Sample size determination for adaptive crossover trial in detecting gene-drug interactions

Abstract

Parallel design and crossover design are two of the most frequently used designs for studying drug-gene interactions. Due to the concerns of statistical power and ethics, it is often more prudent to use the crossover design while allowing the patients to have choices of not switching the treatment if the first stage treatment is effective. This complicates the calculation of the required sample size to achieve pre-specified statistical power. We propose a method to determine the required sample size with a closed-form formula. The proposed approach is applied to determine the sample size of an adaptive crossover trial in studying gene-drug interaction in treating atrial fibrillation, the most common cardiac arrhythmia in clinical practice. Our simulation study confirms the power achieved by the sample size determined using the proposed approach. Issues related to the adaptive crossover trial are also discussed and practical guidelines are provided.

Keywords

Atrial fibrillation interaction pharmacogenomic approach randomization weighted estimator

1 Introduction

The study of gene-drug interactions is important for the development of personalized medicine.^1,2 The clinic Pharmacogenomic approach has been increasingly implemented^3–5 in cancer treatment,⁶ in heart health,^7,8 and in treating and managing chronic diseases.⁹ Randomized trials to determine gene-drug interactions clinically manifested have been frequently performed to generate knowledge on the interactions. To guide the success of such clinical trials, an important consideration is the sample size required for detecting the interactions with good statistical power. The sample size and power can be substantially affected by the study designs in addition to the genetic variation among the patient population.

A simple design for studying the interactions is the parallel design where patients are randomized to receive different treatments.¹⁰ This design is simple both in implementation and in statistical analysis of the treatment effects. However, it is not as efficient as the crossover design^11–13 where patients are randomized to receive one treatment and later switch to another treatment. The crossover design is more powerful in detecting the interactions with the same sample size as the parallel design. The former is however limited by the possible carry-over effect that may confound the treatment effect evaluation. When such effects can be well controlled in a trial, it is often favored over the parallel design. A second issue is the ethical concern of switching from a treatment that is efficacious to one that may not be effective. To address such issues, an adaptive crossover design that leaves the option to not switch when the first treatment is deemed effective may be more attractive.¹⁴ The adaptive crossover design can be more powerful than the parallel design and addresses the ethical concern. However, it is less powerful than the crossover design and requires a sample size calculation adjusting for the adaptive nature of the trial to maintain the same statistical power for detecting gene-drug interactions.

The sample size determination for the parallel design and the crossover design have been well documented respectively in the literature, e.g., Jones and Kenward,¹² Chow et al.,¹⁰ Siyasinghe and Sooriyarachchi.¹³ As far as we are aware, the sample size requirement for detecting the gene-drug interaction for the adaptive crossover design has not been systematically examined. A related but different design termed the dynamic treatment regime^15–17 has been proposed in the literature. The sample size calculation for the dynamic treatment regime has been studied by many authors in different scenarios.^16,18,19 However, as pointed out by Murphy,¹⁶ the dynamic treatment regime serves a different purpose and the sample size calculation is different from the adaptive design.

In this article, we examine in detail the adaptive crossover design for studying gene-drug interactions and propose a sample size determination approach to achieve the require power. We compare the efficiency of the adaptive crossover design with the parallel design and the crossover design in terms of their power achieved with the same sample sizes. We also discuss analytical issues that may be encountered in such a design and provide guidelines on when such a design may be used.

2 Designs for studying gene-drug interaction

2.1 Parallel design for studying gene-drug interaction

The parallel design is the simplest one where subjects are randomized to receive the treatment drug A or drug B for the two genotypes: Risk allele (R) and wild type (W). Table 1 lists the design and outcome model for the parallel randomized trial. The objective of the study is to test the gene-drug interactions. The hypothesis to be tested is

H_{0} : μ_{A \times R}^{T G} - μ_{A \times W}^{T G} - μ_{B \times R}^{T G} + μ_{B \times W}^{T G} = 0.

(1)

Table 1.

Parallel design and the outcome model.

Gene	Treatment	Outcome model	SS
R	A	$Y (R, A) = μ_{0} + μ_{A}^{T} + μ_{R}^{G} + μ_{A \times R}^{T G} + ϵ$	$n_{1}$
R	B	$Y (R, B) = μ_{0} + μ_{B}^{T} + μ_{R}^{G} + μ_{B \times R}^{T G} + ϵ$	$n_{2}$
W	A	$Y (W, A) = μ_{0} + μ_{A}^{T} + μ_{W}^{G} + μ_{A \times W}^{T G} + ϵ$	$n_{3}$
W	B	$Y (W, B) = μ_{0} + μ_{B}^{T} + μ_{W}^{G} + μ_{B \times W}^{T G} + ϵ$	$n_{4}$

SS: Sample size allocated. The error $ϵ$ has mean $0$ and variance $σ^{2}$ .

2.2 Cross-over design for studying gene-drug interaction

The crossover design randomly assigns patients to treatment drug A for a period of time and then switches to treatment drug B possibly after a washout period, or alternatively to treatment drug B for a period of time and then switches to treatment drug A after a washout period. Table 2 lists the design and outcome models for the two outcomes for each patient. The objective of the study is again to test the gene-drug interaction, i.e., to test the hypothesis in (1).

2.3 Adaptive design for studying gene-drug interaction

The parallel randomized design is simple and gives rise to straightforward treatment effects estimates. On the other hand, the crossover design is more efficient in general. In practice, however, if the first stage treatment is effective, patients or doctors may choose to not switch the treatment. In this case, the crossover design is not completely executed. Table 3 lists the possible treatment scenarios under such a situation.

Table 2.
Crossover design and the outcome models.

Gene $T_{1}$ $T_{2}$ Outcome model at time 1 Outcome model at time 2 SS

R A B $Y_{1} (R, A) = μ_{0} + μ_{A}^{T} + μ_{R}^{G} + μ_{A \times R}^{T G} + ϵ_{1}$ $Y_{2} (R, B) = μ_{0} + μ_{B}^{T} + μ_{R}^{G} + μ_{B \times R}^{T G} + ϵ_{2}$ $n_{1}$

B A $Y_{1} (R, B) = μ_{0} + μ_{B}^{T} + μ_{R}^{G} + μ_{B \times R}^{T G} + ϵ_{1}$ $Y_{2} (R, A) = μ_{0} + μ_{A}^{T} + μ_{R}^{G} + μ_{A \times R}^{T G} + ϵ_{2}$ $n_{2}$

W A B $Y_{1} (W, A) = μ_{0} + μ_{A}^{T} + μ_{W}^{G} + μ_{A \times W}^{T G} + ϵ_{1}$ $Y_{2} (W, B) = μ_{0} + μ_{B}^{T} + μ_{W}^{G} + μ_{B \times W}^{T G} + ϵ_{2}$ $n_{3}$

