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 ℓ ( 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?

Question

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  ℓ  (      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?

Darrell Valencia · Accepted Answer

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                    }        .