A note on point estimation and interval estimation of the relative treatment effect under a simple crossover design

Abstract

To increase power or reduce the number of patients needed for a parallel groups design, the crossover design has been often used to study treatments for noncurable chronic diseases. However, in the presence of carry-over effect caused by treatments, the commonly-used estimator which ignores the carry-over effect leads to a biased estimator for estimating the treatment effect difference. A two-stage test approach aimed to address carry-over effect proposed was found to be potentially misleading. In this paper, we propose a weighted average of the commonly-used estimator and an unbiased estimator that uses only the first period of the data. We derive an optimal weight that minimizes the mean squared error (MSE) and its modified estimator. We apply Monte Carlo simulation to evaluate the performance of the proposed estimators in a variety of situations. In the simulations, we examine the estimated MSE (EMSE), percentile interval length, and coverage probability calculated from the percentile intervals among considered estimators. Simulation results show that our proposed weighted average estimator and its modified estimator lead to smaller EMSEs on average comparing to the two commonly used estimators. The coverage probabilities using our proposed estimators are reasonably close to the nominal confidence level and the interval lengths are shorter comparing to the use of the unbiased estimator that uses only the first period of the data. We apply an example that was to evaluate the efficacy of two type of bronchodilators for asthma treatment to demonstrate the use of the proposed estimators.

1 |. INTRODUCTION

To increase power or reduce the number of patients needed for a parallel groups design,^1,2,3 the crossover design has been often used to study treatments for noncurable chronic diseases, because each patient serves as his/her own control. Following Senn³ and Fleiss,^2,4 we should not employ the crossover design if we cannot employ an adequate washout period to assure that patients are weaned off the residual effects from earlier treatments. However, we can never be certain that the washout period has worked. Grizzle⁵ proposed a two-stage test procedure as follows. We first test whether the carry-over effect exists. Because the test itself for the carry-over effect is subject to the response variation between patients, the power of this test is generally low and thereby a high-nominal level of Type I error (10% or 15%) is usually chosen. If the test for the carry-over effect is nonsignificant, we will carry on analyses based on the difference between responses within patients as done for the crossover trial with assuming no carry-over effects. Otherwise, the test procedure using the data at the first period only is carried out as for the parallel groups design and all data obtained at the second period are excluded from data analysis. Freeman⁶ carried out a thorough investigation of the two-stage test approach and concluded that this approach could be potentially misleading. This is because the test for detecting the carry-over effect is highly correlated to that for testing equality based on the data at the first period. Thus, the test based on the data at the first period is likely to be significant as well when the test result for the carry-over effect is significant. This leads that the actual Type I error rate for the two-stage test is higher than the nominal α - level. Due to this concern, we do not want to use the two-stage test procedure in practice.^3,7,8,9 Other notes in use of the two-stage test in a crossover trial can be found elsewhere.^10,11

2 |. NOTATION AND METHODS

Consider comparing an experimental treatment B with a standard treatment A (or a placebo) under an AB/BA crossover design. Suppose that we randomly assign n₁ patients to group g = 1 with the treatment-receipt sequence A-then-B, in which patients receive treatment A at period 1 and then crossover to receive treatment B at period 2, and n₂ patients to group g = 2 with the treatment-receipt sequence B-then-A, in which patients receive treatment B at period 1 and then crossover to receive treatment A at period 2. For patient i (i=1, 2, ⋯, n_g) assigned to group g (g = 1, 2), let $Y_{iz}^{(g)}$ denote the patient response at period z (= 1, 2). We assume that $Y_{iz}^{(g)}$ can be expressed by the following random effects linear additive risk model:

$$Y_{iz}^{(\text{g})}={\mu }_i^{(\text{g})}+{\eta }_{BA}{X}_{iz}^{(\text{g})}+\gamma Z_{iz}^{(\text{g})}+{\lambda }_A{1}_{\left\{g=1\right\}}{Z}_{iz}^{(\text{g})}+{\lambda }_B\left(1-1_{\left\{\text{g}=1\right\}}\right)Z_{iz}^{(\text{g})}+{\epsilon }_{iz}^{(\text{g})},$$ (1)

