# repeated measures factorial design

I have learned about repeated measures ANOVA but is it possible to have a repeated measures factorial design? So if you have factor A with 2 levels and factor B with 2 levels where each combination has 2 units and each unit is measured three times. Is there another name for this design? Would there be a total of 23 degrees of freedom?

• Yes, is the short answer. I'm not sure I understand the design though. Factor A has two levels, and factor B has two levels means we have 4 measures for each unit (person): A1, A2, B1, B2. What does each unit being measured 3 times refer to? – Jeremy Miles Apr 2 '13 at 20:54
• There are 4 possible treatment combinations. 2 subjects are assigned to each treatment combination. Each subject in each combination is measured at three different time periods. – phil12 Apr 2 '13 at 21:01
• What do you do, or want to do with those three measures. Do you want to include time as a factor, or do you just average them? – Jeremy Miles Apr 2 '13 at 22:15
• If I averaged them the design would be plain factorial design. I would like to see the trend in time. – phil12 Apr 2 '13 at 22:30
• I think I see. This is sometimes called a mixed-anova, because some of your factors are between individuals and some are within. So you have a 2 (factor A) x 2 (factor B) x 3 (time) design, where A and B are between subjects, and time is within. So the short answer is definitely yes. If you want to treat time as a trend, I think you're going to need a multilevel model, but your sample is small and that might lead to problems. What software do you use? – Jeremy Miles Apr 2 '13 at 23:14

Yes, it's possible, but it's hard to get a time trend factor, it might be easier as a multilevel model. You can do this with SAS proc mixed:

proc mixed data = mydata;
class  unit A B;
model outcome = A B time ;
repeated /subject = unit type = cs rcorr;
run;


The data should be in long format, so outcome is your outcome variable, and unit identifies the unit - each unit will have three rows in the dataset.

You might want to add A*B to the model line (but you're going to be close to running out of degrees of freedom).

You could also treat time as categorical by adding it to the class line.

Sometimes I like to play with a simpler model, to test that they really are equivalent:

proc mixed data = mydata;
class  unit time;
model outcome =  time ;
repeated /subject = unit type = cs rcorr;
run;


This model is just a repeated measures anova, with time as the only factor. You should get the same (or very nearly the same) answer doing it both ways.

• Would the model with time be $Y_{ijkl} = \mu + A_{i} +B_{j} + A_{i}B_{j}+ T_{k} + E_{ijkl}$? – phil12 Apr 3 '13 at 0:59