gamlss (version 4.2-4)

random: Specify a simple random effect in a GAMLSS Formula

Description

Includes random effect terms in an GAMLSS model. The function is based on the original random() function of Trevor Hastie in the package gam. This version of the function has been modified to allow a "local" maximum likelihood estimation of the smoothing parameter lambda. This method is equivalent to the PQL method of Breslow and Clayton (1993) applied at the local iterations of the algorithm. In fact for a GLM model and a simple random effect it is equivalent to glmmPQL() function in the package MASS see Venables and Ripley (2002). Venables and Ripley (2002) claimed that this iterative method was first introduced by Schall (1991). Note that in order for the "local" maximum likelhood estimation procedure to operate both argument df and lambda has to be NULL.

Usage

random(x, df = NULL, lambda = NULL, start=10)

Arguments

x

a factor

df

the target degrees of freedom

lambda

the smoothing parameter lambda which can be viewed as a shrinkage parameter.

start

starting value for lambda if local Maximul likelihood is used.

Value

• x is returned with class "smooth", with an attribute named "call" which is to be evaluated in the backfitting additive.fit() called by gamlss()

Details

This is a smoother for use with factors in gamlss(). It allows the fitted values for a factor predictor to be shrunk towards the overall mean, where the amount of shrinking depends either on lambda, or on the equivalent degrees of freedom. Similar in spirit to smoothing splines, this fitting method can be justified on Bayesian grounds or by a random effects model.

Note that the behavier of the function is different from the original Hastie function. Here the function behaves as follows: i) if both df and lambda are NULL then the PQL method is used ii) if lambda is not NULL, lambda is used for fitting iii) if lambda is NULL and df is not NULL then df is used for fitting. Since factors are coded by model.matrix() into a set of contrasts, care has been taken to add an appropriate "contrast" attribute to the output of random(). This zero contrast results in a column of zeros in the model matrix, which is aliased with any column and is hence ignored

References

Breslow, N. E. and Clayton, D. G. (1993) Approximate inference in generalized linear mixed models. Journal of the American Statistical Association 88, 9???25.

Chambers, J. M. and Hastie, T. J. (1991). Statistical Models in S, Chapman and Hall, London.

Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape,(with discussion), Appl. Statist., 54, part 3, pp 507-554.

Schall, R. (1991) Estimation in generalized linear models with random effects. Biometrika 78, 719???727.

Stasinopoulos D. M., Rigby R.A. and Akantziliotou C. (2006) Instructions on how to use the GAMLSS package in R. Accompanying documentation in the current GAMLSS help files, (see also http://www.gamlss.org/).

Stasinopoulos D. M. Rigby R.A. (2007) Generalized additive models for location scale and shape (GAMLSS) in R. Journal of Statistical Software, Vol. 23, Issue 7, Dec 2007, http://www.jstatsoft.org/v23/i07.

Venables, W. N. and Ripley, B. D. (2002) Modern Applied Statistics with S. Fourth edition. Springer.

See Also

gamlss, gamlss.random

Examples

data(hodges)
plot(prind~state, data=hodges)
m1<- gamlss(prind~random(state), sigma.fo=~random(state), nu.fo=~random(state), 
             tau.fo=~random(state), family=BCT, data=hodges)
edfAll(m1)
# radnom effect for tau is not needed
m2<- gamlss(prind~random(state), sigma.fo=~random(state), nu.fo=~random(state),  
            family=BCT, data=hodges, start.from=m1)
edfAll(m2)
plot(m2)
#op<-par(mfrow=c(3,1))
#term.plot(m2, se=TRUE)
#term.plot(m2, se=TRUE, what="sigma")
#term.plot(m2, se=TRUE, what="nu")
#par(op)
# the example from Venable and Ripley (2002)
library(MASS)
data(bacteria)
library(nlme)
summary(glmmPQL(y ~ trt + I(week > 2), random = ~ 1 | ID,
                family = binomial, data = bacteria))
s1 <- gamlss(y ~ trt + I(week > 2)+random(ID), family = BI, data = bacteria)
# the esimate of sigma 
sqrt(s1$mu.coefSmo[[1]]$sig2)
# the esimate of random effect  sigma
sqrt(s1$mu.coefSmo[[1]]$tau2)