In studies of dementia, crossover designs are controversial, reflecting concerns about temporal stability of disease, confounding of treatment effects with period by treatment interactions and/or carryover effects. Carryover effects are differences in the lingering effect of treatments (placebo) into subsequent periods. In the context of a trial to study the effect of donepezil on dementia in patients with Parkinson's disease, we examine two-sequence crossover studies with two or four periods, and a four-sequence design with two periods. We quantify bias in estimated treatment effects due to carryover effects and explore the use of biased estimators in hypothesis testing. For hypothesis testing, type I error rates are valid if (1) repeated administration of treatment alters the outcome only for effective treatments and (2) carryover effects due to placebo following treatment periods are nonzero only for effective treatments. For crossover and parallel group designs, sample sizes are adjusted for reduced statistical power due to carryover effects and temporal changes in variance. For the proposed clinical study, we estimate that a single-period parallel group design with baselines would require 104 patients and take about 23 months to complete. A two-sequence, four-period parallel group design with baselines would require about 80 patients and about 20 months to complete. We conservatively assume a carryover effect of 50% of the treatment effect for a two-sequence four-period crossover design. The estimated treatment effect for this model may underestimate the true treatment effect by up to 13%. The sample size/study length requirements are 28 patients or 12.4 months, respectively, a substantial saving over either parallel group design. The cost of allowing for carryover in the sample size calculation is about 1.2 months of study time.