{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Generalized Linear Models (Formula)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This notebook illustrates how you can use R-style formulas to fit Generalized Linear Models.\n",
    "\n",
    "To begin, we load the ``Star98`` dataset and we construct a formula and pre-process the data:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false,
    "execution": {
 
 
 
 
    },
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [],
   "source": [
    "import statsmodels.api as sm\n",
    "import statsmodels.formula.api as smf\n",
    "\n",
    "star98 = sm.datasets.star98.load_pandas().data\n",
    "formula = \"SUCCESS ~ LOWINC + PERASIAN + PERBLACK + PERHISP + PCTCHRT + \\\n",
    "           PCTYRRND + PERMINTE*AVYRSEXP*AVSALK + PERSPENK*PTRATIO*PCTAF\"\n",
    "dta = star98[\n",
    "    [\n",
    "        \"NABOVE\",\n",
    "        \"NBELOW\",\n",
    "        \"LOWINC\",\n",
    "        \"PERASIAN\",\n",
    "        \"PERBLACK\",\n",
    "        \"PERHISP\",\n",
    "        \"PCTCHRT\",\n",
    "        \"PCTYRRND\",\n",
    "        \"PERMINTE\",\n",
    "        \"AVYRSEXP\",\n",
    "        \"AVSALK\",\n",
    "        \"PERSPENK\",\n",
    "        \"PTRATIO\",\n",
    "        \"PCTAF\",\n",
    "    ]\n",
    "].copy()\n",
    "endog = dta[\"NABOVE\"] / (dta[\"NABOVE\"] + dta.pop(\"NBELOW\"))\n",
    "del dta[\"NABOVE\"]\n",
    "dta[\"SUCCESS\"] = endog"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Then, we fit the GLM model:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false,
    "execution": {
 
 
 
 
    },
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "                 Generalized Linear Model Regression Results                  \n",
      "==============================================================================\n",
      "Dep. Variable:                SUCCESS   No. Observations:                  303\n",
      "Model:                            GLM   Df Residuals:                      282\n",
      "Model Family:                Binomial   Df Model:                           20\n",
      "Link Function:                  Logit   Scale:                          1.0000\n",
      "Method:                          IRLS   Log-Likelihood:                -127.33\n",
      "Date:                Thu, 18 Dec 2025   Deviance:                       8.5477\n",
      "Time:                        07:37:30   Pearson chi2:                     8.48\n",
      "No. Iterations:                     4   Pseudo R-squ. (CS):             0.1115\n",
      "Covariance Type:            nonrobust                                         \n",
      "============================================================================================\n",
      "                               coef    std err          z      P>|z|      [0.025      0.975]\n",
      "--------------------------------------------------------------------------------------------\n",
      "Intercept                    0.4037     25.036      0.016      0.987     -48.665      49.472\n",
      "LOWINC                      -0.0204      0.010     -1.982      0.048      -0.041      -0.000\n",
      "PERASIAN                     0.0159      0.017      0.910      0.363      -0.018       0.050\n",
      "PERBLACK                    -0.0198      0.020     -1.004      0.316      -0.058       0.019\n",
      "PERHISP                     -0.0096      0.010     -0.951      0.341      -0.029       0.010\n",
      "PCTCHRT                     -0.0022      0.022     -0.103      0.918      -0.045       0.040\n",
      "PCTYRRND                    -0.0022      0.006     -0.348      0.728      -0.014       0.010\n",
      "PERMINTE                     0.1068      0.787      0.136      0.892      -1.436       1.650\n",
      "AVYRSEXP                    -0.0411      1.176     -0.035      0.972      -2.346       2.264\n",
      "PERMINTE:AVYRSEXP           -0.0031      0.054     -0.057      0.954      -0.108       0.102\n",
      "AVSALK                       0.0131      0.295      0.044      0.965      -0.566       0.592\n",
      "PERMINTE:AVSALK             -0.0019      0.013     -0.145      0.885      -0.028       0.024\n",
      "AVYRSEXP:AVSALK              0.0008      0.020      0.038      0.970      -0.039       0.041\n",
      "PERMINTE:AVYRSEXP:AVSALK  5.978e-05      0.001      0.068      0.946      -0.002       0.002\n",
      "PERSPENK                    -0.3097      4.233     -0.073      0.942      -8.606       7.987\n",
      "PTRATIO                      0.0096      0.919      0.010      0.992      -1.792       1.811\n",
      "PERSPENK:PTRATIO             0.0066      0.206      0.032      0.974      -0.397       0.410\n",
      "PCTAF                       -0.0143      0.474     -0.030      0.976      -0.944       0.916\n",
      "PERSPENK:PCTAF               0.0105      0.098      0.107      0.915      -0.182       0.203\n",
      "PTRATIO:PCTAF               -0.0001      0.022     -0.005      0.996      -0.044       0.044\n",
      "PERSPENK:PTRATIO:PCTAF      -0.0002      0.005     -0.051      0.959      -0.010       0.009\n",
      "============================================================================================\n"
     ]
    }
   ],
   "source": [
    "mod1 = smf.glm(formula=formula, data=dta, family=sm.families.Binomial()).fit()\n",
    "print(mod1.summary())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Finally, we define a function to operate customized data transformation using the formula framework:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false,
    "execution": {
 
