# Proper order of variables in unbalanced ANOVA

If you have unequal sample sizes in cells, then the order in which you enter model terms changes your results for sequential or Type I SS. The first variable to enter the model is allocated its unique variance plus the variance shared by the second variable, then the second variable gets only its unique variance.

If you have two predictor variables, say whether you get a drug or placebo and whether you think you get a drug or placebo, and you want to look at their relationship to some symptom, and you want to ignore the effect of knowing whether you got the drug, because all you are really interested in is the relationship between getting the drug and the symptom, then which predictor variable would enter into the model first?

I have been reading answers to questions on this site, and sometimes people are told to enter what is thought to be the most important variable first, but then in ANCOVA situations they are told to enter the covariate first. Is this contradictory advice, or is the covariate really the most important variable?

Additionally, would an ANCOVA always use Type I SS?

Welcome to the site, Jdub, and +1 for a good question. I've addressed this topic in a general way before on CV; you may want to read some of my previous answers on the topic, although it sounds like you have a good sense of the ideas. (I've talked about this in several places, but it's best summarized here, with some complimentary info here. These ideas are applied in a different, concrete, context here.)

I don't think there's a fixed 'right' answer to which terms you should enter first. One line of thinking is to enter the variable that accounts for the most variance in the response first. (I suspect this is what was meant when people told you to enter the "most important" variable first.) The idea is that it plainly has an effect, and you wouldn't want to mistakenly infer that another variable is related to the response, when in fact it only appears to be because it shares variance with another variable that has a strong relationship with the response. In this vein, you would enter terms in descending order.

Another line of thinking concerns the situation with ANCOVA. Traditionally, one of the ideas behind ANCOVA was that you aren't really interested in the effect of the covariate, it's just that you were already pretty sure it was related to the response variable and wanted check to see that the target variable really does have an effect after controlling for the covariate. As this reason makes clear, it follows that would want to enter the covariate first. Note also, that the thinking here parallels that above.

A second traditional rationale for ANCOVA has been to increase your power, by subtracting variability (more correctly SS) that is known to be due to some other factor from the error term. In this case, the SS will come out of the error term no matter where it is entered into the model, but your power is maximally enhanced if you enter the target variable first, and the covariate last.

I think these reasons show that there are different valid orderings are possible. I would urge you to think about the substance of the hypotheses you want to test, and use that to guide the order you end up using. Statistical analyses simply cannot answer questions in a vacuum. You must have relevant background knowledge in some form and must bring it to bear on the analysis, albeit mostly this happens without people being aware of it. (One possible example of how this plays out in a real-world situation could be seen here.) Ultimately, you are making a judgment call regarding where you think those SS belong, and that's something that only you can decide, as it depends on the question you are asking and how you are thinking about it. You simply want to be able to give an account of why you used a particular ordering, that is, why you think the SS should be given to the variables in that manner.