statsmodels.genmod.generalized_linear_model.GLM¶

class statsmodels.genmod.generalized_linear_model.GLM(endog, exog, family=None, offset=None, exposure=None, freq_weights=None, var_weights=None, missing='none', **kwargs)[source]¶

Generalized Linear Models

GLM inherits from statsmodels.base.model.LikelihoodModel

Parameters:¶

endogarray_like: 1d array of endogenous response variable. This array can be 1d or 2d. Binomial family models accept a 2d array with two columns. If supplied, each observation is expected to be [success, failure].
exogarray_like: A nobs x k array where nobs is the number of observations and k is the number of regressors. An intercept is not included by default and should be added by the user (models specified using a formula include an intercept by default). See statsmodels.tools.add_constant.
familyfamily class instance: The default is Gaussian. To specify the binomial distribution family = sm.family.Binomial() Each family can take a link instance as an argument. See statsmodels.family.family for more information.
offsetarray_like or None: An offset to be included in the model. If provided, must be an array whose length is the number of rows in exog.
exposurearray_like or None: Log(exposure) will be added to the linear prediction in the model. Exposure is only valid if the log link is used. If provided, it must be an array with the same length as endog.
freq_weightsarray_like: 1d array of frequency weights. The default is None. If None is selected or a blank value, then the algorithm will replace with an array of 1’s with length equal to the endog. WARNING: Using weights is not verified yet for all possible options and results, see Notes.
var_weightsarray_like: 1d array of variance (analytic) weights. The default is None. If None is selected or a blank value, then the algorithm will replace with an array of 1’s with length equal to the endog. WARNING: Using weights is not verified yet for all possible options and results, see Notes.
missingstr: Available options are ‘none’, ‘drop’, and ‘raise’. If ‘none’, no nan checking is done. If ‘drop’, any observations with nans are dropped. If ‘raise’, an error is raised. Default is ‘none’.

Attributes:¶

df_modelfloat: Model degrees of freedom is equal to p - 1, where p is the number of regressors. Note that the intercept is not reported as a degree of freedom.
df_residfloat: Residual degrees of freedom is equal to the number of observation n minus the number of regressors p.
endogndarray: See Notes. Note that endog is a reference to the data so that if data is already an array and it is changed, then endog changes as well.
exposurearray_like: Include ln(exposure) in model with coefficient constrained to 1. Can only be used if the link is the logarithm function.
exogndarray: See Notes. Note that exog is a reference to the data so that if data is already an array and it is changed, then exog changes as well.
freq_weightsndarray: See Notes. Note that freq_weights is a reference to the data so that if data is already an array and it is changed, then freq_weights changes as well.
var_weightsndarray: See Notes. Note that var_weights is a reference to the data so that if data is already an array and it is changed, then var_weights changes as well.
iterationint: The number of iterations that fit has run. Initialized at 0.
familyfamily class instance: The distribution family of the model. Can be any family in statsmodels.families. Default is Gaussian.
mundarray: The mean response of the transformed variable. mu is the value of the inverse of the link function at lin_pred, where lin_pred is the linear predicted value of the WLS fit of the transformed variable. mu is only available after fit is called. See statsmodels.families.family.fitted of the distribution family for more information.
n_trialsndarray: See Notes. Note that n_trials is a reference to the data so that if data is already an array and it is changed, then n_trials changes as well. n_trials is the number of binomial trials and only available with that distribution. See statsmodels.families.Binomial for more information.
normalized_cov_paramsndarray: The p x p normalized covariance of the design / exogenous data. This is approximately equal to (X.T X)^(-1)
offsetarray_like: Include offset in model with coefficient constrained to 1.
scalefloat: The estimate of the scale / dispersion of the model fit. Only available after fit is called. See GLM.fit and GLM.estimate_scale for more information.
scaletypestr: The scaling used for fitting the model. This is only available after fit is called. The default is None. See GLM.fit for more information.
weightsndarray: The value of the weights after the last iteration of fit. Only available after fit is called. See statsmodels.families.family for the specific distribution weighting functions.

