I want to predict using residuals.
Let's say we used regression of J against H (assume all positive values), giving us residuals (Jres) that express, in theory, J, with the effect of H removed. We then regress those residuals (Jres) against another variable (K). We get a slope and intercept (let's say it is significant). Now we want to predict an unknown Jres from a known K using our new slope and intercept.
The problem - since residuals are centered on 0, the predicted Jres has to be adjusted to be in the original scale of J. What do we add to the Y-intercept so that the new predicted can be in a scale comparable to the original J?
In other words, let's say you have a population that smokes and lives near a coal-powered factory, and you see cancer cells in their blood. You want to regress away the effect of smoking (from the cancer data) and then use the distance from the factory in a new regression to see if distance predicts tumor cells independent of the smoking. If so, how far away would you have to be to reduce the cancer cells to 0?
It looks like this, using residuals against K (all data are hypothetical):
