Why are these properties important in statistical analysis?
We now investigate statistical bias issues. In other words, does a particular crossover design have any nuisance effects, such as sequence, period, or first-order carryover effects, aliased with direct treatment effects? We consider first-order carryover effects only. If the design incorporates washout periods of inadequate length, then treatment effects could be aliased with higher-order carryover effects as well, but let us assume the washout period was adequate for eliminating carryover beyond 1 treatment period.
The approach is very simple in that the expected value of each cell in the crossover design is expressed in terms of a direct treatment effect and the assumed nuisance effects. Then these expected values are averaged and/or differenced to construct the desired effects.
For example, in the 2 × 2 crossover design in [Design 1], if we include nuisance effects for sequence, period, and first-order carryover, then model for this would look like:
Period 1 | Period 2 | |
Sequence AB | \(\mu_A + \nu + \rho\) | \(\mu_B + \nu - \rho + \lambda_A\) |
Sequence BA | \(\mu_B - \nu + \rho\) | \(\mu_A - \nu - \rho + \lambda_B\) |
where \(\mu_A\) and \(\mu_B\) represent population means for the direct effects of treatments A and B, respectively, \(\nu\) represents a sequence effect, \(\rho\) represents a period effect, and \(\lambda_A\) and \(\lambda_B\) represent carryover effects of treatments A and B, respectively.
A natural choice of an estimate of \(\mu_A\) (or \(\mu_B\)) is simply the average over all cells where treatment A (or B) is assigned: [12]
\(\hat{\mu}_A=\dfrac{1}{2}\left( \bar{Y}_{AB, 1}+ \bar{Y}_{BA, 2}\right) \text{ and } \hat{\mu}_B=\dfrac{1}{2}\left( \bar{Y}_{AB, 2}+ \bar{Y}_{BA, 1}\right)\)
Will this give us a good estimate of the means across the treatment? Not quite...
The mathematical expectations of these estimates are as follows: [13]
\(E(\hat{\mu}_A)=\dfrac{1}{2}\left( \mu_A+\nu+\rho+\mu_A-\nu-\rho+ \lambda_B \right)=\mu_A +\dfrac{1}{2}\lambda_B\)
\(E(\hat{\mu}_B)=\dfrac{1}{2}\left( \mu_B+\nu-\rho+\mu_B-\nu+\rho+ \lambda_A \right)=\mu_B +\dfrac{1}{2}\lambda_A\)
\(E(\hat{\mu}_A-\hat{\mu}_B) = ( \mu_A-\mu_B) - \dfrac{1}{2}( \lambda_A- \lambda_B) \)
From [Design 13] it is observed that the direct treatment effects and the treatment difference are not aliased with sequence or period effects, but are aliased with the carryover effects.
The treatment difference, however, is not aliased with carryover effects when the carryover effects are equal, i.e., \(\lambda_A = \lambda_B\). The results in [13] are due to the fact that the AB|BA crossover design is uniform and balanced with respect to first-order carryover effects. Any crossover design which is uniform and balanced with respect to first-order carryover effects, such as the designs in [Design 5] and [Design 8], also exhibits these results.
Example Section
Consider the ABB|BAA design, which is uniform within periods, not uniform with sequences, and is strongly balanced.
[14] | Period 1 | Period 2 | Period 3 |
Sequence ABB | \(\mu_A + \nu + \rho_1\) | \(\mu_B + \nu + \rho_2 + \lambda_A\) | \(mu_B + \nu - \rho_1 - \rho_2 + \lambda_B\) |
Sequence BAA | \(\mu_B - \nu + \rho_1\) | \(\mu_A - \nu + \rho_2 + \lambda_B\) | \(\mu_A - \nu - \rho_1 - \rho_2 + \lambda_A\) |
A natural choice of an estimate of \(\mu_A\) (or \(\mu_B\)) is simply the average over all cells where treatment A (or B) is assigned: [15]
\(\hat{\mu}_A=\dfrac{1}{3}\left( \bar{Y}_{ABB, 1}+ \bar{Y}_{BAA, 2}+ \bar{Y}_{BAA, 3}\right) \text{ and } \hat{\mu}_B=\dfrac{1}{3}\left( \bar{Y}_{ABB, 2}+ \bar{Y}_{ABB, 3}+ \bar{Y}_{BAA, 1}\right)\)
The mathematical expectations of these estimates are solved to be: [16]
\( E(\hat{\mu}_A)=\mu_A+\dfrac{1}{3}(\lambda_A+ \lambda_B-\nu)\)
\( E(\hat{\mu}_B)=\mu_B+\dfrac{1}{3}(\lambda_A+ \lambda_B+\nu)\)
\( E(\hat{\mu}_A-\hat{\mu}_B)=(\mu_A-\mu_B)-\dfrac{2}{3}\nu\)
From [16], the direct treatment effects are aliased with the sequence effect and the carryover effects, whereas the treatment difference only is aliased with the sequence effect. The results in [16] are due to the ABB|BAA crossover design being uniform within periods and strongly balanced with respect to first-order carryover effects.