{
"cells": [
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [],
"source": [
"import pandas as pd\n",
"import matplotlib.pyplot as plt\n",
"%matplotlib inline\n",
"\n",
"# Hide Numpy warnings from Statsmodels\n",
"import warnings\n",
"warnings.filterwarnings('ignore')"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Load data"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"[Dataset from University of Notre Dame](https://www3.nd.edu/~rwilliam/stats3/L04.pdf) for demonstration of **logistic regression.**\n",
"\n",
"It is being used here to compare output from `appelpy` to output from Stata directly."
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [],
"source": [
"df = pd.read_stata('https://www3.nd.edu/~rwilliam/statafiles/glm-logit.dta')"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<div>\n",
"<style scoped>\n",
" .dataframe tbody tr th:only-of-type {\n",
" vertical-align: middle;\n",
" }\n",
"\n",
" .dataframe tbody tr th {\n",
" vertical-align: top;\n",
" }\n",
"\n",
" .dataframe thead th {\n",
" text-align: right;\n",
" }\n",
"</style>\n",
"<table border=\"1\" class=\"dataframe\">\n",
" <thead>\n",
" <tr style=\"text-align: right;\">\n",
" <th></th>\n",
" <th>grade</th>\n",
" <th>gpa</th>\n",
" <th>tuce</th>\n",
" <th>psi</th>\n",
" </tr>\n",
" </thead>\n",
" <tbody>\n",
" <tr>\n",
" <th>0</th>\n",
" <td>0</td>\n",
" <td>2.06</td>\n",
" <td>22</td>\n",
" <td>1</td>\n",
" </tr>\n",
" <tr>\n",
" <th>1</th>\n",
" <td>1</td>\n",
" <td>2.39</td>\n",
" <td>19</td>\n",
" <td>1</td>\n",
" </tr>\n",
" <tr>\n",
" <th>2</th>\n",
" <td>0</td>\n",
" <td>2.63</td>\n",
" <td>20</td>\n",
" <td>0</td>\n",
" </tr>\n",
" <tr>\n",
" <th>3</th>\n",
" <td>0</td>\n",
" <td>2.92</td>\n",
" <td>12</td>\n",
" <td>0</td>\n",
" </tr>\n",
" <tr>\n",
" <th>4</th>\n",
" <td>0</td>\n",
" <td>2.76</td>\n",
" <td>17</td>\n",
" <td>0</td>\n",
" </tr>\n",
" </tbody>\n",
"</table>\n",
"</div>"
],
"text/plain": [
" grade gpa tuce psi\n",
"0 0 2.06 22 1\n",
"1 1 2.39 19 1\n",
"2 0 2.63 20 0\n",
"3 0 2.92 12 0\n",
"4 0 2.76 17 0"
]
},
"execution_count": 3,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"df.head()"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"<class 'pandas.core.frame.DataFrame'>\n",
"Int64Index: 32 entries, 0 to 31\n",
"Data columns (total 4 columns):\n",
"grade 32 non-null int8\n",
"gpa 32 non-null float32\n",
"tuce 32 non-null int8\n",
"psi 32 non-null int8\n",
"dtypes: float32(1), int8(3)\n",
"memory usage: 480.0 bytes\n"
]
}
],
"source": [
"df.info()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Inspect data"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Let's do some exploratory data analysis on this dataset."