where ${\mu }_i^{(g)}$ denotes the random effect due to the ith patient in group g and all ${\mu }_i^{(\text{g})}\text{ 's }$ are assumed to be independent and identically distributed as an unspecified probability density function f_g(μ) with variance ${\sigma }_u^2;X_{iz}^{(\text{g})}$ denotes the treatment-received covariate for treatment B, and $Z_{iz}^{(g)}=1$ if the ith patient in group g at period z receives treatment B, and = 0, otherwise; $Z_{iz}^{(g)}$ represents the period covariate, and $Z_{iz}^{(g)}=1$ for period z = 2, and = 0, otherwise; 1_{g=1} is the indicator function of group 1, and = 1 for group g = 1 and = 0, otherwise; and the random errors ${\epsilon }_{iz}^{(g)}\text{'}s$ are assumed to be independent and identically distributed as a continuous distribution with mean 0 and variance ${\sigma }_e^2$, and are assumed to be also independent of ${\mu }_i^{(g)}\text{'}s$. Parameters η_BA and γ sin model (1) denote the difference in effects between treatments B and A, and that between periods 2 and 1, respectively. Furthermore, λ_A and λ_B in model (1) represent the carry-over effects due to treatments A and B. Under model (1), we can see that the covariance between $Y_{i1}^{(g)}$ and $Y_{i2}^{(g)}$ is $\mathit{\text{cov}}\left(Y_{i1}^{(g)},Y_{i2}^{(g)}\right)=\mathit{\text{Var}}\left({\mu }_i^{(g)}\right)={\sigma }_u^2>0$ and thereby, $Y_{i1}^{(g)}$ and $Y_{i2}^{(g)}$ are positively correlated with the intraclass correlation $\rho ={\sigma }_u^2/\left({\sigma }_u^2+{\sigma }_e^2\right)$. Thus, the larger the variation ${\sigma }_u^2$ between patients, the higher is the value of the intraclass correlation between responses within patients.

For patient i (= 1, 2, ⋯, n_g) in group g (= 1, 2), we define $d_i^{(g)}=Y_{i2}^{(g)}-Y_{i1}^{(g)}$, representing the difference in responses between two periods. We further define ${\overline{d}}^{(g)}={\sum }_{i=1}^{n_g}{d}_i^{(g)}/n_g={\overline{Y}}_{+2}^{(g)}-{\overline{Y}}_{+1}^{(g)}$, where ${\overline{Y}}_{+z}^{(g)}={\sum }_{i=1}^{n_g}{Y}_{iz}^{(g)}/n_g$ (for z = 1, 2), as the average of response differences for period 2 versus period 1 over patients in group g. When there are no carry-over effects (i.e., λ_A = λ_B), the following estimator is the most commonly-used unbiased estimator for η_BA under model (1):

$${\widehat{\eta }}_{BA}=\left({\overline{d}}^{(1)}-{\overline{d}}^{(2)}\right)/2,$$ (2)

and its variance.

$$\mathit{\text{Var}}\left({\widehat{\eta }}_{BA}\right)={\sigma }_d^2\left(1/n_1+1/n_2\right)/4,$$ (3)

where ${\sigma }_d^2=\mathit{\text{Var}}\left(d_i^{(\text{g})}\right)$. We can estimate the variance ${\sigma }_d^2$ by the unbiased pooled-sample variance.^4,5

$${\widehat{\sigma }}_d^2={\sum }_{g=1}^2{\sum }_{i=1}^{n_g}{\left(d_i^{(g)}-{\overline{d}}^{(g)}\right)}^2/\left(n_{+}-2\right),$$ (4)

where n₊ = n₁ + n₂ denotes the total number of patients in the trial. Therefore, when substituting ${\widehat{\sigma }}_d^2$ (4) for ${\sigma }_d^2$ in $\mathit{\text{Var}}\left({\widehat{\eta }}_{BA}\right)$ (3), we obtain the variance estimator $\widehat{\mathit{\text{Var}}}\left({\widehat{\eta }}_{BA}\right)$. When there are carry-over effects (i.e., λ_A ≠ λ_B), ${\widehat{\eta }}_{BA}$ (2) is known to be a biased estimator of η_BA and has the bias given by.

