2021-11-12/data-analysis.ipynb

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   "source": [
    "# Basic data analysis tasks\n",
    "\n",
    "- Data: Fitness in Arabidopsis recombinant inbred lines\n",
    "- Data input: CSV files, missing data\n",
    "- Data description: Summary statistics, plots\n",
    "- Modeling: Linear regression"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Example: Fitness measured in Arabidopsis recombinant inbred lines\n",
    "\n",
    "Topics: Reading data, JIT compiler, missing data, summary statistics, plots, linear regression "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "using Statistics, CSV, Plots, DataFrames, GLM"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Reading data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
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       "<table class=\"data-frame\"><thead><tr><th></th><th>it09</th><th>it10</th><th>it11</th><th>sw09</th><th>sw10</th><th>sw11</th><th>id</th><th>flc</th></tr><tr><th></th><th>Float64?</th><th>Float64?</th><th>Float64</th><th>Float64?</th><th>Float64?</th><th>Float64</th><th>Int64</th><th>Int64</th></tr></thead><tbody><p>10 rows × 8 columns</p><tr><th>1</th><td>19.7616</td><td>24.1761</td><td>15.6778</td><td>5.66443</td><td>21.4804</td><td>4.47778</td><td>1</td><td>1</td></tr><tr><th>2</th><td>6.28846</td><td>1.77401</td><td>3.23889</td><td>9.54305</td><td>22.1222</td><td>7.76667</td><td>2</td><td>2</td></tr><tr><th>3</th><td>12.034</td><td>12.4605</td><td>10.601</td><td>11.8038</td><td>23.0487</td><td>14.8712</td><td>3</td><td>2</td></tr><tr><th>4</th><td>20.1253</td><td>14.1169</td><td>12.9035</td><td>7.44263</td><td>22.5834</td><td>8.1937</td><td>6</td><td>1</td></tr><tr><th>5</th><td>15.1278</td><td>13.2925</td><td>15.0646</td><td>6.89696</td><td>22.1404</td><td>8.4891</td><td>7</td><td>1</td></tr><tr><th>6</th><td>19.2091</td><td>13.9549</td><td>12.9868</td><td>8.39536</td><td>25.5612</td><td>8.08018</td><td>8</td><td>1</td></tr><tr><th>7</th><td>12.9434</td><td>11.9672</td><td>11.4831</td><td>12.014</td><td>24.298</td><td>18.1587</td><td>11</td><td>2</td></tr><tr><th>8</th><td>14.2244</td><td>5.45326</td><td>5.124</td><td>8.51941</td><td>22.8739</td><td>19.51</td><td>13</td><td>1</td></tr><tr><th>9</th><td>20.5921</td><td>14.8691</td><td>11.1797</td><td>6.51861</td><td>21.5116</td><td>8.03754</td><td>14</td><td>1</td></tr><tr><th>10</th><td>15.6141</td><td>12.0889</td><td>10.2923</td><td>11.8832</td><td>22.879</td><td>10.9994</td><td>16</td><td>1</td></tr></tbody></table>"
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       "\t& Float64? & Float64? & Float64 & Float64? & Float64? & Float64 & Int64 & Int64\\\\\n",
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       "\t2 & 6.28846 & 1.77401 & 3.23889 & 9.54305 & 22.1222 & 7.76667 & 2 & 2 \\\\\n",
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       "\u001b[1m10×8 DataFrame\u001b[0m\n",
       "\u001b[1m Row \u001b[0m│\u001b[1m it09     \u001b[0m\u001b[1m it10     \u001b[0m\u001b[1m it11     \u001b[0m\u001b[1m sw09     \u001b[0m\u001b[1m sw10     \u001b[0m\u001b[1m sw11     \u001b[0m\u001b[1m id    \u001b[0m\u001b[1m flc \u001b[0m ⋯\n",
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       "─────┼──────────────────────────────────────────────────────────────────────────\n",
       "   1 │ 19.7616   24.1761   15.6778    5.66443   21.4804   4.47778      1       ⋯\n",
       "   2 │  6.28846   1.77401   3.23889   9.54305   22.1222   7.76667      2\n",
       "   3 │ 12.034    12.4605   10.601    11.8038    23.0487  14.8712       3\n",
       "   4 │ 20.1253   14.1169   12.9035    7.44263   22.5834   8.1937       6\n",
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       "   8 │ 14.2244    5.45326   5.124     8.51941   22.8739  19.51        13\n",
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       "  10 │ 15.6141   12.0889   10.2923   11.8832    22.879   10.9994      16\n",
