# remlscore

##### REML for Heteroscedastic Regression

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

- Keywords
- regression

##### Usage

`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

##### Details

Write \(\mu_i=E(y_i)\) for the expectation of the \(i\)th response and \(s_i=\var(y_i)\). 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).

##### Value

List with the following components:

vector of regression coefficients for predicting the mean

vector of standard errors for beta

vector of regression coefficients for predicting the variance

vector of standard errors for gamma

estimated means

estimated variances

minus twice the REML log-likelihood

numeric vector of leverages

estimated covariance matrix for beta

estimated covarate matrix for gamma

number of iterations used

##### References

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

##### Examples

```
# NOT RUN {
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)
# }
```

*Documentation reproduced from package statmod, version 1.4.32, License: GPL-2 | GPL-3*