |  7.019070e-17 |  7.703081e-02 |  2.100840e-02 |  1.166201e-16 | -2.801120e-02 |  1.865921e-17 | -4.201681e-02 | -2.239105e-16 | -7.798967e-17 |\n",
       "|  0.00000000   |  0.00000000   |  0.00000000   |  0.00000000   | -1.890756e-02 | -7.563025e-02 | -1.890756e-02 |  2.938826e-17 |  3.781513e-02 |  3.781513e-02 | -3.358658e-17 |  4.329520e-18 |  2.100840e-02 |  8.403361e-02 |  2.100840e-02 | -3.265362e-17 | -4.201681e-02 | -4.201681e-02 |  3.731842e-17 | -4.810578e-18 |\n",
       "|  0.00000000   |  0.00000000   |  0.00000000   |  0.00000000   | -4.198322e-17 | -1.890756e-02 | -7.563025e-02 | -8.396645e-18 |  3.781513e-02 | -1.890756e-02 |  1.469413e-17 |  3.781513e-02 |  4.664803e-17 |  2.100840e-02 |  8.403361e-02 |  9.329605e-18 | -4.201681e-02 |  2.100840e-02 | -1.632681e-17 | -4.201681e-02 |\n",
       "| -0.01019462   |  0.00000000   |  0.03707136   |  0.00000000   |  2.521008e-02 |  6.297484e-17 |  4.828071e-17 | -7.952236e-02 | -1.890756e-02 | -2.518993e-17 |  3.781513e-02 | -9.577423e-18 | -2.801120e-02 | -6.997204e-17 | -5.364523e-17 |  1.141022e-01 |  2.100840e-02 |  2.798882e-17 | -4.201681e-02 |  1.064158e-17 |\n",
       "|  0.00000000   | -0.01019462   |  0.00000000   |  0.03707136   | -4.198322e-18 |  3.781513e-02 |  3.781513e-02 | -1.890756e-02 | -1.047324e-01 |  3.358658e-17 |  3.781513e-02 |  3.673532e-17 |  4.664803e-18 | -4.201681e-02 | -4.201681e-02 |  2.100840e-02 |  1.421134e-01 | -3.731842e-17 | -4.201681e-02 | -4.081702e-17 |\n",
       "|  0.00000000   | -0.01019462   |  0.00000000   |  0.03707136   |  3.781513e-02 |  3.781513e-02 | -1.890756e-02 |  4.408238e-17 |  2.938826e-17 | -1.047324e-01 | -4.408238e-17 |  3.781513e-02 | -4.201681e-02 | -4.201681e-02 |  2.100840e-02 | -4.898043e-17 | -3.265362e-17 |  1.421134e-01 |  4.898043e-17 | -4.201681e-02 |\n",
       "| -0.01019462   |  0.00000000   |  0.03707136   |  0.00000000   | -2.099161e-18 |  1.679329e-17 |  4.198322e-18 |  3.781513e-02 |  3.781513e-02 |  2.099161e-18 | -8.582488e-02 | -5.116705e-18 |  2.332401e-18 | -1.865921e-17 | -4.664803e-18 | -4.201681e-02 | -4.201681e-02 | -2.332401e-18 |  1.211050e-01 |  5.685228e-18 |\n",
       "| -0.01019462   |  0.00000000   |  0.03707136   |  0.00000000   |  4.591915e-17 |  1.449733e-16 |  3.781513e-02 |  9.826698e-17 | -4.381999e-17 |  3.781513e-02 | -9.183830e-17 | -8.582488e-02 | -5.102128e-17 | -1.610815e-16 | -4.201681e-02 | -1.091855e-16 |  4.868888e-17 | -4.201681e-02 |  1.020426e-16 |  1.211050e-01 |\n",
       "\n",
       "\n",
       "$est\n",
       ":   \n",
       "|  4.360866999 |\n",
       "|  3.397261592 |\n",
       "|  6.799897620 |\n",
       "|  5.880295937 |\n",
       "|  0.150915567 |\n",
       "| -0.015392510 |\n",
       "| -0.078391896 |\n",
       "| -0.010238959 |\n",
       "| -0.270331441 |\n",
       "|  0.275808258 |\n",
       "| -0.316117562 |\n",
       "|  0.243755523 |\n",
       "|  0.279597973 |\n",
       "| -0.007610071 |\n",
       "| -0.170341439 |\n",
       "| -0.012670709 |\n",
       "| -0.477830262 |\n",
       "|  0.517238387 |\n",
       "| -0.478983695 |\n",
       "|  0.391961543 |\n",
       "\n",
       "\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "$C\n",
       "             [,1]        [,2]        [,3]        [,4]          [,5]\n",
       " [1,]  0.08341057  0.00000000 -0.03058387  0.00000000  0.000000e+00\n",
