Section 4.6 — Generalized linear models#

This notebook contains the code examples from Section 4.6 Generalized linear models from the No Bullshit Guide to Statistics.

Notebook setup#

# load Python modules
import os
import numpy as np
import pandas as pd
import seaborn as sns
# Figures setup
import matplotlib.pyplot as plt
plt.clf()  # needed otherwise `sns.set_theme` doesn't work
from plot_helpers import RCPARAMS
RCPARAMS.update({"figure.figsize": (5, 3)})   # good for screen
# RCPARAMS.update({'figure.figsize': (10, 3)})   # good for screen
# RCPARAMS.update({'figure.figsize': (4, 2)})  # good for print
sns.set_theme(
    context="paper",
    style="whitegrid",
    palette="colorblind",
    rc=RCPARAMS,
)

# High-resolution please
%config InlineBackend.figure_format = 'retina'

# Where to store figures
DESTDIR = "figures/lm/generalized"
<Figure size 640x480 with 0 Axes>
from ministats.utils import savefigure
#######################################################

Definitions#

Logistic regression#

TODO FORMULA

# FIGURES ONLY
from scipy.stats import bernoulli
from scipy.special import expit

# Define the logistic regression model function
def expit_model(x):
    p = expit(-10 + 2*x)
    return p

xlims = [0, 10]

stem_half_width = 0.03

with sns.axes_style("ticks"):
    fig, ax = plt.subplots(figsize=(5, 3))

    # Plot the logistic regression model
    xs = np.linspace(xlims[0], xlims[1], 200)
    ps = expit_model(xs)
    sns.lineplot(x=xs, y=ps, ax=ax, label=r"$p(x) = \mathrm{expit}(\beta_0 + \beta_1x)$", linewidth=2)

    # Plot Bernoulli distributions at specified x positions
    x_positions = [2,4,5,6,8,10]
    for x_pos in x_positions:
        p_pos = expit_model(x_pos)
        ys = [0,1]
        pmf = bernoulli(p=p_pos).pmf(ys)
        ys_plot = [p_pos-stem_half_width, p_pos+stem_half_width]
        ax.stem(ys_plot, x_pos - pmf, bottom=x_pos, orientation='horizontal')

    # Figure setup
    ax.set_xlabel("$x$")
    ax.set_ylabel("$p$")
    ax.legend(loc="upper left")

    filename = os.path.join(DESTDIR, "logistic_regression_xy_with_stemplots.pdf")
    savefigure(fig, filename)
Saved figure to figures/lm/generalized/logistic_regression_xy_with_stemplots.pdf
Saved figure to figures/lm/generalized/logistic_regression_xy_with_stemplots.png
../_images/7037635c81105b3b63caa1767e533482a4d9d79863285f5b1b3effbb51090082.png
expit(-6)
0.0024726231566347743
expit(10)
0.9999546021312976

Example 1: hiring student interns#

interns = pd.read_csv("../datasets/interns.csv")
print(interns.shape)
interns.head(3)
(100, 2)
work hired
0 42.5 1
1 39.3 0
2 43.2 1
import statsmodels.formula.api as smf

lr1 = smf.logit("hired ~ 1 + work", data=interns).fit()
print(lr1.params)
Optimization terminated successfully.
         Current function value: 0.138101
         Iterations 10
Intercept   -78.693205
work          1.981458
dtype: float64
lr1.summary()
Logit Regression Results
Dep. Variable: hired No. Observations: 100
Model: Logit Df Residuals: 98
Method: MLE Df Model: 1
Date: Thu, 05 Sep 2024 Pseudo R-squ.: 0.8005
Time: 15:49:30 Log-Likelihood: -13.810
converged: True LL-Null: -69.235
Covariance Type: nonrobust LLR p-value: 6.385e-26
coef std err z P>|z| [0.025 0.975]
Intercept -78.6932 19.851 -3.964 0.000 -117.600 -39.787
work 1.9815 0.500 3.959 0.000 1.001 2.962


Possibly complete quasi-separation: A fraction 0.32 of observations can be
perfectly predicted. This might indicate that there is complete
quasi-separation. In this case some parameters will not be identified.
ax = sns.scatterplot(data=interns, x="work", y="hired")
wgrid = np.linspace(27, 50, 100)
hired_preds = lr1.predict({"work": wgrid})
sns.lineplot(x=wgrid, y=hired_preds, ax=ax);
../_images/177d412a04ceb10fa9c484d9973a2e95022cecb8b2182df55c77a05d85cdff8b.png
from ministats import plot_reg
plot_reg(lr1)

# FIGURES ONLY
filename = os.path.join(DESTDIR, "logistic_regression_interns_hired_vs_work.pdf")
savefigure(plt.gcf(), filename)
Saved figure to figures/lm/generalized/logistic_regression_interns_hired_vs_work.pdf
Saved figure to figures/lm/generalized/logistic_regression_interns_hired_vs_work.png
../_images/77b8dddda53b94cfbb8c1ab358072b2fbff2eb97114c00aa7b281f5ed3b0ebc3.png
lr1.summary()
Logit Regression Results
Dep. Variable: hired No. Observations: 100
Model: Logit Df Residuals: 98
Method: MLE Df Model: 1
Date: Thu, 05 Sep 2024 Pseudo R-squ.: 0.8005
Time: 15:49:31 Log-Likelihood: -13.810
converged: True LL-Null: -69.235
Covariance Type: nonrobust LLR p-value: 6.385e-26
coef std err z P>|z| [0.025 0.975]
Intercept -78.6932 19.851 -3.964 0.000 -117.600 -39.787
work 1.9815 0.500 3.959 0.000 1.001 2.962


Possibly complete quasi-separation: A fraction 0.32 of observations can be
perfectly predicted. This might indicate that there is complete
quasi-separation. In this case some parameters will not be identified.

