Comparisons with a control group using order restricted estimators based on KKT conditions in crossover study with Latin squares

In crossover studies employing the Latin square method, this paper proposes CORKCL, a method that performs group comparisons against the control group using estimators with order restrictions that are monotonically increasing or decreasing. In this method, a Lagrangian function is constructed that takes into account the log-likelihood function of the mixed model and the order restrictions. The optimization problem is then solved under inequality constraints based on the KKT conditions (Karush-Kuhn-Tucker conditions). Furthermore, this optimization problem is reformulated to include multiple equality constraints, and estimates of the treatment effects are derived. To calculate the significance probability, the probability of obtaining a difference greater than the observed difference in the order-restricted estimates under the null hypothesis is calculated for each pattern of KKT conditions. These probabilities are then summed up. Afterward, a fixed-sequence procedure is applied. Simulation studies confirmed that the actual significance level was maintained at the nominal level. Regarding power, the results showed that CORKCL achieved high power in many settings, making it suitable for crossover studies. Therefore, CORKCL is appropriate for crossover studies using the Latin square method.