       " [2,]  0.00000000  0.05560704  0.00000000 -0.02038925  0.000000e+00\n",
       " [3,] -0.03058387  0.00000000  0.11121409  0.00000000  0.000000e+00\n",
       " [4,]  0.00000000 -0.02038925  0.00000000  0.07414272  0.000000e+00\n",
       " [5,]  0.00000000  0.00000000  0.00000000  0.00000000  1.540616e-01\n",
       " [6,]  0.00000000  0.00000000  0.00000000  0.00000000  4.201681e-02\n",
       " [7,]  0.00000000  0.00000000  0.00000000  0.00000000  9.329605e-17\n",
       " [8,]  0.02780352  0.00000000 -0.01019462  0.00000000 -5.602241e-02\n",
       " [9,]  0.00000000  0.02780352  0.00000000 -0.01019462  9.329605e-18\n",
       "[10,]  0.00000000  0.02780352  0.00000000 -0.01019462 -8.403361e-02\n",
       "[11,]  0.02780352  0.00000000 -0.01019462  0.00000000  4.664803e-18\n",
       "[12,]  0.02780352  0.00000000 -0.01019462  0.00000000 -1.020426e-16\n",
       "[13,]  0.00000000  0.00000000  0.00000000  0.00000000 -6.932773e-02\n",
       "[14,]  0.00000000  0.00000000  0.00000000  0.00000000 -1.890756e-02\n",
       "[15,]  0.00000000  0.00000000  0.00000000  0.00000000 -4.198322e-17\n",
       "[16,] -0.01019462  0.00000000  0.03707136  0.00000000  2.521008e-02\n",
       "[17,]  0.00000000 -0.01019462  0.00000000  0.03707136 -4.198322e-18\n",
       "[18,]  0.00000000 -0.01019462  0.00000000  0.03707136  3.781513e-02\n",
       "[19,] -0.01019462  0.00000000  0.03707136  0.00000000 -2.099161e-18\n",
       "[20,] -0.01019462  0.00000000  0.03707136  0.00000000  4.591915e-17\n",
       "               [,6]          [,7]          [,8]          [,9]         [,10]\n",
       " [1,]  0.000000e+00  0.000000e+00  2.780352e-02  0.000000e+00  0.000000e+00\n",
       " [2,]  0.000000e+00  0.000000e+00  0.000000e+00  2.780352e-02  2.780352e-02\n",
       " [3,]  0.000000e+00  0.000000e+00 -1.019462e-02  0.000000e+00  0.000000e+00\n",
       " [4,]  0.000000e+00  0.000000e+00  0.000000e+00 -1.019462e-02 -1.019462e-02\n",
       " [5,]  4.201681e-02  2.332401e-16 -5.602241e-02  3.731842e-17 -8.403361e-02\n",
       " [6,]  1.680672e-01  4.201681e-02 -6.530724e-17 -8.403361e-02 -8.403361e-02\n",
       " [7,]  4.201681e-02  1.680672e-01  1.865921e-17 -8.403361e-02  4.201681e-02\n",
       " [8,] -1.399441e-16 -1.072905e-16  1.818651e-01  4.201681e-02  5.597763e-17\n",
       " [9,] -8.403361e-02 -8.403361e-02  4.201681e-02  2.378876e-01 -7.463684e-17\n",
       "[10,] -8.403361e-02  4.201681e-02 -9.796086e-17 -6.530724e-17  2.378876e-01\n",
       "[11,] -3.731842e-17 -9.329605e-18 -8.403361e-02 -8.403361e-02 -4.664803e-18\n",
       "[12,] -3.221629e-16 -8.403361e-02 -2.183711e-16  9.737775e-17 -8.403361e-02\n",
       "[13,] -1.890756e-02 -1.049581e-16  2.521008e-02 -1.679329e-17  3.781513e-02\n",
       "[14,] -7.563025e-02 -1.890756e-02  2.938826e-17  3.781513e-02  3.781513e-02\n",
       "[15,] -1.890756e-02 -7.563025e-02 -8.396645e-18  3.781513e-02 -1.890756e-02\n",
       "[16,]  6.297484e-17  4.828071e-17 -7.952236e-02 -1.890756e-02 -2.518993e-17\n",
       "[17,]  3.781513e-02  3.781513e-02 -1.890756e-02 -1.047324e-01  3.358658e-17\n",
       "[18,]  3.781513e-02 -1.890756e-02  4.408238e-17  2.938826e-17 -1.047324e-01\n",
       "[19,]  1.679329e-17  4.198322e-18  3.781513e-02  3.781513e-02  2.099161e-18\n",
       "[20,]  1.449733e-16  3.781513e-02  9.826698e-17 -4.381999e-17  3.781513e-02\n",
       "              [,11]         [,12]         [,13]         [,14]         [,15]\n",
