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231RegressionDiagnostics - Stat 231 Handout on Regression...

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Stat 231 Handout on Regression Diagnostics There are various kinds of residuals, ways of plotting them and measures of "in°uence" on a regression that are meant to help in the black art of model building. We have already alluded to the fact that under the MLR model, we expect ordinary residuals e i = y i ° b y i to look like mean 0 normal random noise and that standardized or studentized residuals e ° i = e i standard error of e i should like standard normal random noise. Deleted Residuals and the PRESS Statistic There is also the notion of deleted residuals. These are built on the idea that a model should not be terribly sensitive to individual data points used to ±t it, or equivalently that one ought to be able to predict a response even without using that response to ±t the model. Beginning with a particular form for a MLR model and n data points, let b y ( i ) = the value of y i predicted by a model ±t to the other ( n ° 1) data points (note that this is not necessarily b y i ). The i th deleted residual is e ( i ) = y i ° b y ( i ) and the hope that if a model is a good one and not overly sensitive to the exact data vectors used to ±t it, these shouldn²t be ridiculously larger in magnitude than the regular residuals, e i . The "prediction sum of squares" is a single number summary of these PRESS = n X i =1 ( y i ° b y ( i ) ) 2 and one wants small values of this. (Note that PRESS ± SSE , but one hopes that it is not too much larger.) This does not exhaust the ways in which people have suggested using the residual idea. It is possible to invent standardized/Studentized deleted residuals e ° ( i ) = e ( i ) standard error of e ( i ) and there are yet other possibilities.
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