B A $Y_{1} (W, B) = μ_{0} + μ_{B}^{T} + μ_{W}^{G} + μ_{B \times W}^{T G} + ϵ_{1}$ $Y_{2} (W, A) = μ_{A}^{T} + μ_{W}^{G} + μ_{A \times W}^{T G} + ϵ_{2}$ $n_{4}$

Gene	$T_{1}$	$T_{2}$	Outcome model at time 1	Outcome model at time 2	SS
R	A	B	$Y_{1} (R, A) = μ_{0} + μ_{A}^{T} + μ_{R}^{G} + μ_{A \times R}^{T G} + ϵ_{1}$	$Y_{2} (R, B) = μ_{0} + μ_{B}^{T} + μ_{R}^{G} + μ_{B \times R}^{T G} + ϵ_{2}$	$n_{1}$
B	A	$Y_{1} (R, B) = μ_{0} + μ_{B}^{T} + μ_{R}^{G} + μ_{B \times R}^{T G} + ϵ_{1}$	$Y_{2} (R, A) = μ_{0} + μ_{A}^{T} + μ_{R}^{G} + μ_{A \times R}^{T G} + ϵ_{2}$	$n_{2}$
W	A	B	$Y_{1} (W, A) = μ_{0} + μ_{A}^{T} + μ_{W}^{G} + μ_{A \times W}^{T G} + ϵ_{1}$	$Y_{2} (W, B) = μ_{0} + μ_{B}^{T} + μ_{W}^{G} + μ_{B \times W}^{T G} + ϵ_{2}$	$n_{3}$
B	A	$Y_{1} (W, B) = μ_{0} + μ_{B}^{T} + μ_{W}^{G} + μ_{B \times W}^{T G} + ϵ_{1}$	$Y_{2} (W, A) = μ_{A}^{T} + μ_{W}^{G} + μ_{A \times W}^{T G} + ϵ_{2}$	$n_{4}$

$T_{1}$ means treatment at time 1 and $T_{2}$ means treatment at time 2. $ϵ_{1} = e_{0} + e_{1}$ , $ϵ_{2} = e_{0} + e_{2}$ , and $e_{0}$ is the between patient error and $e_{1}$ and $e_{2}$ are within patient errors. SS: Sample size allocated.

Table 3.

Adaptive design and decision rule.

Gene	Treatment I	Treatment II	Decision rule
R	A	A*	Treatment A success
	A	B	Treatment A failure
	B	B*	Treatment B success
	B	A	Treatment B failure
W	A	A*	Treatment A success
	A	B	Treatment A failure
	B	B*	Treatment B success
	B	A	Treatment B failure

A* and B* indicate maintaining the previous treatment.

If we ignored the second stage trials, i.e., the crossover, the trial can still be analyzed as a parallel trial. One issue is that the trial may be substantially under-powered. For ethical consideration, we may design a study that allow patients to not switch if the first stage treatment is effective. The altered trial is often termed the adaptive crossover trial. We may use the second-stage data under reasonable conditions to increase the power, especially when the patients are difficult to recruit. Our main objective is to provide sample size and power calculation under the adaptive trial design for testing the gene-drug interaction.

3 Sample sizes for the designs

3.1 Sample size requirement in the parallel design

In the parallel design, let the sample size be allocated as shown in Table 1. The mean difference for the risk allele genotype carriers under two different treatments,

\begin{aligned} μ_{A}^{T} - μ_{B}^{T} + μ_{A \times R}^{T G} - μ_{B \times R}^{T G}, \end{aligned}

(2)

can be estimated by

\begin{aligned} \frac{1}{n_{1}} \sum_{i = 1}^{n_{1}} Y_{i} (R, A) - \frac{1}{n_{2}} \sum_{i = 1}^{n_{2}} Y_{i} (R, B) . \end{aligned}

(3)

The mean difference for subjects with wild-type genotype under the two different treatments,

\begin{aligned} μ_{A}^{T} - μ_{B}^{T} + μ_{A \times W}^{T G} - μ_{B \times W}^{T G}, \end{aligned}

(4)

can be estimated by

\begin{aligned} \frac{1}{n_{3}} \sum_{i = 1}^{n_{3}} Y_{i} (W, A) - \frac{1}{n_{4}} \sum_{i = 1}^{n_{4}} Y_{i} (W, B) . \end{aligned}

(5)

An estimator of the interaction effect

d = μ_{A \times R}^{T G} - μ_{A \times W}^{T G} - μ_{B \times R}^{T G} + μ_{B \times W}^{T G}

\begin{aligned} \hat{d} = \frac{1}{n_{1}} \sum_{i = 1}^{n_{1}} Y_{i} (R, A) - \frac{1}{n_{2}} \sum_{i = 1}^{n_{2}} Y_{i} (R, B) - \frac{1}{n_{3}} \sum_{i = 1}^{n_{3}} Y_{i} (W, A) + \frac{1}{n_{4}} \sum_{i = 1}^{n_{4}} Y_{i} (W, B) . \end{aligned}

(6)

The variance estimate of

\hat{d}

\begin{aligned} \hat{v} & = \frac{1}{n_{1} (n_{1} - 1)} \sum_{i = 1}^{n_{1}} (Y_{i} (R, A) - \bar{Y} (R, A))^{2} + \frac{1}{n_{2} (n_{2} - 1)} \sum_{i = 1}^{n_{2}} (Y_{i} (R, B) - \bar{Y} (R, B))^{2} \\ + \frac{1}{n_{3} (n_{3} - 1)} \sum_{i = 1}^{n_{3}} (Y_{i} (W, A) - \bar{Y} (W, A))^{2} + \frac{1}{n_{4} (n_{4} - 1)} \sum_{i = 1}^{n_{4}} (Y_{i} (W, B) - \bar{Y} (W, B))^{2}, \end{aligned}

(7)

where

\bar{Y}

is the sample mean of the corresponding Y sample. Alternatively, a partially pooled variance estimator is

\begin{aligned} \begin{array}{l} (\frac{1}{n_{1}} + \frac{1}{n_{2}}) \frac{1}{(n_{1} + n_{2} - 2)} [\sum_{i = 1}^{n_{1}} (Y_{i} (R, A) - \bar{Y} (R, A))^{2} + \sum_{i = 1}^{n_{2}} (Y_{i} (R, B) - \bar{Y} (R, B))^{2}] \\ + (\frac{1}{n_{3}} + \frac{1}{n_{4}}) \frac{1}{(n_{3} + n_{4} - 2)} [\sum_{i = 1}^{n_{3}} {(Y_{i} (W, A) - \bar{Y} (W, A))}^{2} + \sum_{i = 1}^{n_{4}} {(Y_{i} (W, B) - \bar{Y} (W, B))}^{2}] . \end{array} \end{aligned}

(8)

