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.
# S3 method for glmmTMB
sigma(object, ...)a “glmmTMB” fitted object
(ignored; for method compatibility)
The value returned varies by family:
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)
returns a dispersion parameter (usually denoted \(\alpha\) as in Hardin and Hilbe (2007)): such that the variance equals \(\mu(1+\alpha)\).
returns a dispersion parameter (usually denoted \(\theta\) or \(k\)); in contrast to most other families, larger \(\theta\) corresponds to a lower variance which is \(\mu(1+\mu/\theta)\).
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)
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):
This family uses the same parameterization (governing
the Beta distribution that underlies the binomial probabilities) as beta.
returns the index of dispersion \(\phi^2\), where the variance is \(\mu\phi^2\) (Consul & Famoye 1992)
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 underdispersed when \(1/\nu <1\) and approximately overdispersed when \(1/\nu >1\). In this implementation, \(\mu\) is exactly equal to the mean (Huang 2017), which differs from the COMPoissonReg package (Sellers & Lotze 2015).
returns the value of \(\phi\),
where the variance is \(\phi\mu^p\).
The value of \(p\) can be extracted using family_params
The most commonly used GLM families
(binomial, poisson) have fixed dispersion parameters which are
internally ignored.
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