Impact of incorrect designation of working correlation structure matrix on sample size estimation in 2×2 cross design: a simulation study

doi:10.12122/j.issn.1673-4254.2025.11.22

Abstract

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

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.

Figures/Tables 8

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
$x i 11 … x i 1 M$	$x i j 1 * 01 … x i j M * 01$	-	$x i j 1 * 01 … x i j M * 01$	$x i j 1 * 00 … x i j M * 00$	-	$x i j 1 * 00 … x i j M * 00$
$x i 21 … x i 2 M$	-	$x i j 1 * 10 … x i j M * 10$	$x i j 1 * 10 … x i j M * 10$	-	$x i j 1 * 11 … x i j M * 11$	$x i j 1 * 11 … x i j M * 11$

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
$x i 11 … x i 1 M$	$x i j 1 * 01 … x i j M * 01$	-	$x i j 1 * 01 … x i j M * 01$	$x i j 1 * 00 … x i j M * 00$	-	$x i j 1 * 00 … x i j M * 00$
$x i 21 … x i 2 M$	-	$x i j 1 * 10 … x i j M * 10$	$x i j 1 * 10 … x i j M * 10$	-	$x i j 1 * 11 … x i j M * 11$	$x i j 1 * 11 … x i j M * 11$

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

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

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

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

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

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

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

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