Interpreting the model parameters#

Parameters as changes in the log-odds#

lr1.params["work"]
1.9814577697476699

Parameters as ratios of odds#

np.exp(lr1.params["work"])
7.25330893626573

Differences in probabilities#

What is the marginal effect of the predictor work for an intern who invests 40 hours of effort?

# using `statsmodels`
lr1.get_margeff(atexog={1:40}).summary_frame()
dy/dx Std. Err. z Pr(>|z|) Conf. Int. Low Cont. Int. Hi.
work 0.45783 0.112623 4.065157 0.000048 0.237093 0.678567
lr1.get_margeff(atexog={1:42}).summary_frame()
dy/dx Std. Err. z Pr(>|z|) Conf. Int. Low Cont. Int. Hi.
work 0.020949 0.021358 0.98084 0.326672 -0.020912 0.06281
# # ALT. manual calculation plugging into derivative of `expit`
# p40 = lr1.predict({"work":40}).item()
# marg_effect_at_40 = p40 * (1 - p40) * lr1.params['work']
# marg_effect_at_40

Prediction#

p42 = lr1.predict({"work":42})[0]
p42
0.9893134055105761

Poisson regression#

# FIGURES ONLY
from scipy.stats import poisson

# Define the linear model function
def exp_model(x):
    lam = np.exp(1 + 0.2*x)
    return lam

onepixel = 0.07

xlims = [0, 20]
ylims = [0, 100]

with sns.axes_style("ticks"):
    fig, ax = plt.subplots(figsize=(5, 3))

    # Plot the linear model
    xs = np.linspace(xlims[0], xlims[1], 200)
    lams = exp_model(xs)
    sns.lineplot(x=xs, y=lams, ax=ax, label=r"$\mu_Y(x) = \exp(\beta_0 + \beta_1x)$", linewidth=2)
    
    # Plot Gaussian distributions at specified x positions and add sigma lines
    x_positions = range(2, xlims[1]-1, 3)
    for x_pos in x_positions:
        lam_pos = exp_model(x_pos)
        sigma = np.sqrt(lam_pos)
        ys_lower = int(lam_pos-2.5*sigma)
        ys_upper = int(lam_pos+3.4*sigma)
        ys = np.arange(ys_lower, ys_upper, 3)
        pmf = poisson(mu=lam_pos).pmf(ys)
        # ax.fill_betweenx(ys, x_pos - 2 * pmf * sigma, x_pos, color="grey", alpha=0.5)
        ax.stem(ys, x_pos- 2 * pmf * sigma, bottom=x_pos, orientation='horizontal')
        # Draw vertical sigma line and label it on the opposite side of the Gaussian shape
        # ax.plot([x_pos+onepixel, x_pos+onepixel], [lam_pos, lam_pos - sigma], "k", lw=1)
        # ax.text(x_pos + 0.1, lam_pos - sigma / 2 - 3*onepixel, r"$\sigma$", fontsize=12, va="center")

    # y-intercept
    ax.text(0 - 0.6, np.exp(1), r"$\exp(\beta_0)$", fontsize=10, va="center", ha="right")

    # Set up x-axis
    ax.set_xlim(xlims)
    ax.set_xlabel("$x$")
    ax.set_xticks(range(2, xlims[1], 3))
    ax.set_xticklabels([])
    
    # Set up y-axis
    ax.set_ylim([ylims[0]-4,ylims[1]])
    ax.set_ylabel("$y$")
    ax.set_yticks(list(range(ylims[0],ylims[1],20)) + [np.exp(1)] )
    ax.set_yticklabels([])
    
    ax.legend(loc="upper left")

    filename = os.path.join(DESTDIR, "poisson_regression_xy_with_stemplots.pdf")
    savefigure(fig, filename)
Saved figure to figures/lm/generalized/poisson_regression_xy_with_stemplots.pdf
Saved figure to figures/lm/generalized/poisson_regression_xy_with_stemplots.png
../_images/d7a4ae8a998f4cb89c56d709ecf787c03178bb6afea6843720bb0a8c4435568b.png

Example 2: hard disk failures over time#

hdisks = pd.read_csv("../datasets/hdisks.csv")
hdisks.head(3)
age failures
0 1.7 3
1 14.6 46
2 10.9 23
import statsmodels.formula.api as smf

pr2 = smf.poisson("failures ~ 1 + age", data=hdisks).fit()
pr2.params
Optimization terminated successfully.
         Current function value: 2.693129
         Iterations 6
Intercept    1.075999
age          0.193828
dtype: float64
pr2.summary()
Poisson Regression Results
Dep. Variable: failures No. Observations: 100
Model: Poisson Df Residuals: 98
Method: MLE Df Model: 1
Date: Thu, 05 Sep 2024 Pseudo R-squ.: 0.6412
Time: 15:49:32 Log-Likelihood: -269.31
converged: True LL-Null: -750.68
Covariance Type: nonrobust LLR p-value: 2.271e-211
coef std err z P>|z| [0.025 0.975]
Intercept 1.0760 0.076 14.114 0.000 0.927 1.225
age 0.1938 0.007 28.603 0.000 0.181 0.207
from ministats import plot_reg
plot_reg(pr2)

