Denoting yi = I(gi = 1), the log-likelihood part of (13) can be written in the more explicit form (14)ℓ(β0,β)=1N∑i=1Nyi·(β0+xiTβ)−log(1+e(β0+xiTβ)), a concave function of the parameters. The Newton algorithm for maximizing the (unpenalized) log-likelihood (14) amounts to iteratively reweighted least squares. Hence if the current estimates of the parameters are (β̃0, β̃), we form a quadratic approximation to the log-likelihood (Taylor expansion about current estimates), which is (15)ℓQ(β0,β)=−12N∑i=1Nwi(zi−β0−xiTβ)2+C(β˜0,β˜)2 where (16)zi=β˜0+xiTβ˜+yi−p˜(xi)p˜(xi)(1−p˜(xi)), (working response) (17)wi=p˜(xi)(1−p˜(xi)),(weights) and p̃(xi) is evaluated at the current parameters. The last term is constant. The Newton update is obtained by minimizing ℓQ.
