Now that we have examined statistical biases that can arise in crossover designs, we next examine statistical precision.

During the design phase of a trial, the question may arise as to which crossover design provides the best precision. For our purposes, we label one design as more precise than another if it yields a smaller variance for the estimated treatment mean difference.

Although a comparison of treatment means may be the primary interest of the experimenter, there may be other circumstances that affect the choice of an appropriate design. For example, later we will compare designs with respect to which designs are best for estimating and comparing variances.

At the moment, however, we focus on differences in estimated treatment means in two-period, two-treatment designs.

The two-period, two-treatment designs we consider here are the 2 × 2 crossover design AB|BA in [Design 1], Balaam's design AB|BA|AA|BB in [Design 6], and the two-period parallel design AA|BB.

In order for the resources to be equitable across designs, we assume that the total sample size, n, is a positive integer divisible by 4. Then:

- \(\dfrac{1}{2}\)n patients will be randomized to each sequence in the AB|BA design
- \(\dfrac{1}{2}\)n patients will be randomized to each sequence in the AA|BB design, and
- \(\dfrac{1}{4}\)n patients will be randomized to each sequence in the AB|BA|AA|BB design.

Because the designs we are considering involve repeated measurements on patients, the statistical modeling must account for between-patient variability and within-patient variability.

Between-patient variability accounts for the dispersion in measurements from one patient to another. Within-patient variability accounts for the dispersion in measurements from one time point to another within a patient. Within-patient variability tends to be smaller than between-patient variability.

The variance components we model are as follows:

- \(W_{AA}\) = between-patient variance for treatment A;
- \(W_{BB}\) = between-patient variance for treatment B;
- \(W_{AB}\) = between-patient covariance between treatments A and B;
- \(\sigma_{AA}\) = within-patient variance for treatment A;
- \(\sigma_{BB}\) = within-patient variance for treatment B.

The following table provides expressions for the variance of the estimated treatment mean difference for each of the two-period, two-treatment designs:

Design |
Variance |

Crossover | \(\dfrac{\sigma^2}{n} = \dfrac{1.0(W_{AA} + W_{BB}) - 2.0(W_{AB}) + (\sigma_{AA} + \sigma_{BB})}{n}\) |

Balaam | \(\dfrac{\sigma^2}{n} = \dfrac{1.5(W_{AA} + W_{BB}) - 1.0(W_{AB}) + (\sigma_{AA} + \sigma_{BB})}{n}\) |

Parallel | \(\dfrac{\sigma^2}{n} = \dfrac{2.0(W_{AA} + W_{BB}) - 0.0(W_{AB}) + (\sigma_{AA} + \sigma_{BB})}{n}\) |

Under most circumstances, \(W_{AB}\) will be positive, so we assume this is so for the sake of comparison. Not surprisingly, the 2 × 2 crossover design yields the smallest variance for the estimated treatment mean difference, followed by Balaam's design and then the parallel design.

The investigator needs to consider other design issues, however, prior to selecting the 2 × 2 crossover. In particular, if there is any concern over the possibility of differential first-order carryover effects, then the 2 × 2 crossover is not recommended. In this situation, the parallel design would be a better choice than the 2 × 2 crossover design. Balaam's design is strongly balanced so that the treatment difference is not aliased with differential first-order carryover effects, so it also is a better choice than the 2 × 2 crossover design.

With respect to a sample size calculation, the total sample size, n, required for a two-sided, \(\alpha\) significance level test with \(100 \left(1 - \beta \right)\%\) statistical power and effect size \(\mu_A - \mu_B\) is:

\(n=(z_{1-\alpha/2}+z_{1-\beta})^2 \sigma2/(\mu_A -\mu_B)^2 \)

Suppose that an investigator wants to conduct a two-period trial but is not sure whether to invoke a parallel design, a crossover design, or Balaam's design. He wants to use a 0.05 significance level test with 90% statistical power for detecting the effect size of \(\mu_A - \mu_B= 10\). From published results, the investigator assumes that:

\(W_{AA} = W_{BB} = W_{AB} = 400\), and

\(\sigma_{AA} = \sigma_{BB}\) = 100

The sample sizes for the three different designs are as follows:

Parallel n = 190

Balaam n = 105

Crossover n = 21

The crossover design yields a much smaller sample size because the within-patient variances are one-fourth that of the inter-patient variances (which is not unusual).

Another issue in selecting a design is whether the experimenter wishes to compare the within-patient variances\(\sigma_{AA}\) and \(\sigma_{BB}\).

For the 2 × 2 crossover design, the within-patient variances can be estimated by imposing restrictions on the between-patient variances and covariances. The resultant estimators of\(\sigma_{AA}\) and \(\sigma_{BB}\), however, may lack precision and be unstable. Hence, the 2 × 2 crossover design is not recommended when comparing\(\sigma_{AA}\) and \(\sigma_{BB}\) is an objective.

The parallel design provides an optimal estimation of the within-unit variances because it has ½ n patients who can provide data in estimating each of\(\sigma_{AA}\) and \(\sigma_{BB}\), whereas Balaam's design has ¼ n patients who can provide data in estimating each of\(\sigma_{AA}\) and \(\sigma_{BB}\). Again, Balaam's design is a compromise between the 2 × 2 crossover design and the parallel design.