 
 
 
    },
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "                 Generalized Linear Model Regression Results                  \n",
      "==============================================================================\n",
      "Dep. Variable:                SUCCESS   No. Observations:                  303\n",
      "Model:                            GLM   Df Residuals:                      282\n",
      "Model Family:                Binomial   Df Model:                           20\n",
      "Link Function:                  Logit   Scale:                          1.0000\n",
      "Method:                          IRLS   Log-Likelihood:                -127.33\n",
      "Date:                Thu, 18 Dec 2025   Deviance:                       8.5477\n",
      "Time:                        07:37:30   Pearson chi2:                     8.48\n",
      "No. Iterations:                     4   Pseudo R-squ. (CS):             0.1115\n",
      "Covariance Type:            nonrobust                                         \n",
      "============================================================================================\n",
      "                               coef    std err          z      P>|z|      [0.025      0.975]\n",
      "--------------------------------------------------------------------------------------------\n",
      "Intercept                    0.4037     25.036      0.016      0.987     -48.665      49.472\n",
      "double_it(LOWINC)           -0.0102      0.005     -1.982      0.048      -0.020      -0.000\n",
      "PERASIAN                     0.0159      0.017      0.910      0.363      -0.018       0.050\n",
      "PERBLACK                    -0.0198      0.020     -1.004      0.316      -0.058       0.019\n",
      "PERHISP                     -0.0096      0.010     -0.951      0.341      -0.029       0.010\n",
      "PCTCHRT                     -0.0022      0.022     -0.103      0.918      -0.045       0.040\n",
      "PCTYRRND                    -0.0022      0.006     -0.348      0.728      -0.014       0.010\n",
      "PERMINTE                     0.1068      0.787      0.136      0.892      -1.436       1.650\n",
      "AVYRSEXP                    -0.0411      1.176     -0.035      0.972      -2.346       2.264\n",
      "PERMINTE:AVYRSEXP           -0.0031      0.054     -0.057      0.954      -0.108       0.102\n",
      "AVSALK                       0.0131      0.295      0.044      0.965      -0.566       0.592\n",
      "PERMINTE:AVSALK             -0.0019      0.013     -0.145      0.885      -0.028       0.024\n",
      "AVYRSEXP:AVSALK              0.0008      0.020      0.038      0.970      -0.039       0.041\n",
      "PERMINTE:AVYRSEXP:AVSALK  5.978e-05      0.001      0.068      0.946      -0.002       0.002\n",
      "PERSPENK                    -0.3097      4.233     -0.073      0.942      -8.606       7.987\n",
      "PTRATIO                      0.0096      0.919      0.010      0.992      -1.792       1.811\n",
      "PERSPENK:PTRATIO             0.0066      0.206      0.032      0.974      -0.397       0.410\n",
      "PCTAF                       -0.0143      0.474     -0.030      0.976      -0.944       0.916\n",
      "PERSPENK:PCTAF               0.0105      0.098      0.107      0.915      -0.182       0.203\n",
      "PTRATIO:PCTAF               -0.0001      0.022     -0.005      0.996      -0.044       0.044\n",
      "PERSPENK:PTRATIO:PCTAF      -0.0002      0.005     -0.051      0.959      -0.010       0.009\n",
      "============================================================================================\n"
     ]
    }
   ],
   "source": [
    "def double_it(x):\n",
    "    return 2 * x\n",
    "\n",
    "\n",
    "formula = \"SUCCESS ~ double_it(LOWINC) + PERASIAN + PERBLACK + PERHISP + PCTCHRT + \\\n",
    "           PCTYRRND + PERMINTE*AVYRSEXP*AVSALK + PERSPENK*PTRATIO*PCTAF\"\n",
    "mod2 = smf.glm(formula=formula, data=dta, family=sm.families.Binomial()).fit()\n",
    "print(mod2.summary())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As expected, the coefficient for ``double_it(LOWINC)`` in the second model is half the size of the ``LOWINC`` coefficient from the first model:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false,
    "execution": {
 
 
 
 
    },
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-0.020395987154755987\n",
      "-0.02039598715475644\n"
     ]
    }
   ],
   "source": [
    "print(mod1.params.iloc[1])\n",
    "print(mod2.params.iloc[1] * 2)"
   ]
  }
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