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

In [2]: data = sm.datasets.scotland.load(as_pandas=False)

In [3]: data.exog = sm.add_constant(data.exog)

# 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
Link Function:          inverse_power   Scale:                       0.0035843
Method:                          IRLS   Log-Likelihood:                -83.017
Date:                Tue, 02 Feb 2021   Deviance:                     0.087389
Time:                        07:07:06   Pearson chi2:                   0.0860
No. Iterations:                     6                                         
Covariance Type:            nonrobust                                         
==============================================================================
                 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:

• GLM

• Formula

Technical Documentation

The statistical model for each observation $i$ is assumed to be

$Y_i \sim F_{EDM}(\cdot|\theta,\phi,w_i)$ and $\mu_i = E[Y_i|x_i] = g^{-1}(x_i^\prime\beta)$.

where $g$ is the link function and $F_{EDM}(\cdot|\theta,\phi,w)$ is a distribution of the family of exponential dispersion models (EDM) with natural parameter $\theta$, scale parameter $\phi$ and weight $w$. Its density is given by

$f_{EDM}(y|\theta,\phi,w) = c(y,\phi,w) \exp\left(\frac{y\theta-b(\theta)}{\phi}w\right)\,.$

It follows that $\mu = b'(\theta)$ and $Var[Y|x]=\frac{\phi}{w}b''(\theta)$. The inverse of the first equation gives the natural parameter as a function of the expected value $\theta(\mu)$ such that

$Var[Y_i|x_i] = \frac{\phi}{w_i} v(\mu_i)$

with $v(\mu) = b''(\theta(\mu))$. Therefore it is said that a GLM is determined by link function $g$ and variance function $v(\mu)$ alone (and $x$ of course).

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

Distribution	Domain	$\mu=E[Y|x]$	$v(\mu)$	$\theta(\mu)$	$b(\theta)$	$\phi$
Binomial $B(n,p)$	$0,1,\ldots,n$	$np$	$\mu-\frac{\mu^2}{n}$	$\log\frac{p}{1-p}$	$n\log(1+e^\theta)$	1
Poisson $P(\mu)$	$0,1,\ldots,\infty$	$\mu$	$\mu$	$\log(\mu)$	$e^\theta$	1
Neg. Binom. $NB(\mu,\alpha)$	$0,1,\ldots,\infty$	$\mu$	$\mu+\alpha\mu^2$	$\log(\frac{\alpha\mu}{1+\alpha\mu})$	$-\frac{1}{\alpha}\log(1-\alpha e^\theta)$	1
Gaussian/Normal $N(\mu,\sigma^2)$	$(-\infty,\infty)$	$\mu$	$1$	$\mu$	$\frac{1}{2}\theta^2$	$\sigma^2$
Gamma $N(\mu,\nu)$	$(0,\infty)$	$\mu$	$\mu^2$	$-\frac{1}{\mu}$	$-\log(-\theta)$	$\frac{1}{\nu}$
Inv. Gauss. $IG(\mu,\sigma^2)$	$(0,\infty)$	$\mu$	$\mu^3$	$-\frac{1}{2\mu^2}$	$-\sqrt{-2\theta}$	$\sigma^2$
Tweedie $p\geq 1$	depends on $p$	$\mu$	$\mu^p$	$\frac{\mu^{1-p}}{1-p}$	$\frac{\alpha-1}{\alpha}\left(\frac{\theta}{\alpha-1}\right)^{\alpha}$	$\phi$

The Tweedie distribution has special cases for $p=0,1,2$ not listed in the table and uses $\alpha=\frac{p-2}{p-1}$.

Correspondence of mathematical variables to code:

• $Y$ and $y$ are coded as endog, the variable one wants to model

• $x$ is coded as exog, the covariates alias explanatory variables

• $\beta$ is coded as params, the parameters one wants to estimate

• $\mu$ is coded as mu, the expectation (conditional on $x$) of $Y$

• $g$ is coded as link argument to the class Family

• $\phi$ is coded as scale, the dispersion parameter of the EDM

• $w$ is not yet supported (i.e. $w=1$), in the future it might be var_weights

• $p$ is coded as var_power for the power of the variance function $v(\mu)$ of the Tweedie distribution, see table

• $\alpha$ is either

  • Negative Binomial: the ancillary parameter alpha, see table

  • Tweedie: an abbreviation for $\frac{p-2}{p-1}$ of the power $p$ 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

Results Class

GLMResults(model, params, …[, cov_type, …])	Class to contain GLM results.
PredictionResults(predicted_mean, var_pred_mean)	Attributes

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, eql])	Tweedie family.

Link Functions

The link functions currently implemented are the following. Not all link functions are available for each distribution family. The list of available link functions can be obtained by

>>> sm.families.family.<familyname>.links

Link()	A generic link function for one-parameter exponential family.
CDFLink([dbn])	The use the CDF of a scipy.stats distribution
CLogLog()	The complementary log-log transform
Log()	The log transform
Logit()	The logit transform
NegativeBinomial([alpha])	The negative binomial link function
Power([power])	The power transform
cauchy()	The Cauchy (standard Cauchy CDF) transform
cloglog()	The CLogLog transform link function.
identity()	The identity transform
inverse_power()	The inverse transform
inverse_squared()	The inverse squared transform
log()	The log transform
logit()	Methods
nbinom([alpha])	The negative binomial link function.
probit([dbn])	The probit (standard normal CDF) transform

Variance Functions

Each of the families has an associated variance function. You can access the variance functions here:

>>> sm.families.<familyname>.variance

VarianceFunction()	Relates the variance of a random variable to its mean.
constant	The call method of constant returns a constant variance, i.e., a vector of ones.
Power([power])	Power variance function
mu	Returns np.fabs(mu)
mu_squared	Returns np.fabs(mu)**2
mu_cubed	Returns np.fabs(mu)**3
Binomial([n])	Binomial variance function
binary	The binomial variance function for n = 1
NegativeBinomial([alpha])	Negative binomial variance function
nbinom	Negative Binomial variance function.