Generalized Linear Models (Formula)ΒΆ
This notebook illustrates how you can use R-style formulas to fit Generalized Linear Models.
To begin, we load the Star98 dataset and we construct a formula and pre-process the data:
[1]:
import statsmodels.api as sm
import statsmodels.formula.api as smf
star98 = sm.datasets.star98.load_pandas().data
formula = "SUCCESS ~ LOWINC + PERASIAN + PERBLACK + PERHISP + PCTCHRT + \
PCTYRRND + PERMINTE*AVYRSEXP*AVSALK + PERSPENK*PTRATIO*PCTAF"
dta = star98[
[
"NABOVE",
"NBELOW",
"LOWINC",
"PERASIAN",
"PERBLACK",
"PERHISP",
"PCTCHRT",
"PCTYRRND",
"PERMINTE",
"AVYRSEXP",
"AVSALK",
"PERSPENK",
"PTRATIO",
"PCTAF",
]
].copy()
endog = dta["NABOVE"] / (dta["NABOVE"] + dta.pop("NBELOW"))
del dta["NABOVE"]
dta["SUCCESS"] = endog
Then, we fit the GLM model:
[2]:
mod1 = smf.glm(formula=formula, data=dta, family=sm.families.Binomial()).fit()
print(mod1.summary())
Generalized Linear Model Regression Results
==============================================================================
Dep. Variable: SUCCESS No. Observations: 303
Model: GLM Df Residuals: 282
Model Family: Binomial Df Model: 20
Link Function: Logit Scale: 1.0000
Method: IRLS Log-Likelihood: -127.33
Date: Wed, 02 Nov 2022 Deviance: 8.5477
Time: 17:02:03 Pearson chi2: 8.48
No. Iterations: 4 Pseudo R-squ. (CS): 0.1115
Covariance Type: nonrobust
============================================================================================
coef std err z P>|z| [0.025 0.975]
--------------------------------------------------------------------------------------------
Intercept 0.4037 25.036 0.016 0.987 -48.665 49.472
LOWINC -0.0204 0.010 -1.982 0.048 -0.041 -0.000
PERASIAN 0.0159 0.017 0.910 0.363 -0.018 0.050
PERBLACK -0.0198 0.020 -1.004 0.316 -0.058 0.019
PERHISP -0.0096 0.010 -0.951 0.341 -0.029 0.010
PCTCHRT -0.0022 0.022 -0.103 0.918 -0.045 0.040
PCTYRRND -0.0022 0.006 -0.348 0.728 -0.014 0.010
PERMINTE 0.1068 0.787 0.136 0.892 -1.436 1.650
AVYRSEXP -0.0411 1.176 -0.035 0.972 -2.346 2.264
PERMINTE:AVYRSEXP -0.0031 0.054 -0.057 0.954 -0.108 0.102
AVSALK 0.0131 0.295 0.044 0.965 -0.566 0.592
PERMINTE:AVSALK -0.0019 0.013 -0.145 0.885 -0.028 0.024
AVYRSEXP:AVSALK 0.0008 0.020 0.038 0.970 -0.039 0.041
PERMINTE:AVYRSEXP:AVSALK 5.978e-05 0.001 0.068 0.946 -0.002 0.002
PERSPENK -0.3097 4.233 -0.073 0.942 -8.606 7.987
PTRATIO 0.0096 0.919 0.010 0.992 -1.792 1.811
PERSPENK:PTRATIO 0.0066 0.206 0.032 0.974 -0.397 0.410
PCTAF -0.0143 0.474 -0.030 0.976 -0.944 0.916
PERSPENK:PCTAF 0.0105 0.098 0.107 0.915 -0.182 0.203
PTRATIO:PCTAF -0.0001 0.022 -0.005 0.996 -0.044 0.044
PERSPENK:PTRATIO:PCTAF -0.0002 0.005 -0.051 0.959 -0.010 0.009
============================================================================================
Finally, we define a function to operate customized data transformation using the formula framework:
[3]:
def double_it(x):
return 2 * x
formula = "SUCCESS ~ double_it(LOWINC) + PERASIAN + PERBLACK + PERHISP + PCTCHRT + \
PCTYRRND + PERMINTE*AVYRSEXP*AVSALK + PERSPENK*PTRATIO*PCTAF"
mod2 = smf.glm(formula=formula, data=dta, family=sm.families.Binomial()).fit()
print(mod2.summary())
Generalized Linear Model Regression Results
==============================================================================
Dep. Variable: SUCCESS No. Observations: 303
Model: GLM Df Residuals: 282
Model Family: Binomial Df Model: 20
Link Function: Logit Scale: 1.0000
Method: IRLS Log-Likelihood: -127.33
Date: Wed, 02 Nov 2022 Deviance: 8.5477
Time: 17:02:03 Pearson chi2: 8.48
No. Iterations: 4 Pseudo R-squ. (CS): 0.1115
Covariance Type: nonrobust
============================================================================================
coef std err z P>|z| [0.025 0.975]
--------------------------------------------------------------------------------------------
Intercept 0.4037 25.036 0.016 0.987 -48.665 49.472
double_it(LOWINC) -0.0102 0.005 -1.982 0.048 -0.020 -0.000
PERASIAN 0.0159 0.017 0.910 0.363 -0.018 0.050
PERBLACK -0.0198 0.020 -1.004 0.316 -0.058 0.019
PERHISP -0.0096 0.010 -0.951 0.341 -0.029 0.010
PCTCHRT -0.0022 0.022 -0.103 0.918 -0.045 0.040
PCTYRRND -0.0022 0.006 -0.348 0.728 -0.014 0.010
PERMINTE 0.1068 0.787 0.136 0.892 -1.436 1.650
AVYRSEXP -0.0411 1.176 -0.035 0.972 -2.346 2.264
PERMINTE:AVYRSEXP -0.0031 0.054 -0.057 0.954 -0.108 0.102
AVSALK 0.0131 0.295 0.044 0.965 -0.566 0.592
PERMINTE:AVSALK -0.0019 0.013 -0.145 0.885 -0.028 0.024
AVYRSEXP:AVSALK 0.0008 0.020 0.038 0.970 -0.039 0.041
PERMINTE:AVYRSEXP:AVSALK 5.978e-05 0.001 0.068 0.946 -0.002 0.002
PERSPENK -0.3097 4.233 -0.073 0.942 -8.606 7.987
PTRATIO 0.0096 0.919 0.010 0.992 -1.792 1.811
PERSPENK:PTRATIO 0.0066 0.206 0.032 0.974 -0.397 0.410
PCTAF -0.0143 0.474 -0.030 0.976 -0.944 0.916
PERSPENK:PCTAF 0.0105 0.098 0.107 0.915 -0.182 0.203
PTRATIO:PCTAF -0.0001 0.022 -0.005 0.996 -0.044 0.044
PERSPENK:PTRATIO:PCTAF -0.0002 0.005 -0.051 0.959 -0.010 0.009
============================================================================================
As expected, the coefficient for double_it(LOWINC) in the second model is half the size of the LOWINC coefficient from the first model:
[4]:
print(mod1.params[1])
print(mod2.params[1] * 2)
-0.02039598715475585
-0.020395987154756844