# FIGURES ONLY
filename = os.path.join(DESTDIR, "poisson_regression_hdisks_failures_vs_age.pdf")
savefigure(plt.gcf(), filename)
Saved figure to figures/lm/generalized/poisson_regression_hdisks_failures_vs_age.pdf
Saved figure to figures/lm/generalized/poisson_regression_hdisks_failures_vs_age.png
../_images/984dd7f8d91a8ed7563622bc958a47fb16fcb896b19c2135f4653157d253f488.png
pr2.summary()
Poisson Regression Results
Dep. Variable: failures No. Observations: 100
Model: Poisson Df Residuals: 98
Method: MLE Df Model: 1
Date: Thu, 05 Sep 2024 Pseudo R-squ.: 0.6412
Time: 15:49:33 Log-Likelihood: -269.31
converged: True LL-Null: -750.68
Covariance Type: nonrobust LLR p-value: 2.271e-211
coef std err z P>|z| [0.025 0.975]
Intercept 1.0760 0.076 14.114 0.000 0.927 1.225
age 0.1938 0.007 28.603 0.000 0.181 0.207

Interpreting the model parameters#

Log-counts#

pr2.params["age"]
0.19382784821454072

Incidence rate ratio (IRR)#

np.exp(pr2.params["age"])
1.213887292102993

Marginal effect#

What is the marginal effect of the predictor age for a 10 year old hard disk installation?

# using `statsmodels` .get_margeff() method
pr2.get_margeff(atexog={1:10}).summary_frame()
dy/dx Std. Err. z Pr(>|z|) Conf. Int. Low Cont. Int. Hi.
age 3.94912 0.151882 26.001165 4.804151e-149 3.651435 4.246804
# # ALT. manual calculation of the slope by evaluating the derivative
# b_0 = pr2.params['Intercept']
# b_age = pr2.params['age']
# np.exp(b_0 + b_age*10)*b_age

Predictions#

lam10 = pr2.predict({"age":10})[0]
lam10
20.374365915173986
from scipy.stats import poisson
Hhat = poisson(mu=lam10)
Hhat.ppf(0.05), Hhat.ppf(0.95)
(13.0, 28.0)

Explanations#

The exponential family of distributions#

  • exponential

  • Gaussian (normal)

  • Poisson

  • Binomial

The generalized linear model template#

  • choose

Generalized linear models using statsmodels#

import statsmodels.api as sm

Norm = sm.families.Gaussian()
Bin = sm.families.Binomial()
Pois = sm.families.Poisson()

Linear model#

students = pd.read_csv('../datasets/students.csv')
formula0 = "score ~ 1 + effort"
glm0 = smf.glm(formula0, data=students, family=Norm).fit()
glm0.params
Intercept    32.465809
effort        4.504850
dtype: float64

Logistic regression#

formula1 = "hired ~ 1 + work"
glm1 = smf.glm(formula1, data=interns, family=Bin).fit()
glm1.params
# glm1.summary()
Intercept   -78.693205
work          1.981458
dtype: float64

Poisson regression#

formula2 = "failures ~ 1 + age"
glm2 = smf.glm(formula2, data=hdisks, family=Pois).fit()
glm2.params
# glm2.summary()
Intercept    1.075999
age          0.193828
dtype: float64

Fitting generalized linear models#

Standardization of predictors#

from scipy.stats import zscore

efforts = students["effort"]
zefforts = zscore(efforts)
zefforts.head(3)
0    1.092044
1   -0.114057
2   -0.161876
Name: effort, dtype: float64
# # ALT. define custom function equivalent to zscore(data)
# def standardize(data):
#     datamean = data.mean()
#     datastd = data.std(ddof=1)
#     zdata = (data - datamean) / datastd
#     return zdata
students["zeffort"] = zefforts
lm1s = smf.ols("score~1+zeffort", data=students).fit()
lm1s.params
Intercept    72.580000
zeffort       8.478568
dtype: float64
lm1s.params["zeffort"] / students["effort"].std(ddof=0)
4.504850344209072
glm0.params["effort"]
4.504850344209074

Marginal effects#

It can be difficult to interpret GLM parameters directly, but we can always ask the question about “slopes” the rate of change of interesting parameters.

import marginaleffects as me
---------------------------------------------------------------------------
ModuleNotFoundError                       Traceback (most recent call last)
Cell In[50], line 1
----> 1 import marginaleffects as me

ModuleNotFoundError: No module named 'marginaleffects'

Raw parameters#

lr1.params["work"], np.exp(lr1.params["work"])
(1.9814577697476699, 7.25330893626573)

The parameter tells us the log-odds ratio changes by 1.981, or equivalently, that the odd-ratio changes by a factor of 7.253 for each additional hour of work.

Marginal effect at a user-specified value#

Example To calculate the marginal effect of the predictor work for an intern who invests 40 hours of effort.

