On 29 Mar 2004 at 14:59, Rich Ulrich wrote: > On 29 Mar 2004 09:47:30 -0800, [EMAIL PROTECTED] (Paul R Swank) > wrote: > > > The size of the error in prediction is smallest at the mean of the > > predictors and increases as the values for the predictors move > > toward the extremes. One woul need to take this into account. Any > > good regression book > > Now we can talk about two different 'predictions' -- Paul S. > is describing the nature of actual predictions for new values, > where extreme predictions are increasingly inaccurate. > > Up to now, I have assumed that the 'error in prediction' was > a reference to the error of the fitted values, concerning only > the sample in hand. Here, the errors are supposed to be > (by assumption) homogeneous, across the range of the > prediction;
But this is just plain wrong! The error term in the theoretical model assumes homogeneous errors, but this does not imply that the residuals have homogeneous variance! The leverage comes in, the variance of residual i is (from memory) sigma^2*(1-h_i), (or something proportional to this), where h_i is the leverage of observation i. So in the extreme case, leverage 1, the variance is actually zero, and the predicted value will always equal the observation. and that is one of the assumptions that you can > check by looking at the right plots of the residuals. AND using the right type of residual, some standardized version, typically divided by sqrt(1-h_i) to stabilize the variance Kjetil Halvorsen > > > > (like Draper and Smith) will have the formula, which is difficult to > > represent in text format. Var(mean Y hat) = X(0)'(X'X)inv X(0)sigma > > hat squared: where X(0) represents the vector of X's you are > > predicting from. Assuming normality, then a t could be determined > > that would give an approximate idea of how far the observation was > > from the prediction. > > -- > Rich Ulrich, [EMAIL PROTECTED] > http://www.pitt.edu/~wpilib/index.html > - I need a new job, after March 31. Openings? - > . > . > ================================================================= > Instructions for joining and leaving this list, remarks about the > problem of INAPPROPRIATE MESSAGES, and archives are available at: . > http://jse.stat.ncsu.edu/ . > ================================================================= . . ================================================================= Instructions for joining and leaving this list, remarks about the problem of INAPPROPRIATE MESSAGES, and archives are available at: . http://jse.stat.ncsu.edu/ . =================================================================