$$E\left({\widehat{\eta }}_{BA}-{\eta }_{BA}\right)={\lambda }_D/2,$$ (5)

where λ_D = λ_A − λ_B. We can estimate λ_D by the unbiased estimator.

$${\widehat{\lambda }}_D=\left({\overline{Y}}_{+1}^{(1)}+{\overline{Y}}_{+2}^{(1)}\right)-\left({\overline{Y}}_{+1}^{(2)}+{\overline{Y}}_{+2}^{(2)}\right).$$ (6)

The variance of ${\widehat{\lambda }}_D$ is given by

$$\mathit{\text{Var}}\left({\widehat{\lambda }}_D\right)=\left(4{\sigma }_u^2+2{\sigma }_e^2\right)\left(\frac{1}{n_1}+\frac{1}{n_2}\right).$$ (7)

Note the variance can be estimated by the unbiased pooled-sample variance^2,3 and we can obtain the variance estimator $\widehat{\mathit{\text{Var}}}\left({\widehat{\lambda }}_D\right)$. When there are carry-over effects, we can use ${\widehat{\lambda }}_D$ (6) to adjust the bias in (5) to obtain the unbiased estimator only based on the data at period 1 as done for the parallel groups design.⁵ This leads us to consider the following estimator ${\widehat{\eta }}_{PAR}$.

$${\widehat{\eta }}_{\mathit{\text{PAR}}}={\overline{Y}}_{+1}^{(2)}-{\overline{Y}}_{+1}^{(1)},$$ (8)

which is an unbiased estimator for η_BA even in the presence of carry-over effects. The variance for ${\widehat{\eta }}_{PAR}$ (8) is

$$\mathit{\text{Var}}\left({\widehat{\eta }}_{\mathit{\text{PAR}}}\right)={\sigma }^2\left(1/n_1+1/n_2\right),$$ (9)

where ${\sigma }^2={\sigma }_u^2+{\sigma }_e^2$. We can estimate σ² by

$${\widehat{\sigma }}^2={\sum }_{g=1}^2{{\sum }_{i=1}^{n_g}\left(Y_{i1}^{\left(g\right)}-{\overline{Y}}_{+1}^{(g)}\right)}^2/\left(n_{+}-2\right),$$ (10)

and obtain the variance estimator $\widehat{\mathit{\text{Var}}}\left({\widehat{\eta }}_{PAR}\right)={\widehat{\sigma }}^2\left(1/n_1+1/n_2\right)$. Note that Willan and Pater¹² stated that the mean squared error (MSE) $E{\left({\widehat{\eta }}_{BA}-{\eta }_{BA}\right)}^2$ is smaller than $E{\left({\widehat{\eta }}_{PAR}-{\eta }_{BA}\right)}^2$ under the balanced case (i.e., n₁ = n₂ = n) if and only if

$${\lambda }_D^2/4<(1+\rho ){\sigma }^2/n.$$ (11)

However, this equality (11) is generally difficult to apply in practice due to the lack of prior knowledge of the difference λ_D, the intraclass correlation ρ, or the variance σ². Also, the two-stage test procedure¹ has been suggested to determine which estimator ${\widehat{\eta }}_{BA}$ or ${\widehat{\eta }}_{PAR}$ for use. As noted elsewhere,^2,5,12,13 there are concerns, such as the lack of power or the bias of the estimator for the relative treatment effect, in application of this two-stage test procedure as well.

