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Maximum likelihood estimation in logistic regression with mixed effects isknown to often result in estimates on the boundary of the parameter space. Suchestimates, which include infinite values for fixed effects and singular orinfinite variance components, can cause havoc to numerical estimationprocedures and inference. We introduce an appropriately scaled additive penaltyto the log-likelihood function, or an approximation thereof, which penalizesthe fixed effects by the Jeffreys' invariant prior for the model with no randomeffects and the variance components by a composition of negative Huber lossfunctions. The resulting maximum penalized likelihood estimates are shown tolie in the interior of the parameter space. Appropriate scaling of the penaltyguarantees that the penalization is soft enough to preserve the optimalasymptotic properties expected by the maximum likelihood estimator, namelyconsistency, asymptotic normality, and Cram\'er-Rao efficiency. Our choice ofpenalties and scaling factor preserves equivariance of the fixed effectsestimates under linear transformation of the model parameters, such ascontrasts. Maximum softly-penalized likelihood is compared to competingapproaches on two real-data examples, and through comprehensive simulationstudies that illustrate its superior finite sample performance.

Maximum softly-penalized likelihood for mixed effects logistic regression