Distribution families in mgcv

Description

As well as the standard families (of class family) documented in family (see also glm) which can be used with functions gam, bam and gamm, mgcv also supplies some extra families, most of which are currently only usable with gam, although some can also be used with bam. These are described here.

Details

The following families (class family) are in the exponential family given the value of a single parameter. They are usable with all modelling functions.

• Tweedie An exponential family distribution for which the variance of the response is given by the mean response to the power p. p is in (1,2) and must be supplied. Alternatively, see tw to estimate p (gam/bam only).

• negbin The negative binomial. Alternatively see nb to estimate the theta parameter of the negative binomial (gam/bam only).

The following families (class extended.family) are for regression type models dependent on a single linear predictor, and with a log likelihood which is a sum of independent terms, each corresponding to a single response observation. Usable with gam, with smoothing parameter estimation by "NCV", "REML" or "ML" (the latter does not integrate the unpenalized and parameteric effects out of the marginal likelihood optimized for the smoothing parameters). Also usable with bam.

• betar for proportions data on (0,1) when the binomial is not appropriate.

• cnorm censored normal distribution, for log normal accelerated failure time models, Tobit regression and rounded data, for example.

• nb for negative binomial data when the theta parameter is to be estimated.

• ocat for ordered categorical data.

• scat scaled t for heavy tailed data that would otherwise be modelled as Gaussian.

• tw for Tweedie distributed data, when the power parameter relating the variance to the mean is to be estimated.

• ziP for zero inflated Poisson data, when the zero inflation rate depends simply on the Poisson mean.

The above families of class family and extended.family can be combined to model data where different response observations come from different distributions. For example, when modelling the combination of presence-absence and abundance data, binomial and nb families might be used.

• gfam creates a 'grouped family' (or 'family group') from a list of families. The response is supplied as a two column matrix, the first containing the response observations, and the second an index of the family to which each observation relates.

The following families (class general.family) implement more general model classes. Usable only with gam and only with REML or NCV smoothing parameter estimation.

• cox.ph the Cox Proportional Hazards model for survival data (no NCV).

• gammals a gamma location-scale model, where the mean and standared deviation are modelled with separate linear predictors.

• gaulss a Gaussian location-scale model where the mean and the standard deviation are both modelled using smooth linear predictors.

• gevlss a generalized extreme value (GEV) model where the location, scale and shape parameters are each modelled using a linear predictor.

• gumbls a Gumbel location-scale model (2 linear predictors).

• multinom: multinomial logistic regression, for unordered categorical responses.

• mvn: multivariate normal additive models (no NCV).

• shash Sinh-arcsinh location scale and shape model family (4 linear predicors).

• twlss Tweedie location scale and variance power model family (3 linear predicors). Can only be fitted using EFS method.

• ziplss a ‘two-stage’ zero inflated Poisson model, in which 'potential-presence' is modelled with one linear predictor, and Poisson mean abundance given potential presence is modelled with a second linear predictor.

Author(s)

Simon N. Wood (s.wood@r-project.org) & Natalya Pya

References

Wood, S.N., N. Pya and B. Saefken (2016), Smoothing parameter and model selection for general smooth models. Journal of the American Statistical Association 111, 1548-1575 doi:10.1080/01621459.2016.1180986