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:                Sun, 19 May 2019   Deviance:                     0.087389
Time:                        07:42:55   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:

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 class

Results Class

GLMResults(model, params, …[, cov_type, …])

Class to contain GLM results.

PredictionResults(predicted_mean, var_pred_mean)

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.

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.