2×2交叉设计中错误指定工作相关结构矩阵对样本量估计值的影响

doi:10.12122/j.issn.1673-4254.2025.11.22

摘要/Abstract

摘要：

目的在2×2交叉设计中基于广义估计模型，探究错误指定工作相关结构矩阵对样本量估计值的影响。方法基于蒙特卡罗模拟，控制模拟时总样本量大小n，分配到AB序列（s=1）的受试者比例 $θ$ ，相关系数 $ρ$ 和安慰剂效果OR四个因素后，探究不同条件下错误指定工作相关结构矩阵对样本量估计值的影响。评价样本量估计值和理论值之间差异的指标采用偏差和均方误差。结果当正确工作相关结构矩阵为独立时，正确指定工作相关结构矩阵的样本量估计效果优于错误指定工作相关结构矩阵的样本量估计效果。但是当正确工作相关结构矩阵为等相关时，相关系数越接近0的前提下，其他因素越小（本文中 $n ≤ 50, θ ≤ 0.5, O R = 2$ ），出现了正确指定工作相关结构矩阵后样本量估计值对理论值的bias大于错误指定工作相关结构矩阵的bias的情况。结论大多数条件下错误指定工作相关结构矩阵会使样本量估计值远偏离理论值，但是一定条件下错误指定工作相关结构矩阵对样本量估计值的影响较小。

关键词: 2×2交叉试验, 广义估计设计, 工作相关结构矩阵, 样本量估计

Abstract:

Objective To investigate the impact of incorrect specification of the working correlation structure matrix on estimated sample size in a 2×2 crossover design based on the generalized estimating equation (GEE). Methods Based on Monte Carlo simulation, the influence of incorrect specification of the work-related structure matrix on the sample size estimation under different conditions was evaluated after controlling the total sample size n, the proportion of subjects assigned to AB sequence (s=1) θ, the correlation coefficient ρ, and the placebo effect OR. Bias and mean square error (MSE) were used to assess the difference between the sample size estimates and the theoretical values. Results When the correctly specified working correlation structure matrix is independent, the sample size estimation effect of correctly specifying the working correlation structure matrix is better than that of incorrect specification. But when the correctly specified working correlation structure matrix is equal and the correlation coefficient is closer to 0, with other factors being smaller (n≤50, θ≤0.5, OR=2 in this article), there is a situation where the bias of the sample size estimation value for the correctly specified working correlation structure matrix is greater than the bias for the incorrectly specified working correlation structure matrix. Conclusion Under most conditions, incorrectly specifying the working correlation structure matrix can cause the estimated sample size to deviate significantly from the theoretical value, but under certain conditions, the impact of incorrectly specifying the working correlation structure matrix can be small on the estimated sample size.

Key words: 2×2 crossover trials, generalized estimation equation design, working correlation structure matrix, sample size estimation

张佩瑜, 谢子恒, 庄严. 2×2交叉设计中错误指定工作相关结构矩阵对样本量估计值的影响[J]. 南方医科大学学报, 2025, 45(11): 2495-2503.

Peiyu ZHANG, Ziheng XIE, Yan ZHUANG. Impact of incorrect designation of working correlation structure matrix on sample size estimation in 2×2 cross design: a simulation study[J]. Journal of Southern Medical University, 2025, 45(11): 2495-2503.

图/表 8

表1 每个时期的单个时间点收集结果的交叉试验中的权重和协变量搭配

Tab.1 Weights and covariate configurations in crossover trials with outcomes collected at a single time point in each period

Item	Sequence AB (s=1)			Sequence BA (s=2)
Item	Period 1	Period 2	Both period	Period 1	Period 2	Both period
Weights^a
	$w 11 = θ 1 π 1 (1)$	$w 12 = θ 1 π 2 (1)$	$w 13 = θ 1 π 12 (1)$	$w 21 = θ 2 π 1 (2)$	$w 22 = θ 2 π 2 (2)$	$w 23 = θ 2 π 12 (2)$
Covariates for binary
$x i 1$	$101$	-	$101$	$100$	-	$100$
$x i 2$	-	$110$	$110$	-	$111$	$111$
Covariates for ordinal outcome with $L$ levels^b
$x i 11 … x i 1 M$	$x i j 1 * 01 … x i j M * 01$	-	$x i j 1 * 01 … x i j M * 01$	$x i j 1 * 00 … x i j M * 00$	-	$x i j 1 * 00 … x i j M * 00$
$x i 21 … x i 2 M$	-	$x i j 1 * 10 … x i j M * 10$	$x i j 1 * 10 … x i j M * 10$	-	$x i j 1 * 11 … x i j M * 11$	$x i j 1 * 11 … x i j M * 11$

表1 每个时期的单个时间点收集结果的交叉试验中的权重和协变量搭配

Tab.1 Weights and covariate configurations in crossover trials with outcomes collected at a single time point in each period

Item	Sequence AB (s=1)			Sequence BA (s=2)
Item	Period 1	Period 2	Both period	Period 1	Period 2	Both period
Weights^a
	$w 11 = θ 1 π 1 (1)$	$w 12 = θ 1 π 2 (1)$	$w 13 = θ 1 π 12 (1)$	$w 21 = θ 2 π 1 (2)$	$w 22 = θ 2 π 2 (2)$	$w 23 = θ 2 π 12 (2)$
Covariates for binary
$x i 1$	$101$	-	$101$	$100$	-	$100$
$x i 2$	-	$110$	$110$	-	$111$	$111$
Covariates for ordinal outcome with $L$ levels^b
$x i 11 … x i 1 M$	$x i j 1 * 01 … x i j M * 01$	-	$x i j 1 * 01 … x i j M * 01$	$x i j 1 * 00 … x i j M * 00$	-	$x i j 1 * 00 … x i j M * 00$
$x i 21 … x i 2 M$	-	$x i j 1 * 10 … x i j M * 10$	$x i j 1 * 10 … x i j M * 10$	-	$x i j 1 * 11 … x i j M * 11$	$x i j 1 * 11 … x i j M * 11$

表2 数据结构

Tab.2 Data structure

Id	Period	Treatment	s	Y
1	0	1	1	0
1	1	0	1	0
2	0	1	1	0
2	1	0	1	0
︙	︙	︙	︙	︙
49	0	0	2	0
49	1	1	2	0
50	0	0	2	0
50	1	1	2	0

图1 关于分别固定n和θ后随着ρ的变化bias的变化趋势

Fig.1 Changes of bias with ρ after fixing n and θ.

图2 关于分别固定n和θ后随着ρ的变化log(MSE)的变化趋势

Fig.2 Changes of log(MSE) with ρ after fixing n and θ.

图3 关于分别固定ρ和OR后随θ的变化bias的变化趋势

Fig.3 Changed of bias with θ after fixing ρ and OR.

图4 关于分别固定ρ和OR后随θ的变化log(MSE)的变化趋势

Fig.4 Changes of log(MSE) with θ after fixing ρ and OR.

图5 关于分别固定n和θ后随着ρ的变化bias的变化趋势

Fig.5 Change of bias with ρ after fixing n and θ.

图6 关于分别固定n和θ后随着ρ的变化log(MSE)的变化趋势

Fig.6 Changes of log(MSE) with ρ after fixing n and θ.

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