Modeling Rates and Proportions in SAS – 8
7. FRACTIONAL LOGIT MODEL
Different from all models introduced previously that assume specific distributional families for the proportional outcomes of interests, the fractional logit model proposed by Papke and Wooldridge (1996) is a quasi-likelihood method that does not specify the full distribution but only requires the conditional mean to be correctly specified for consistent parameter estimates. Under the assumption E(Y|X) = G(X`B) = 1 / (1 + EXP(-X`B)), the fractional logit has the identical likelihood function to the one for a Bernoulli distribution such that
F(Y) = (G(X`B) ** Y) * (1 – G(X`B)) ** (1 – Y) with 1 >= Y >= 0
Based upon the above formulation, parameter estimates are calculated in the same manner as in the binary logistic regression by maximizing the log likelihood.
In SAS, the most convenient way to implement the fractional logit model is with GLIMMIX procedure. In addition, we can also use NLMIXED procedure by explicitly specifying the likelihood function as shown above.