The paper focuses on sequential experiments whose outcomes, success or failure, can be expressed using a binomial distribution, and in which whether or not a further experiment is made depends on the prior outcomes. Examples include subsequent medical interventions until a target response is achieved. The maximum likelihood estimate of the treatment success probability is the ratio of the number of observed successes and the total number of times the treatment was administered. Despite a prespecified initial sample size, the total number of conducted interventions cannot be determined in advance, and the estimated total number of successes may not be equal to the one observed, so the estimated variance of success probability is not that of a binomial distribution. These results generalize to designs with a tree structure consisting of groups of the experiments with group-specific outcome distributions. The multiplicative models entailed are studied using the tools from algebraic statistics and the theory of relational models. The estimates of success probabilities and their covariance are derived and shown to depend on the structural properties of the relevant exponential family.

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