# using `statsmodels`
lr1.get_margeff(atexog={1:40}).summary_frame()
dy/dx Std. Err. z Pr(>|z|) Conf. Int. Low Cont. Int. Hi.
work 0.45783 0.112623 4.065157 0.000048 0.237093 0.678567
# ALT. using `marginaleffects`
dg40 = me.datagrid(lr1, work=[40])
me.slopes(lr1, newdata=dg40).to_pandas()
term contrast estimate std_error statistic p_value s_value conf_low conf_high
0 work dY/dX 0.45783 0.112263 4.078207 0.000045 14.427447 0.237799 0.677861
# ALT2. manual calculation plugging into derivative of `expit`
p40 = lr1.predict({"work":40}).item()
marg_effect_at_40 = p40 * (1 - p40) * lr1.params['work']
marg_effect_at_40
0.45782997989918905

Average marginal effect (AME)#

For each observation \((w_i,h_i)\), compute the marginal effect at \(w=w_i\), then average them together.

lr1.get_margeff().summary_frame()
dy/dx Std. Err. z Pr(>|z|) Conf. Int. Low Cont. Int. Hi.
work 0.08077 0.00422 19.139418 1.185874e-81 0.072499 0.089041
# ALT. using the `marginaleffects` package
me.avg_slopes(lr1).to_pandas()
term contrast estimate std_error statistic p_value s_value conf_low conf_high
0 work mean(dY/dX) 0.08077 0.00422 19.138224 0.0 inf 0.072498 0.089042
# ALT2. manual computation using a for-loop
meffects = []
for i, row in interns.iterrows():
    p = lr1.predict({"work":row["work"]}).item()
    meffect = p * (1-p) * lr1.params['work']
    meffects.append(meffect)
AME = np.mean(meffects)
AME
0.08076985840552156

Marginal effect at the mean (MEM)#

lr1.get_margeff(at="mean").summary_frame()
dy/dx Std. Err. z Pr(>|z|) Conf. Int. Low Cont. Int. Hi.
work 0.47034 0.121697 3.864838 0.000111 0.231818 0.708862
# ALT. using the `marginaleffects` package
me.slopes(lr1, newdata="mean").to_pandas()
term contrast estimate std_error statistic p_value s_value conf_low conf_high
0 work dY/dX 0.47034 0.121818 3.861021 0.000113 13.112488 0.231582 0.709098
# ALT2. manual computation
meanwork = interns["work"].mean()
p_at_mean = lr1.predict({"work":meanwork}).item()
MEM = p_at_mean * (1-p_at_mean) * lr1.params['work']
MEM
0.47034017780957577

The marginaleffects package provides some useful plots to visualize the predictions and slopes.

# me.plot_predictions(lr1, condition="work")
# me.plot_slopes(lr1, condition="work")

Links to learn more about marginal effects

Discussion#

Model diagnostics and validation#

# Dispersion from GLM attributes
# glm2.pearson_chi2 / glm2.df_resid
# Calculate Pearson chi-squared statistic
observed = hdisks['failures']
predicted = pr2.predict()
pearson_residuals = (observed - predicted) / np.sqrt(predicted)
pearson_chi2 = np.sum(pearson_residuals**2)
df_resid = pr2.df_resid
dispersion = pearson_chi2 / df_resid
print(f'Dispersion: {dispersion}')
# If dispersion > 1, consider Negative Binomial regression
Dispersion: 0.9869289289681199

ScikitLearn models#

# Cross check with sklearn
from sklearn.linear_model import LogisticRegression
X1_skl = interns[["work"]]
y1_skl = interns["hired"]
lr1_skl = LogisticRegression(penalty=None).fit(X1_skl, y1_skl)
lr1_skl.intercept_, lr1_skl.coef_
(array([-78.69320824]), array([[1.98145785]]))
lr1.params
Intercept   -78.693205
work          1.981458
dtype: float64
hdisks.dtypes
age         float64
failures      int64
dtype: object
from sklearn.linear_model import PoissonRegressor
X2_skl = hdisks[["age"]]
y2_skl = hdisks["failures"]
pr2_skl = PoissonRegressor(alpha=0).fit(X2_skl, y2_skl)
pr2_skl.intercept_, pr2_skl.coef_
(1.0759942750890905, array([0.19382823]))
pr2.params
Intercept    1.075999
age          0.193828
dtype: float64

Logistic regression as a building blocks for neural networks#

The operation of the perceptron, which is the basic building block of neural networks, is essentially the same as linear regression model:

  • constant intercept (bias term)

  • linear combination of inputs

  • nonlinear function used to force the output to be between 0 and 1

Limitations of GLMs#

  • GLMs assume observations are independent

  • Assumes distribution \(\mathcal{M}\) is one of the exponential family

  • Outliers can be problematic

  • Interpretability

Exercises#

Exercise 1: probabilities to odds and log-odds#

0.3/(1-0.3), 0.99/(1-0.99), 0.7/(1-0.7)
(0.4285714285714286, 98.99999999999991, 2.333333333333333)
logit(0.3), logit(0.99), logit(0.7)
(-0.8472978603872036, 4.595119850134589, 0.8472978603872034)

Exercise 2: log-odds to probabilities#

expit(-1), expit(1), expit(2)
(0.2689414213699951, 0.7310585786300049, 0.8807970779778823)

Exercise 3: students pass or fail#

a) Load the dataset students.csv and add a column passing that contains 1 or 0, based on the above threshold score of 70.

students = pd.read_csv('../datasets/students.csv')
students["passing"] = (students["score"] > 70).astype(int)
# students.head()

b) Fit a logistic regression model for passing based on effort variable.

