{ "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": [ "
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gradegpatucepsi
002.06221
112.39191
202.63200
302.92120
402.76170
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" ], "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": [ "\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": [ "
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meanvarskewkurtosis
grade0.343750.2328630.657952-1.5671
gpa3.117190.2178210.122658-0.429932
tuce21.937515.2218-0.525110.0483051
psi0.43750.2540320.251976-1.93651
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" ], "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": [ "\n", "\n", "\n", " \n", "\n", "\n", " \n", "\n", "\n", " \n", "\n", "\n", " \n", "\n", "\n", " \n", "\n", "\n", " \n", "\n", "\n", " \n", "\n", "
Logit Regression Results
Dep. Variable: grade No. Observations: 32
Model: Logit Df Residuals: 28
Method: MLE Df Model: 3
Date: Fri, 03 Jan 2020 Pseudo R-squ.: 0.3740
Time: 21:39:27 Log-Likelihood: -12.890
converged: True LL-Null: -20.592
Covariance Type: nonrobust LLR p-value: 0.001502
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coef std err z P>|z| [0.025 0.975]
const -13.0213 4.931 -2.641 0.008 -22.687 -3.356
gpa 2.8261 1.263 2.238 0.025 0.351 5.301
tuce 0.0952 0.142 0.672 0.501 -0.182 0.373
psi 2.3787 1.065 2.234 0.025 0.292 4.465
" ], "text/plain": [ "\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": [ "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
Unstandardized and Standardized Estimates
coef z P>|z| coef_stdX coef_stdXy stdev_X
grade
gpa+2.8261+2.2380.025+1.3190+0.49120.4667
tuce+0.0952+0.6720.501+0.3713+0.13833.9015
psi+2.3787+2.2340.025+1.1989+0.44650.5040
" ], "text/plain": [ "" ] }, "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": [ "
" ] }, "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 }