.. currentmodule:: statsmodels.genmod.generalized_linear_model .. _glm: Generalized Linear Models ========================= Generalized linear models currently supports estimation using the one-parameter exponential families. See `Module Reference`_ for commands and arguments. Examples -------- .. ipython:: python :okwarning: # Load modules and data import statsmodels.api as sm data = sm.datasets.scotland.load() data.exog = sm.add_constant(data.exog) # Instantiate a gamma family model with the default link function. gamma_model = sm.GLM(data.endog, data.exog, family=sm.families.Gamma()) gamma_results = gamma_model.fit() print(gamma_results.summary()) Detailed examples can be found here: * `GLM `_ * `Formula `_ Technical Documentation ----------------------- .. ..glm_techn1 .. ..glm_techn2 The statistical model for each observation :math:`i` is assumed to be :math:`Y_i \sim F_{EDM}(\cdot|\theta,\phi,w_i)` and :math:`\mu_i = E[Y_i|x_i] = g^{-1}(x_i^\prime\beta)`. where :math:`g` is the link function and :math:`F_{EDM}(\cdot|\theta,\phi,w)` is a distribution of the family of exponential dispersion models (EDM) with natural parameter :math:`\theta`, scale parameter :math:`\phi` and weight :math:`w`. Its density is given by :math:`f_{EDM}(y|\theta,\phi,w) = c(y,\phi,w) \exp\left(\frac{y\theta-b(\theta)}{\phi}w\right)\,.` It follows that :math:`\mu = b'(\theta)` and :math:`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 :math:`\theta(\mu)` such that :math:`Var[Y_i|x_i] = \frac{\phi}{w_i} v(\mu_i)` with :math:`v(\mu) = b''(\theta(\mu))`. Therefore it is said that a GLM is determined by link function :math:`g` and variance function :math:`v(\mu)` alone (and :math:`x` of course). Note that while :math:`\phi` is the same for every observation :math:`y_i` and therefore does not influence the estimation of :math:`\beta`, the weights :math:`w_i` might be different for every :math:`y_i` such that the estimation of :math:`\beta` depends on them. ================================================= ============================== ============================== ======================================== =========================================== ============================================================================ ===================== Distribution Domain :math:`\mu=E[Y|x]` :math:`v(\mu)` :math:`\theta(\mu)` :math:`b(\theta)` :math:`\phi` ================================================= ============================== ============================== ======================================== =========================================== ============================================================================ ===================== Binomial :math:`B(n,p)` :math:`0,1,\ldots,n` :math:`np` :math:`\mu-\frac{\mu^2}{n}` :math:`\log\frac{p}{1-p}` :math:`n\log(1+e^\theta)` 1 Poisson :math:`P(\mu)` :math:`0,1,\ldots,\infty` :math:`\mu` :math:`\mu` :math:`\log(\mu)` :math:`e^\theta` 1 Neg. Binom. :math:`NB(\mu,\alpha)` :math:`0,1,\ldots,\infty` :math:`\mu` :math:`\mu+\alpha\mu^2` :math:`\log(\frac{\alpha\mu}{1+\alpha\mu})` :math:`-\frac{1}{\alpha}\log(1-\alpha e^\theta)` 1 Gaussian/Normal :math:`N(\mu,\sigma^2)` :math:`(-\infty,\infty)` :math:`\mu` :math:`1` :math:`\mu` :math:`\frac{1}{2}\theta^2` :math:`\sigma^2` Gamma :math:`N(\mu,\nu)` :math:`(0,\infty)` :math:`\mu` :math:`\mu^2` :math:`-\frac{1}{\mu}` :math:`-\log(-\theta)` :math:`\frac{1}{\nu}` Inv. Gauss. :math:`IG(\mu,\sigma^2)` :math:`(0,\infty)` :math:`\mu` :math:`\mu^3` :math:`-\frac{1}{2\mu^2}` :math:`-\sqrt{-2\theta}` :math:`\sigma^2` Tweedie :math:`p\geq 1` depends on :math:`p` :math:`\mu` :math:`\mu^p` :math:`\frac{\mu^{1-p}}{1-p}` :math:`\frac{\alpha-1}{\alpha}\left(\frac{\theta}{\alpha-1}\right)^{\alpha}` :math:`\phi` ================================================= ============================== ============================== ======================================== =========================================== ============================================================================ ===================== The Tweedie distribution has special cases for :math:`p=0,1,2` not listed in the table and uses :math:`\alpha=\frac{p-2}{p-1}`. Correspondence of mathematical variables to code: * :math:`Y` and :math:`y` are coded as ``endog``, the variable one wants to model * :math:`x` is coded as ``exog``, the covariates alias explanatory variables * :math:`\beta` is coded as ``params``, the parameters one wants to estimate * :math:`\mu` is coded as ``mu``, the expectation (conditional on :math:`x`) of :math:`Y` * :math:`g` is coded as ``link`` argument to the ``class Family`` * :math:`\phi` is coded as ``scale``, the dispersion parameter of the EDM * :math:`w` is not yet supported (i.e. :math:`w=1`), in the future it might be ``var_weights`` * :math:`p` is coded as ``var_power`` for the power of the variance function :math:`v(\mu)` of the Tweedie distribution, see table * :math:`\alpha` is either * Negative Binomial: the ancillary parameter ``alpha``, see table * Tweedie: an abbreviation for :math:`\frac{p-2}{p-1}` of the power :math:`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 ---------------- .. module:: statsmodels.genmod.generalized_linear_model :synopsis: Generalized Linear Models (GLM) Model Class ^^^^^^^^^^^ .. autosummary:: :toctree: generated/ GLM Results Class ^^^^^^^^^^^^^ .. autosummary:: :toctree: generated/ GLMResults PredictionResultsMean .. _families: Families ^^^^^^^^ The distribution families currently implemented are .. module:: statsmodels.genmod.families.family .. currentmodule:: statsmodels.genmod.families.family .. autosummary:: :toctree: generated/ Family Binomial Gamma Gaussian InverseGaussian NegativeBinomial Poisson Tweedie .. _links: Link Functions ^^^^^^^^^^^^^^ Note: The lower case link classes have been deprecated and will be removed in future. Link classes now follow the Python class name convention. 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..links .. module:: statsmodels.genmod.families.links .. currentmodule:: statsmodels.genmod.families.links .. autosummary:: :toctree: generated/ Link CDFLink CLogLog LogLog LogC Log Logit NegativeBinomial Power Cauchy Identity InversePower InverseSquared Probit .. _varfuncs: Variance Functions ^^^^^^^^^^^^^^^^^^ Each of the families has an associated variance function. You can access the variance functions here: :: >>> sm.families..variance .. module:: statsmodels.genmod.families.varfuncs .. currentmodule:: statsmodels.genmod.families.varfuncs .. autosummary:: :toctree: generated/ VarianceFunction constant Power mu mu_squared mu_cubed Binomial binary NegativeBinomial nbinom