# How to analyze multiple treatments and measurements?

I've been asked to analyze data from an experiment measuring blood hormone levels from rats after a surgery. Hormone measurements were taken 10 days after surgery and 20 days after surgery. To make it more complicated, some of the rats in the surgery group were also subjected to a modified diet. So, there are three groups of rats (sham surgery + regular diet, surgery + regular diet, surgery + modified diet), and hormone measurements were taken at two time points.

What is the correct way to analyze the difference in hormone levels between these groups of rats, both within each time point and between time points? I'm having trouble figuring out the correct ANOVA and/or other method to use. I'm using R.

• Were there no baseline measurements taken in the rates before surgery? Do you only have 2 measurements at most taken on each rat? – StatsStudent Jan 27 '16 at 19:09
• That's correct. No baseline measurements, and only 2 measurements per rat. I think it's not the best experimental design, but the person who did the experiment left the lab before I got here, so I'm just trying to figure out how to make sense of it all! – ricompute Jan 27 '16 at 23:53

There are a few ways to approach this. In the simplest case (and the one I'd use and recommend), you can use a simple paired t-test with an adjustment for modified diet to compare the the differences in the measurements between the two time points. So, you will create a new variable for each of the $n$ subjects, called $y_{diff}\equiv y_{20 day}-y_{10 day}.$ Then you will use regular OLS regression to model:
$y_{diff_i}=SurgeryX_{1i} + DietX_{2i} +\epsilon_i$
where $i=1, 2, . . . n$ and $X_{1i} = 1$ if the subject received the normal surgery, 0 otherwise; and $X_{2i} = 1$ is the subject received the modified diet; 0 otherwise. You can then use standard regression analysis to analyze the results -- since you've already taken the correlation due to the repeated measurements on each subject into account, there is no need to use any fancy machinery.
• Thanks so much for your answer. Maybe I'm overthinking this, because we're reaching the boundaries of my statistics knowledge here, but the part that I'm still having trouble wrapping my brain around is the fact that the modified diet only happens in the rats w/ surgery, not the sham rats. Does that affect the interaction term in the model you gave? When I try this with lm() in R, it returns NA for the interaction coefficient and says that 1 coefficient is not defined because of singularities. Is this a problem I need to worry about? – ricompute Jan 27 '16 at 23:52
• Does it make sense to analyze the difference between days 10 and 20, at all? Is it then possible that any difference between those days is just noise and says nothing about the treatments? A change from baseline (usually with baseline also in the model as a covariate) makes sense, but why look at $y_\text{diff}$? I would have thought a model for each timepoint or a joint model across timepoints (taking into account the correlation) would make more sense. I guess the other issue is missing data, which there may be (e.g. a MMRM would implictily impute it under MAR). – Björn Jan 28 '16 at 7:33