statmod (version 1.4.32)

remlscoregamma: Approximate REML for gamma regression with structured dispersion

Description

Estimates structured dispersion effects using approximate REML with gamma responses.

Usage

remlscoregamma(y,X,Z,mlink="log",dlink="log",trace=FALSE,tol=1e-5,maxit=40)

Arguments

y

numeric vector of responses

X

design matrix for predicting the mean

Z

design matrix for predicting the variance

mlink

character string or numeric value specifying link for mean model

dlink

character string or numeric value specifying link for dispersion model

trace

Logical variable. If true then output diagnostic information at each iteration.

tol

Convergence tolerance

maxit

Maximum number of iterations allowed

Value

List with the following components:

beta

Vector of regression coefficients for predicting the mean

se.beta

<Standard errors for beta

gamma

Vector of regression coefficients for predicting the variance

se.gam

Standard errors for gamma

mu

Estimated means

phi

Estimated dispersions

deviance

Minus twice the REML log-likelihood

h

Leverages

Details

Write ${\mu }_i=E(y_i)$ for the expectation of the $i$th response and $s_i=\text{var}(y_i)$. We assume the heteroscedastic regression model $${\mu }_i=\text{bold}{x}_i^T\text{bold}\beta$$ $$\log ({\sigma }_i^2)=\text{bold}{z}_i^T\text{bold}\gamma ,$$ where $\text{bold}{x}_i$ and $\text{bold}{z}_i$ are vectors of covariates, and $\text{bold}\beta$ and $\text{bold}\gamma$ are vectors of regression coefficients affecting the mean and variance respectively.

Parameters are estimated by maximizing the REML likelihood using REML scoring as described in Smyth and Verbyla (2001). See also Smyth and Verbyla (1999a,b).

References

Smyth, G. K., and Verbyla, A. P. (1999a). Adjusted likelihood methods for modelling dispersion in generalized linear models. Environmetrics 10, 695-709. http://www.statsci.org/smyth/pubs/ties98tr.html

Smyth, G. K., and Verbyla, A. P. (1999b). Double generalized linear models: approximate REML and diagnostics. In Statistical Modelling: Proceedings of the 14th International Workshop on Statistical Modelling, Graz, Austria, July 19-23, 1999, H. Friedl, A. Berghold, G. Kauermann (eds.), Technical University, Graz, Austria, pages 66-80. http://www.statsci.org/smyth/pubs/iwsm99-Preprint.pdf

Smyth, GK, and Verbyla, AP (2009). Leverage adjustments for dispersion modelling in generalized nonlinear models. Australian and New Zealand Journal of Statistics 51, 433-448.

Examples

# NOT RUN {
data(welding)
attach(welding)
y <- Strength
X <- cbind(1,(Drying+1)/2,(Material+1)/2)
colnames(X) <- c("1","B","C")
Z <- cbind(1,(Material+1)/2,(Method+1)/2,(Preheating+1)/2)
colnames(Z) <- c("1","C","H","I")
out <- remlscoregamma(y,X,Z)
# }