Or use the fully pooled variance estimator

\begin{aligned} (\frac{1}{n_{1}} + \frac{1}{n_{2}} + \frac{1}{n_{3}} + \frac{1}{n_{4}}) \frac{1}{(n_{1} + n_{2} + n_{3} + n_{4} - 4)} [\sum_{i = 1}^{n_{1}} (Y_{i} (R, A) - \bar{Y} (R, A))^{2} + \sum_{i = 1}^{n_{2}} (Y_{i} (R, B) - \bar{Y} (R, B))^{2}] \\ + [\sum_{i = 1}^{n_{3}} (Y_{i} (W, A) - \bar{Y} (W, A))^{2} + \sum_{i = 1}^{n_{4}} (Y_{i} (W, B) - \bar{Y} (W, B))^{2}] . \end{aligned}

(9)

The test statistic for the hypothesis of no interaction is

\begin{aligned} \frac{\hat{d}}{\sqrt{\hat{v}}} \sim N (0, 1) . \end{aligned}

(10)

Under

H_{0}

. Sample size or power can be determined by

\begin{aligned} power = P (\frac{\hat{d} - d}{\sqrt{\hat{v}}} > z_{1 - \frac{α}{2}} - \frac{d}{\sqrt{\hat{v}}} | H_{A}) + P (\frac{\hat{d} - d}{\sqrt{\hat{v}}} < - z_{1 - \frac{α}{2}} - \frac{d}{\sqrt{\hat{v}}} | H_{A}) . \end{aligned}

(11)

\begin{aligned} power = 1 - Φ (z_{1 - \frac{α}{2}} - \frac{d}{\sqrt{\hat{v}}}) + Φ (- z_{1 - \frac{α}{2}} - \frac{d}{\sqrt{\hat{v}}}), \end{aligned}

(12)

where

α

is the type I error of the test and

1 - β

is the power of the test. Let the sample size for drug A be

n r

and the sample size for drug B be

n (1 - r)

, where n is the total sample size. Then

\begin{aligned} n_{1} & = n r P (G = R), \end{aligned}

(13)

\begin{aligned} n_{2} & = n (1 - r) P (G = R), \end{aligned}

(14)

\begin{aligned} n_{3} & = n r P (G = W), \end{aligned}

(15)

\begin{aligned} n_{4} & = n (1 - r) P (G = W), \end{aligned}

(16)

The sample size can be determined approximately by

\begin{aligned} n = {(\frac{z_{1 - \frac{α}{2}} + z_{1 - β}}{d})}^{2} v_{1} (\frac{1}{r (1 - r) P (G = R)} + \frac{1}{r (1 - r) P (G = W)}), \end{aligned}

(17)

where

v_{1}

is the variance of

ϵ

3.2 Sample size requirement in the crossover design

In the cross-over design, let the sample size allocation be as given in Table 2. Assume no residual effects nor treatment order effects. The mean difference for the risk allele genotype carriers under two different treatments,

\begin{aligned} μ_{A}^{T} - μ_{B}^{T} + μ_{A \times R}^{T G} - μ_{B \times R}^{T G}, \end{aligned}

(18)

can be estimated by

\begin{aligned} η_{1} = \frac{1}{n_{1} + n_{2}} (\sum_{i = 1}^{n_{1}} [Y_{1 i} (R, A) - Y_{2 i} (R, B)] + \sum_{i = 1}^{n_{2}} [Y_{2 i} (R, A) - Y_{1 i} (R, B)]), \end{aligned}

(19)

The mean difference for subjects with wild-type genotype under the two different treatments,

\begin{aligned} μ_{A}^{T} - μ_{B}^{T} + μ_{A \times W}^{T G} - μ_{\times W}^{T G}, \end{aligned}

(20)

can be estimated by

\begin{aligned} η_{2} = \frac{1}{n_{3} + n_{4}} (\sum_{i = 1}^{n_{3}} [Y_{1 i} (W, A) - Y_{2 i} (W, B)] + \sum_{i = 1}^{n_{4}} [Y_{2 i} (W, A) - Y_{1 i} (W, B)]), \end{aligned}

(21)

An estimator of the interaction effect

d = μ_{A \times R}^{T G} - μ_{A \times W}^{T G} - μ_{B \times R}^{T G} + μ_{B \times W}^{T G}

\hat{d} = η_{1} - η_{2}

. The variance estimate of

\hat{d}

\begin{aligned} \hat{v} & = \frac{1}{(n_{1} + n_{2}) (n_{1} + n_{2} - 1)} (\sum_{i = 1}^{n_{1}} {[Y_{1 i} (R, A) - Y_{2 i} (R, B) - η_{1}]}^{2} + \sum_{i = 1}^{n_{2}} {[Y_{2 i} (R, A) - Y_{1 i} (R, B) - η_{1}]}^{2}) \\ + \frac{1}{(n_{3} + n_{4}) (n_{3} + n_{4} - 1)} (\sum_{i = 1}^{n_{3}} {[Y_{1 i} (W, A) - Y_{2 i} (W, B) - η_{2}]}^{2} + \sum_{i = 1}^{n_{4}} {[Y_{2 i} (W, A) - Y_{1 i} (W, B) - η_{2}]}^{2}), \end{aligned}

(22)

Or use the fully pooled variance estimator

\begin{aligned} (\frac{1}{n_{1} + n_{2}} + \frac{1}{n_{3} + n_{4}}) \frac{1}{(n_{1} + n_{2} + n_{3} + n_{4} - 2)} (\sum_{i = 1}^{n_{1}} {[Y_{1 i} (R, A) - Y_{2 i} (R, B) - η_{1}]}^{2} + \sum_{i = 1}^{n_{2}} {[Y_{2 i} (R, A) - Y_{1 i} (R, B) - η_{1}]}^{2} \\ + \sum_{i = 1}^{n_{3}} {[Y_{1 i} (W, A) - Y_{2 i} (W, B) - η_{2}]}^{2} + \sum_{i = 1}^{n_{4}} {[Y_{2 i} (W, A) - Y_{1 i} (W, B) - η_{2}]}^{2}) \end{aligned}

(23)

The test statistic for the interaction hypothesis is

\begin{aligned} \frac{\hat{d}}{\sqrt{\hat{v}}} \sim N (0, 1), \end{aligned}

(24)

Under

H_{0}

. Sample size or power can be determined by

\begin{aligned} power = P (\frac{\hat{d} - d}{\sqrt{\hat{v}}} > z_{1 - \frac{α}{2}} - \frac{d}{\sqrt{\hat{v}}} | H_{A}) + P (\frac{\hat{d} - d}{\sqrt{\hat{v}}} < - z_{1 - \frac{α}{2}} - \frac{d}{\sqrt{\hat{v}}} | H_{A}) . \end{aligned}

(25)

\begin{aligned} power = 1 - Φ (z_{1 - \frac{α}{2}} - \frac{d}{\sqrt{\hat{v}}}) + Φ (- z_{1 - \frac{α}{2}} - \frac{d}{\sqrt{\hat{v}}}), \end{aligned}

(26)

where

α

is the type I error of the test and

1 - β

is the power of the test. The sample size can be determined approximately by

\begin{aligned} n = {(\frac{z_{1 - \frac{α}{2}} + z_{1 - β}}{d})}^{2} v_{2} (\frac{1}{P (G = R)} + \frac{1}{P (G = W)}), \end{aligned}

(27)

where

v_{2} = var (e_{1}) + var (e_{2})

. The variance

v_{1} = var (ϵ) = var (ϵ_{1}) = var (e_{1}) + var (e_{0})

represents both the between patients and the within-patient variations, while

v_{2}

represents twice the within-patient variation. It follows

v_{2} = 2 (1 - ρ) v_{1}

, where

ρ = var (e_{0}) / v a r (ϵ_{1})

is the intra-patient correlation.

