Generalized Linear Models =========================== .. _glm_notebook: `Link to Notebook GitHub `_ .. raw:: html
In [ ]:
from __future__ import print_function
   import numpy as np
   import statsmodels.api as sm
   from scipy import stats
   from matplotlib import pyplot as plt

GLM: Binomial response data

Load data

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:

In [ ]:
print(sm.datasets.star98.NOTE)

Load the data and add a constant to the exogenous (independent) variables:

In [ ]:
data = sm.datasets.star98.load()
   data.exog = sm.add_constant(data.exog, prepend=False)
   
::
   
       Number of Observations - 303 (counties in California).
   
       Number of Variables - 13 and 8 interaction terms.
   
       Definition of variables names::
   
           NABOVE   - Total number of students above the national median for the
                      math section.
           NBELOW   - Total number of students below the national median for the
                      math section.
           LOWINC   - Percentage of low income students
           PERASIAN - Percentage of Asian student
           PERBLACK - Percentage of black students
           PERHISP  - Percentage of Hispanic students
           PERMINTE - Percentage of minority teachers
           AVYRSEXP - Sum of teachers' years in educational service divided by the
                   number of teachers.
           AVSALK   - Total salary budget including benefits divided by the number
                      of full-time teachers (in thousands)
           PERSPENK - Per-pupil spending (in thousands)
           PTRATIO  - Pupil-teacher ratio.
           PCTAF    - Percentage of students taking UC/CSU prep courses
           PCTCHRT  - Percentage of charter schools
           PCTYRRND - Percentage of year-round schools
   
           The below variables are interaction terms of the variables defined
           above.
   
           PERMINTE_AVYRSEXP
           PEMINTE_AVSAL
           AVYRSEXP_AVSAL
           PERSPEN_PTRATIO
           PERSPEN_PCTAF
           PTRATIO_PCTAF
           PERMINTE_AVTRSEXP_AVSAL
           PERSPEN_PTRATIO_PCTAF

The dependent variable is N by 2 (Success: NABOVE, Failure: NBELOW):

In [ ]:
print(data.endog[:5,:])

The independent variables include all the other variables described above, as well as the interaction terms:

In [ ]:
print(data.exog[:2,:])
   
[[ 452.  355.]
    [ 144.   40.]
    [ 337.  234.]
    [ 395.  178.]
    [   8.   57.]]

Fit and summary

In [ ]:
glm_binom = sm.GLM(data.endog, data.exog, family=sm.families.Binomial())
   res = glm_binom.fit()
   print(res.summary())
   
[[  3.43973000e+01   2.32993000e+01   1.42352800e+01   1.14111200e+01
       1.59183700e+01   1.47064600e+01   5.91573200e+01   4.44520700e+00
       2.17102500e+01   5.70327600e+01   0.00000000e+00   2.22222200e+01
       2.34102872e+02   9.41688110e+02   8.69994800e+02   9.65065600e+01
       2.53522420e+02   1.23819550e+03   1.38488985e+04   5.50403520e+03
       1.00000000e+00]
    [  1.73650700e+01   2.93283800e+01   8.23489700e+00   9.31488400e+00
       1.36363600e+01   1.60832400e+01   5.95039700e+01   5.26759800e+00
       2.04427800e+01   6.46226400e+01   0.00000000e+00   0.00000000e+00
       2.19316851e+02   8.11417560e+02   9.57016600e+02   1.07684350e+02
       3.40406090e+02   1.32106640e+03   1.30502233e+04   6.95884680e+03
       1.00000000e+00]]

Quantities of interest

In [ ]:
print('Total number of trials:',  data.endog[0].sum())
   print('Parameters: ', res.params)
   print('T-values: ', res.tvalues)
   
                 Generalized Linear Model Regression Results                  
   ==============================================================================
   Dep. Variable:           ['y1', 'y2']   No. Observations:                  303
   Model:                            GLM   Df Residuals:                      282
   Model Family:                Binomial   Df Model:                           20
   Link Function:                  logit   Scale:                             1.0
   Method:                          IRLS   Log-Likelihood:                -2998.6
   Date:                Mon, 20 Jul 2015   Deviance:                       4078.8
   Time:                        17:43:33   Pearson chi2:                 4.05e+03
   No. Iterations:                     7                                         
   ==============================================================================
                    coef    std err          z      P>|z|      [95.0% Conf. Int.]
   ------------------------------------------------------------------------------
   x1            -0.0168      0.000    -38.749      0.000        -0.018    -0.016
   x2             0.0099      0.001     16.505      0.000         0.009     0.011
   x3            -0.0187      0.001    -25.182      0.000        -0.020    -0.017
   x4            -0.0142      0.000    -32.818      0.000        -0.015    -0.013
   x5             0.2545      0.030      8.498      0.000         0.196     0.313
   x6             0.2407      0.057      4.212      0.000         0.129     0.353
   x7             0.0804      0.014      5.775      0.000         0.053     0.108
   x8            -1.9522      0.317     -6.162      0.000        -2.573    -1.331
   x9            -0.3341      0.061     -5.453      0.000        -0.454    -0.214
   x10           -0.1690      0.033     -5.169      0.000        -0.233    -0.105
   x11            0.0049      0.001      3.921      0.000         0.002     0.007
   x12           -0.0036      0.000    -15.878      0.000        -0.004    -0.003
   x13           -0.0141      0.002     -7.391      0.000        -0.018    -0.010
   x14           -0.0040      0.000     -8.450      0.000        -0.005    -0.003
   x15           -0.0039      0.001     -4.059      0.000        -0.006    -0.002
   x16            0.0917      0.015      6.321      0.000         0.063     0.120
   x17            0.0490      0.007      6.574      0.000         0.034     0.064
   x18            0.0080      0.001      5.362      0.000         0.005     0.011
   x19            0.0002   2.99e-05      7.428      0.000         0.000     0.000
   x20           -0.0022      0.000     -6.445      0.000        -0.003    -0.002
   const          2.9589      1.547      1.913      0.056        -0.073     5.990
   ==============================================================================

