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Spring 2015 Statistics 151a (Linear Models) : Lecture TwentyAditya Guntuboyina09 April 20151Generalized Linear ModelsWe have so far studied linear models. We havenobservations on a response variabley1, . . . , ynand oneach ofpexplanatory variablesxijfori= 1, . . . , nandj= 1, . . . , p.The linear model that we have seen modelsμi:=Eyi=β0+β1xi1+· · ·+βpxip.What this model implies is that when there is a unit increase in the explanatory variablexj, the meanof the response variable changes by the amountβj. This may not always be a reasonable assumption.For example, if the responseyiis a binary variable, then its meanμiis a probability which is alwaysconstrained to stay between 0 and 1. Therefore, the amount by whichμichanges per unit change inxjwould now depend on the value ofμi(for example, the change whenμi= 0.9 may not be the same aswhenμi= 0.5). Therefore, modelingμias a linear combination ofx1, . . . , xpmay not be the best ideaalways.A more general model might beg(μi) :=β0+β1xi1+· · ·+βpxip(1)for a functiongthat is not necessarily the identity function.Another feature of the linear model that people do not always like is that some aspects of the theory aretied to the normal distribution. Indeed, most of the results on hypothesis testing rely on the assumptionof normality. The assumption thaty1, . . . , ynare normal may not always be appropriate. Examples arisewheny1, . . . , ynare binary or when they represent counts. It might therefore be nice to generalize thetheory of linear models to include these other distributional assumptions for the response values.