See also

statsmodels.genmod.families.family.Family
Families
Link Functions

Notes

Note: PerfectSeparationError exception has been converted to a PerfectSeparationWarning and perfect separation or perfect prediction will not raise an exception by default. (changed in version 0.14)

Only the following combinations make sense for family and link:

Family

ident

log

logit

probit

cloglog

pow

opow

nbinom

loglog

logc

Gaussian

x

x

x

x

x

x

x

x

x

inv Gaussian

x

x

x

binomial

x

x

x

x

x

x

x

x

x

Poisson

x

x

x

neg binomial

x

x

x

x

gamma

x

x

x

Tweedie

x

x

x

Family	ident	log	logit	probit	cloglog	pow	opow	nbinom	loglog	logc
Gaussian	x	x	x	x	x	x	x	x	x
inv Gaussian	x	x				x
binomial	x	x	x	x	x	x	x		x	x
Poisson	x	x				x
neg binomial	x	x				x		x
gamma	x	x				x
Tweedie	x	x				x

Not all of these link functions are currently available.

Endog and exog are references so that if the data they refer to are already arrays and these arrays are changed, endog and exog will change.

statsmodels supports two separate definitions of weights: frequency weights and variance weights.

Frequency weights produce the same results as repeating observations by the frequencies (if those are integers). Frequency weights will keep the number of observations consistent, but the degrees of freedom will change to reflect the new weights.

Variance weights (referred to in other packages as analytic weights) are used when endog represents an an average or mean. This relies on the assumption that that the inverse variance scales proportionally to the weight–an observation that is deemed more credible should have less variance and therefore have more weight. For the Poisson family–which assumes that occurrences scale proportionally with time–a natural practice would be to use the amount of time as the variance weight and set endog to be a rate (occurrences per period of time). Similarly, using a compound Poisson family, namely Tweedie, makes a similar assumption about the rate (or frequency) of occurrences having variance proportional to time.

Both frequency and variance weights are verified for all basic results with nonrobust or heteroscedasticity robust cov_type. Other robust covariance types have not yet been verified, and at least the small sample correction is currently not based on the correct total frequency count.

Currently, all residuals are not weighted by frequency, although they may incorporate n_trials for Binomial and var_weights

Residual Type	Applicable weights
Anscombe	`var_weights`
Deviance	`var_weights`
Pearson	`var_weights` and `n_trials`
Reponse	`n_trials`
Working	`n_trials`

WARNING: Loglikelihood and deviance are not valid in models where scale is equal to 1 (i.e., Binomial, NegativeBinomial, and Poisson). If variance weights are specified, then results such as loglike and deviance are based on a quasi-likelihood interpretation. The loglikelihood is not correctly specified in this case, and statistics based on it, such AIC or likelihood ratio tests, are not appropriate.

Examples

>>> 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()
>>> gamma_results.params
array([-0.01776527,  0.00004962,  0.00203442, -0.00007181,  0.00011185,
       -0.00000015, -0.00051868, -0.00000243])
>>> gamma_results.scale
0.0035842831734919055
>>> gamma_results.deviance
0.087388516416999198
>>> gamma_results.pearson_chi2
0.086022796163805704
>>> gamma_results.llf
-83.017202161073527

Methods

`estimate_scale`(mu)	Estimate the dispersion/scale.
`estimate_tweedie_power`(mu[, method, low, high])	Tweedie specific function to estimate scale and the variance parameter.
`fit`([start_params, maxiter, method, tol, ...])	Fits a generalized linear model for a given family.
`fit_constrained`(constraints[, start_params])	fit the model subject to linear equality constraints
`fit_regularized`([method, alpha, ...])	Return a regularized fit to a linear regression model.
`from_formula`(formula, data[, subset, drop_cols])	Create a Model from a formula and dataframe.
`get_distribution`(params[, scale, exog, ...])	Return a instance of the predictive distribution.
`hessian`(params[, scale, observed])	Hessian, second derivative of loglikelihood function
`hessian_factor`(params[, scale, observed])	Weights for calculating Hessian
`information`(params[, scale])	Fisher information matrix.
`initialize`()	Initialize a generalized linear model.
`loglike`(params[, scale])	Evaluate the log-likelihood for a generalized linear model.
`loglike_mu`(mu[, scale])	Evaluate the log-likelihood for a generalized linear model.
`predict`(params[, exog, exposure, offset, ...])	Return predicted values for a design matrix
`score`(params[, scale])	score, first derivative of the loglikelihood function
`score_factor`(params[, scale])	weights for score for each observation
`score_obs`(params[, scale])	score first derivative of the loglikelihood for each observation.
`score_test`(params_constrained[, ...])	score test for restrictions or for omitted variables

Properties

`endog_names`	Names of endogenous variables.
`exog_names`	Names of exogenous variables.
`exposure_name`	Name of the exposure variable if available.
`freq_weights_name`	Name of the freq weights variable if available.
`offset_name`	Name of the offset variable if available.
`var_weights_name`	Name of var weights variable if available.

Last update: Oct 29, 2024