Here are some data:
Run y
1 1 4.970
2 1 5.040
3 1 5.060
4 1 4.975
5 1 4.990
6 1 5.045
7 1 5.005
8 1 5.000
9 1 5.075
10 1 5.080
11 2 4.665
12 2 4.885
13 2 4.840
14 2 4.925
15 2 5.235
16 2 4.970
17 2 5.020
18 2 4.905
19 2 4.890
20 2 5.150
This dataset is called dat
. Then I fit the model introduced here : a common mean for $y$ but a variance depending on the run (the data are much more dispersed in run 2). The SAS code is:
PROC MIXED DATA=dat;
CLASS Run;
MODEL y= / solution CL outp=predicted ;
REPEATED / group=Run;
RUN;QUIT;
I use PROC MIXED
whereas the model has no random effect, because I don't know another SAS procedure to handle heteroscedasticity. For those more familiar with R the equivalent R code is:
library(nlme)
gls(y~1, data=dat, weights=varIdent(form= ~1|Run)))
As you have seen I have added the option outp=predicted
in the SAS code in order to get prediction intervals for individual values. The result is strange: each observation has the same prediction interval, and this one coincides with the confidence interval about the mean. What's going on ?
EDIT: (later - also change the title). How could we define a prediction interval for this model ? We could imagine a new observation in run 1 or in run 2, but if we consider a third run we would need to assign a variance to the new observation, and there's no natural choice for that. So SAS should return nothing or it should return something derived from a particular convention which gives a sense to the "new observation".