Description

For Gaussian models, sigma returns the value of the residual standard deviation; for other families, it returns the dispersion parameter, however it is defined for that particular family. See details for each family below.

Usage

Arguments

Details

The value returned varies by family:

gaussian

returns the maximum likelihood estimate of the standard deviation (i.e., smaller than the results of sigma(lm(...)) by a factor of (n-1)/n)

nbinom1

returns an overdispersion parameter (usually denoted alpha as in Hardin and Hilbe (2007)): such that the variance equals mu(1+alpha).

nbinom2

returns an overdispersion parameter (usually denoted theta or k); in contrast to most other families, larger theta corresponds to a lower variance which is mu(1+mu/theta).

Gamma

Internally, glmmTMB fits Gamma responses by fitting a mean and a shape parameter; sigma is estimated as (1/sqrt(shape)), which will typically be close (but not identical to) that estimated by stats:::sigma.default, which uses sqrt(deviance/df.residual)

beta

returns the value of phi, where the conditional variance is mu*(1-mu)/(1+phi) (i.e., increasing phi decreases the variance.) This parameterization follows Ferrari and Cribari-Neto (2004) (and the betareg package):

betabinomial

This family uses the same parameterization (governing the Beta distribution that underlies the binomial probabilities) as beta.

genpois

returns the index of dispersion phi^2, where the variance is mu*phi^2 (Consul & Famoye 1992)

compois

returns the value of 1/nu, When nu=1, compois is equivalent to the Poisson distribution. There is no closed form equation for the variance, but it is approximately undersidpersed when 1/nu <1 and approximately oversidpersed when 1/nu>1. In this implementation, mu is exactly the mean (Huang 2017), which differs from the COMPoissonReg package (Sellers & Lotze 2015).

tweedie

returns the value of phi, where the variance is phi*mu^p. The value of p can be extracted using the internal function glmmTMB:::.tweedie_power.

The most commonly used GLM families (binomial, poisson) have fixed dispersion parameters which are internally ignored.

References

• Consul PC, and Famoye F (1992). "Generalized Poisson regression model. Communications in Statistics: Theory and Methods" 21:89–109.

• Ferrari SLP, Cribari-Neto F (2004). "Beta Regression for Modelling Rates and Proportions." J. Appl. Stat. 31(7), 799-815.

• Hardin JW & Hilbe JM (2007). "Generalized linear models and extensions." Stata press.

• Huang A (2017). "Mean-parametrized Conway–Maxwell–Poisson regression models for dispersed counts. " Statistical Modelling 17(6), 1-22.

• Sellers K & Lotze T (2015). "COMPoissonReg: Conway-Maxwell Poisson (COM-Poisson) Regression". R package version 0.3.5. https://CRAN.R-project.org/package=COMPoissonReg