Instead of deciding to choose either ${\widehat{\eta }}_{BA}$ or ${\widehat{\eta }}_{PAR}$ for use based on nonexistent prior knowledge or a powerless hypothesis testing procedure, we may consider a weighted average $W{\widehat{\eta }}_{PAR}+(1-W){\widehat{\eta }}_{BA}$ (where 0 ≤ W ≤ 1), that includes ${\widehat{\eta }}_{PAR}$ and ${\widehat{\eta }}_{BA}$ as special cases. We can show that the optimal weight W₀ to minimize the MSE $E{\left(W{\widehat{\eta }}_{PAR}+(1-W){\widehat{\eta }}_{BA}-{\eta }_{BA}\right)}^2$ is given by

$$W_o=\left[E{\left({\widehat{\eta }}_{BA}-{\eta }_{BA}\right)}^2-E\left({\widehat{\eta }}_{BA}-{\eta }_{BA}\right)\left({\widehat{\eta }}_{PAR}-{\eta }_{BA}\right)\right]/\left[E{\left({\widehat{\eta }}_{BA}-{\eta }_{BA}\right)}^2+E{\left({\widehat{\eta }}_{PAR}-{\eta }_{BA}\right)}^2-2E\left({\widehat{\eta }}_{BA}-{\eta }_{BA}\right)\left({\widehat{\eta }}_{PAR}-{\eta }_{BA}\right)\right].$$ (12)

We can further show that

$$E\left({\widehat{\eta }}_{BA}-{\eta }_{BA}\right)\left({\widehat{\eta }}_{PAR}-{\eta }_{BA}\right)=\left({\sigma }_e^2/2\right)\left(1/n_1+1/n_2\right).$$ (13)

From (3), (7), (9) and (13), we can see that the optimal weight W₀ (12) simplifies to

$$W_0=\left({\lambda }_D^2/4\right)/\left[{\lambda }_D^2/4+{\sigma }^2((1+\rho )/2)\left(1/n_1+1/n_2\right)\right].$$ (14)

Note that the optimal weight W₀ is a function of unknown parameters and hence we cannot apply $W_0{\widehat{\eta }}_{PAR}+\left(1-W_o\right){\widehat{\eta }}_{BA}$ to reduce the MSE of ${\widehat{\eta }}_{PAR}$ and ${\widehat{\eta }}_{BA}$ directly.

We can estimate ${\lambda }_D^2$ by the unbiased estimator

$${\widehat{\lambda }}_D^2-\widehat{\mathit{\text{Var}}}\left({\widehat{\lambda }}_D\right)\mathrm{..}$$ (15)

We can further estimate σ²((1+ρ)/2) by the unbiased estimator

$${\widehat{\sigma }}^2-{\widehat{\sigma }}_d^2/4.$$ (16)

On the basis of (15)–(16), we can estimate the optimal weight W_o by

$${\widehat{W}}_0=\left[\left({\widehat{\lambda }}_D^2-\widehat{\mathit{\text{Var}}}\left({\widehat{\lambda }}_D\right)\right)/4\right]/\left[\left({\widehat{\lambda }}_D^2-\widehat{\mathit{\text{Var}}}\left({\widehat{\lambda }}_D\right)\right)/4+\left({\widehat{\sigma }}^2-{\widehat{\sigma }}_d^2/4\right)\left(1/n_1+1/n_2\right)\right]$$ (17)

and define

$${\widehat{\eta }}_{OP}={\widehat{W}}_0{\widehat{\eta }}_{PAR}+\left(1-{\widehat{W}}_0\right){\widehat{\eta }}_{BA}.$$ (18)

