Estimating the cell probability for the joint cross-classification of zero responses among multivariate binary data arises in medicine, for example, when Bernoulli outcomes are recorded for subsets of anatomical sites within individuals resulting in partially-sampled clusters. A simple beta-binomial model of within-cluster exchangeability utilizes the purposively incomplete clusters to estimate disease prevalence based on complete clusters where disease is defined as the presence of one or more relevant sites affected with the condition, -the complement of the zero cell. For more realistic and complex anatomical models, where the propensity for the condition varies across sites and pairwise correlations follow a spatial clustering model, alternative prevalence estimators are derived under a conditional linear family of multivariate Bernoulli distributions. Properties of the estimators are investigated with the aim of introducing estimators of periodontitis prevalence using random partial-mouth samples into oral epidemiological research. Alternative definitions of disease, e.g., two or more sites affected and study designs, e.g., fixed-site selection methods, are briefly discussed.