/Statsmodels

# Generalized Linear Models

Generalized linear models currently supports estimation using the one-parameter exponential families.

See Module Reference for commands and arguments.

## Examples

# Load modules and data
In [1]: import statsmodels.api as sm

# Instantiate a gamma family model with the default link function.
In [4]: gamma_model = sm.GLM(data.endog, data.exog, family=sm.families.Gamma())

In [5]: gamma_results = gamma_model.fit()

In [6]: print(gamma_results.summary())
Generalized Linear Model Regression Results
==============================================================================
Dep. Variable:                      y   No. Observations:                   32
Model:                            GLM   Df Residuals:                       24
Model Family:                   Gamma   Df Model:                            7
Method:                          IRLS   Log-Likelihood:                -83.017
Date:                Tue, 28 Feb 2017   Deviance:                     0.087389
Time:                        21:38:03   Pearson chi2:                   0.0860
No. Iterations:                     4
==============================================================================
coef    std err          z      P>|z|      [0.025      0.975]
------------------------------------------------------------------------------
const         -0.0178      0.011     -1.548      0.122      -0.040       0.005
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
==============================================================================


Detailed examples can be found here:

## Technical Documentation

The statistical model for each observation is assumed to be

and .

where is the link function and is a distribution of the family of exponential dispersion models (EDM) with natural parameter , scale parameter and weight . Its density is given by

It follows that and . The inverse of the first equation gives the natural parameter as a function of the expected value such that

with . Therefore it is said that a GLM is determined by link function and variance function alone (and of course).

Note that while is the same for every observation and therefore does not influence the estimation of , the weights might be different for every such that the estimation of depends on them.

Distribution Domain
Binomial 1
Poisson 1
Neg. Binom. 1
Gaussian/Normal
Gamma
Inv. Gauss.
Tweedie depends on

The Tweedie distribution has special cases for not listed in the table and uses .

Correspondence of mathematical variables to code:

• and are coded as endog, the variable one wants to model
• is coded as exog, the covariates alias explanatory variables
• is coded as params, the parameters one wants to estimate
• is coded as mu, the expectation (conditional on ) of
• is coded as link argument to the class Family
• is coded as scale, the dispersion parameter of the EDM
• is not yet supported (i.e. ), in the future it might be var_weights
• is coded as var_power for the power of the variance function of the Tweedie distribution, see table
• is either
• Negative Binomial: the ancillary parameter alpha, see table
• Tweedie: an abbreviation for of the power of the variance function, see table

### References

• Gill, Jeff. 2000. Generalized Linear Models: A Unified Approach. SAGE QASS Series.
• Green, PJ. 1984. “Iteratively reweighted least squares for maximum likelihood estimation, and some robust and resistant alternatives.” Journal of the Royal Statistical Society, Series B, 46, 149-192.
• Hardin, J.W. and Hilbe, J.M. 2007. “Generalized Linear Models and Extensions.” 2nd ed. Stata Press, College Station, TX.
• McCullagh, P. and Nelder, J.A. 1989. “Generalized Linear Models.” 2nd ed. Chapman & Hall, Boca Rotan.

## Module Reference

### Model Class

 GLM(endog, exog[, family, offset, exposure, ...]) Generalized Linear Models class

### Results Class

 GLMResults(model, params, ...[, cov_type, ...]) Class to contain GLM results.

### Families

The distribution families currently implemented are

 Family(link, variance) The parent class for one-parameter exponential families. Binomial([link]) Binomial exponential family distribution. Gamma([link]) Gamma exponential family distribution. Gaussian([link]) Gaussian exponential family distribution. InverseGaussian([link]) InverseGaussian exponential family. NegativeBinomial([link, alpha]) Negative Binomial exponential family. Poisson([link]) Poisson exponential family. Tweedie([link, var_power, link_power]) Tweedie family.
>>> sm.families.family.<familyname>.links