       "\u001b[36m                                                                1 column omitted\u001b[0m"
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     "execution_count": 2,
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     "output_type": "execute_result"
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   ],
   "source": [
    "agrenURL = \"https://raw.githubusercontent.com/sens/smalldata/master/arabidopsis/agren2013.csv\"\n",
    "agren = CSV.read(download(agrenURL),DataFrame,missingstring=\"NA\");\n",
    "first(agren,10)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A set of Arabidopsis recombinant inbred lines were derived from a cross of two\n",
    "parents from Italy and Sweden.  They were grown in three years\n",
    "(2009-2011) in Italy and Sweden.  The phenotype is the average number\n",
    "of seeds per plant for each line which is a measure of fitness.\n",
    "\n",
    "Column names:\n",
    "\n",
    "- id: ID of RI line (first two are the parents)\n",
    "- it09: Italy 2009\n",
    "- it10: Italy 2010\n",
    "- it11: Italy 2011\n",
    "- sw09: Sweden 2009\n",
    "- sw10: Sweden 2010\n",
    "- sw11: Sweden 2011\n",
    "\n",
    "\n",
    "Reference:\n",
    "\n",
    "J Ågren, CG Oakley, JK McKay, JT Lovell, DW Schemske. 2013. Genetic\n",
    "mapping of adaptation reveals fitness trade-offs in Arabidopsis\n",
    "thaliana. Proceedings of the National Academy of Sciences 110:\n",
    "21077-21082.\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Data description"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
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       "<table class=\"data-frame\"><thead><tr><th></th><th>variable</th><th>mean</th><th>min</th><th>median</th><th>max</th><th>nmissing</th><th>eltype</th></tr><tr><th></th><th>Symbol</th><th>Float64</th><th>Real</th><th>Float64</th><th>Real</th><th>Int64</th><th>Type</th></tr></thead><tbody><p>8 rows × 7 columns</p><tr><th>1</th><td>it09</td><td>12.2106</td><td>6.28846</td><td>12.0582</td><td>20.5921</td><td>6</td><td>Union{Missing, Float64}</td></tr><tr><th>2</th><td>it10</td><td>8.08256</td><td>1.77401</td><td>7.53993</td><td>24.1761</td><td>2</td><td>Union{Missing, Float64}</td></tr><tr><th>3</th><td>it11</td><td>8.92073</td><td>3.23889</td><td>8.52447</td><td>19.3107</td><td>0</td><td>Float64</td></tr><tr><th>4</th><td>sw09</td><td>10.8503</td><td>5.66443</td><td>10.832</td><td>15.0012</td><td>6</td><td>Union{Missing, Float64}</td></tr><tr><th>5</th><td>sw10</td><td>23.3658</td><td>19.2318</td><td>23.0893</td><td>34.1327</td><td>2</td><td>Union{Missing, Float64}</td></tr><tr><th>6</th><td>sw11</td><td>13.4934</td><td>4.47778</td><td>12.8102</td><td>32.1889</td><td>0</td><td>Float64</td></tr><tr><th>7</th><td>id</td><td>325.637</td><td>1</td><td>313.5</td><td>700</td><td>0</td><td>Int64</td></tr><tr><th>8</th><td>flc</td><td>1.535</td><td>1</td><td>2.0</td><td>2</td><td>0</td><td>Int64</td></tr></tbody></table>"
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       "\\begin{tabular}{r|ccccccc}\n",
       "\t& variable & mean & min & median & max & nmissing & eltype\\\\\n",
       "\t\\hline\n",
       "\t& Symbol & Float64 & Real & Float64 & Real & Int64 & Type\\\\\n",
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       "\t1 & it09 & 12.2106 & 6.28846 & 12.0582 & 20.5921 & 6 & Union\\{Missing, Float64\\} \\\\\n",
       "\t2 & it10 & 8.08256 & 1.77401 & 7.53993 & 24.1761 & 2 & Union\\{Missing, Float64\\} \\\\\n",
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       "\t4 & sw09 & 10.8503 & 5.66443 & 10.832 & 15.0012 & 6 & Union\\{Missing, Float64\\} \\\\\n",
       "\t5 & sw10 & 23.3658 & 19.2318 & 23.0893 & 34.1327 & 2 & Union\\{Missing, Float64\\} \\\\\n",
       "\t6 & sw11 & 13.4934 & 4.47778 & 12.8102 & 32.1889 & 0 & Float64 \\\\\n",
       "\t7 & id & 325.637 & 1 & 313.5 & 700 & 0 & Int64 \\\\\n",