lmpass = smf.logit("passing ~ 1 + effort", data=students).fit()
print(lmpass.params)

efforts = np.linspace(0, 13, 100)
passing_preds = lmpass.predict({"effort": efforts})
ax = sns.scatterplot(data=students, x="effort", y="passing", alpha=0.3)
sns.lineplot(x=efforts, y=passing_preds, ax=ax);

filename = os.path.join(DESTDIR, "logistic_regression_students_passing_vs_effort.pdf")
savefigure(plt.gcf(), filename)
Optimization terminated successfully.
         Current function value: 0.276583
         Iterations 8
Intercept   -16.257302
effort        2.047882
dtype: float64
Saved figure to figures/lm/generalized/logistic_regression_students_passing_vs_effort.pdf
Saved figure to figures/lm/generalized/logistic_regression_students_passing_vs_effort.png
../_images/e70db10c70033c34fc1facb75f3d6602d8207518bcee0cc9feea7b876fa2204c.png

c) Use your model to predict the probability of a new student with effort 10 will pass.

lmpass.predict({"effort":10})
0    0.985536
dtype: float64
# ALT. compute prediction manually
intercept, b_effort = lmpass.params
expit(intercept + b_effort*10)
0.9855358765361845

Exercise 4: titanic survival data#

Fit a logistic regression model that calculates the probability of survival for people who were on the Titanic, based on the data in datasets/exercises/titanic.csv. Use the variables age, sex, and pclass as predictors.

cf. Titanic_Logistic_Regression.ipynb

titanic = pd.read_csv('../datasets/exercises/titanic.csv')
formula = "survived ~ age + C(sex) + C(pclass)"
lrtitanic = smf.logit(formula, data=titanic).fit()
lrtitanic.params
Optimization terminated successfully.
         Current function value: 0.453279
         Iterations 6
Intercept         3.777013
C(sex)[T.M]      -2.522781
C(pclass)[T.2]   -1.309799
C(pclass)[T.3]   -2.580625
age              -0.036985
dtype: float64

Use your logistic regression model to estimate the probability of survival for a 30 year old female traveling in second class.

pass30Fpclass2 = {"age":30, "sex":"F", "pclass":2}
lrtitanic.predict(pass30Fpclass2)
0    0.795378
dtype: float64
# # Cross check with sklearn
# from sklearn.linear_model import LogisticRegression
# df = pd.get_dummies(titanic, columns=['sex', 'pclass'], drop_first=True)
# X, y = df.drop('survived', axis=1), df['survived']
# sktitanic = LogisticRegression(penalty=None)
# sktitanic.fit(X, y)
# sktitanic.intercept_, sktitanic.coef_

Exercise 5: asthma attacks#

Fit a Poisson regression model to the ../datasets/exercises/asthma.csv dataset.

data source drkamarul/multivar_data_analysis

asthma = pd.read_csv("../datasets/exercises/asthma.csv")
asthma
gender res_inf ghq12 attack
0 female yes 21 6
1 male no 17 4
2 male yes 30 8
3 female yes 22 5
4 male yes 27 2
... ... ... ... ...
115 male yes 0 2
116 female yes 31 2
117 female yes 18 2
118 female yes 21 3
119 female yes 11 2

120 rows × 4 columns

formula_asthma = "attack ~ 1 + C(gender) + C(res_inf) + ghq12"
prasthma = smf.poisson(formula_asthma, data=asthma).fit()
prasthma.params
Optimization terminated successfully.
         Current function value: 1.707281
         Iterations 6
Intercept           -0.315387
C(gender)[T.male]   -0.041905
C(res_inf)[T.yes]    0.426431
ghq12                0.049508
dtype: float64

cf. https://bookdown.org/drki_musa/dataanalysis/poisson-regression.html#multivariable-analysis-1

Exercise 6: student admissions dataset#

The dataset datasets/exercises/binary.csv contains information about the acceptance decision for 400 students to a prestigious school. Try to fit a logistic regression model for the variable admit using the variables gre, gpa, and rank as predictors.

# ORIGINAL https://stats.idre.ucla.edu/stat/data/binary.csv
binary = pd.read_csv('../datasets/exercises/binary.csv')
binary.head(3)
admit gre gpa rank
0 0 380 3.61 3
1 1 660 3.67 3
2 1 800 4.00 1
lrbinary = smf.logit('admit ~ gre + gpa + C(rank)', data=binary).fit()
lrbinary.params
Optimization terminated successfully.
         Current function value: 0.573147
         Iterations 6
Intercept      -3.989979
C(rank)[T.2]   -0.675443
C(rank)[T.3]   -1.340204
C(rank)[T.4]   -1.551464
gre             0.002264
gpa             0.804038
dtype: float64

The above model uses the rank=1 as the reference category an the log odds reported are with respect to this catrgory

\[ \log p(accept|rank=1) / \log p(accept|rank=2) = \texttt{C(rank)rank[T.2]} = -0.675443 \]

etc. for others rank[T.3] -1.340204 rank[T.4] -1.551464

See LogisticRegressionChangeOfReferenceCategoricalValue.ipynb for exercise recodign relative to different refrence level.

