REML for Heteroscedastic Regression

Fits a heteroscedastic regression model using residual maximum likelihood (REML).

remlscore(y, X, Z, 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

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

vector of standard errors for beta

gamma

vector of regression coefficients for predicting the variance

se.gam

vector of standard errors for gamma

mu

estimated means

phi

estimated variances

deviance

minus twice the REML log-likelihood

h

numeric vector of leverages

cov.beta

estimated covariance matrix for beta

cov.gam

estimated covarate matrix for gamma

iter

number of iterations used

Details

Write \(\mu_i=E(y_i)\) and \(\sigma^2_i=\mbox{var}(y_i)\) for the expectation and variance of the \(i\)th response. We assume the heteroscedastic regression model $$\mu_i=\bold{x}_i^T\bold{\beta}$$ $$\log(\sigma^2_i)=\bold{z}_i^T\bold{\gamma},$$ where \(\bold{x}_i\) and \(\bold{z}_i\) are vectors of covariates, and \(\bold{\beta}\) and \(\bold{\gamma}\) 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 (2002).

References

Smyth, G. K. (2002). An efficient algorithm for REML in heteroscedastic regression. Journal of Computational and Graphical Statistics 11, 836-847. doi:10.1198/106186002871

Author

Gordon Smyth

Examples

data(welding)
attach(welding)
y <- Strength
# Reproduce results from Table 1 of Smyth (2002)
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 <- remlscore(y,X,Z)
cbind(Estimate=out$gamma,SE=out$se.gam)
#>                         SE
#> [1,] -3.15886017 0.8313270
#> [2,] -2.73542576 0.8224828
#> [3,] -0.08588713 0.8351308
#> [4,]  3.33238821 0.8250499