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       "\u001b[1m8×7 DataFrame\u001b[0m\n",
       "\u001b[1m Row \u001b[0m│\u001b[1m variable \u001b[0m\u001b[1m mean      \u001b[0m\u001b[1m min     \u001b[0m\u001b[1m median    \u001b[0m\u001b[1m max     \u001b[0m\u001b[1m nmissing \u001b[0m\u001b[1m eltype     \u001b[0m ⋯\n",
       "\u001b[1m     \u001b[0m│\u001b[90m Symbol   \u001b[0m\u001b[90m Float64   \u001b[0m\u001b[90m Real    \u001b[0m\u001b[90m Float64   \u001b[0m\u001b[90m Real    \u001b[0m\u001b[90m Int64    \u001b[0m\u001b[90m Type       \u001b[0m ⋯\n",
       "─────┼──────────────────────────────────────────────────────────────────────────\n",
       "   1 │ it09       12.2106   6.28846   12.0582   20.5921         6  Union{Missi ⋯\n",
       "   2 │ it10        8.08256  1.77401    7.53993  24.1761         2  Union{Missi\n",
       "   3 │ it11        8.92073  3.23889    8.52447  19.3107         0  Float64\n",
       "   4 │ sw09       10.8503   5.66443   10.832    15.0012         6  Union{Missi\n",
       "   5 │ sw10       23.3658   19.2318   23.0893   34.1327         2  Union{Missi ⋯\n",
       "   6 │ sw11       13.4934   4.47778   12.8102   32.1889         0  Float64\n",
       "   7 │ id        325.637          1  313.5          700         0  Int64\n",
       "   8 │ flc         1.535          1    2.0            2         0  Int64\n",
       "\u001b[36m                                                                1 column omitted\u001b[0m"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "describe(agren)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Calculating summary statistics"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "12.210566789309647"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# calculate mean skipping any missing values\n",
    "mean(skipmissing(agren.it09))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "8-element Vector{Float64}:\n",
       "  12.210566789309647\n",
       "   8.082563855927145\n",
       "   8.920732948515003\n",
       "  10.850313666266498\n",
       "  23.36584553505025\n",
       "  13.493410758197498\n",
       " 325.6375\n",
       "   1.535"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# calculate the mean of all columns skipping missing values\n",
    "mean.(skipmissing.(eachcol(agren)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Visualization: histogram"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
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     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# histogram of one trait\n",
    "histogram(Float64.(skipmissing(agren.it09)),lab=\"\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Visualization: scatterplot"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
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      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# scatterplot of two traits\n",
    "scatter(log2.(agren.it09),log2.(agren.it10),lab=\"\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Modeling: linear regression\n",
    "\n",
    "We now perform linear regression of the fitness values in Italy in 2011 on the fitness in \n",
    "Italy in 2009, adjusting for FLC genotype.\n",
    "\n",
    "The syntax is similar to R; the formula has to be indicated using the `@formula` macro."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "StatsModels.TableRegressionModel{LinearModel{GLM.LmResp{Vector{Float64}}, GLM.DensePredChol{Float64, LinearAlgebra.CholeskyPivoted{Float64, Matrix{Float64}}}}, Matrix{Float64}}\n",
       "\n",
       "it11 ~ 1 + it09 + flc\n",
       "\n",
       "Coefficients:\n",
       "─────────────────────────────────────────────────────────────────────────\n",
       "                 Coef.  Std. Error      t  Pr(>|t|)  Lower 95%  Upper 95%\n",
       "─────────────────────────────────────────────────────────────────────────\n",
