Donald Burrill wrote: > It may help to bear in mind that the PREDICTED value is a function > ONLY of the INDEPENDENT variables. (Many people seem to lose sight > of this fact, or not to notice it in the first place: perhaps a > confusion arising from labelling the predicted value "Y-hat", which > might lead the naive to suppose that it was somehow a function of the > dependent variable; of course, it is not, whatever is left of the > dependent variable being contained in the residuals.)
This is not strictly true, since the predicted values are functions of the regression coeffs, which are functions of the observations as well as of the independent variables. Of course the functional dependence of predicted values on individual observations is not strong, but strong enough that, for example, the sample variance of the residuals is usually adjusted to take account of the number of regression coefficients estimated. > > It would be possible to argue further that the only information of > interest (for these diagnostic purpopses -- heteroscedasticity and > all) about the predictors is contained in the fitted function of them > that is the predicted value. One is not so much interested in > observing, say, heteroscedasticity of the residuals w.r.t. X_1, as in > whether any such heteroscedasticity survives (or persists?) into the > predicted-value variable. But if "heteroscedasticity of the residuals w.r.t. X_1" were found, the regression model could be adjusted to take account of this, for example by weighting using this specific variable. David Jones . . ================================================================= Instructions for joining and leaving this list, remarks about the problem of INAPPROPRIATE MESSAGES, and archives are available at: . http://jse.stat.ncsu.edu/ . =================================================================