       " [1,]  2.780352e-02  2.780352e-02  0.000000e+00  0.000000e+00  0.000000e+00\n",
       " [2,]  0.000000e+00  0.000000e+00  0.000000e+00  0.000000e+00  0.000000e+00\n",
       " [3,] -1.019462e-02 -1.019462e-02  0.000000e+00  0.000000e+00  0.000000e+00\n",
       " [4,]  0.000000e+00  0.000000e+00  0.000000e+00  0.000000e+00  0.000000e+00\n",
       " [5,] -4.478211e-16 -1.559793e-16 -6.932773e-02 -1.890756e-02 -1.049581e-16\n",
       " [6,]  7.463684e-17 -9.621155e-18 -1.890756e-02 -7.563025e-02 -1.890756e-02\n",
       " [7,] -3.265362e-17 -8.403361e-02 -4.198322e-17 -1.890756e-02 -7.563025e-02\n",
       " [8,] -8.403361e-02  2.128316e-17  2.521008e-02  6.297484e-17  4.828071e-17\n",
       " [9,] -8.403361e-02 -8.163405e-17 -4.198322e-18  3.781513e-02  3.781513e-02\n",
       "[10,]  9.796086e-17 -8.403361e-02  3.781513e-02  3.781513e-02 -1.890756e-02\n",
       "[11,]  1.958707e-01  1.137046e-17 -2.099161e-18  1.679329e-17  4.198322e-18\n",
       "[12,]  2.040851e-16  1.958707e-01  4.591915e-17  1.449733e-16  3.781513e-02\n",
       "[13,]  2.015195e-16  7.019070e-17  7.703081e-02  2.100840e-02  1.166201e-16\n",
       "[14,] -3.358658e-17  4.329520e-18  2.100840e-02  8.403361e-02  2.100840e-02\n",
       "[15,]  1.469413e-17  3.781513e-02  4.664803e-17  2.100840e-02  8.403361e-02\n",
       "[16,]  3.781513e-02 -9.577423e-18 -2.801120e-02 -6.997204e-17 -5.364523e-17\n",
       "[17,]  3.781513e-02  3.673532e-17  4.664803e-18 -4.201681e-02 -4.201681e-02\n",
       "[18,] -4.408238e-17  3.781513e-02 -4.201681e-02 -4.201681e-02  2.100840e-02\n",
       "[19,] -8.582488e-02 -5.116705e-18  2.332401e-18 -1.865921e-17 -4.664803e-18\n",
       "[20,] -9.183830e-17 -8.582488e-02 -5.102128e-17 -1.610815e-16 -4.201681e-02\n",
       "              [,16]         [,17]         [,18]         [,19]         [,20]\n",
       " [1,] -1.019462e-02  0.000000e+00  0.000000e+00 -1.019462e-02 -1.019462e-02\n",
       " [2,]  0.000000e+00 -1.019462e-02 -1.019462e-02  0.000000e+00  0.000000e+00\n",
       " [3,]  3.707136e-02  0.000000e+00  0.000000e+00  3.707136e-02  3.707136e-02\n",
       " [4,]  0.000000e+00  3.707136e-02  3.707136e-02  0.000000e+00  0.000000e+00\n",
       " [5,]  2.521008e-02 -1.679329e-17  3.781513e-02  2.015195e-16  7.019070e-17\n",
       " [6,]  2.938826e-17  3.781513e-02  3.781513e-02 -3.358658e-17  4.329520e-18\n",
       " [7,] -8.396645e-18  3.781513e-02 -1.890756e-02  1.469413e-17  3.781513e-02\n",
       " [8,] -7.952236e-02 -1.890756e-02 -2.518993e-17  3.781513e-02 -9.577423e-18\n",
       " [9,] -1.890756e-02 -1.047324e-01  3.358658e-17  3.781513e-02  3.673532e-17\n",
       "[10,]  4.408238e-17  2.938826e-17 -1.047324e-01 -4.408238e-17  3.781513e-02\n",
       "[11,]  3.781513e-02  3.781513e-02  2.099161e-18 -8.582488e-02 -5.116705e-18\n",
       "[12,]  9.826698e-17 -4.381999e-17  3.781513e-02 -9.183830e-17 -8.582488e-02\n",
       "[13,] -2.801120e-02  1.865921e-17 -4.201681e-02 -2.239105e-16 -7.798967e-17\n",
       "[14,] -3.265362e-17 -4.201681e-02 -4.201681e-02  3.731842e-17 -4.810578e-18\n",
       "[15,]  9.329605e-18 -4.201681e-02  2.100840e-02 -1.632681e-17 -4.201681e-02\n",
       "[16,]  1.141022e-01  2.100840e-02  2.798882e-17 -4.201681e-02  1.064158e-17\n",
       "[17,]  2.100840e-02  1.421134e-01 -3.731842e-17 -4.201681e-02 -4.081702e-17\n",
