We consider point estimation and inference based on modifications of the profile likelihood in models for dyadic interactions between n agents featuring agent-specific parameters. This setup covers the β-model of network formation and generalizations thereof. The maximum-likelihood estimator of such models has bias and standard deviation of O(n−1) and so is asymptotically biased. Estimation based on modified likelihoods leads to estimators that are asymptotically unbiased and likelihood-ratio tests that exhibit correct size. We apply the modifications to versions of the β-model for network formation and of the Bradley-Terry model for paired comparisons.