Approximate REML for Gamma Regression with Structured Dispersion
Estimates structured dispersion effects using approximate REML with gamma responses.
remlscoregamma(y, X, Z, mlink="log", dlink="log", trace=FALSE, tol=1e-5, maxit=40)
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 value. If |
tol |
convergence tolerance. |
maxit |
maximum number of iterations allowed. |
This function fits a double generalized linear model (glm) with gamma responses.
As for ordinary gamma glms, a link-linear model is assumed for the expected values.
The double glm assumes a separate link-linear model for the dispersions as well.
The responses y are assumed to follow a gamma generalized linear model with link mlink and design matrix
X.
The dispersions follow a link-linear model with link dlink and design matrix Z.
Write y_i for the ith response. The y_i are assumed to be independent and gamma distributed with E(y_i) = μ_i and var(y_i)=φ_iμ_i^2. The link-linear model for the means can be written as
g(μ)=Xβ
where g is the mean-link function defined by mlink and μ is the vector of means.
The dispersion link-linear model can be written as
h(φ)=Zγ
where h is the dispersion-link function defined by dlink and φ is the vector of dispersions.
The parameters γ are estimated by approximate REML likelihood using an adaption of the algorithm described by Smyth (2002). See also Smyth and Verbyla (1999a,b) and Smyth and Verbyla (2009). Having estimated γ and φ, the β are estimated as usual for a gamma glm.
The estimated values for β, μ, γ and φ are return as beta, mu, gamma and phi respectively.
List with the following components:
beta |
numeric vector of regression coefficients for predicting the mean. |
se.beta |
numeric vector of standard errors for beta. |
gamma |
numeric vector of regression coefficients for predicting the variance. |
se.gam |
numeric vector of standard errors for gamma. |
mu |
numeric vector of estimated means. |
phi |
numeric vector of estimated dispersions. |
deviance |
minus twice the REML log-likelihood. |
h |
numeric vector of leverages. |
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, G. K. (2002). An efficient algorithm for REML in heteroscedastic regression. Journal of Computational and Graphical Statistics 11, 836-847.
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.
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)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.