To compare the MSE of ${\widehat{\eta }}_{OP}$ (18) with ${\widehat{\eta }}_{BA}$ (2) and ${\widehat{\eta }}_{PAR}$ (8), we plot MSE versus λ_D. Figure 1 and Figure 2 show the relationship (with true W₀) between MSE and λ_D when n = 50, ρ = 0.2, and σ² = 2 (Figure 1) and n = 50, ρ = 0.4, and σ² = 4 (Figure 2) for ${\widehat{\eta }}_{OP}$, ${\widehat{\eta }}_{BA}$, and ${\widehat{\eta }}_{PAR}$. As expected, both graphs show that the MSE of ${\widehat{\eta }}_{OP}$, is smaller than the MSE of ${\widehat{\eta }}_{PAR}$ and ${\widehat{\eta }}_{BA}$. The value of λ_D does not affect the MSE of ${\widehat{\eta }}_{PAR}$ since ${\widehat{\eta }}_{PAR}$ uses only the first period of the data. When the absolute value of λ_D is large (large carry-over effect difference), the MSE of ${\widehat{\eta }}_{BA}$ increases substantially. In a situation that the carry-over effect difference is 0 (λ_D = 0), the MSE of ${\widehat{\eta }}_{OP}$ and ${\widehat{\eta }}_{BA}$ is the same and the MSE is the smallest. This can be seen from (14) that when λ_D = 0, the value of W₀ = 0 as well. This leads to the equivalence of ${\widehat{\eta }}_{OP}$ and ${\widehat{\eta }}_{BA}$. We further notice that when the MSEs of ${\widehat{\eta }}_{BA}$ and ${\widehat{\eta }}_{PAR}$ are equal (the crossing of the two curves), the reduction of the MSE using ${\widehat{\eta }}_{OP}$ is the largest. In Figure 1, the two MSE curves of ${\widehat{\eta }}_{BA}$ and ${\widehat{\eta }}_{PAR}$ crossed each other when λ_D is near 0.4 or −0.4. This is the λ_D value that the MSE will reduce the most when ${\widehat{\eta }}_{OP}$ is applied.

FIGURE 1.

The relationship (with true W₀) between MSE of ${\widehat{\eta }}_{OP}$ (20), ${\widehat{\eta }}_{BA}$ (2), and ${\widehat{\eta }}_{PAR}$ (9) and λ_D when n = 50 per group, intraclass correlation ρ = 0.2, and random error variance σ² = 2

FIGURE 2.

The relationship (with true W₀) between MSE of ${\widehat{\eta }}_{OP}$ (20), ${\widehat{\eta }}_{BA}$ (2), and ${\widehat{\eta }}_{PAR}$ (9) and λ_D when n = 50 per group, intraclass correlation ρ = 0.4, and random error variance σ² = 4

3 |. MONTE CARLO SIMULATION

To compare the performance of ${\widehat{\eta }}_{OP}$ (18) with ${\widehat{\eta }}_{BA}$ (2) and ${\widehat{\eta }}_{PAR}$ (8), we use Monte Carlo simulation. We consider the situations in which the random errors ${\epsilon }_{iz}^{(g)}$ are assumed to be independently identically distributed as the normal distribution with mean 0 and variance σ² = 1–3 with an increment of 0.5; the effect for treatment A versus treatment B = 0 to 1 with an increment of 0.2; the effect for period 2 versus period 1, γ = 1; the intraclass correlation ρ = 0.1–0.8 with an increment of 0.1; the carry-over effect difference λ_D = λ_A − λ_B = 0–1.5 with an increment of 0.25; and the number of patients per group n (= n1 = n2) = 15, 25, 50, 75, 100, 200. For each configuration determined by a combination of the above parameters, we apply SAS 9.4¹⁴ to generate 1000 simulated samples, each consisting of n observations per group g (= 1, 2). We use these settings to compare the estimated MSEs. From simulation results, we found that the estimate W₀ tends to underestimate when the true W₀ is close to 1 and overestimate when the true W₀ is close to 0. To reduce the cause due to the overestimate (underestimate) near the boundary, we propose a modified estimator of ${\lambda }_D^2$ when we estimate W₀. If $\frac{{\widehat{\lambda }}_D}{{\widehat{\sigma }}^2}<.1$, we use ${\widehat{\lambda }}_D^2$ instead of the unbiased estimator ${\widehat{\lambda }}_D^2-V\widehat{A}R\left({\widehat{\lambda }}_D\right)$ (15) to estimate ${\lambda }_D^2$. On the other hand, if $\frac{{\widehat{\lambda }}_D}{{\widehat{\sigma }}^2}\ge .1$, we adopt the unbiased estimator ${\widehat{\lambda }}_D^2-V\widehat{A}R\left({\widehat{\lambda }}_D\right)$ (15) to estimate ${\lambda }_D^2$. This is to account for potential over-correction (when ${\widehat{\lambda }}_D$ is small) and we may observe a negative ${\widehat{W}}_0$. That is,