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [],
"source": [
"from appelpy.eda import statistical_moments"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"CPU times: user 15.1 ms, sys: 165 µs, total: 15.2 ms\n",
"Wall time: 13.1 ms\n"
]
},
{
"data": {
"text/html": [
"<div>\n",
"<style scoped>\n",
" .dataframe tbody tr th:only-of-type {\n",
" vertical-align: middle;\n",
" }\n",
"\n",
" .dataframe tbody tr th {\n",
" vertical-align: top;\n",
" }\n",
"\n",
" .dataframe thead th {\n",
" text-align: right;\n",
" }\n",
"</style>\n",
"<table border=\"1\" class=\"dataframe\">\n",
" <thead>\n",
" <tr style=\"text-align: right;\">\n",
" <th></th>\n",
" <th>mean</th>\n",
" <th>var</th>\n",
" <th>skew</th>\n",
" <th>kurtosis</th>\n",
" </tr>\n",
" </thead>\n",
" <tbody>\n",
" <tr>\n",
" <th>grade</th>\n",
" <td>0.34375</td>\n",
" <td>0.232863</td>\n",
" <td>0.657952</td>\n",
" <td>-1.5671</td>\n",
" </tr>\n",
" <tr>\n",
" <th>gpa</th>\n",
" <td>3.11719</td>\n",
" <td>0.217821</td>\n",
" <td>0.122658</td>\n",
" <td>-0.429932</td>\n",
" </tr>\n",
" <tr>\n",
" <th>tuce</th>\n",
" <td>21.9375</td>\n",
" <td>15.2218</td>\n",
" <td>-0.52511</td>\n",
" <td>0.0483051</td>\n",
" </tr>\n",
" <tr>\n",
" <th>psi</th>\n",
" <td>0.4375</td>\n",
" <td>0.254032</td>\n",
" <td>0.251976</td>\n",
" <td>-1.93651</td>\n",
" </tr>\n",
" </tbody>\n",
"</table>\n",
"</div>"
],
"text/plain": [
" mean var skew kurtosis\n",
"grade 0.34375 0.232863 0.657952 -1.5671\n",
"gpa 3.11719 0.217821 0.122658 -0.429932\n",
"tuce 21.9375 15.2218 -0.52511 0.0483051\n",
"psi 0.4375 0.254032 0.251976 -1.93651"
]
},
"execution_count": 6,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"%%time\n",
"statistical_moments(df)"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"grade 2\n",
"gpa 29\n",
"tuce 14\n",
"psi 2\n",
"dtype: int64"
]
},
"execution_count": 7,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"df.nunique()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"`grade` is a binary outcome that we would like to predict in this dataset, so logistic regression is a natural choice for modelling it."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Logistic regression"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Let's do a regression of `grade` on `gpa`, `tuce`, `psi`."
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {},
"outputs": [],
"source": [
"from appelpy.discrete_model import Logit"
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {},
"outputs": [],
"source": [
"y_list = ['grade']\n",
"X_list = ['gpa', 'tuce', 'psi']"
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"CPU times: user 103 ms, sys: 6.05 ms, total: 109 ms\n",
"Wall time: 121 ms\n"
]
}
],
"source": [
"%%time\n",
"model1 = Logit(df, y_list, X_list).fit()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Standardized and unstandardized estimates"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The **unstandardized estimates** can be returned in two formats: the **coefficients** themselves and their **odds ratio** equivalents."
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<table class=\"simpletable\">\n",
"<caption>Logit Regression Results</caption>\n",
"<tr>\n",
" <th>Dep. Variable:</th> <td>grade</td> <th> No. Observations: </th> <td> 32</td> \n",
"</tr>\n",
"<tr>\n",
" <th>Model:</th> <td>Logit</td> <th> Df Residuals: </th> <td> 28</td> \n",
"</tr>\n",
"<tr>\n",
" <th>Method:</th> <td>MLE</td> <th> Df Model: </th> <td> 3</td> \n",
"</tr>\n",
"<tr>\n",
" <th>Date:</th> <td>Fri, 03 Jan 2020</td> <th> Pseudo R-squ.: </th> <td>0.3740</td> \n",
"</tr>\n",
"<tr>\n",
" <th>Time:</th> <td>21:39:27</td> <th> Log-Likelihood: </th> <td> -12.890</td>\n",