# # Cross check with sklearn
# from sklearn.linear_model import LogisticRegression
# df = pd.get_dummies(binary, columns=['rank'], drop_first=True)
# X, y = df.drop("admit", axis=1), df["admit"]
# lr = LogisticRegression(solver="lbfgs", penalty=None, max_iter=1000)
# lr.fit(X, y)
# lr.intercept_,  lr.coef_

Exercise 7: ship accidents#

https://rdrr.io/cran/AER/man/ShipAccidents.html

https://pages.stern.nyu.edu/~wgreene/Text/tables/tablelist5.htm

https://pages.stern.nyu.edu/~wgreene/Text/tables/TableF21-3.txt

# TODO

Bonus Exercises#

Bonus exercise A: honors class#

https://stats.oarc.ucla.edu/other/mult-pkg/faq/general/faq-how-do-i-interpret-odds-ratios-in-logistic-regression/

honors = pd.read_csv("../datasets/exercises/honors.csv")
honors.sample(4)
female read write math hon femalexmath
128 1 50 49 56 0 56
41 0 55 59 62 0 0
182 1 52 67 57 1 57
103 1 63 52 54 0 54

Constant model#

lrhon1 = smf.logit("hon ~ 1", data=honors).fit()
lrhon1.params
Optimization terminated successfully.
         Current function value: 0.556775
         Iterations 5
Intercept   -1.12546
dtype: float64
expit(lrhon1.params["Intercept"])
0.24500000000000005
honors["hon"].value_counts(normalize=True)
hon
0    0.755
1    0.245
Name: proportion, dtype: float64

Using only a categorical variable#

lrhon2 = smf.logit("hon ~ 1 + female", data=honors).fit()
lrhon2.params
Optimization terminated successfully.
         Current function value: 0.549016
         Iterations 5
Intercept   -1.470852
female       0.592782
dtype: float64
pd.crosstab(honors["hon"], honors["female"], margins=True)
female 0 1 All
hon
0 74 77 151
1 17 32 49
All 91 109 200
b0 = lrhon2.params["Intercept"]
b_female = lrhon2.params["female"]
# male prob
expit(b0), 17/91
(0.18681318681318684, 0.18681318681318682)
# male odds
np.exp(b0), 17/74  # = (17/91) / (74/91)
(0.2297297297297298, 0.22972972972972974)
# male log-odds
b0, np.log(17/74)
(-1.4708517491479534, -1.4708517491479536)
# female prob
expit(b0 + b_female), 32/109
(0.2935779816513761, 0.29357798165137616)
# female odds
np.exp(b0 + b_female), 32/77 # = (32/109) / (77/109)
(0.4155844155844155, 0.4155844155844156)
b0 + b_female, np.log(32/77)
(-0.8780695190539575, -0.8780695190539572)
# odds female relative to male
np.exp(b_female)
1.809014514896867

Logistic regression with a single continuous predictor variable#

lrhon3 = smf.logit("hon ~ 1 + math", data=honors).fit()
lrhon3.params
Optimization terminated successfully.
         Current function value: 0.417683
         Iterations 7
Intercept   -9.793942
math         0.156340
dtype: float64

So the model equation is

\[ \log(p/(1-p)) = \texttt{logit}(p) = -9.793942 + .1563404 \cdot \texttt{math} \]
# Increase in log-odds between math=54 and math=55
p54 = lrhon3.predict({"math":[54]}).item()
p55 = lrhon3.predict({"math":[55]}).item()
logit(p55) - logit(p54), lrhon3.params["math"]
(0.1563403555859233, 0.15634035558592282)

We can say now that the coefficient for math is the difference in the log odds. In other words, for a one-unit increase in the math score, the expected change in log odds is .1563404.

# Increase (multiplicative) in odds for unit increase in math
np.exp(lrhon3.params["math"]),  (p55/(1-p55)) / (p54/(1-p54))
(1.1692240873242836, 1.1692240873242843)

So we can say for a one-unit increase in math score, we expect to see about 17% increase in the odds of being in an honors class. This 17% of increase does not depend on the value that math is held at.

Logistic regression with multiple predictor variables and no interaction terms#

lrhon4 = smf.logit("hon ~ 1 + math + female + read", data=honors).fit()
lrhon4.params
Optimization terminated successfully.
         Current function value: 0.390424
         Iterations 7
Intercept   -11.770246
math          0.122959
female        0.979948
read          0.059063
dtype: float64

Logistic regression with an interaction term of two predictor variables#

lrhon5 = smf.logit("hon ~ 1 + math + female + femalexmath", data=honors).fit()
lrhon5.params
Optimization terminated successfully.
         Current function value: 0.399417
         Iterations 7
Intercept     -8.745841
math           0.129378
female        -2.899863
femalexmath    0.066995
dtype: float64
# ALT. without using `femalexmath` column
# lrhon5 = smf.logit("hon ~ 1 + math + female + female*math", data=honors).fit()
# lrhon5.params

Bonus exercise: LA high schools (NOT A VERY GOOD FIT FOR POISSON MODEL)#

Dataset info: http://www.philender.com/courses/intro/assign/data.html

This dataset consists of data from computer exercises collected from two high school in the Los Angeles area.

http://www.philender.com/courses/intro/code.html

lahigh_raw = pd.read_stata("https://stats.idre.ucla.edu/stat/stata/notes/lahigh.dta")
lahigh = lahigh_raw.convert_dtypes()

lahigh["gender"] = lahigh["gender"].astype(object).replace({1:"F", 2:"M"})
lahigh["ethnic"] = lahigh["ethnic"].astype(object).replace({
    1:"Native American",
    2:"Asian",
    3:"African-American",
    4:"Hispanic",
    5:"White",
    6:"Filipino",
    7:"Pacific Islander"})
lahigh["school"] = lahigh["school"].astype(object).replace({1:"Alpha", 2:"Beta"})
lahigh.head()
---------------------------------------------------------------------------
gaierror                                  Traceback (most recent call last)
File ~/.pyenv/versions/3.9.4/lib/python3.9/urllib/request.py:1346, in AbstractHTTPHandler.do_open(self, http_class, req, **http_conn_args)
   1345 try:
-> 1346     h.request(req.get_method(), req.selector, req.data, headers,
   1347               encode_chunked=req.has_header('Transfer-encoding'))
   1348 except OSError as err: # timeout error

