Discussion Board for Issues Relating to the Analysis of Serial, Longitudinal, and Repeated Measures Data

Karl Knoblick (karlknoblich@yahoo.de) asked several good questions on the R-help list on 17May07

I have two groups (placebo/verum), every subject is measured at 5 times, the first time t0 is the baseline measurement, t1 to t4 are the measurements after applying the medication (placebo or verum). The question is, if there is a significant difference in the two groups and how large the differnce is (95% confidence intervals).

Let me give sample data

# Data
ID <- factor(rep(1:50,each=5)) # 50 subjects
GROUP <- factor(c(rep("Verum", 115), rep("Placebo", 135)))
TIME <- factor(rep(paste("t",0:4,sep=""), 50))
set.seed(1234)
Y <- rnorm(250)
# to have an effect:
Y[GROUP=="Verum" & TIME=="t1"] <- Y[GROUP=="Verum" & TIME=="t1"] + 0.6
Y[GROUP=="Verum" & TIME=="t2"] <- Y[GROUP=="Verum" & TIME=="t2"] + 0.3
Y[GROUP=="Verum" & TIME=="t3"] <- Y[GROUP=="Verum" & TIME=="t3"] + 0.9
Y[GROUP=="Verum" & TIME=="t4"] <- Y[GROUP=="Verum" & TIME=="t4"] + 0.9
DF <- data.frame(Y, ID, GROUP, TIME)

1. Comparing the endpoint t4 between the groups (t-test), ignoring baseline
2. Comparing the difference t4 minus t0 between the two groups (t-test)
3. Comparing the endpoint t4 with t0 as a covariate between the groups (ANOVA - how can this model be calculated in R?)
4. Taking a summary score (im not sure but this may be a suggestion of Altman) istead of t4
5. ANOVA (repeated measurements) times t0 to t4, group placebo/verum), subject as random factor - interested in interaction times*groups (How to do this in R?)
6. As 5) but times t1 to t4, ignoring baseline (How to do this in R?)
7. As 6) but additional covariate baseline t0 (How to do this in R?)

FrankHarrell's response:

1. Don't even consider t-tests ignoring baseline.
2. Comparing differences from baseline over the two groups is not optimal.
3. Using t0 as a covariate is the way to go. A question is whether to just use t4. Generally this is not optimum.
4. It's not obvious that random effects are needed if you take the correlation into account in a good way. Generalized least squares using for example an AR1 correlation structure (and there are many others) is something I often prefer. A detailed case study with R code (similar to your situation) is in FrankHarrellGLS?. This includes details about why t0 is best to consider as a covariate. One reason is that the t0 effect may not be linear.
5. If you want to focus on t4 it is easy to specify a contrast (after fitting is completed) that tests t4. If time is continuous this contrast would involve predicted values at the 4th time, otherwise testing single parameters.
6. Like mixed effects models, GLS is fairly robust to non-random dropouts and allows you to use all available data without the need to impute missing response values.