Residual maximum likelihood (REML) estimation is adapted to certain logistic mixed models for which representation of the unconditional mean as a linear function of the fixed effects is possible. Only the first two moments of the unconditional distribution need be evaluated, and except for the form of the covariance, the maximization algorithm carries over directly from linear models. The exact unconditional covariance is computed from the logistic-normal mixture for input into the algorithm. Residual log-likelihood plots provide a means of inference for the dispersion components. The Taylor series approximation to this covariance, besides exhibiting insufficient accuracy, fails to be positive definite for large values of the dispersion components. As a consequence, REML loglikelihood plots based on this approximate covariance attribute misleadingly high precision to the dispersion component estimates. The method is presented in the context of a salamander mating experiment in which random effects corresponding to male and female animals occur in a crossed design. Analyses of these data by several methods are compared.