How do I visualize autocorrelation within subjects? My response is $y_{ij}$ where $j$ denotes a patient, and $i$ denotes a time.
If I just use the residuals of my regression model to do an ACF plot, then how will it know that I am only interested in the correlation within each subject?
 A: Simple. Calculate ACFs for each patient separately, by using that patient's residuals only. Then plot these ACF values against lags, as for a "normal" ACFs plot, in a scatterplot.
I personally like to overlay this with a beanplot, to show an idea of an estimated kernel density, but this may not be necessary.
Below is an example with 10 patients and 20 data points each, where I simulate residuals obs from an AR(2) process.
library(forecast)
library(beanplot)

n.time <- 20
n.patients <- 10

model <- Arima(WWWusage,order=c(2,0,0)) # just to get something to simulate from

set.seed(1)
obs <- replicate(n.patients,simulate(model,nsim=n.time))
acfs <- t(apply(obs,2,function(xx)acf(xx,plot=FALSE)$acf))

beanplot(data.frame(acfs[,-1]),what=c(0,1,0,0),col="grey",border=NA,at=2:ncol(acfs),
    ylim=range(acfs),xlim=c(1,ncol(acfs)),main="ACFs",xlab="Lag",ylab="",xaxt="n")
for ( ii in 1:ncol(acfs) ) points(runif(n.patients,ii-.3,ii+.3),acfs[,ii],pch=19)
axis(1,at=1:ncol(acfs),labels=0:(ncol(acfs)-1))


You could even tie together the ACFs from each patient by using the same symbol, or the same color, or by connecting them with lines. But I'd only do that for a small number of patients, e.g., five. In which case I wouldn't add the beans, either.