"</tr>\n",
"<tr>\n",
" <th>converged:</th> <td>True</td> <th> LL-Null: </th> <td> -20.592</td>\n",
"</tr>\n",
"<tr>\n",
" <th>Covariance Type:</th> <td>nonrobust</td> <th> LLR p-value: </th> <td>0.001502</td>\n",
"</tr>\n",
"</table>\n",
"<table class=\"simpletable\">\n",
"<tr>\n",
" <td></td> <th>coef</th> <th>std err</th> <th>z</th> <th>P>|z|</th> <th>[0.025</th> <th>0.975]</th> \n",
"</tr>\n",
"<tr>\n",
" <th>const</th> <td> -13.0213</td> <td> 4.931</td> <td> -2.641</td> <td> 0.008</td> <td> -22.687</td> <td> -3.356</td>\n",
"</tr>\n",
"<tr>\n",
" <th>gpa</th> <td> 2.8261</td> <td> 1.263</td> <td> 2.238</td> <td> 0.025</td> <td> 0.351</td> <td> 5.301</td>\n",
"</tr>\n",
"<tr>\n",
" <th>tuce</th> <td> 0.0952</td> <td> 0.142</td> <td> 0.672</td> <td> 0.501</td> <td> -0.182</td> <td> 0.373</td>\n",
"</tr>\n",
"<tr>\n",
" <th>psi</th> <td> 2.3787</td> <td> 1.065</td> <td> 2.234</td> <td> 0.025</td> <td> 0.292</td> <td> 4.465</td>\n",
"</tr>\n",
"</table>"
],
"text/plain": [
"<class 'statsmodels.iolib.summary.Summary'>\n",
"\"\"\"\n",
" Logit Regression Results \n",
"==============================================================================\n",
"Dep. Variable: grade No. Observations: 32\n",
"Model: Logit Df Residuals: 28\n",
"Method: MLE Df Model: 3\n",
"Date: Fri, 03 Jan 2020 Pseudo R-squ.: 0.3740\n",
"Time: 21:39:27 Log-Likelihood: -12.890\n",
"converged: True LL-Null: -20.592\n",
"Covariance Type: nonrobust LLR p-value: 0.001502\n",
"==============================================================================\n",
" coef std err z P>|z| [0.025 0.975]\n",
"------------------------------------------------------------------------------\n",
"const -13.0213 4.931 -2.641 0.008 -22.687 -3.356\n",
"gpa 2.8261 1.263 2.238 0.025 0.351 5.301\n",
"tuce 0.0952 0.142 0.672 0.501 -0.182 0.373\n",
"psi 2.3787 1.065 2.234 0.025 0.292 4.465\n",
"==============================================================================\n",
"\"\"\""
]
},
"execution_count": 11,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"model1.results_output"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"There are two significant regressors in the model."
]
},
{
"cell_type": "code",
"execution_count": 12,
"metadata": {
"scrolled": true
},
"outputs": [
{
"data": {
"text/plain": [
"['gpa', 'psi']"
]
},
"execution_count": 12,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"model1.significant_regressors(0.05)"
]
},
{
"cell_type": "code",
"execution_count": 13,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"gpa 16.879721\n",
"tuce 1.099832\n",
"psi 10.790733\n",
"Name: odds_ratios, dtype: float64"
]
},
"execution_count": 13,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"model1.odds_ratios"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The `results_output_standardized` object contains the **standardized estimates** of the regressors (and the unstandardized ones).\n",
"\n",
"Standardized coefficients are sometimes called **beta coefficients**.\n",
"\n",
"The output is similar to what would be returned by Stata's _`listcoef` command_.\n",
"\n",
"These are the standardized estimates listed:\n",
"- `coef_stdX`: x-standardized coefficient, i.e. how much does `y` increase with a one-standard deviation increase in `x`.\n",
"- `coef_stdXy`: fully standardized coefficient, i.e. by how many standard deviations does `y` increase with a one=standard deviation increase in `x`.\n",
"- `stdev_X`: standard deviation of regressor"
]
},
{
"cell_type": "code",
"execution_count": 14,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<style type=\"text/css\" >\n",
