Sequential Testing of Multinomial Hypotheses with Applications to Detecting Implementation Errors and Missing Data in Randomized Experiments
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Simply randomized designs are one of the most common controlled experiments used to study causal effects. Failure of the assignment mechanism, to provide proper randomization of units across treatments, or the data collection mechanism, when data is missing not at random, can render subsequent analysis invalid if not properly identified. In this paper we demonstrate that such practical implementation errors can often be identified, fortunately, through consideration of the total unit counts resulting in each treatment group. Based on this observation, we introduce a sequential hypothesis test constructed from Bayesian multinomial-Dirichlet families for detecting practical implementation errors in simply randomized experiments. By establishing a Martingale property of the posterior odds under the null hypothesis, frequentist Type-I error is controlled under both optional stopping and continuation via maximal inequalities, preventing practitioners from potentially inflating false positive probabilities through continuous monitoring. In contrast to other statistical tests that are performed once all data collection is completed, the proposed test is sequential - frequently rejecting the null during the process of data collection itself, saving further units from entering an improperly-executed experiment. We illustrate the utility of this test in the context of online controlled experiments (OCEs), where the assignment is automated through code and data collected through complex processing pipelines, often in the presence of unintended bugs and logical errors. Confidence sequences possessing desired sequential frequentist coverage probabilities are provided and their connection to the Bayesian support interval is examined. The differences between pure Bayesian and sequential frequentist testing procedures are finally discussed through a conditional frequentist testing perspective.
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