       "[18,] -4.898043e-17 -3.265362e-17  1.421134e-01  4.898043e-17 -4.201681e-02\n",
       "[19,] -4.201681e-02 -4.201681e-02 -2.332401e-18  1.211050e-01  5.685228e-18\n",
       "[20,] -1.091855e-16  4.868888e-17 -4.201681e-02  1.020426e-16  1.211050e-01\n",
       "\n",
       "$est\n",
       "              [,1]\n",
       " [1,]  4.360866999\n",
       " [2,]  3.397261592\n",
       " [3,]  6.799897620\n",
       " [4,]  5.880295937\n",
       " [5,]  0.150915567\n",
       " [6,] -0.015392510\n",
       " [7,] -0.078391896\n",
       " [8,] -0.010238959\n",
       " [9,] -0.270331441\n",
       "[10,]  0.275808258\n",
       "[11,] -0.316117562\n",
       "[12,]  0.243755523\n",
       "[13,]  0.279597973\n",
       "[14,] -0.007610071\n",
       "[15,] -0.170341439\n",
       "[16,] -0.012670709\n",
       "[17,] -0.477830262\n",
       "[18,]  0.517238387\n",
       "[19,] -0.478983695\n",
       "[20,]  0.391961543\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "results"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>20</li>\n",
       "\t<li>20</li>\n",
       "</ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item 20\n",
       "\\item 20\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 20\n",
       "2. 20\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "[1] 20 20"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "dim(results$C)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>20</li>\n",
       "\t<li>1</li>\n",
       "</ol>\n"
      ],
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       "\\begin{enumerate*}\n",
       "\\item 20\n",
       "\\item 1\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 20\n",
       "2. 1\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "[1] 20  1"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "dim(results$est)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>0.150915567312792</li>\n",
       "\t<li>-0.0153925096914866</li>\n",
       "\t<li>-0.0783918961641912</li>\n",
       "\t<li>-0.0102389585292879</li>\n",
       "\t<li>-0.270331441389241</li>\n",
       "\t<li>0.275808257580568</li>\n",
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       "\\begin{enumerate*}\n",
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       "\\item -0.0153925096914866\n",
       "\\item -0.0783918961641912\n",
       "\\item -0.0102389585292879\n",
       "\\item -0.270331441389241\n",
       "\\item 0.275808257580568\n",
       "\\item -0.316117561639436\n",
       "\\item 0.243755523005399\n",
       "\\end{enumerate*}\n"
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      "text/markdown": [
       "1. 0.150915567312792\n",
       "2. -0.0153925096914866\n",
       "3. -0.0783918961641912\n",
       "4. -0.0102389585292879\n",
       "5. -0.270331441389241\n",
       "6. 0.275808257580568\n",
       "7. -0.316117561639436\n",
       "8. 0.243755523005399\n",
       "\n",
       "\n"
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      "text/plain": [
       "[1]  0.15091557 -0.01539251 -0.07839190 -0.01023896 -0.27033144  0.27580826\n",
       "[7] -0.31611756  0.24375552"
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       "\t<li>-0.0126707086241202</li>\n",
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       "\\item -0.0126707086241202\n",
       "\\item -0.477830261602723\n",
       "\\item 0.517238387038388\n",
       "\\item -0.478983695055083\n",
