In logistic regression, the regression coefficients ( β 0 ^ , β 1 ^ )...

Sonia Ayers

Sonia Ayers

Answered

2022-07-09

In logistic regression, the regression coefficients ( β 0 ^ , β 1 ^ ) are calculated via the general method of maximum likelihood. For a simple logistic regression, the maximum likelihood function is given as
( β 0 , β 1 ) = i : y i = 1 p ( x i ) i : y i = 0 ( 1 p ( x i ) ) .
What is the maximum likelihood function for 2 predictors? Or 3 predictors?

Answer & Explanation

Darrell Valencia

Darrell Valencia

Expert

2022-07-10Added 10 answers

The same for any finite number of predictors, i.e.,
p ( y i ; x i ) = 1 1 + exp { x i β } ,
where x i = ( 1 , x 1 i , . . . , x p i ) and β = ( β 0 , . . . , β p ). Thus the likelihood function is Binomial
L ( β ) = i = 1 m p ( y i ; x i ) y i ( 1 p ( y i ; x i ) ) 1 y i .
E.g., if you have two predictors, i.e., x 1 and x 2 , and total of n observations, i.e., { ( y i , x 1 i , x 2 i } then the likelihood function is given by
i = 1 n p ( y i ; x 1 i , x 2 i ) y i ( 1 p ( y i ; x 1 i , x 2 i ) ) 1 y i ,
where
p ( y i ; x 1 i , x 2 i ) = 1 1 + exp { β 0 + β 1 x 1 i + β 2 x 2 i } .

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