# Hierarchical regression using residuals

Edit: Perhaps my first asking (see below) was a little too ambitious. The crux of the question is this: in a linear regression setting, is there anything statistically invalid about regressing the residuals of a previous model on new predictors (assuming the observations are IID, plenty of degrees of freedom, etc)? There's a bit more about why one might want to do this below, along with some other possible alternatives to achieving the same aim; but since the above is something I actually did, I'd be happy to start with knowing how wrong it was.

Suppose a standard regression situation, if there is such a thing: IID data, a set of (not necessarily orthogonal) predictors, and the desire to figure out how some of the latter relate to the former. For sake of argument, let's say there are 9 predictors and 90+ independent observations, with the 9 predictors falling into 3 separate groups, based on some theoretical ordering of their relative importance, $y = a_1 + a_2 + a_3 + b_1 + b_2 + b_3 + c_1 + c_2 + c_3$. A standard approach to assessing the impact of the $b$ group after controlling for the $a$ group would be to use hierarchical regression, and compare the fits with an ANOVA; and likewise for the $c$ group.

However, when you do this, the actual parameter estimates you end up looking at are those that come from whichever stage makes the most sense--so, if none of the $c$ group variables were significant (or more precisely, if there was a nonsignificant change in model fit when adding those variables), but some of the $b$ group were, you might consider your "official" estimates for the $b$ variables as those coming from the model $y = a_{1:3} + b_{1:3}$ (adopting a bit of shorthand). The issue is that those $b$ estimates were estimated simultaneously with the $a$ variables, so the $b$ estimates themselves and their individual significances aren't being strictly controlled for $a$, as far as I can tell. In any situation where you want to be fairly conservative in ascribing explanatory power to those $b$ variables, it seems like this is only giving you partial information--that is, you know the $b$ group as a whole is improving the fit, but not much beyond that.

Off the top of my head, I can think of three possible solutions (there are surely more, and some or all of these might be completely idiotic). The first is to run multiple hierarchical models: for instance, maybe $y=a_{1:3}$ followed by each of $y=a_{1:3}+b_i, i\in \{1:3\}$ (or $y=a_{1:3}+b_{i}+b_{j}, i\ne{}j \in \{1:3\}$ followed by $y=a_{1:3}+b_{1:3}$) and drawing conclusions about $b_i$ from the model in which $b_i$ was first added. However, that doesn't solve the fact that the coefficients are all estimated simultaneously, so the $b$ estimates aren't "pure".

Second is to orthogonalize groups of regressors w.r.t. one another: for instance, $y=a_{1:3}+b_{1\perp{}a_{1:3}}+b_{2\perp{}a_{1:3}}+b_{3\perp{}a_{1:3}}$. Now the estimates for the $b_i$s must reflect variance distinct from the $a_i$s, and presumably their significances are correct. For some reason this approach worried me, just because it feels a little voodoo to see your predictors changing around (especially when it comes to interpreting results), but the more I've thought about it, the more it's seemed possibly okay.

Third, and the approach I settled on because of those concerns, is to fit $y=a_{1:3}$, then fit $y-\hat{y}_{a}=b_{1:3}$, and so forth--that is, to fit each subsequent model to the residuals of the previous model, of course always including only the next group of predictors. I can imagine there might be a problem with the degrees of freedom, insofar as each model after the first presumably has less and less residual variance (assuming earlier stages explain at least something), but the degrees of freedom only reflect the current batch of variables. That issue notwithstanding, I have three basic questions: 1) is there anything fundamentally wrong with this procedure? 2) does it address the question we're trying to address? 3) how does it compare to the previous two suggested approaches? Thanks!

## 1 Answer

After a bit more digging, I suppose I'll take a stab at answering this myself. Each of the last two solutions I proposed is one half of a partial regression plot (aka added variable plot). That is, to construct a partial regression plot, the residuals from regressing the response variable against all predictor variables but one is plotted against the residuals from regressing the left-out predictor variable against all others. So actually, neither solution was quite right alone, but together, they make some sense—although, it seems that this is usually used in a much more qualitative way than what I was planning, but that's fine, since this was all basically meant to assuage as-yet-nonexistant reviewer concerns.