.. 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 PredictionResults .. _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 ^^^^^^^^^^^^^^ 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 Log Logit NegativeBinomial Power cauchy cloglog loglog identity inverse_power inverse_squared log logit nbinom 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