from __future__ import print_function
import numpy as np
import statsmodels.api as sm
from scipy import stats
from matplotlib import pyplot as plt
In this example, we use the Star98 dataset which was taken with permission from Jeff Gill (2000) Generalized linear models: A unified approach. Codebook information can be obtained by typing:
print(sm.datasets.star98.NOTE)
Load the data and add a constant to the exogenous (independent) variables:
data = sm.datasets.star98.load()
data.exog = sm.add_constant(data.exog, prepend=False)
The dependent variable is N by 2 (Success: NABOVE, Failure: NBELOW):
print(data.endog[:5,:])
The independent variables include all the other variables described above, as well as the interaction terms:
print(data.exog[:2,:])
glm_binom = sm.GLM(data.endog, data.exog, family=sm.families.Binomial())
res = glm_binom.fit()
print(res.summary())
print('Total number of trials:', data.endog[0].sum())
print('Parameters: ', res.params)
print('T-values: ', res.tvalues)
First differences: We hold all explanatory variables constant at their means and manipulate the percentage of low income households to assess its impact on the response variables:
means = data.exog.mean(axis=0)
means25 = means.copy()
means25[0] = stats.scoreatpercentile(data.exog[:,0], 25)
means75 = means.copy()
means75[0] = lowinc_75per = stats.scoreatpercentile(data.exog[:,0], 75)
resp_25 = res.predict(means25)
resp_75 = res.predict(means75)
diff = resp_75 - resp_25
The interquartile first difference for the percentage of low income households in a school district is:
print("%2.4f%%" % (diff*100))
We extract information that will be used to draw some interesting plots:
nobs = res.nobs
y = data.endog[:,0]/data.endog.sum(1)
yhat = res.mu
Plot yhat vs y:
from statsmodels.graphics.api import abline_plot
fig, ax = plt.subplots()
ax.scatter(yhat, y)
line_fit = sm.OLS(y, sm.add_constant(yhat, prepend=True)).fit()
abline_plot(model_results=line_fit, ax=ax)
ax.set_title('Model Fit Plot')
ax.set_ylabel('Observed values')
ax.set_xlabel('Fitted values');
Plot yhat vs. Pearson residuals:
fig, ax = plt.subplots()
ax.scatter(yhat, res.resid_pearson)
ax.hlines(0, 0, 1)
ax.set_xlim(0, 1)
ax.set_title('Residual Dependence Plot')
ax.set_ylabel('Pearson Residuals')
ax.set_xlabel('Fitted values')
Histogram of standardized deviance residuals:
from scipy import stats
fig, ax = plt.subplots()
resid = res.resid_deviance.copy()
resid_std = stats.zscore(resid)
ax.hist(resid_std, bins=25)
ax.set_title('Histogram of standardized deviance residuals');
QQ Plot of Deviance Residuals:
from statsmodels import graphics
graphics.gofplots.qqplot(resid, line='r')
In the example above, we printed the NOTE
attribute to learn about the
Star98 dataset. Statsmodels datasets ships with other useful information. For
example:
print(sm.datasets.scotland.DESCRLONG)
Load the data and add a constant to the exogenous variables:
data2 = sm.datasets.scotland.load()
data2.exog = sm.add_constant(data2.exog, prepend=False)
print(data2.exog[:5,:])
print(data2.endog[:5])
glm_gamma = sm.GLM(data2.endog, data2.exog, family=sm.families.Gamma())
glm_results = glm_gamma.fit()
print(glm_results.summary())
nobs2 = 100
x = np.arange(nobs2)
np.random.seed(54321)
X = np.column_stack((x,x**2))
X = sm.add_constant(X, prepend=False)
lny = np.exp(-(.03*x + .0001*x**2 - 1.0)) + .001 * np.random.rand(nobs2)
gauss_log = sm.GLM(lny, X, family=sm.families.Gaussian(sm.families.links.log))
gauss_log_results = gauss_log.fit()
print(gauss_log_results.summary())