A crossover design is an experimental design in which each experimental unit receives a

series of experimental treatments over time. The order that an experimental unit receives

its treatments is called a sequence (example, the sequence AB means that treatment A is given first, and then followed by treatment B). A period is the time interval during which a treatment is administered to the experimental unit. A period could range from a few minutes to several months depending on the study. Sequences usually involve subjects receiving a different treatment in each successive period. However, treatments may occur more than once in any sequence (example, ABAB).

Treatments and periods are compared within subjects, i.e. each subject serves as his/her

own control. Therefore, any effect that is related to subject differences is removed from treatment and period comparisons.

Carryover effects are residual effects from a previous treatment manifesting themselves

in subsequent periods. Crossover designs both with and without carryover are traditionally analyzed assuming that the response due to different treatments have equal variances. The effects of unequal variances on traditional tests for treatment and carryover difference were recently considered in crossover designs assuming that the response due to treatments have

unequal variances with a compound symmetry correlation structure.

The likelihood function for the two treatment/two sequence crossover design has closed form maximum likelihood solutions for the parameters at both the null hypothesis, H0 : sigma_A^2 =\sigma_B^2, and at alternative hypothesis, HA : not H0. Under HA : not H0, the method of moment estimators and the maximum likelihood estimators of sigma_A^2,sigma_B^2 and rho are identical. The dual balanced design, ABA=BAB, which is balanced for carryover effects is also considered. The dual balanced design has a closed form solution that maximizes the likelihood function under the null hypothesis, H0 :sigma_A^2=sigma_B^2, but not for the alternative hypothesis, HA : not H0. Similarly, the three treatment/three sequence crossover design, ABC=BCA=CAB, has a closed form solution that maximizes the likelihood function at the null hypothesis, H0 : sigma_A^2=sigma_B^2 = sigma_C^2, but not for the alternative hypothesis, HA : not H0.

An iterative procedure is introduced to estimate the parameters for the two and three

treatment crossover designs. To check the performance of the likelihood ratio tests, Type I error rates and power comparisons are explored using simulations.