File ~/.pyenv/versions/3.9.4/lib/python3.9/http/client.py:1253, in HTTPConnection.request(self, method, url, body, headers, encode_chunked)
   1252 """Send a complete request to the server."""
-> 1253 self._send_request(method, url, body, headers, encode_chunked)

File ~/.pyenv/versions/3.9.4/lib/python3.9/http/client.py:1299, in HTTPConnection._send_request(self, method, url, body, headers, encode_chunked)
   1298     body = _encode(body, 'body')
-> 1299 self.endheaders(body, encode_chunked=encode_chunked)

File ~/.pyenv/versions/3.9.4/lib/python3.9/http/client.py:1248, in HTTPConnection.endheaders(self, message_body, encode_chunked)
   1247     raise CannotSendHeader()
-> 1248 self._send_output(message_body, encode_chunked=encode_chunked)

File ~/.pyenv/versions/3.9.4/lib/python3.9/http/client.py:1008, in HTTPConnection._send_output(self, message_body, encode_chunked)
   1007 del self._buffer[:]
-> 1008 self.send(msg)
   1010 if message_body is not None:
   1011 
   1012     # create a consistent interface to message_body

File ~/.pyenv/versions/3.9.4/lib/python3.9/http/client.py:948, in HTTPConnection.send(self, data)
    947 if self.auto_open:
--> 948     self.connect()
    949 else:

File ~/.pyenv/versions/3.9.4/lib/python3.9/http/client.py:1415, in HTTPSConnection.connect(self)
   1413 "Connect to a host on a given (SSL) port."
-> 1415 super().connect()
   1417 if self._tunnel_host:

File ~/.pyenv/versions/3.9.4/lib/python3.9/http/client.py:919, in HTTPConnection.connect(self)
    918 """Connect to the host and port specified in __init__."""
--> 919 self.sock = self._create_connection(
    920     (self.host,self.port), self.timeout, self.source_address)
    921 self.sock.setsockopt(socket.IPPROTO_TCP, socket.TCP_NODELAY, 1)

File ~/.pyenv/versions/3.9.4/lib/python3.9/socket.py:822, in create_connection(address, timeout, source_address)
    821 err = None
--> 822 for res in getaddrinfo(host, port, 0, SOCK_STREAM):
    823     af, socktype, proto, canonname, sa = res

File ~/.pyenv/versions/3.9.4/lib/python3.9/socket.py:953, in getaddrinfo(host, port, family, type, proto, flags)
    952 addrlist = []
--> 953 for res in _socket.getaddrinfo(host, port, family, type, proto, flags):
    954     af, socktype, proto, canonname, sa = res

gaierror: [Errno 8] nodename nor servname provided, or not known

During handling of the above exception, another exception occurred:

URLError                                  Traceback (most recent call last)
Cell In[102], line 1
----> 1 lahigh_raw = pd.read_stata("https://stats.idre.ucla.edu/stat/stata/notes/lahigh.dta")
      2 lahigh = lahigh_raw.convert_dtypes()
      4 lahigh["gender"] = lahigh["gender"].astype(object).replace({1:"F", 2:"M"})

File ~/Projects/Minireference/STATSbook/noBSstatsnotebooks/venv/lib/python3.9/site-packages/pandas/io/stata.py:2109, in read_stata(filepath_or_buffer, convert_dates, convert_categoricals, index_col, convert_missing, preserve_dtypes, columns, order_categoricals, chunksize, iterator, compression, storage_options)
   2106     return reader
   2108 with reader:
-> 2109     return reader.read()

File ~/Projects/Minireference/STATSbook/noBSstatsnotebooks/venv/lib/python3.9/site-packages/pandas/io/stata.py:1683, in StataReader.read(self, nrows, convert_dates, convert_categoricals, index_col, convert_missing, preserve_dtypes, columns, order_categoricals)
   1671 @Appender(_read_method_doc)
   1672 def read(
   1673     self,
   (...)
   1681     order_categoricals: bool | None = None,
   1682 ) -> DataFrame:
-> 1683     self._ensure_open()
   1685     # Handle options
   1686     if convert_dates is None:

File ~/Projects/Minireference/STATSbook/noBSstatsnotebooks/venv/lib/python3.9/site-packages/pandas/io/stata.py:1175, in StataReader._ensure_open(self)
   1171 """
   1172 Ensure the file has been opened and its header data read.
   1173 """
   1174 if not hasattr(self, "_path_or_buf"):
-> 1175     self._open_file()