"</style><table id=\"T_8433509e_2e71_11ea_b701_5f623a1f415f\" ><caption>Unstandardized and Standardized Estimates</caption><thead> <tr> <th class=\"blank level0\" ></th> <th class=\"col_heading level0 col0\" >coef</th> <th class=\"col_heading level0 col1\" >z</th> <th class=\"col_heading level0 col2\" >P>|z|</th> <th class=\"col_heading level0 col3\" >coef_stdX</th> <th class=\"col_heading level0 col4\" >coef_stdXy</th> <th class=\"col_heading level0 col5\" >stdev_X</th> </tr> <tr> <th class=\"index_name level0\" >grade</th> <th class=\"blank\" ></th> <th class=\"blank\" ></th> <th class=\"blank\" ></th> <th class=\"blank\" ></th> <th class=\"blank\" ></th> <th class=\"blank\" ></th> </tr></thead><tbody>\n",
" <tr>\n",
" <th id=\"T_8433509e_2e71_11ea_b701_5f623a1f415flevel0_row0\" class=\"row_heading level0 row0\" >gpa</th>\n",
" <td id=\"T_8433509e_2e71_11ea_b701_5f623a1f415frow0_col0\" class=\"data row0 col0\" >+2.8261</td>\n",
" <td id=\"T_8433509e_2e71_11ea_b701_5f623a1f415frow0_col1\" class=\"data row0 col1\" >+2.238</td>\n",
" <td id=\"T_8433509e_2e71_11ea_b701_5f623a1f415frow0_col2\" class=\"data row0 col2\" >0.025</td>\n",
" <td id=\"T_8433509e_2e71_11ea_b701_5f623a1f415frow0_col3\" class=\"data row0 col3\" >+1.3190</td>\n",
" <td id=\"T_8433509e_2e71_11ea_b701_5f623a1f415frow0_col4\" class=\"data row0 col4\" >+0.4912</td>\n",
" <td id=\"T_8433509e_2e71_11ea_b701_5f623a1f415frow0_col5\" class=\"data row0 col5\" >0.4667</td>\n",
" </tr>\n",
" <tr>\n",
" <th id=\"T_8433509e_2e71_11ea_b701_5f623a1f415flevel0_row1\" class=\"row_heading level0 row1\" >tuce</th>\n",
" <td id=\"T_8433509e_2e71_11ea_b701_5f623a1f415frow1_col0\" class=\"data row1 col0\" >+0.0952</td>\n",
" <td id=\"T_8433509e_2e71_11ea_b701_5f623a1f415frow1_col1\" class=\"data row1 col1\" >+0.672</td>\n",
" <td id=\"T_8433509e_2e71_11ea_b701_5f623a1f415frow1_col2\" class=\"data row1 col2\" >0.501</td>\n",
" <td id=\"T_8433509e_2e71_11ea_b701_5f623a1f415frow1_col3\" class=\"data row1 col3\" >+0.3713</td>\n",
" <td id=\"T_8433509e_2e71_11ea_b701_5f623a1f415frow1_col4\" class=\"data row1 col4\" >+0.1383</td>\n",
" <td id=\"T_8433509e_2e71_11ea_b701_5f623a1f415frow1_col5\" class=\"data row1 col5\" >3.9015</td>\n",
" </tr>\n",
" <tr>\n",
" <th id=\"T_8433509e_2e71_11ea_b701_5f623a1f415flevel0_row2\" class=\"row_heading level0 row2\" >psi</th>\n",
" <td id=\"T_8433509e_2e71_11ea_b701_5f623a1f415frow2_col0\" class=\"data row2 col0\" >+2.3787</td>\n",
" <td id=\"T_8433509e_2e71_11ea_b701_5f623a1f415frow2_col1\" class=\"data row2 col1\" >+2.234</td>\n",
" <td id=\"T_8433509e_2e71_11ea_b701_5f623a1f415frow2_col2\" class=\"data row2 col2\" >0.025</td>\n",
" <td id=\"T_8433509e_2e71_11ea_b701_5f623a1f415frow2_col3\" class=\"data row2 col3\" >+1.1989</td>\n",
" <td id=\"T_8433509e_2e71_11ea_b701_5f623a1f415frow2_col4\" class=\"data row2 col4\" >+0.4465</td>\n",
" <td id=\"T_8433509e_2e71_11ea_b701_5f623a1f415frow2_col5\" class=\"data row2 col5\" >0.5040</td>\n",
" </tr>\n",
" </tbody></table>"
],
"text/plain": [
"<pandas.io.formats.style.Styler at 0x7f5cd17bcd90>"
]
},
"execution_count": 14,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"model1.results_output_standardized"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Compare the results from the Stata command `listcoef, std help` below:"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"```\n",
"logit (N=32): Unstandardized and standardized estimates\n",
"\n",
" Observed SD: 0.4826\n",
" Latent SD: 2.6851\n",
"\n",
"--------------------------------------------------------------------------------\n",
" | b z P>|z| bStdX bStdY bStdXY SDofX\n",
