What is "Adjusted Resonse"

I am working with a multiple linear regression.

In estimating the effect sizes (mdl.plotEffects) Matlab uses the "getAdjustedResponse" function. I have read the explanation of the adjusted response (and tried to work my way through the code) but neither has been very helpful in explaining precisely what the adjusted response actually does. To me, the wording is quite confusing:

" The adjusted response plot shows the fitted response as a function of var, with the other predictors averaged out by averaging the fitted values over the data used in the fit. Adjusted data points are computed by adding the residual to the adjusted fitted value for each observation."

Huh???

I understand what an added variable plot is, where you regress the dependent against all but one independent variable ( var), then also regress that variable ( var) against the other independent variables. Then plot the resultant " leave one out regression" against the predicted of var. However, this doesn't seem to be exactly what's happening in getAdjustedResponse.

I would greatly appreciate it if someone could give a better explanation of what the adjusted response actually is.

Answer 1

[Tom Lane]

The adjusted response function isn't a diagnostic plot like an added variable plot, where you try to investigate or isolate the effect of a single predictor or term. Instead, it's a summary plot intended to give an idea of the response as a function of one predictor, averaged over the others.

Suppose you have predictors X1, X2, and X3. Suppose the fitted model is a function of those predictors:

   Yhat = f(X1, X2, X3)

Now suppose you want to look at this as a function of X2. You have N data points all together. Then you could compute

   f(x) = average over i=1:N of      f(X1(i), x, X3(i))

So you are computing the fitted values at X2=x, averaged over all of the sets of observed values of the other predictors X1 and X3. Notice that the function f() could include main effects like X1, squared terms like X1^2, and interaction effects like X2*X3. That's different from an added variable plot where the term is the focus. Here it's the predictor variable, which could contribute to multiple terms.

This was proposed by Bill DuMouchel on page 127 of "Computer Science and Statistics: Proceedings of the 20th Symposium on the Interface" in 1988.

[Michael Schwartz]

Hi Tom.

Thank you very much for taking the time to answer my question. While I generally follow what you have written, I am a bit confused with regards to two things:

1) How is this different from what you would get from simply fitting the bivariate/marginal distribution yhat = f(X2)? I can see from the results Matlab gives that it is different, but following what you wrote (which mostly parallels the doc on plotAdjusted), it seems that it simply collapses the other dimensions, which, to my mind would just recover the bivariate regression.

2) The doc file states " Adjusted data points are computed by adding the residual to the adjusted fitted value for each observation." I don't see where, in your description, residuals play any role.

Again, thanks for taking the time to explain. I'll also have a look at the DuMouchel reference if I can get a hold of it.