$${\widehat{W}}_{01}=\{\begin{array}{l}\left[\left({\widehat{\lambda }}_D^2\right)/4\right]/[\left({\widehat{\lambda }}_D^2\right)/4+\left({\widehat{\sigma }}^2-{\widehat{\sigma }}_d^2/4\right)\left(1/n_1+1/n_2\right),\frac{{\widehat{\lambda }}_D}{{\widehat{\sigma }}^2}<.1\hfill \\ \left[\left({\widehat{\lambda }}_D^2-\widehat{\mathit{\text{Var}}}\left({\widehat{\lambda }}_D\right)\right)/4\right]/\left[\left({\widehat{\lambda }}_D^2-\widehat{\mathit{\text{Var}}}\left({\widehat{\lambda }}_D\right)\right)/4+\left({\widehat{\sigma }}^2-{\widehat{\sigma }}_d^2/4\right)\left(1/n_1+1/n_2\right)\right],\frac{{\widehat{\lambda }}_D}{{\widehat{\sigma }}^2}\ge .1\hfill \end{array}.$$ (19)

Based on ${\widehat{W}}_{01}$, we define the modified estimator as

$${\widehat{\eta }}_{O{P}_{-}a}={\widehat{W}}_{01}{\widehat{\eta }}_{PAR}+\left(1-{\widehat{W}}_{01}\right){\widehat{\eta }}_{BA}.$$ (20)

On top of comparing the performance of EMSE of ${\widehat{\eta }}_{PAR}$, ${\widehat{\eta }}_{BA}$, ${\widehat{\eta }}_{OP}$, and ${\widehat{\eta }}_{OP\_a}$, we further examine the efficiency of the four estimators. It is known that in the presence of the carry-over effects, ${\widehat{\eta }}_{BA}$ is a biased estimator of η_BA and hence, ${\widehat{\eta }}_{OP}$ and ${\widehat{\eta }}_{OP\_a}$ are biased in estimating η_BA as well. We use interval estimation and coverage probability to evaluate the efficiency. We generate bootstrap samples to construct 95% percentile intervals and the coverage probabilities of testing a hypothesized treatment effect difference η_BA based on the constructed 95% percentile intervals through simulations. We generate 500 bootstrap samples for each simulation and calculate the average 95% interval length and the coverage probability of the 1000 simulated samples. We consider the situations in which the random errors ${\epsilon }_{iz}^{(g)}$ are assumed to be independently identically distributed as the normal distribution with mean 0 and variance σ² = 0.5 to 1.5 with an increment of 0.5; the effect for treatment A = 1 versus treatment B = 0; the effect for period 2 versus period 1, γ = 1; the intraclass correlation ρ = 0.3, 0.7; the carry-over effect difference λ_D = λ_A − λ_B = 0 to 1 with an increment of 0.25; and the number of patients per group n (= n1 = n2) = 10, 20, 30. For each configuration determined by a combination of the above parameters, we apply SAS 9.4¹⁴ to generate 1000 simulated samples, each consisting of n observations per group g (= 1, 2).

4 |. RESULTS

To compare simulation results, we summarize in Table 1 the estimated MSE (EMSE) for the four estimators we considered (${\widehat{\eta }}_{BA}$ (2), ${\widehat{\eta }}_{PAR}$ (8), ${\widehat{\eta }}_{OP}$ (18), and ${\widehat{\eta }}_{OP\_a}$ (20)) categorizing by the sample size. Note ${\widehat{\eta }}_{OP\_a}$ has the smallest EMSE in all sample sizes we considered here. When sample size is large, the first period of the data contains enough information and thus, the advantage of adopting either ${\widehat{\eta }}_{OP}$ or ${\widehat{\eta }}_{OP\_a}$ gets smaller. For ${\widehat{\eta }}_{BA}$, the reduction of the EMSE is limited even when n gets larger. This is due to the existence of the carry-over effects and it will not disappear even n becomes bigger. When the sample size is large, the difference of the EMSE among ${\widehat{\eta }}_{OP}$, ${\widehat{\eta }}_{OP\_a}$, and ${\widehat{\eta }}_{PAR}$ becomes smaller.

TABLE 1.