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:

In [ ]:
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
   
Total number of trials: 807.0
   Parameters:  [ -1.68150366e-02   9.92547661e-03  -1.87242148e-02  -1.42385609e-02
      2.54487173e-01   2.40693664e-01   8.04086739e-02  -1.95216050e+00
     -3.34086475e-01  -1.69022168e-01   4.91670212e-03  -3.57996435e-03
     -1.40765648e-02  -4.00499176e-03  -3.90639579e-03   9.17143006e-02
      4.89898381e-02   8.04073890e-03   2.22009503e-04  -2.24924861e-03
      2.95887793e+00]
   T-values:  [-38.74908321  16.50473627 -25.1821894  -32.81791308   8.49827113
      4.21247925   5.7749976   -6.16191078  -5.45321673  -5.16865445
      3.92119964 -15.87825999  -7.39093058  -8.44963886  -4.05916246
      6.3210987    6.57434662   5.36229044   7.42806363  -6.44513698
      1.91301155]

The interquartile first difference for the percentage of low income households in a school district is:

In [ ]:
print("%2.4f%%" % (diff*100))

Plots

We extract information that will be used to draw some interesting plots:

In [ ]:
nobs = res.nobs
   y = data.endog[:,0]/data.endog.sum(1)
   yhat = res.mu
   
-11.8753%

Plot yhat vs y:

In [ ]:
from statsmodels.graphics.api import abline_plot
In [ ]:
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:

In [ ]:
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:

In [ ]:
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:

In [ ]:
from statsmodels import graphics
   graphics.gofplots.qqplot(resid, line='r')

GLM: Gamma for proportional count response

Load data

In the example above, we printed the NOTE attribute to learn about the Star98 dataset. Statsmodels datasets ships with other useful information. For example:

In [ ]:
print(sm.datasets.scotland.DESCRLONG)

Load the data and add a constant to the exogenous variables:

In [ ]:
data2 = sm.datasets.scotland.load()
   data2.exog = sm.add_constant(data2.exog, prepend=False)
   print(data2.exog[:5,:])
   print(data2.endog[:5])
   
   This data is based on the example in Gill and describes the proportion of
   voters who voted Yes to grant the Scottish Parliament taxation powers.
   The data are divided into 32 council districts.  This example's explanatory
   variables include the amount of council tax collected in pounds sterling as
   of April 1997 per two adults before adjustments, the female percentage of
   total claims for unemployment benefits as of January, 1998, the standardized
   mortality rate (UK is 100), the percentage of labor force participation,
   regional GDP, the percentage of children aged 5 to 15, and an interaction term
   between female unemployment and the council tax.
   
   The original source files and variable information are included in
   /scotland/src/

Fit and summary

In [ ]:
glm_gamma = sm.GLM(data2.endog, data2.exog, family=sm.families.Gamma())
   glm_results = glm_gamma.fit()
   print(glm_results.summary())
   
[[  7.12000000e+02   2.10000000e+01   1.05000000e+02   8.24000000e+01
       1.35660000e+04   1.23000000e+01   1.49520000e+04   1.00000000e+00]
    [  6.43000000e+02   2.65000000e+01   9.70000000e+01   8.02000000e+01
       1.35660000e+04   1.53000000e+01   1.70395000e+04   1.00000000e+00]
    [  6.79000000e+02   2.83000000e+01   1.13000000e+02   8.63000000e+01
       9.61100000e+03   1.39000000e+01   1.92157000e+04   1.00000000e+00]
    [  8.01000000e+02   2.71000000e+01   1.09000000e+02   8.04000000e+01
       9.48300000e+03   1.36000000e+01   2.17071000e+04   1.00000000e+00]
    [  7.53000000e+02   2.20000000e+01   1.15000000e+02   6.47000000e+01
       9.26500000e+03   1.46000000e+01   1.65660000e+04   1.00000000e+00]]
   [ 60.3  52.3  53.4  57.   68.7]

Artificial data

In [ ]:
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)
   
                 Generalized Linear Model Regression Results                  
   ==============================================================================
   Dep. Variable:                      y   No. Observations:                   32
   Model:                            GLM   Df Residuals:                       24
   Model Family:                   Gamma   Df Model:                            7
   Link Function:          inverse_power   Scale:                0.00358428317349
   Method:                          IRLS   Log-Likelihood:                -83.017
   Date:                Mon, 20 Jul 2015   Deviance:                     0.087389
   Time:                        17:43:34   Pearson chi2:                   0.0860
   No. Iterations:                     6                                         
   ==============================================================================
                    coef    std err          z      P>|z|      [95.0% Conf. Int.]
   ------------------------------------------------------------------------------
   x1          4.962e-05   1.62e-05      3.060      0.002      1.78e-05  8.14e-05
   x2             0.0020      0.001      3.824      0.000         0.001     0.003
   x3         -7.181e-05   2.71e-05     -2.648      0.008        -0.000 -1.87e-05
   x4             0.0001   4.06e-05      2.757      0.006      3.23e-05     0.000
   x5         -1.468e-07   1.24e-07     -1.187      0.235     -3.89e-07  9.56e-08
   x6            -0.0005      0.000     -2.159      0.031        -0.001 -4.78e-05
   x7         -2.427e-06   7.46e-07     -3.253      0.001     -3.89e-06 -9.65e-07
   const         -0.0178      0.011     -1.548      0.122        -0.040     0.005
   ==============================================================================

Fit and summary

In [ ]:
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())