LectureTWENTY151aSpring2015 - Spring 2015 Statistics 151a(Linear Models Lecture Twenty Aditya Guntuboyina 09 April 2015 1 Generalized Linear Models We

LectureTWENTY151aSpring2015 - Spring 2015 Statistics...

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Spring 2015 Statistics 151a (Linear Models) : Lecture Twenty Aditya Guntuboyina 09 April 2015 1 Generalized Linear Models We have so far studied linear models. We have n observations on a response variable y 1 , . . . , y n and on each of p explanatory variables x ij for i = 1 , . . . , n and j = 1 , . . . , p . The linear model that we have seen models μ i := E y i = β 0 + β 1 x i 1 + · · · + β p x ip . What this model implies is that when there is a unit increase in the explanatory variable x j , the mean of the response variable changes by the amount β j . This may not always be a reasonable assumption. For example, if the response y i is a binary variable, then its mean μ i is a probability which is always constrained to stay between 0 and 1. Therefore, the amount by which μ i changes per unit change in x j would now depend on the value of μ i (for example, the change when μ i = 0 . 9 may not be the same as when μ i = 0 . 5). Therefore, modeling μ i as a linear combination of x 1 , . . . , x p may not be the best idea always. A more general model might be g ( μ i ) := β 0 + β 1 x i 1 + · · · + β p x ip (1) for a function g that 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 are tied to the normal distribution. Indeed, most of the results on hypothesis testing rely on the assumption of normality. The assumption that y 1 , . . . , y n are normal may not always be appropriate. Examples arise when y 1 , . . . , y n are binary or when they represent counts. It might therefore be nice to generalize the theory of linear models to include these other distributional assumptions for the response values.

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