Table 4.

Parallel part of the adaptive crossover design and the outcome model.

Gene	$T_{1}$	$T_{2}$	S	Outcome model	Averaged SS
R	A	A*	1	$Y (R, A) = μ_{0} + μ_{A}^{T} + μ_{R}^{G} + μ_{A \times R}^{T G} + ϵ$	$n_{1} P (S = 1 \| R, A)$
R	B	B*	1	$Y (R, B) = μ_{0} + μ_{B}^{T} + μ_{R}^{G} + μ_{B \times R}^{T G} + ϵ$	$n_{2} P (S = 1 \| R, B)$
W	A	A*	1	$Y (W, A) = μ_{0} + μ_{A}^{T} + μ_{W}^{G} + μ_{A \times W}^{T G} + ϵ$	$n_{3} P (S = 1 \| W, A)$
W	B	B*	1	$Y (W, B) = μ_{0} + μ_{B}^{T} + μ_{W}^{G} + μ_{B \times W}^{T G} + ϵ$	$n_{4} P (S = 1 \| W, B)$

S: Sample selection. SS: Sample size.

3.3 Sample size requirement in the adaptive design

For the adaptive design, the analysis ignoring the possible crossover part is simple for analysis. It can yield a causal effect under general conditions. However, the power can be low and the required sample size is large. Under stronger conditions, power can be gained by including all the data in the crossover part of the analysis. In the following, we use $S = 1$ to denote the success of treatment in the first stage. A failure would instead have $S = 0$ . The analysis taking the whole crossover part into consideration needs to adjust for the biased sample in the second stage.

The two-stage analysis approach poses substantial difficulty for the variance estimate due to the overlap sample issue of the two-stage analysis. We take the following alternative approach to simplify the analysis. The alternative approach separates patients without crossover from patients with crossover. The patients without crossover form a parallel design and patients with crossover form a crossover design, and the adaptive design is a mixture of the parallel design and the crossover design. Note that both the parallel design sample and the crossover design sample are biased samples with known sampling probability depending on the first-stage outcomes.

For the parallel design part shown in Table 4, the mean difference $μ_{A}^{T} - μ_{B}^{T} + μ_{A \times R}^{T G} - μ_{B \times R}^{T G}$ can be estimated by

\begin{aligned} {\hat{d}}_{11} = \frac{1}{n_{1}^{*}} \sum_{i = 1}^{n_{1}} \frac{S_{i} Y_{i} (R, A)}{P (S = 1 | Y_{i} (R, A))} - \frac{1}{n_{2}^{*}} \sum_{i = 1}^{n_{2}} \frac{S_{i} Y_{i} (R, B)}{P (S_{i} = 1 | Y_{i} (R, B))}, \end{aligned}

(28)

where

\begin{aligned} n_{1}^{*} = \sum_{i = 1}^{n_{1}} \frac{S_{i}}{P (S = 1 | Y_{i} (R, A))}, n_{2}^{*} = \sum_{i = 1}^{n_{2}} \frac{S_{i}}{P (S = 1 | Y_{i} (R, B))} . \end{aligned}

(29)

The mean difference

μ_{A}^{T} - μ_{B}^{T} + μ_{A \times W}^{T G} - μ_{B \times W}^{T G}

can be estimated by

\begin{aligned} {\hat{d}}_{12} = \frac{1}{n_{3}^{*}} \sum_{i = 1}^{n_{3}} \frac{S_{i} Y_{i} (W, A)}{P (S_{i} = 1 | Y_{i} (W, A))} - \frac{1}{n_{4}^{*}} \sum_{i = 1}^{n_{4}} \frac{S_{i} Y_{i} (W, B)}{P (S_{i} = 1 | Y_{i} (W, B))}, \end{aligned}

(30)

where

\begin{aligned} n_{3}^{*} = \sum_{i = 1}^{n_{3}} \frac{S_{i}}{P (S = 1 | Y_{i} (W, A))}, n_{4}^{*} = \sum_{i = 1}^{n_{4}} \frac{S_{i}}{P (S = 1 | Y_{i} (W, B))} . \end{aligned}

(31)

An estimator of the interaction effect is

\begin{aligned} {\hat{d}}_{1} = {\hat{d}}_{11} - {\hat{d}}_{12} . \end{aligned}

(32)

The variance

{\hat{d}}_{1}

can be approximated by

\begin{aligned} {\hat{v}}_{1} & = \frac{1}{n_{1}^{* 2}} \sum_{i = 1}^{n_{1}} \frac{S_{i} {[Y_{i} (R, A) - \bar{Y} (R, A)]}^{2}}{P^{2} (S_{i} = 1 | Y_{i} (R, A))} + \frac{1}{n_{2}^{* 2}} \sum_{i = 1}^{n_{2}} \frac{S_{i} {[Y_{i} (R, B) - \bar{Y} (R, B)]}^{2}}{P^{2} (S_{i} = 1 | Y_{i} (R, B))} \\ + \frac{1}{n_{3}^{* 2}} \sum_{i = 1}^{n_{3}} \frac{S_{i} {[Y_{i} (W, A) - \bar{Y} (W, A)]}^{2}}{P^{2} (S_{i} = 1 | Y_{i} (W, A))} + \frac{1}{n_{4}^{* 2}} \sum_{i = 1}^{n_{4}} \frac{S_{i} {[Y_{i} (W, B) - \bar{Y} (W, B)]}^{2}}{P^{2} (S_{i} = 1 | Y_{i} (W, B))}, \end{aligned}

(33)

Table 5.

Crossover part of the adaptive crossover design and the outcome models.