       "\\item 0.391961542524088\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 0.27959797274487\n",
       "2. -0.00761007081989917\n",
       "3. -0.170341438593946\n",
       "4. -0.0126707086241202\n",
       "5. -0.477830261602723\n",
       "6. 0.517238387038388\n",
       "7. -0.478983695055083\n",
       "8. 0.391961542524088\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "[1]  0.279597973 -0.007610071 -0.170341439 -0.012670709 -0.477830262\n",
       "[6]  0.517238387 -0.478983695  0.391961543"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "(est_1 = results$est[5:12])\n",
    "(est_2 = results$est[13:20])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "est_1_new = 10*(est_1-mean(est_1))/sd(est_1) + 100\n",
    "est_2_new = 10*(est_2-mean(est_2))/sd(est_2) + 100"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "100"
      ],
      "text/latex": [
       "100"
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      "text/markdown": [
       "100"
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      "text/plain": [
       "[1] 100"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "10"
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       "10"
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       "10"
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      "text/plain": [
       "[1] 10"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "mean(est_1_new)\n",
    "sd(est_1_new)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<thead><tr><th scope=col>est_1_new</th><th scope=col>est_2_new</th></tr></thead>\n",
       "<tbody>\n",
       "\t<tr><td>106.98465</td><td>107.31162</td></tr>\n",
       "\t<tr><td> 99.41299</td><td> 99.65949</td></tr>\n",
       "\t<tr><td> 96.54476</td><td> 95.32381</td></tr>\n",
       "\t<tr><td> 99.64762</td><td> 99.52466</td></tr>\n",
       "\t<tr><td> 87.80615</td><td> 87.13134</td></tr>\n",
       "\t<tr><td>112.67075</td><td>113.64312</td></tr>\n",
       "\t<tr><td> 85.72161</td><td> 87.10061</td></tr>\n",
       "\t<tr><td>111.21146</td><td>110.30535</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{ll}\n",
       " est\\_1\\_new & est\\_2\\_new\\\\\n",
       "\\hline\n",
       "\t 106.98465 & 107.31162\\\\\n",
       "\t  99.41299 &  99.65949\\\\\n",
       "\t  96.54476 &  95.32381\\\\\n",
       "\t  99.64762 &  99.52466\\\\\n",
       "\t  87.80615 &  87.13134\\\\\n",
       "\t 112.67075 & 113.64312\\\\\n",
       "\t  85.72161 &  87.10061\\\\\n",
       "\t 111.21146 & 110.30535\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "| est_1_new | est_2_new |\n",
       "|---|---|\n",
       "| 106.98465 | 107.31162 |\n",
       "|  99.41299 |  99.65949 |\n",
       "|  96.54476 |  95.32381 |\n",
       "|  99.64762 |  99.52466 |\n",
       "|  87.80615 |  87.13134 |\n",
       "| 112.67075 | 113.64312 |\n",
       "|  85.72161 |  87.10061 |\n",
       "| 111.21146 | 110.30535 |\n",
       "\n"
      ],
      "text/plain": [
       "     est_1_new est_2_new\n",
       "[1,] 106.98465 107.31162\n",
       "[2,]  99.41299  99.65949\n",
       "[3,]  96.54476  95.32381\n",
       "[4,]  99.64762  99.52466\n",
       "[5,]  87.80615  87.13134\n",
       "[6,] 112.67075 113.64312\n",
       "[7,]  85.72161  87.10061\n",
       "[8,] 111.21146 110.30535"
      ]
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    {