       "(Intercept)   1.32384    0.617786    2.14    0.0327   0.109246   2.53844\n",
       "it09          0.644665   0.0408089  15.80    <1e-43   0.564433   0.724897\n",
       "flc          -0.191375   0.19962    -0.96    0.3383  -0.583837   0.201088\n",
       "─────────────────────────────────────────────────────────────────────────"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "out0 = lm(@formula(it11~it09+flc),agren)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Extracting information"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3-element Vector{Float64}:\n",
       "  1.3238442023474766\n",
       "  0.6446651610115701\n",
       " -0.19137472009472786"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "coef(out0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3×3 Matrix{Float64}:\n",
       "  0.38166    -0.0215339    -0.0708066\n",
       " -0.0215339   0.00166536    0.000779437\n",
       " -0.0708066   0.000779437   0.0398481"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "vcov(out0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Residual plots"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
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       "<circle clip-path=\"url(#clip5702)\" cx=\"941.099\" cy=\"1031.69\" r=\"14\" fill=\"#009af9\" fill-rule=\"evenodd\" fill-opacity=\"1\" stroke=\"#000000\" stroke-opacity=\"1\" stroke-width=\"3.2\"/>\n",
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       "</svg>\n"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "scatter(residuals(out0),predict(out0),lab=\"\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## GLM"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "StatsModels.TableRegressionModel{GeneralizedLinearModel{GLM.GlmResp{Vector{Float64}, Normal{Float64}, LogLink}, GLM.DensePredChol{Float64, LinearAlgebra.Cholesky{Float64, Matrix{Float64}}}}, Matrix{Float64}}\n",
       "\n",
       "it11 ~ 1 + :(log(it09))\n",
       "\n",
       "Coefficients:\n",
       "──────────────────────────────────────────────────────────────────────────\n",
       "                  Coef.  Std. Error      z  Pr(>|z|)  Lower 95%  Upper 95%\n",
       "──────────────────────────────────────────────────────────────────────────\n",
       "(Intercept)  -0.0167844   0.14326    -0.12    0.9067  -0.29757    0.264001\n",
       "log(it09)     0.88134     0.0559823  15.74    <1e-55   0.771617   0.991063\n",
       "──────────────────────────────────────────────────────────────────────────"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "out1 = glm(@formula( it11 ~ log(it09) ),agren,Normal(),LogLink())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Chaining and automation of model fitting\n",
    "\n",
    "Suppose we want to regress the fitness in Sweden on fitness in Italy in the \n",
    "same year adjusting for FLC.  We can do this using chaining.  First we get the\n",
    "variable names as strings."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2-element Vector{Vector{String}}:\n",
       " [\"sw09\", \"sw10\", \"sw11\"]\n",
       " [\"it09\", \"it10\", \"it11\"]"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# the years\n",
    "yr = [\"09\",\"10\",\"11\"]\n",
    "# construct names by concatnating strings\n",
    "sw = (\"sw\" .* yr);\n",
    "it = (\"it\" .* yr);\n",
    "[sw, it]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3-element Vector{FormulaTerm{Term, Tuple{Term, Term}}}:\n",
       " sw09 ~ it09 + flc\n",
       " sw10 ~ it10 + flc\n",
       " sw11 ~ it11 + flc"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# construct models from the strings\n",
    "models = Term.(Symbol.(sw)) .~ Term.(Symbol.(it)) .+ Term.(:flc)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3-element Vector{StatsBase.CoefTable}:\n",
       " ────────────────────────────────────────────────────────────────────────────\n",
       "                  Coef.  Std. Error      t  Pr(>|t|)   Lower 95%   Upper 95%\n",