Gene	$T_{1}$	$T_{2}$	Outcome model at time 1	Outcome model at time 2	Averaged SS
R	A	B	$Y_{1} (R, A) = μ_{0} + μ_{A}^{T} + μ_{R}^{G} + μ_{A \times R}^{T G} + ϵ_{1}$	$Y_{2} (R, B) = μ_{0} + μ_{B}^{T} + μ_{R}^{G} + μ_{B \times R}^{T G} + ϵ_{2}$	$n_{1} P (S = 0 \| R, A)$
R	B	A	$Y_{1} (R, B) = μ_{0} + μ_{B}^{T} + μ_{R}^{G} + μ_{B \times R}^{T G} + ϵ_{1}$	$Y_{2} (R, A) = μ_{0} + μ_{A}^{T} + μ_{R}^{G} + μ_{A \times R}^{T G} + ϵ_{2}$	$n_{2} P (S = 0 \| R, B)$
W	A	B	$Y_{1} (W, A) = μ_{0} + μ_{A}^{T} + μ_{W}^{G} + μ_{A \times W}^{T G} + ϵ_{1}$	$Y_{2} (W, B) = μ_{0} + μ_{B}^{T} + μ_{W}^{G} + μ_{B \times W}^{T G} + ϵ_{2}$	$n_{3} P (S = 0 \| W, A)$
W	B	A	$Y_{1} (W, B) = μ_{0} + μ_{B}^{T} + μ_{W}^{G} + μ_{B \times W}^{T G} + ϵ_{1}$	$Y_{2} (W, A) = μ_{0} + μ_{A}^{T} + μ_{W}^{G} + μ_{A \times W}^{T G} + ϵ_{2}$	$n_{4} P (S = 0 \| W, B)$

S: Sample selection. SS: Sample size.

For the crossover design part shown in Table 5, the mean difference $μ_{A}^{T} - μ_{B}^{T} + μ_{A \times R}^{T G} - μ_{B \times R}^{T G}$ can be estimated by

\begin{aligned} {\hat{η}}_{1} = \frac{1}{n_{1}^{* *} + n_{2}^{* *}} (\sum_{i = 1}^{n_{1}} \frac{(1 - S_{i}) [Y_{1 i} (R, A) - Y_{2 i} (R, B)]}{P (S_{i} = 0 | Y_{i} (R, A))} + \sum_{i = 1}^{n_{2}} \frac{(1 - S_{i}) [Y_{2 i} (R, A) - Y_{1 i} (R, B)]}{P (S_{i} = 0 | Y_{i} (R, B))}), \end{aligned}

(34)

where

\begin{aligned} n_{1}^{* *} = \sum_{i = 1}^{n_{1}} \frac{1 - S_{i}}{P (S = 0 | Y_{i} (R, A))}, n_{2}^{* *} = \sum_{i = 1}^{n_{2}} \frac{1 - S_{i}}{P (S = 0 | Y_{i} (R, B))} . \end{aligned}

(35)

The mean difference

μ_{A}^{T} - μ_{B}^{T} + μ_{A \times W}^{T G} - μ_{B \times W}^{T G}

can be estimated by

\begin{aligned} {\hat{η}}_{2} = \frac{1}{n_{3}^{* *} + n_{4}^{* *}} (\sum_{i = 1}^{n_{3}} \frac{(1 - S_{i}) [Y_{1 i} (W, A) - Y_{2 i} (W, B)]}{P (S_{i} = 0 | Y_{i} (W, A))} + \sum_{i = 1}^{n_{4}} \frac{(1 - S_{i}) [Y_{2 i} (W, A) - Y_{1 i} (W, B)]}{P (S_{i} = 0 | Y_{i} (W, B))}), \end{aligned}

(36)

where

\begin{aligned} n_{3}^{* *} = \sum_{i = 1}^{n_{3}} \frac{1 - S_{i}}{P (S = 0 | Y_{i} (W, A))}, n_{4}^{* *} = \sum_{i = 1}^{n_{4}} \frac{1 - S_{i}}{P (S = 0 | Y_{i} (W, B))} . \end{aligned}

(37)

An estimator of the interaction effect

d = μ_{A \times R}^{T G} - μ_{A \times W}^{T G} - μ_{B \times R}^{T G} + μ_{B \times W}^{T G}

{\hat{d}}_{2} = {\hat{η}}_{1} - {\hat{η}}_{2}

. The variance of

{\hat{d}}_{2}

can be approximated by

\begin{aligned} {\hat{v}}_{2} & = \frac{1}{(n_{1}^{* *} + n_{2}^{* *}) (n_{1}^{* *} + n_{2}^{* *} - 1)} (\sum_{i = 1}^{n_{1}} \frac{(1 - S_{i}) {[Y_{1 i} (R, A) - Y_{2 i} (R, B) - {\hat{η}}_{1}]}^{2}}{P^{2} (S_{i} = 0 | Y_{i} (R, A))} \\ + \sum_{i = 1}^{n_{2}} \frac{(1 - S_{i}) {[Y_{2 i} (R, A) - Y_{1 i} (R, B) - {\hat{η}}_{1}]}^{2}}{P^{2} (S_{i} = 0 | Y_{i} (R, B))}) \\ + \frac{1}{(n_{3}^{* *} + n_{4}^{* *}) (n_{3}^{* *} + n_{4}^{* *} - 1)} (\sum_{i = 1}^{n_{3}} \frac{(1 - S_{i}) {[Y_{1 i} (W, A) - Y_{2 i} (W, B) - {\hat{η}}_{2}]}^{2}}{P^{2} (S_{i} = 0 | Y_{i} (W, A))} \\ + \sum_{i = 1}^{n_{4}} \frac{(1 - S_{i}) {[Y_{2 i} (W, A) - Y_{1 i} (W, B) - {\hat{η}}_{2}]}^{2}}{P^{2} (S_{i} = 0 | Y_{i} (W, B))}) . \end{aligned}

(38)

We next combine the two estimates and their respective variance to test the hypothesis as follow,

\begin{aligned} \hat{d} = \frac{\frac{{\hat{d}}_{1}}{{\hat{v}}_{1}} + \frac{{\hat{d}}_{2}}{{\hat{v}}_{2}}}{\frac{1}{{\hat{v}}_{1}} + \frac{1}{{\hat{v}}_{2}}} . \end{aligned}

(39)

The variance of

\hat{d}

can be estimated by

\begin{aligned} \hat{v} = \frac{{\hat{v}}_{1} {\hat{v}}_{2}}{{\hat{v}}_{1} + {\hat{v}}_{2}} . \end{aligned}

(40)

The test statistic is

\begin{aligned} \frac{\hat{d}}{\sqrt{\hat{v}}} \sim N (0, 1), \end{aligned}

(41)

Under

H_{0}

. Sample size or power can be determined by

\begin{aligned} power = P (\frac{\hat{d} - d}{\sqrt{\hat{v}}} > z_{1 - \frac{α}{2}} - \frac{d}{\sqrt{\hat{v}}} | H_{A}) + P (\frac{\hat{d} - d}{\sqrt{\hat{v}}} < - z_{1 - \frac{α}{2}} - \frac{d}{\sqrt{\hat{v}}} | H_{A}) . \end{aligned}