     "data": {
      "text/html": [
       "0.995859246739469"
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       "0.995859246739469"
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       "[1] 0.9958592"
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   "source": [
    "cbind(est_1_new, est_2_new)\n",
    "cor(est_1_new, est_2_new)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Dokładność oceny wielocechowej"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
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       "\n",
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       "1. 0.102455353618369\n",
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       "\n",
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       "\n",
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       "[1] 0.2641274 0.1420607 0.3245854 0.4180517 0.4158686 0.3773957 0.3798000\n",
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       "1. 0.320086478343539\n",
       "2. 0.175525574154548\n",
       "3. 0.393724403255812\n",
       "4. 0.506830428344145\n",
       "5. 0.503373177464353\n",
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       "\n",
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       "[1] 0.3200865 0.1755256 0.3937244 0.5068304 0.5033732 0.4608890 0.4646624\n",
       "[8] 0.5205888"
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   ],
   "source": [
    "C = as.matrix(mme3(y, X, Z, A, G, R)$C)\n",
    "invC = ginv(C)\n",
    "\n",
    "invC22 = invC[5:20, 5:20]\n",
    "\n",
    "trait1 = diag(invC22)[1:8]\n",
    "trait2 = diag(invC22)[9:16]\n",
    "\n",
    "(r2_1 = (20-trait1) / 20)\n",
    "(r2_2 = (40-trait2) / 40)\n",
    "\n",
    "(r1 = sqrt(r2_1))\n",
    "(r2 = sqrt(r2_2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
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       "\t<tr><td>0.264</td><td>0.320</td></tr>\n",
       "\t<tr><td>0.142</td><td>0.176</td></tr>\n",
       "\t<tr><td>0.325</td><td>0.394</td></tr>\n",
       "\t<tr><td>0.418</td><td>0.507</td></tr>\n",
       "\t<tr><td>0.416</td><td>0.503</td></tr>\n",
       "\t<tr><td>0.377</td><td>0.461</td></tr>\n",
       "\t<tr><td>0.380</td><td>0.465</td></tr>\n",
       "\t<tr><td>0.431</td><td>0.521</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{ll}\n",
       "\t 0.264 & 0.320\\\\\n",
       "\t 0.142 & 0.176\\\\\n",
       "\t 0.325 & 0.394\\\\\n",
       "\t 0.418 & 0.507\\\\\n",
       "\t 0.416 & 0.503\\\\\n",
       "\t 0.377 & 0.461\\\\\n",
       "\t 0.380 & 0.465\\\\\n",
       "\t 0.431 & 0.521\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "| 0.264 | 0.320 |\n",
       "| 0.142 | 0.176 |\n",
       "| 0.325 | 0.394 |\n",
       "| 0.418 | 0.507 |\n",
       "| 0.416 | 0.503 |\n",
       "| 0.377 | 0.461 |\n",
       "| 0.380 | 0.465 |\n",
       "| 0.431 | 0.521 |\n",
       "\n"
      ],
      "text/plain": [
       "     [,1]  [,2] \n",
       "[1,] 0.264 0.320\n",
       "[2,] 0.142 0.176\n",
       "[3,] 0.325 0.394\n",
       "[4,] 0.418 0.507\n",
       "[5,] 0.416 0.503\n",
       "[6,] 0.377 0.461\n",
       "[7,] 0.380 0.465\n",
       "[8,] 0.431 0.521"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "cbind(round(r1, digits = 3), round(r2, digits = 3))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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