       "────────────────────────────────────────────────────────────────────────────\n",
       "(Intercept)  11.5007      0.439196   26.19    <1e-87  10.6372     12.3642\n",
       "it09         -0.0323098   0.0290118  -1.11    0.2661  -0.0893484   0.0247287\n",
       "flc          -0.166348    0.141913   -1.17    0.2418  -0.445357    0.112661\n",
       "────────────────────────────────────────────────────────────────────────────\n",
       " ───────────────────────────────────────────────────────────────────────────\n",
       "                  Coef.  Std. Error      t  Pr(>|t|)   Lower 95%  Upper 95%\n",
       "───────────────────────────────────────────────────────────────────────────\n",
       "(Intercept)  22.9014      0.409458   55.93    <1e-99  22.0964     23.7064\n",
       "it10          0.0709767   0.0286537   2.48    0.0137   0.0146438   0.12731\n",
       "flc          -0.0711718   0.179746   -0.40    0.6923  -0.424549    0.282206\n",
       "───────────────────────────────────────────────────────────────────────────\n",
       " ──────────────────────────────────────────────────────────────────────────\n",
       "                 Coef.  Std. Error      t  Pr(>|t|)  Lower 95%   Upper 95%\n",
       "──────────────────────────────────────────────────────────────────────────\n",
       "(Intercept)  17.6783     1.03194    17.13    <1e-48  15.6496    19.707\n",
       "it11         -0.339387   0.0812381  -4.18    <1e-04  -0.499098  -0.179677\n",
       "flc          -0.753946   0.411693   -1.83    0.0678  -1.56332    0.0554242\n",
       "──────────────────────────────────────────────────────────────────────────"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# apply the lm function on each of the models, and then pipe to get the coefficient table\n",
    "broadcast( m->lm(m,agren), models )  .|> coeftable"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3-element Vector{StatsBase.CoefTable}:\n",
       " ──────────────────────────────────────────────────────────────────────────────\n",
       "                   Coef.  Std. Error      z  Pr(>|z|)    Lower 95%   Upper 95%\n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "(Intercept)   2.44399     0.0403901   60.51    <1e-99   2.36482     2.52315\n",
       "it09         -0.00292266  0.00267998  -1.09    0.2755  -0.00817533  0.00233001\n",
       "flc          -0.0157048   0.0130643   -1.20    0.2293  -0.0413104   0.00990072\n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       " ────────────────────────────────────────────────────────────────────────────────\n",
       "                   Coef.  Std. Error       z  Pr(>|z|)     Lower 95%   Upper 95%\n",
       "────────────────────────────────────────────────────────────────────────────────\n",
       "(Intercept)   3.13196     0.0174432   179.55    <1e-99   3.09777      3.16615\n",
       "it10          0.00297075  0.0012085     2.46    0.0140   0.000602132  0.00533937\n",
       "flc          -0.00309161  0.00768858   -0.40    0.6876  -0.018161     0.0119777\n",
       "────────────────────────────────────────────────────────────────────────────────\n",
       " ────────────────────────────────────────────────────────────────────────────\n",
       "                  Coef.  Std. Error      z  Pr(>|z|)   Lower 95%   Upper 95%\n",
       "────────────────────────────────────────────────────────────────────────────\n",
       "(Intercept)   2.91438    0.0762161   38.24    <1e-99   2.765       3.06376\n",
       "it11         -0.0253012  0.00628377  -4.03    <1e-04  -0.0376172  -0.0129853\n",
       "flc          -0.057773   0.0303786   -1.90    0.0572  -0.117314    0.001768\n",
       "────────────────────────────────────────────────────────────────────────────"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# a more complex example using glm, and using the @. macro for chaining\n",