(42)

\begin{aligned} power = 1 - Φ (z_{1 - \frac{α}{2}} - \frac{d}{\sqrt{\hat{v}}}) + Φ (- z_{1 - \frac{α}{2}} - \frac{d}{\sqrt{\hat{v}}}) . \end{aligned}

(43)

Approximately,

\begin{aligned} \frac{1}{\hat{v}} = \frac{1}{{\hat{v}}_{1}} + \frac{1}{{\hat{v}}_{2}} = {(\frac{z_{1 - \frac{α}{2}} + z_{1 - β}}{d})}^{2} . \end{aligned}

(44)

The total sample size is determined by

\begin{aligned} n = {(\frac{z_{1 - \frac{α}{2}} + z_{1 - β}}{d})}^{2} {(\frac{1}{n {\hat{v}}_{1}} + \frac{1}{n {\hat{v}}_{2}})}^{- 1} = {(\frac{z_{1 - \frac{α}{2}} + z_{1 - β}}{d})}^{2} \frac{n {\hat{v}}_{1} n {\hat{v}}_{2}}{n {\hat{v}}_{1} + n {\hat{v}}_{2}} . \end{aligned}

(45)

For a fixed genotype G, and a fixed first-stage treatment

T_{1}

, let

\begin{aligned} P (S = 1 ∣ G, T_{1}) = E [P {S = 1 ∣ Y (G, T_{1})}] . \end{aligned}

(46)

We have

\begin{aligned} \sum_{i = 1}^{n_{1}} \frac{S_{i} {[Y_{i} (G, T_{1}) - \bar{Y} (G, T_{1})]}^{2}}{P^{2} (S_{i} = 1 | Y_{i} (G, T_{1}))} = \frac{1}{P (S = 1 | G, T_{1})} \sum_{i = 1}^{n_{1}} \frac{S_{i}}{P (S_{i} = 1 | Y_{i} (G, T_{1}))} \frac{P (S_{i} = 1 | G, T_{1}) {[Y_{i} (G, T_{1}) - \bar{Y} (G, T_{1})]}^{2}}{P (S_{i} = 1 | Y_{i} (G, T_{1}))}, \end{aligned}

(47)

which approximates

\begin{aligned} \frac{1}{P (S = 1 | G, T_{1})} \sum_{i = 1}^{n_{1}} {[Y_{i} (G, T_{1}) - \bar{Y} (G, T_{1})]}^{2} \end{aligned}

(48)

when

P (S = 1 ∣ G, T_{1})

approximates

P {S = 1 ∣ Y (G, T_{1})}

. Similarly, let

\begin{aligned} P (S = 0 ∣ G, T_{1}) = E [P {S = 0 ∣ Y (G, T_{1})}] . \end{aligned}

(49)

We have

\begin{aligned} \sum_{i = 1}^{n_{1}} \frac{(1 - S_{i}) {[Y_{1 i} (G, T_{1}) - Y_{2 i} (G, T_{1}) - \hat{η}]}^{2}}{P^{2} (S_{i} = 0 | Y_{i} (G, T_{1}))} \end{aligned}

(50)

is approximated by

\begin{aligned} \frac{1}{P (S = 0 | G, T_{1})} \sum_{i = 1}^{n_{1}} {[Y_{1 i} (G, T_{1}) - Y_{2 i} (G, T_{1}) - \hat{η}]}^{2} \end{aligned}

(51)

when

P (S = 0 ∣ G, T)

approximates

P {S = 0 ∣ Y (G, T)}

, where

\hat{η}

is either

{\hat{η}}_{1}

{\hat{η}}_{2}

according to the values

G, T

take. With these approximations, we have

\begin{aligned} n {\hat{v}}_{1} & \approx v_{1} [\frac{1}{r P (G = R) P (S = 1 | R, A)} + \frac{1}{(1 - r) P (G = R) P (S = 1 | R, B)} + \frac{1}{r P (G = W) P (S = 1 | W, A)} \\ + \frac{1}{(1 - r) P (G = W) P (S = 1 | W, B)}] \end{aligned}

(52)

and

\begin{aligned} n {\hat{v}}_{2} & \approx v_{2} [\frac{r}{P (G = R) P (S = 0 | R, A)} + \frac{1 - r}{P (G = R) P (S = 0 | R, B)} \\ + \frac{r}{P (G = W) P (S = 0 | W, A)} + \frac{1 - r}{P (G = W) P (S = 0 | W, B)}] . \end{aligned}

(53)

To carry out sample size or power calculation, we need to specify

(1)

$d$ : The interaction effect $μ_{A \times R}^{T G} - μ_{B \times R}^{T G} - μ_{A \times W}^{T G} + μ_{B \times W}^{T G}$ ,

(2)

$α$ : Type I error

(3)

$β$ : Type II error for sample size calculation, o r

(4)

$n$ : Total sample size for power calculation.

(5)

$r$ : The proportion of subjects assigned to treatment A,

(6)

$P (G = R)$ : The prevalence of the risk allele,

(7)

$P (S = 1 | R, A), P (S = 1 | R, B), P (S = 1 | W, A), P (S = 1 | W, B)$ : Proportions of not switching for each treatment-by-gene group,

(8)

$v_{1}$ : The error variance for $ϵ$

(9)

$ρ$ : The intra-patient correlation for the calculation of $v_{2} = 2 (1 - ρ) v_{1}$ .

Items 5–9 allow us to compute

n {\hat{v}}_{1}

and

n {\hat{v}}_{2}

using expressions (52) and (53), respectively. Items 1–3 along with

n {\hat{v}}_{1}

and

n {\hat{v}}_{2}

allow us to calculate the sample size by expression (45).

n {\hat{v}}_{1}

and

n {\hat{v}}_{2}

allow us to compute v using expression (40). Items 1–2 and 4 along with v allow us to calculate power by expression (43).

