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"$p(x) = \text{expit}(\beta_0 + \beta_1x)$", 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/e07388f26065ab89313423889d8bb54043efa1b3bc97aa6974c407819bd11d17.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, 18 Jul 2024 Pseudo R-squ.: 0.8005
Time: 10:18:00 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/1b9740419120924fb9e3bfd519a61117ca1fbfc349dfd752b92786a539efbb19.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/bd2bd272719ce132266d37adb340c89f3c3d5c06a3434e7adfbcb0084e373a28.png 
lr1.summary()
Logit Regression Results
Dep. Variable: hired No. Observations: 100
Model: Logit Df Residuals: 98
Method: MLE Df Model: 1
Date: Thu, 18 Jul 2024 Pseudo R-squ.: 0.8005
Time: 10:18:01 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#

expit(lr1.params["work"])

0.8788364754400606

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 
# # 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/c961c561af9d17021ad227e137826aa831fa034f668dec63cd26fc29dc3aaa07.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, 18 Jul 2024 Pseudo R-squ.: 0.6412
Time: 10:18:02 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/d39e6b60d9ac4d4790c082b3889e86bb422068c5aff44e4fcc075fc60a1a6008.png 
pr2.summary()
Poisson Regression Results
Dep. Variable: failures No. Observations: 100
Model: Poisson Df Residuals: 98
Method: MLE Df Model: 1
Date: Thu, 18 Jul 2024 Pseudo R-squ.: 0.6412
Time: 10:18:03 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#

pr2.predict({"age":10})[0]

20.374365915173986

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

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().fit(X2_skl, y2_skl)
pr2_skl.intercept_, pr2_skl.coef_

(1.0854501273030226, array([0.1929434]))

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: students pass or fail#

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

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 
lmpass.predict({"effort":8})

0    0.531396
dtype: float64

intercept, b_effort = lmpass.params
expit(intercept + b_effort*8)

0.5313963881248303

Exercise: titanic survival data#

cf. Titanic_Logistic_Regression.ipynb

titanic_raw = pd.read_csv('../datasets/exercises/titanic.csv')
titanic = titanic_raw[['Survived', 'Age', 'Sex', 'Pclass']]
titanic = titanic.dropna()
titanic.head()
Survived Age Sex Pclass
0 0 22.0 male 3
1 1 38.0 female 1
2 1 26.0 female 3
3 1 35.0 female 1
4 0 35.0 male 3 
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.male]   -2.522781
C(Pclass)[T.2]   -1.309799
C(Pclass)[T.3]   -2.580625
Age              -0.036985
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_

(array([3.77702703]),
 array([[-0.03698571, -2.52276365, -1.30981349, -2.58063585]]))

Exercise: student admissions dataset#

# data = whether students got admitted (admit=1) or not (admit=0) based on their gre and gpa scores, and the rank of their instutution
# raw_data = pd.read_csv('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_

(array([-3.99001587]),
 array([[ 0.00226442,  0.80404719, -0.67543916, -1.3401993 , -1.5514551 ]]))

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
id gender ethnic school mathpr langpr mathnce langnce biling daysabs
0 1001 M Hispanic Alpha 63 36 56.988831 42.450859 2 4
1 1002 M Hispanic Alpha 27 44 37.094158 46.820587 2 4
2 1003 F Hispanic Alpha 20 38 32.275455 43.566574 2 2
3 1004 F Hispanic Alpha 16 38 29.056717 43.566574 2 3
4 1005 F Hispanic Alpha 2 14 6.748048 27.248474 3 3
... ... ... ... ... ... ... ... ... ... ...
311 2153 M Hispanic Beta 26 46 36.451145 47.884865 2 1
312 2154 F White Beta 79 81 66.983231 68.488495 2 3
313 2155 F Hispanic Beta 59 56 54.792099 53.179413 0 0
314 2156 F White Beta 90 82 76.989479 69.277588 0 0
315 2157 F White Beta 77 84 65.560112 70.943283 0 2

316 rows × 10 columns

lahigh
id gender ethnic school mathpr langpr mathnce langnce biling daysabs
0 1001 M Hispanic Alpha 63 36 56.988831 42.450859 2 4
1 1002 M Hispanic Alpha 27 44 37.094158 46.820587 2 4
2 1003 F Hispanic Alpha 20 38 32.275455 43.566574 2 2
3 1004 F Hispanic Alpha 16 38 29.056717 43.566574 2 3
4 1005 F Hispanic Alpha 2 14 6.748048 27.248474 3 3
... ... ... ... ... ... ... ... ... ... ...
311 2153 M Hispanic Beta 26 46 36.451145 47.884865 2 1
312 2154 F White Beta 79 81 66.983231 68.488495 2 3
313 2155 F Hispanic Beta 59 56 54.792099 53.179413 0 0
314 2156 F White Beta 90 82 76.989479 69.277588 0 0
315 2157 F White Beta 77 84 65.560112 70.943283 0 2

316 rows × 10 columns

formula = "daysabs ~ 1 + mathnce + langnce + C(gender)"
prlahigh = smf.poisson(formula, data=lahigh).fit()
prlahigh.params

Optimization terminated successfully.
         Current function value: 4.898642
         Iterations 5

Intercept         2.687666
C(gender)[T.M]   -0.400921
mathnce          -0.003523
langnce          -0.012152
dtype: float64

# IRR
np.exp(prlahigh.params[1:])

C(gender)[T.M]    0.669703
mathnce           0.996483
langnce           0.987921
dtype: float64

# CI for IRR F 
np.exp(prlahigh.conf_int().loc["C(gender)[T.M]"])

0    0.609079
1    0.736361
Name: C(gender)[T.M], dtype: float64

# prlahigh.summary()
# prlahigh.aic, prlahigh.bic

prlahigh.summary()
Poisson Regression Results
Dep. Variable: daysabs No. Observations: 316
Model: Poisson Df Residuals: 312
Method: MLE Df Model: 3
Date: Thu, 18 Jul 2024 Pseudo R-squ.: 0.05358
Time: 10:18:06 Log-Likelihood: -1548.0
converged: True LL-Null: -1635.6
Covariance Type: nonrobust LLR p-value: 9.246e-38
coef std err z P>|z| [0.025 0.975]
Intercept 2.6877 0.073 36.994 0.000 2.545 2.830
C(gender)[T.M] -0.4009 0.048 -8.281 0.000 -0.496 -0.306
mathnce -0.0035 0.002 -1.934 0.053 -0.007 4.66e-05
langnce -0.0122 0.002 -6.623 0.000 -0.016 -0.009

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

Optimization terminated successfully.
         Current function value: 4.898642
         Iterations 5

Intercept                       2.286745
C(gender, Treatment(1))[T.F]    0.400921
mathnce                        -0.003523
langnce                        -0.012152
dtype: float64

Exercise: asthma attacks#

data 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

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

Exercise: 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

Exercise: 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
88 0 50 31 40 0 0
41 0 55 59 62 0 0
93 1 39 44 52 0 52
178 1 50 52 45 0 45

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

\[ \log(p/(1-p)) = \text{logit}(p) = -9.793942 + .1563404 \cdot \tt{math} \]
# 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

CUT MATERIAL#