Optimal allocations for two treatment comparisons within the proportional odds cumulative logits model

Mirjam Moerbeek, Mirjam Moerbeek

Abstract

This paper studies optimal treatment allocations for two treatment comparisons when the outcome is ordinal and analyzed by a proportional odds cumulative logits model. The variance of the treatment effect estimator is used as optimality criterion. The optimal design is sought so that this variance is minimal for a given total sample size or a given budget, meaning that the power for the test on treatment effect is maximal, or it is sought so that a required power level is achieved at a minimal total sample size or budget. Results are presented for three, five and seven ordered response categories, three treatment effect sizes and a skewed, bell-shaped or polarized distribution of the response probabilities. The optimal proportion subjects in the intervention condition decreases with the number of response categories and the costs for the intervention relative to those for the control. The relation between the optimal proportion and effect size depends on the distribution of the response probabilities. The widely used balanced design is not always the most efficient; its efficiency as compared to the optimal design decreases with increasing cost ratio. The optimal design is highly robust to misspecification of the response probabilities and treatment effect size. The optimal design methodology is illustrated using two pharmaceutical examples. A Shiny app is available to find the optimal treatment allocation, to evaluate the efficiency of the balanced design and to study the relation between budget or sample size and power.

Conflict of interest statement

The author has declared that no competing interests exist.

Figures

Fig 1. Response probabilities in the control…
Fig 1. Response probabilities in the control condition.
Fig 2. Estimated probabilities in the placebo…
Fig 2. Estimated probabilities in the placebo and drug condition for the schizophrenia example.
Fig 3. Efficiency graph for the schizophrenia…
Fig 3. Efficiency graph for the schizophrenia example.
Fig 4. Power graph for the schizophrenia…
Fig 4. Power graph for the schizophrenia example.
Fig 5. Estimated probabilities in the placebi…
Fig 5. Estimated probabilities in the placebi and medium dose group in the trauma example.
Fig 6. Efficiency graph for the trauma…
Fig 6. Efficiency graph for the trauma example.
Fig 7. Power graph for the trauma…
Fig 7. Power graph for the trauma example.

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