The estimated MSE and standard error (in bracket) organized by sample size n for ${\widehat{\eta }}_{PAR}$, ${\widehat{\eta }}_{BA}$, ${\widehat{\eta }}_{OP}$, and ${\widehat{\eta }}_{OP\_a}$ in situations in which random errors ${\epsilon }_{iz}^{(g)}$ are assumed to be independently identically distributed as the normal distribution with mean 0 and variance σ² = 1–3 with an increment of 0.5; the effect for treatment A versus treatment B = 0–1 with an increment of 0.2; the effect for period 2 versus period 1, γ = 1; the intraclass correlation ρ = 0.1 to 0.8 with an increment of 0.1; the carry-over effect difference λ_D = λ_A − λ_B = 0–1.5 with an increment of 0.25; and the number of patients per group n (= n1 = n2) = 15, 25, 50, 75, 100, 200

N	EMSE_npar (SE)	EMSE_nBA (SE)	EMSE_nOP (SE)	EMSE_nOP_a (SE)
15	0.233(0.115)	0.267(0.200)	0.197(0.104)	0.182(0.090)
25	0.140(0.069)	0.242(0.197)	0.131(0.073)	0.119(0.062)
50	0.070(0.034)	0.222(0.196)	0.072(0.042)	0.065(0.035)
75	0.047(0.023)	0.216(0.195)	0.049(0.029)	0.045(0.025)
100	0.035(0.017)	0.213(0.195)	0.037(0.022)	0.034(0.019)
200	0.018(0.009)	0.208(0.195)	0.018(0.011)	0.018(0.010)

Note: Each entry is calculated on the basis of 1000 repeated samples.

Table 2 shows the EMSE of the four estimators grouped by different λ_D. When the λ_D (the carry-over effect) is relative small (< 0.5), the estimate ${\widehat{\eta }}_{BA}$ performs the best. However, the EMSE of ${\widehat{\eta }}_{BA}$ increases rather quickly when λ_D becomes large. On the other hand, the EMSE of ${\widehat{\eta }}_{PAR}$ stays the same. Note both ${\widehat{\eta }}_{OP\_a}$ and ${\widehat{\eta }}_{OP}$ perform moderately well. They can be used to protect the worst scenario when the carry-over effect difference is large and they are more efficient than the ${\widehat{\eta }}_{PAR}$ when the sample size is small.

TABLE 2.

The estimated MSE and standard error (in bracket) organized by λ_D for ${\widehat{\eta }}_{PAR}$, ${\widehat{\eta }}_{BA}$, ${\widehat{\eta }}_{OP}$, and ${\widehat{\eta }}_{OP\_a}$ in situations in which random errors ${\epsilon }_{iz}^{(g)}$ are assumed to be independently identically distributed as the normal distribution with mean 0 and variance σ² = 1–3 with an increment of 0.5; the effect for treatment A versus treatment B = 0 to 1 with an increment of 0.2; the effect for period 2 versus period 1, γ = 1; the intraclass correlation ρ = 0.1–0.8 with an increment of 0.1; the carry-over effect difference λ_D = λ_A − λ_B = 0–1.5 with an increment of 0.25; and the number of patients per group n (= n1 = n2) = 15, 25, 50, 75, 100, 200

λD	EMSE_npar (SE)	EMSE_nBA (SE)	EMSE_nOP (SE)	EMSE_nOP_a (SE)
0	0.091 (0.095)	0.025 (0.030)	0.048 (0.051)	0.052 (0.056)
0.25	0.090 (0.094)	0.040 (0.030)	0.056 (0.052)	0.057 (0.055)
0.5	0.090 (0.095)	0.088 (0.031)	0.074 (0.059)	0.069 (0.059)
0.75	0.090 (0.094)	0.165 (0.030)	0.090 (0.072)	0.080 (0.067)
1	0.090 (0.094)	0.275 (0.030)	0.101 (0.088)	0.089 (0.077)
1.25	0.090 (0.095)	0.416 (0.031)	0.108 (0.103)	0.095 (0.088)
1.5	0.090 (0.095)	0.587 (0.031)	0.112 (0.115)	0.099 (0.097)

Note: Each entry is calculated on the basis of 1000 repeated samples.

