Is Grouping Always Detrimental to Monitoring Multinomial Data?

Monitoring processes with multi-category outcomes is critical in many domains, including manufacturing, healthcare, and public health. Two natural approaches for such monitoring are: (1) using a full multinomial CUSUM chart, or (2) grouping the categories and applying Bernoulli CUSUM charts to the resulting binary outcomes. A common belief is that grouping would lead to a loss of information and thus inferior monitoring performance. In this paper, we critically examine this belief by asking: Is grouping always detrimental to monitoring multinomial data? Our findings reveal a nuanced answer that depends on the degree of prior knowledge about the true post-change category probabilities. When strong prior knowledge is available, grouping does lead to a loss in efficiency, with the multinomial CUSUM chart outperforming the Bernoulli CUSUM charts. However, in more realistic scenarios where such knowledge is limited or unavailable, grouping can actually enhance monitoring performance. In particular, adaptive Bernoulli CUSUM charts often outperform their multinomial counterparts under these conditions. Through both theoretical insights and extensive simulations, we demonstrate that grouping, when paired with adaptivity, can unexpectedly improve detection performance. These results challenge long-held assumptions in statistical process control and point to new strategies for effective and robust monitoring of multinomial processes in real-world applications.