4 Application to a trial for gene-drug interaction in treating atrial fibrillation

4.1 Basic assumption for the trial

Atrial fibrillation (AF), the most common sustained cardiac arrhythmia in clinical practice, is associated with significant morbidity and increased mortality. Antiarrhythmic drugs (AADs) are commonly used to treat symptomatic AF. However, response in an individual patient is highly variable and AADs can be associated with adverse events. The limited success of AADs is related to heterogeneity of the electrical substrate, limited understanding of the underlying mechanisms, and our inability to predict responses to therapy in individual patients. Nonetheless, recent advances in our understanding of the genetic mechanisms of AF support the hypothesis that variability in response to antiarrhythmic therapy is modulated by common genetic variants associated with AF. Genetic approaches to AF have revealed that susceptibility to and response to therapy is modulated in part by the underlying genetic substrate. We showed that a common chromosome (chr) 4q25 single nucleotide polymorphism (SNP) (rs10033464) not only predicted successful control of AF symptoms but that patients who carried the risk allele responded better to Na ⁺-channel (class I) than K ⁺-channel blocker (class III) AADs. Our ongoing study is testing the hypothesis that a common chr4q25 SNP modulates response to AADs in patients with frequent symptomatic AF assessing burden as a measure of therapeutic efficacy. We propose a prospective randomized pharmacogenomic cross-over study that will determine if there is an interaction between a chr4q25 SNP and reduction in AF burden when patients are treated with flecainide (class I) versus sotalol (class III) AADs for 6-months each across race-ethnicities. The class III drug serves as a baseline and the class I drug is more effective for patients carrying the risk allele. Based on the pilot study,²⁰ the mean AF burden on flecainide (class I) was 20 ± 14% (SD). A mean reduction in AF burden of 10% in chr4q25 AF risk allele versus wild-type allele carriers may be clinically meaningful assuming that half of the patients will receive flecainide first and the other half sotalol first. As the minor allele frequency of rs10033464 is 12%, the ratio of carriers to non-carriers of the risk allele is 1:8. The variation of the AF burden over time within a patient is assumed to be 50% of the inter-patient variation.

Based on the assumptions, we can derive the following quantities for the sample size calculation.

(1)
$P (G = R) = 0.12$ and $P (G = W) = 0.88$ .
(2)
$μ_{I}^{T} + μ_{W}^{G} + μ_{I \times W}^{T G} = 0.2$ with $v_{1} = V a r (ϵ) = {0.14}^{2}$ . Furthermore, $μ_{I I I}^{T} + μ_{W}^{G} + μ_{I I I \times W}^{T G} = 0.2$ and $μ_{I I I}^{T} + μ_{R}^{G} + μ_{I I I \times R}^{T G} = 0.2$ .
(3)
$d = μ_{I \times R}^{T G} - μ_{I I I \times R}^{T G} - μ_{I \times W}^{T G} + μ_{I I I \times W}^{T G} = 0.1$ as clinically effective treatment genotype interactions .
(4)
Equal sample size is assumed for each treatment group, i.e., $r = 0.5$ .
(5)
Intra-patient correlation is assumed to be $ρ = 0.5$ .

4.2 Sample size calculation for the parallel design

The formula for the total sample size is

\begin{aligned} n = {(\frac{z_{1 - \frac{α}{2}} + z_{1 - β}}{d})}^{2} v_{1} (\frac{1}{r (1 - r) P (G = R)} + \frac{1}{r (1 - r) P (G = W)}) . \end{aligned}

(54)

With type I error 0.05, power 0.80, and variance

v_{1} = {0.14}^{2}

and equal sample size, i.e.,

r = 1 / 2

\begin{aligned} n = {(\frac{1.96 + 0.84}{0.1})}^{2} {0.14}^{2} * 4 * (\frac{1}{0.12} + \frac{1}{0.88}) = 582. \end{aligned}

(55)

4.3 Sample size calculation for the cross-over design

When the intra-patient correlation is $ρ = 0.5$ , we have $v_{2} = v_{1}$ . The sample can be determined by

\begin{aligned} n = {(\frac{z_{1 - \frac{α}{2}} + z_{1 - β}}{d})}^{2} v_{2} (\frac{1}{P (G = R)} + \frac{1}{P (G = W)}) . \end{aligned}

(56)

With type I error 0.05, power 0.80, and variance

v_{2} = {0.14}^{2}

and equal sample size, i.e.,

r = 1 / 2

\begin{aligned} n = {(\frac{1.96 + 0.84}{0.1})}^{2} * {0.14}^{2} * (\frac{1}{0.12} + \frac{1}{0.88}) = 146. \end{aligned}

(57)

4.4 Sample size calculation for the adaptive design

We need to obtain the success probability of each drug under different genotypes. This is estimated from the preliminary data on the treatment response rates. Table 6 lists the hypothetical response rates for different treatment and genotype combinations.

With the given response rates, we have

\begin{aligned} n {\hat{v}}_{1} & = (\frac{1}{0.12 * 0.50} + \frac{1}{0.12 * 0.50} + \frac{1}{0.88 * 0.6} + \frac{1}{0.88 * 0.3}) * {0.14}^{2} * 2 = 1.5294. \end{aligned}

(58)

\begin{aligned} n {\hat{v}}_{2} & = (\frac{1}{0.12 * 0.50} + \frac{1}{0.12 * 0.50} + \frac{1}{0.88 * 0.4} + \frac{1}{0.88 * 0.7}) * \frac{{0.14}^{2}}{2} = 0.3704. \end{aligned}

(59)

The sample size is

\begin{aligned} n = {(\frac{1.96 + 0.84}{0.1})}^{2} {(\frac{1}{1.5294} + \frac{1}{0.3704})}^{- 1} = 234. \end{aligned}

(60)

Table 6.

Response rates of treatment-gene combinations.

Cohort	Genotype = W	Genotype = R
Class I drug	50%	60%
Class III drug	50%	30%

Table 7.

Simulation settings and results for $(p_{r}, p_{a}) = (0.12, 0.88)$ .