File ~/Projects/Minireference/STATSbook/noBSstatsnotebooks/venv/lib/python3.9/site-packages/pandas/io/stata.py:1188, in StataReader._open_file(self)
   1181 if not self._entered:
   1182     warnings.warn(
   1183         "StataReader is being used without using a context manager. "
   1184         "Using StataReader as a context manager is the only supported method.",
   1185         ResourceWarning,
   1186         stacklevel=find_stack_level(),
   1187     )
-> 1188 handles = get_handle(
   1189     self._original_path_or_buf,
   1190     "rb",
   1191     storage_options=self._storage_options,
   1192     is_text=False,
   1193     compression=self._compression,
   1194 )
   1195 if hasattr(handles.handle, "seekable") and handles.handle.seekable():
   1196     # If the handle is directly seekable, use it without an extra copy.
   1197     self._path_or_buf = handles.handle

File ~/Projects/Minireference/STATSbook/noBSstatsnotebooks/venv/lib/python3.9/site-packages/pandas/io/common.py:728, in get_handle(path_or_buf, mode, encoding, compression, memory_map, is_text, errors, storage_options)
    725     codecs.lookup_error(errors)
    727 # open URLs
--> 728 ioargs = _get_filepath_or_buffer(
    729     path_or_buf,
    730     encoding=encoding,
    731     compression=compression,
    732     mode=mode,
    733     storage_options=storage_options,
    734 )
    736 handle = ioargs.filepath_or_buffer
    737 handles: list[BaseBuffer]

File ~/Projects/Minireference/STATSbook/noBSstatsnotebooks/venv/lib/python3.9/site-packages/pandas/io/common.py:384, in _get_filepath_or_buffer(filepath_or_buffer, encoding, compression, mode, storage_options)
    382 # assuming storage_options is to be interpreted as headers
    383 req_info = urllib.request.Request(filepath_or_buffer, headers=storage_options)
--> 384 with urlopen(req_info) as req:
    385     content_encoding = req.headers.get("Content-Encoding", None)
    386     if content_encoding == "gzip":
    387         # Override compression based on Content-Encoding header

File ~/Projects/Minireference/STATSbook/noBSstatsnotebooks/venv/lib/python3.9/site-packages/pandas/io/common.py:289, in urlopen(*args, **kwargs)
    283 """
    284 Lazy-import wrapper for stdlib urlopen, as that imports a big chunk of
    285 the stdlib.
    286 """
    287 import urllib.request
--> 289 return urllib.request.urlopen(*args, **kwargs)

File ~/.pyenv/versions/3.9.4/lib/python3.9/urllib/request.py:214, in urlopen(url, data, timeout, cafile, capath, cadefault, context)
    212 else:
    213     opener = _opener
--> 214 return opener.open(url, data, timeout)

File ~/.pyenv/versions/3.9.4/lib/python3.9/urllib/request.py:517, in OpenerDirector.open(self, fullurl, data, timeout)
    514     req = meth(req)
    516 sys.audit('urllib.Request', req.full_url, req.data, req.headers, req.get_method())
--> 517 response = self._open(req, data)
    519 # post-process response
    520 meth_name = protocol+"_response"

File ~/.pyenv/versions/3.9.4/lib/python3.9/urllib/request.py:534, in OpenerDirector._open(self, req, data)
    531     return result
    533 protocol = req.type
--> 534 result = self._call_chain(self.handle_open, protocol, protocol +
    535                           '_open', req)
    536 if result:
    537     return result

File ~/.pyenv/versions/3.9.4/lib/python3.9/urllib/request.py:494, in OpenerDirector._call_chain(self, chain, kind, meth_name, *args)
    492 for handler in handlers:
    493     func = getattr(handler, meth_name)
--> 494     result = func(*args)
    495     if result is not None:
    496         return result

File ~/.pyenv/versions/3.9.4/lib/python3.9/urllib/request.py:1389, in HTTPSHandler.https_open(self, req)
   1388 def https_open(self, req):
-> 1389     return self.do_open(http.client.HTTPSConnection, req,
   1390         context=self._context, check_hostname=self._check_hostname)

File ~/.pyenv/versions/3.9.4/lib/python3.9/urllib/request.py:1349, in AbstractHTTPHandler.do_open(self, http_class, req, **http_conn_args)
   1346         h.request(req.get_method(), req.selector, req.data, headers,
   1347                   encode_chunked=req.has_header('Transfer-encoding'))
   1348     except OSError as err: # timeout error
-> 1349         raise URLError(err)
   1350     r = h.getresponse()
   1351 except:

URLError: <urlopen error [Errno 8] nodename nor servname provided, or not known>
formula = "daysabs ~ 1 + mathnce + langnce + C(gender)"
prlahigh = smf.poisson(formula, data=lahigh).fit()
prlahigh.params
# IRR
np.exp(prlahigh.params[1:])
# CI for IRR F 
np.exp(prlahigh.conf_int().loc["C(gender)[T.M]"])
# prlahigh.summary()
# prlahigh.aic, prlahigh.bic

Diagnostics#

via https://www.statsmodels.org/dev/examples/notebooks/generated/postestimation_poisson.html

prdiag = prlahigh.get_diagnostic()
# Plot observed versus predicted frequencies for entire sample
# prdiag.plot_probs();

# Other:
# ['plot_probs',
#  'probs_predicted',
#  'results',
#  'test_chisquare_prob',
#  'test_dispersion',
#  'test_poisson_zeroinflation',
#  'y_max']
# Code to get exactly the same numbers as in
# https://stats.oarc.ucla.edu/stata/output/poisson-regression/  
formula2 = "daysabs ~ 1 + mathnce + langnce + C(gender, Treatment(1))"
prlahigh2 = smf.poisson(formula2, data=lahigh).fit()
prlahigh2.params