15.5 - Higher-order Carryover Effects
15.5 - Higher-order Carryover EffectsThe lack of aliasing between the treatment difference and the first-order carryover effects does not guarantee that the treatment difference and higher-order carryover effects also will not be aliased or confounded. For example, let \(\lambda_{2A}\) and \(\lambda_{2B}\) denote the second-order carryover effects of treatments A and B, respectively, for the design in [Design 2] (Second-order carryover effects looks at the carryover effects of the treatment that took place previous to the prior treatment.):
Period 1 | Period 2 | Period 3 | |
Sequence ABB | \(mu_A + \nu + rho_1\) | \(\mu_B + \nu + \rho_2 + \lambda_A\) | \(\mu_B + \nu - \rho_1 - \rho_2 + \lambda_B + \lambda_{2A}\) |
Sequence BAA | \(mu_B - \nu + \rho_1\) | \(\mu_A - \nu + \rho_2 + \lambda_B\) | \(\mu_A - \nu - \rho_1 - \rho_2 + \lambda_A + \lambda_{2B}\) |
[18] \( E(\hat{\mu}_A-\hat{\mu}_B)=(\mu_A-\mu_B)-\dfrac{2}{3}\nu-\dfrac{1}{3}(\lambda_{2A}-\lambda_{2B}) \)
The expectation of the treatment mean difference indicates that it is aliased with second-order carryover effects.
Summary of Impacts of Design Types
The ensuing remarks summarize the impact of various design features on the aliasing of direct treatment and nuisance effects.
- If the crossover design is uniform within sequences, then sequence effects are not aliased with treatment differences.
- If the crossover design is uniform within periods, then period effects are not aliased with treatment differences.
- If the crossover design is balanced with respect to first-order carryover effects, then carryover effects are aliased with treatment differences. If the carryover effects are equal, then carryover effects are not aliased with treatment differences.
- If the crossover design is strongly balanced with respect to first- order carryover effects, then carryover effects are not aliased with treatment differences.
Complex Carryover
The type of carryover effects we modeled here is called “simple carryover” because it is assumed that the treatment in the current period does not interact with the carryover from the previous period. “Complex carryover” refers to the situation in which such an interaction is modeled. For example, suppose we have a crossover design and want to model carryover effects. With simple carryover in a two-treatment design, there are two carryover parameters, namely, \(\lambda_A\) and \(\lambda_B\).
With complex carryover, however, there are four carryover parameters, namely, \(\lambda_{AB}, \lambda_{BA}, \lambda_{AA}\) and \(\lambda_{BB}\), where \(\lambda_{AB}\) represents the carryover effect of treatment A into a period in which treatment B is administered, \(\lambda_{BA}\) represents the carryover effect of treatment B into a period in which treatment A is administered, etc. As you might imagine, this will certainly complicate things!