Given a set of data with many independent variables and a single dependent variable how can I find the "experimental region" of the resulting model? This is the region within which we can make valid predictions and doesn't go outside of the bounds of the data we were provided. Also what is a good way to articulate this region?

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    $\begingroup$ Many people use the convex hull of the independent values for this purpose. It's a stretch to claim that all predictions within it are "valid," but its signal--and defining--property is that every point within the convex hull can be reached by linearly interpolating among the data; no extrapolation is required. But this is by no means the only valid construction of an "experimental region." For instance, if I vary $X_1$ while fixing $X_2$ at $0$ and then vary $X_2$ while fixing $X_1$ at $0$, observing $Y$ all the while, I can regress $Y$ on the $X_i$ but maybe my region is just a cross shape. $\endgroup$
    – whuber
    Aug 16, 2013 at 1:51
  • $\begingroup$ Thanks! Two other questions... Is there a simple way to build one of these? Second, why do so many people answer questions in the comments here instead of in the actual answer area? $\endgroup$ Aug 16, 2013 at 3:25
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    $\begingroup$ @JustinBozonier My perception is that most users only submit their responses as "answers" when they feel that their response is at least a somewhat definitive and direct answer to the question that was asked. If instead a user has a response that amounts to essentially just a few relevant thoughts that occurred to them, but they don't think these thoughts entail a complete, satisfactory answer to the question, they are more likely to submit these responses as "comments." Of course, different people have different standards about what constitutes a complete, satisfactory answer. $\endgroup$ Aug 16, 2013 at 4:07
  • $\begingroup$ Thank @Jake: that sounds like a good characterization. In this case I was seeking a clarification (but ran out of room after explaining why clarification is desirable): Justin, could you make more precise what you mean by "bounds of the data," by "valid" predictions, and by "articulating" a region? $\endgroup$
    – whuber
    Aug 16, 2013 at 13:44
  • $\begingroup$ @whuber Here's a concrete simple example. Say I have the following linear model: E(y) = 2 + 1.12 * X1 ... and the data it was fitted on for X1 ranged from 2 to 100... I can't plug in 101 and use that to predict the response variable. So for this simple problem my experimental region would just be 2 to 100 (I think!). Now how do I do something like this for a multiple regression model? $\endgroup$ Aug 16, 2013 at 14:12


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