$(μ_{0}, μ_{r}^{G}, μ_{a}^{T}, μ_{r a}^{G T})$	$(σ_{0}^{2}, σ_{1}^{2})$	$(α_{0}, α_{1})$	SS	Selection probability	Power achieved
(0.1, 0.1, 0, −0.025)	$({0.14}^{2}, {0.14}^{2}) / 2$	(0, 0)	234	(0.5, 0.5, 0.5, 0.5)	(0.45, 0.95, 0.82)
		(0, 1)	242	(0.54, 0.51, 0.56, 0.49)	(0.47, 0.96, 0.82)
		(0, −1)	228	(0.46, 0.49, 0.44, 0.51)	(0.46, 0.94, 0.82)
		(1, −1)	314	(0.69, 0.73, 0.68, 0.74)	(0.56, 0.99, 0.82)
		(1, 0)	324	(0.73, 0.73, 0.73, 0.73)	(0.57, 0.99, 0.81)
		(−1, 1)	188	(0.31, 0.27, 0.32, 0.26)	(0.39, 0.88, 0.80)
		(−1, 2)	192	(0.35, 0.28, 0.37, 0.26)	(0.40, 0.91, 0.83)
		(−2, 2)	164	(0.16, 0.13, 0.18, 0.12)	(0.37, 0.86, 0.82)
	$({0.14}^{2}, 3 * {0.14}^{2}) / 4$	(0, 0)	318	(0.5, 0.5, 0.5, 0.5)	(0.57, 0.92, 0.82)
		(0, 1)	326	(0.54, 0.51, 0.56, 0.49)	(0.59, 0.93, 0.82)
		(0, −1)	312	(0.46, 0.49, 0.44, 0.51)	(0.55, 0.92, 0.82)
		(1, −1)	394	(0.69, 0.73, 0.68, 0.74)	(0.65, 0.96, 0.82)
		(1, 0)	404	(0.73, 0.73, 0.73, 0.73)	(0.66, 0.97, 0.81)
		(−1, 1)	268	(0.31, 0.27, 0.32, 0.26)	(0.50, 0.87, 0.81)
		(−1, 2)	274	(0.35, 0.28, 0.37, 0.26)	(0.51, 0.87, 0.80)
		(−2, 2)	242	(0.16, 0.13 0.18, 0.12)	(0.47, 0.86, 0.82)
(0.2, 0.1, 0.1, 0.025)	$({0.14}^{2}, {0.14}^{2}) / 2$	(0, 0)	234	(0.5, 0.5, 0.5, 0.5)	(0.47, 0.94, 0.81)
		(0, 1)	256	(0.60, 0.54, 0.54, 0.51)	(0.49, 0.96, 0.81)
		(0, −1)	218	(0.40, 0.46, 0.46, 0.49)	(0.45, 0.92, 0.82)
		(1, −1)	296	(0.64, 0.69, 0.69, 0.73)	(0.54, 0.98, 0.82)
		(1, 0)	324	(0.73, 0.73, 0.73, 0.73)	(0.57, 0.99, 0.82)
		(−1, 1)	194	(0.36, 0.31, 0.31, 0.27)	(0.41, 0.91, 0.81)
		(−1, 2)	210	(0.46, 0.35, 0.35, 0.28)	(0.43, 0.92, 0.81)
		(−2, 2)	172	(0.24, 0.16, 0.16, 0.13)	(0.37, 0.86, 0.81)
	$({0.14}^{2}, 3 * {0.14}^{2}) / 4$	(0, 0)	318	(0.5, 0.5, 0.5, 0.5)	(0.57, 0.93, 0.82)
		(0, 1)	342	(0.60, 0.54, 0.54, 0.51)	(0.61, 0.95, 0.82)
		(0, −1)	302	(0.40, 0.46, 0.46, 0.49)	(0.55, 0.91, 0.81)
		(1, −1)	380	(0.64, 0.69, 0.69, 0.73)	(0.65, 0.96, 0.82)
		(1, 0)	404	(0.73, 0.73, 0.73, 0.73)	(0.65, 0.96, 0.81)
		(−1, 1)	276	(0.36, 0.31, 0.31, 0.27)	(0.52, 0.89, 0.82)
		(−1, 2)	294	(0.46, 0.35, 0.35, 0.28)	(0.54, 0.91, 0.81)
		(−2, 2)	252	(0.24, 0.16, 0.16, 0.13)	(0.48, 0.85, 0.79)

SS: Sample size determined by the formula for the adaptive design with type I error 0.05 and power 80%; selection probability: The averaged selection probability for the four treatment categories; Power achieved: Empirical power in the simulation respectively for the parallel design, the crossover design, and the adaptive design.

5 Simulation study to validate the power of the adaptive design

We perform a simulation study to examine the accuracy of the sample size determined by the approximate formula for practical use. The simulation generates data using the same models as given in Table 2 for the two outcomes. We assume that $e_{0} \sim N (0, σ_{0}^{2})$ , $e_{1} \sim N (0, σ_{1}^{2})$ , $e_{2} \sim N (0, σ_{1}^{2})$ , and are independent. We also assume that the decision on whether to switch the treatment depends on the first-stage outcomes in the following way:

\begin{aligned} P (S = 1 ∣ Y_{i 1}, Y_{i 2}) = \frac{e x p (α_{0} + α_{1} Y_{i 1})}{1 + e x p (α_{0} + α_{1} Y_{i 1})} . \end{aligned}

(61)

We first use the sample size formula given in the previous section to determine the required sample size for achieving the required power under alternative hypothesis. Next, we generate data with the sample size as determined in the simulation and perform the test based on the simulated data to compute the power actually achieved. By comparing the planned power and the actual power achieved, we evaluate the reliability of the proposed sample size formula.

Assuming a two-sided test with type I error 0.05 and targeted power of 80%, the results of the simulation study are shown in Table 7. The simulation results are based on 5000 replicates of each scenario. From the table, we see that in almost all the cases, the actual power achieved based on the sample sizes determined by the formula is very close to the planned power for the adaptive design. The cross-over design is more powerful than the adaptive design, while the parallel design is less powerful than the adaptive design. The simulation results validate the reliability of the sample size determination formula.

6 Discussion

The adaptive crossover trial provides patients the reassurance that should the initial AAD be efficacious in treating their AF, there is no necessity for the other class of ADD. While the design is less powerful than a crossover design, it is more powerful than the parallel trial in detecting a gene-drug interaction.

Depending on the success rate in the first stage trial, the sample size reduction can be substantial in comparison with that required by a parallel design. The adaptive design, however, requires stronger conditions to make the analysis valid. The primary reason is that patients making the treatment switch can be systematically different from patients who choose not to switch. For the analysis, we assume that the first-stage potential outcomes do not affect the potential outcomes in the second stage. The estimated interaction effect can be biased if the condition is violated.

The analysis of the trial can be fit into the longitudinal data framework through the likelihood estimation approach. Such an analysis framework can be more efficient than the multiple samples approach taken in this article. This means the required sample size can be smaller to achieve the same level of statistical power. However, the likelihood approach requires stronger modeling assumptions than those for the multiple-sample approach taken in this article. The latter is more robust against possible model misspecification.

The sample size formula is an approximation. When concerns about the impact of the approximation $P (S = s ∣ G, T_{1}) \approx P {S = s ∣ Y (G, T_{1})}$ on the variance estimate arise, a Monte Carlo simulation approach may be used instead. However, the Monte Carlo simulation approach is not necessarily more reliable than the approximation because more specific models need to be assumed for the simulation. There is no guarantee that such models hold in practice. It is usually not a concern if the condition for the approximation is not radically violated as the simulation results in this paper provide much assurance for the reliability of the proposed formula.

Supplemental Material

sj-r-1-smm-10.1177_09622802231181704 - Supplemental material for Sample size determination for adaptive crossover trial in detecting gene-drug interactions

Supplemental material, sj-r-1-smm-10.1177_09622802231181704 for Sample size determination for adaptive crossover trial in detecting gene-drug interactions by Hua Yun Chen and Dawood Darbar in Statistical Methods in Medical Research

Supplemental Material

sj-r-2-smm-10.1177_09622802231181704 - Supplemental material for Sample size determination for adaptive crossover trial in detecting gene-drug interactions

Supplemental material, sj-r-2-smm-10.1177_09622802231181704 for Sample size determination for adaptive crossover trial in detecting gene-drug interactions by Hua Yun Chen and Dawood Darbar in Statistical Methods in Medical Research

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Foundation for the National Institutes of Health (grant number 5R01HL148444).

ORCID iD

Hua Yun Chen

Supplemental material

Supplemental material for this article is available online.

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