R: Approximate REML for Gamma Regression with Structured...

remlscoregamma {statmod} | R Documentation

## 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 value. If `TRUE` then diagnostic information is output at each iteration.
`tol` | convergence tolerance.
`maxit` | maximum number of iterations allowed.

### Details

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.

### Value

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.

### 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, 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.

### Examples

    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)

[Package statmod version 1.4.36 Index]