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Additional info for Practical Regression and Anova using R
We can decompose the usual model as follows: y Xβ ε ☎ ☎ Xβ Zγ δ where Z are unincluded predictors and δ is measurement error in the response. We can assume that Eε ☎ 0 without any loss of generality, because if Eε ☎ c, we could simply redefine β 0 as β0 c and the error would again have expectation zero. This is another reason why it is generally unwise to remove the intercept term from the model since it acts as a sink for the mean effect of unincluded variables. So we see that ε incorporates both measurement error and the effect of other variables.
Suppose you are studying the behavior of alcoholics and advertise in the media for study subjects. It seems very likely that such a sample will be biased perhaps in unpredictable ways. In cases such as this, a sample of convenience is clearly biased in which case conclusions must be limited to the sample itself. This situation reduces to the next case, where the sample is the population. Sometimes, researchers may try to select a “representative” sample by hand. Quite apart from the obvious difficulties in doing this, the logic behind the statistical inference depends on the sample being random.
Models that derive directly from physical theory are relatively uncommon so that usually the linear model can only be regarded as an approximation to a reality which is very complex. Most statistical theory rests on the assumption that the model is correct. In practice, the best one can hope for is that the model is a fair representation of reality. A model can be no more than a good portrait. All models are wrong but some are useful. George Box is only a slight exaggeration. Einstein said So far as theories of mathematics are about reality; they are not certain; so far as they are certain, they are not about reality.