Prediction interval for the one-way random effect ANOVA

Question

I don't find a textbook presenting prediction intervals for mixed models. Moreover I'm only looking for a prediction interval for the balanced one-way random effect ANOVA model. Do you know a formula for this interval ?

Answer 1

I have now find some answers in Lin & Liao's paper.

I have applying method 2 (the one using Satterwaithe degrees of freedom) for the one-way random effect ANOVA. Below is the R code. I have checked with some simulations that it works quite well.

ranovapred <- function(y, group, conf=0.95){
    group <- factor(group)
    means <- aggregate(y~group, FUN=mean)$y  # groups means
    I <- length(levels(group))
    J <- length(y)/I
    sizes <- table(group) # groups sizes
    if(!all(as.numeric(sizes)==J)){ stop("balanced only!") }
    ssw <- crossprod(y-rep(means, times=sizes))  # within sum of squares
    ssb <- J*crossprod(means-mean(y)) # between sum of squares
    a <- (1/J*(1+1/I))/(I-1)
    b <- 1/I/J 
    v <- a*ssb+b*ssw # estimates the variance of (Ynew-Ybar)
    nu <- v^2/((a*ssb)^2/(I-1)+(b*ssw)^2/I/(J-1)) # Satterthwaite degrees of freedom
    alpha <- 1-conf
    bounds <- mean(y) + c(-1,1)*sqrt(v)*qt(1-alpha/2,nu)
return(bounds)
}

# test using simulated data
I <- 7 # number of groups
J <- 10 # number of replicates per group
mu <- 0 # overall mean
sigmab <- 1 # between standard deviation
sigmaw <- 1 # within standard deviation
group <- LETTERS[gl(I,J)] # factor levels
x <- c(sapply(rnorm(I,mu,sigmab), function(mui) rnorm(J, mui, sigmaw))) # responses
ranovapred(y=x, group=group)

Enjoy.

EDIT: By the way I have compared with SAS prediction intervals... these ones are totally wrong. It's frightening that some SAS customers build engines for aircraft ;-)