    "( @. Term(Symbol(sw)) ~ Term(Symbol(it)) + Term(:flc) ) .|> ( m->glm(m,agren,Normal(),LogLink()) ) .|> coeftable"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Generating random numbers"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "using Random\n",
    "using Distributions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4×4 Matrix{Float64}:\n",
       " -7.58382      0.581556   0.620802  0.360067\n",
       " -5.55343      7.20588    9.40243   1.21663\n",
       " -8.91005     -9.16916   -1.55499   0.878696\n",
       "  0.00442865   1.25381   -2.64062   0.371543"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# initialize random number generator\n",
    "rnd = MersenneTwister(100)\n",
    "# Draw uniform (0,1) numbers in 4x4 matrix\n",
    "rand(Cauchy(),4,4)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Calculating probabilities and densities"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Calculating the CDF of the normal distribution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.8498285541119834"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cdf(Cauchy(0,1),1.96)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Quantiles of the normal distribution."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.6448536269514717"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "quantile(Normal(),0.95)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Generating random normal variables with mean 0.3 and standard deviation 0.5."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1000-element Vector{Float64}:\n",
       " -0.45046598127981746\n",
       "  0.23134421258882393\n",
       "  0.2517271102039687\n",
       " -0.5728252933073521\n",
       "  0.43493558464975157\n",
       "  0.213012013933084\n",
       "  0.3421748432166778\n",
       " -0.27338211103715276\n",
       "  0.5047833624899034\n",
       " -0.0755106919393046\n",
       "  1.4504780925337375\n",
       "  0.13648212828722928\n",
       " -0.6281694855786075\n",
       "  ⋮\n",
       " -0.2556554701982759\n",
       "  0.12291775475404931\n",
       "  0.35515495957438026\n",
       "  1.1177627331209108\n",
       " -0.13532811887806623\n",
       "  0.43476002987259715\n",
       "  0.06298287074397868\n",
       "  0.07067957927768084\n",
       "  0.9479217516381206\n",
       " -0.27746754253213607\n",
       " -0.6880452360452243\n",
       "  0.934102133119902"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = rand(Normal(0.3,0.5),1000)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We now fit the MLE assuming a normal distribution."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Laplace{Float64}(μ=0.29707392015615663, θ=0.5209103192451021)"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "normFit = fit_mle(Laplace,x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We generate normal variables with parameters equal to the estimated ones."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "10-element Vector{Float64}:\n",
       "  0.5546055049393865\n",
       "  1.433570096640013\n",
       "  0.1813174020298847\n",
       " -0.8250784796119491\n",
       " -0.7375658870463838\n",
       " -0.11567066405894966\n",
       "  0.46309400524163624\n",
       "  1.3986439200195688\n",
       "  0.22150302351203688\n",
       " -0.5009202916969959"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rand(normFit,10)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Version info"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Julia Version 1.6.3\n",
      "Commit ae8452a9e0 (2021-09-23 17:34 UTC)\n",
      "Platform Info:\n",
      "  OS: macOS (x86_64-apple-darwin19.5.0)\n",
      "  CPU: Intel(R) Core(TM) i5-5287U CPU @ 2.90GHz\n",
      "  WORD_SIZE: 64\n",
      "  LIBM: libopenlibm\n",
      "  LLVM: libLLVM-11.0.1 (ORCJIT, broadwell)\n"
     ]
    }
   ],
   "source": [
    "versioninfo()"
   ]
  }
 ],
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