# Is this really suited for 2-way ANOVA?

My dataset is a biological analysis of a disease which takes 3 months to develop, and we're looking at treatment of that disease. The data is gene expression data and we want to identify changes in gene expression over time. This resembles a 2x2 design in which I have equal sized groups as:

          trtmnt     no-trtmnt
1 month     10           10
3 month     10           10


After thinking about it, I'm not sure if this is really valid for a 2-way ANOVA. The disease develops over the period of 3 months, so the 1 month subjects should look the same regardless of treatment. There are multiple ways to look at this:

1) I could compare trtmnt and no-trtmnt at the 3-month stage. Simple two-group analysis. 2) I could look at 1 month vs 3 month for trtmnt, and then 1 month vs 3 month for no-trtmnt. This would be 2, two-group analyses. But then I would have to compare the two different analyses to look for gene change in common or different. This is where it starts to resemble a 2-way ANOVA, but not really.

I appreciate any thoughts.

EDIT: I should have pointed out that these are animal studies, so we have 10 animals in each group. Rather than taking samples from each animal twice - once at 1 month and once at 3 months - we have 4 different groups, all of which were used for sampling. I hope that makes sense.

You're right, 2-way ANOVA is not appropriate, but for a different reason: The gene expressions from the same subject are dependent. The (expected) identical distribution between both expression levels among the healthy subjects is not the reason.

You should take the difference of the expression levels (baseline - 3 months) and compare them, similar to your suggestion (1), but for the differences. The baseline values are informative. They should not be discarded. (In fact, discarding any measurements is alway suspicious.)

EDIT due to @KirkDCO's edit: If the four groups are independent, it is a 2-way ANOVA: Hypothesis is that there is no interaction.

• Thanks for the comment. I should have pointed out that these are animal studies, so we have 10 animals in each group. Rather than taking samples from each animal twice, we have 4 different groups, all of which were used for sampling. I hope that makes sense. Commented Jan 16, 2014 at 21:34
• Both answers are great and help me with other questions in which we do have matched samples. Commented Jan 17, 2014 at 0:23

Doing a randomized-block-factorial ANOVA design (aka., repeated measures ANOVA with two conditions) will allow you to look at within-person change under both treatment and no-treatment conditions and to quantify the amount of that change very cleanly. This is sometimes known as a "mixed" design because it combines between-subjects and within-subjects variables and it deals with the non-independence of error terms.

You would hypothesize that in the 1-month condition, there should be no difference between treatment and no-treatment, but that the difference exacerbates (interacts) with time. Simple effects (or even better, treatment-by-contrast interactions) tests will allow you to do this nicely in your repeated-measures design. Thus, you would hypothesize (1) a main effect of treatment vs. control, (2) a main effect of time, and (3) a time-by-treatment interaction. This is completely appropriate so long as you use a block (rather than a pure factorial) ANOVA design.

Another approach (my favored) is to use a hierarchical linear (or nonlinear) model. If you can get baseline measures, then take measures at several timepoints along the way (time 0, 1 month, 2-month, 3 month), you can generate a model that essentially looks at the trajectories of the disease for people who have (or don't have) your treatment. But I know that is considerably more money, obviously. One advantage to the HLM approach is that you can more precisely estimate the shape of the trajectory.

Other approaches that might suffice include latent growth models, but you are starting to look at fairly complex approaches that require large sample sizes when you go down that road. For what you want, if you don't want to add additional timepoints, I would recommend a randomized-block (a.k.a., repeated-measures) ANOVA. Note that there are additional assumptions for this ANOVA design that must be considered.

A 2x2 ANOVA is fine here. You are looking for an interaction. That means that the treatment effect has changed across months.

Do not test simple effects because it's a pointless increase in tests that would increase Type I error or cause corrections and increase Type II. The interaction tells you what you need to know, whether the effect of treatment varies over months.

(If you had records of outcome measures at 1 month for your 3 month trt group then you could do a mixed ANOVA on that group. That would be more powerful. Your N here is very small.)