"-------------+------------------------------------------------------------------\n",
" gpa | 2.8261 2.238 0.025 1.319 1.053 0.491 0.467\n",
" tuce | 0.0952 0.672 0.501 0.371 0.035 0.138 3.902\n",
" 1.psi | 2.3787 2.234 0.025 1.199 0.886 0.447 0.504\n",
" constant | -13.0213 -2.641 0.008 . . . .\n",
"--------------------------------------------------------------------------------\n",
" b = raw coefficient\n",
" z = z-score for test of b=0\n",
" P>|z| = p-value for z-test\n",
" bStdX = x-standardized coefficient\n",
" bStdY = y-standardized coefficient\n",
" bStdXY = fully standardized coefficient\n",
" SDofX = standard deviation of X\n",
" ```"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Model selection"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Some model selection statistics are stored in the `model_selection_stats` attribute."
]
},
{
"cell_type": "code",
"execution_count": 15,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"{'log_likelihood': -12.88963346533348,\n",
" 'r_squared_pseudo': 0.3740383321251376,\n",
" 'aic': 33.779266930666964,\n",
" 'bic': 39.642210541865865}"
]
},
"execution_count": 15,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"model1.model_selection_stats"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The log-likelihood is also stored in its own attribute `log_likelihood` for ease of reference."
]
},
{
"cell_type": "code",
"execution_count": 16,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"-12.88963346533348"
]
},
"execution_count": 16,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"model1.log_likelihood"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Prediction"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The `predict` method can be used to return estimated probabilities for observations."
]
},
{
"cell_type": "code",
"execution_count": 17,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"CPU times: user 2.27 ms, sys: 23 µs, total: 2.29 ms\n",
"Wall time: 2.16 ms\n"
]
},
{
"data": {
"text/plain": [
"array([0.06137582, 0.11103084, 0.02447037, 0.02590164, 0.02650405,\n",
" 0.02657799, 0.24177247, 0.03018375, 0.03485825, 0.03858832,\n",
" 0.30721867, 0.05155897, 0.0536264 , 0.05950126, 0.48113296,\n",
" 0.52911713, 0.11112664, 0.58987241, 0.63542067, 0.6607859 ,\n",
" 0.18725992, 0.18999741, 0.19321116, 0.32223953, 0.83829052,\n",
" 0.84170419, 0.8520909 , 0.36098992, 0.90484726, 0.56989301,\n",
" 0.94534027, 0.69351141])"
]
},
"execution_count": 17,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"%%time\n",
"preds = model1.predict(df[X_list].to_numpy())\n",
"preds"
]
},
{
"cell_type": "code",
"execution_count": 18,
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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\n",
"text/plain": [
"<Figure size 792x504 with 1 Axes>"
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
}
],
"source": [
"fig, ax = plt.subplots(figsize=(11,7))\n",
"pd.Series(preds).hist(ax=ax, range=(0,1))\n",
"ax.set_xlim([0, 1])\n",
"ax.set_title('Histogram of probabilities predicted from the original data.')\n",
"ax.set_ylabel('Frequency')\n",
"ax.set_xlabel('Probability')\n",
"plt.show()"
]
}
],
"metadata": {
"kernelspec": {
"display_name": "Python 3",
"language": "python",
"name": "python3"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 3
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.7.5"
},
"toc": {
"base_numbering": 1,
"nav_menu": {},
"number_sections": true,
"sideBar": true,
"skip_h1_title": false,
"title_cell": "Table of Contents",
"title_sidebar": "Contents",
"toc_cell": false,
"toc_position": {},
"toc_section_display": true,
"toc_window_display": true
}
},
"nbformat": 4,
"nbformat_minor": 2
}