Let Y be the response variable that takes values . Let j index all observations and be the value of response for the jth observation. Consider the multinomial variable such that
and . With denoting the probability that the jth observation has response value i, the expected value of is , and . The covariance matrix of is , which is the covariance matrix of a multinomial random variable for one trial with parameter vector . Let be the vector of regression parameters—for example, for cumulative logit model. Let be the matrix of partial derivatives of with respect to . The estimating equation for the regression parameters is
where , and and are the WEIGHT and FREQ values of the jth observation.
With a starting value of , the pseudo-estimate of is obtained iteratively as
where , , and are evaluated at the ith iteration . The expression after the plus sign is the step size. If the log likelihood evaluated at is less than that evaluated at , then is recomputed by step-halving or ridging. The iterative scheme continues until convergence is obtained—that is, until is sufficiently close to . Then the maximum likelihood estimate of is .
By default, starting values are zero for the slope parameters, and starting values are the observed cumulative logits (that is, logits of the observed cumulative proportions of response) for the intercept parameters. Alternatively, the starting values can be specified with the INEST= option in the PROC SURVEYLOGISTIC statement.