lec21_slides.pdf - Generalized Linear Models 1 GLMs I We have so far studied linear models We have n observations on a response variable y1 yn and on

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Generalized Linear Models 1 April 18, 2019
GLMs I 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 .
GLMs I 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 . I The linear model that we have seen models μ i := E y i = β 0 + β 1 x i 1 + · · · + β p x ip .
GLMs I 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 . I The linear model that we have seen models μ i := E y i = β 0 + β 1 x i 1 + · · · + β p x ip . I 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 .
GLMs I 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 . I The linear model that we have seen models μ i := E y i = β 0 + β 1 x i 1 + · · · + β p x ip . I 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 . I This may not always be a reasonable assumption.
GLMs I 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.
GLMs I 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. I 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).
GLMs I 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. I 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). I Therefore, modeling μ i as a linear combination of x 1 , . . . , x p may not be the best idea always.
GLMs I 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. I 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). I Therefore, modeling μ i as a linear combination of x 1 , . . . , x p may not be the best idea always. I 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.
I 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.
I Another feature of the linear model that people do not