In Table 3, we summarize the coverage probabilities and 95% percentile interval lengths grouped by the sample size. It can be seen that the percentile interval lengths using ${\widehat{\eta }}_{BA}$ are the shortest since ${\widehat{\eta }}_{BA}$ uses all available observations. However, since we consider simulations including nonzero carry-over effects, the coverage probabilities using ${\widehat{\eta }}_{BA}$ do not close to the nominal 0.95 confidence level. The other three estimates have similar coverage probabilities that are close to the nominal 0.95 level, but ${\widehat{\eta }}_{PAR}$ produces larger interval lengths. Table 4 shows the coverage probabilities and 95% percentile interval lengths grouped by different λ_D. Note that λ_D does not impact the percentile interval lengths. The interval lengths remain about the same for each estimate. The interval lengths generated using ${\widehat{\eta }}_{BA}$ are still the shortest. However, the coverage probabilities using ${\widehat{\eta }}_{BA}$ do not close to the nominal 0.95 confidence level when λ_D is not 0. Both ${\widehat{\eta }}_{OP}$ and ${\widehat{\eta }}_{OP\_a}$ are able to maintain reasonable coverage probabilities even λ_D is large while produce shorter interval lengths comparing to the lengths produced by the use of ${\widehat{\eta }}_{PAR}$. The simulation results demonstrate that our proposed estimators ${\widehat{\eta }}_{OP}$ and ${\widehat{\eta }}_{OP\_a}$ are able to factor in the existence of cross-over effects and are more efficient comparing to the use of ${\widehat{\eta }}_{PAR}$. Furthermore, since the coverage probabilities using both ${\widehat{\eta }}_{OP}$ and ${\widehat{\eta }}_{OP\_a}$ are closer to the nominal 95% confidence level, the large Type I error rate issue caused by the two-stage test procedure is not a concern by applying our proposed estimators.

TABLE 3.

The coverage probability based on the 95% percentile intervals and the average interval length (in bracket) organized by sample size n for ${\widehat{\eta }}_{PAR}$, ${\widehat{\eta }}_{BA}$, ${\widehat{\eta }}_{OP}$, and ${\widehat{\eta }}_{OP\_a}$ in situations in which random errors ${\epsilon }_{iz}^{(g)}$ are assumed to be independently identically distributed as the normal distribution with mean 0 and variance σ² = 0.5–1.5 with an increment of 0.5; the effect for treatment A = 1 versus treatment B = 0; the effect for period 2 versus period 1, γ = 1; the intraclass correlation ρ = 0.3, 0.7; the carry-over effect difference λ_D = λ_A − λ_B = 0 to 1 with an increment of 0.25; the number of patients per group n (= n1 = n2) = 10, 20, 30, and 500 bootstrap samples

n	${\widehat{\eta }}_{PAR}$	${\widehat{\eta }}_{BA}$	${\widehat{\eta }}_{OP}$	${\widehat{\eta }}_{OP\_a}$
10	0.918 (1.598)	0.644 (0.780)	0.918 (1.425)	0.918 (1.418)
20	0.934 (1.172)	0.535 (0.574)	0.931 (1.059)	0.933 (1.050)
30	0.937 (0.967)	0.448 (0.466)	0.930 (0.890)	0.934 (0.879)

Note: Each entry is calculated on the basis of 1000 repeated samples.

TABLE 4.

The coverage probability based on the 95% percentile intervals and the average interval length (in bracket) organized by λ_D for ${\widehat{\eta }}_{PAR}$, ${\widehat{\eta }}_{BA}$, ${\widehat{\eta }}_{OP}$, and ${\widehat{\eta }}_{OP\_a}$ in situations in which random errors ${\epsilon }_{iz}^{(g)}$ are assumed to be independently identically distributed as the normal distribution with mean 0 and variance σ² = 0.5–1.5 with an increment of 0.5; the effect for treatment A = 1 versus treatment B = 0; the effect for period 2 versus period 1, γ = 1; the intraclass correlation ρ = 0.3, 0.7; the carry-over effect difference λ_D = λ_A − λ_B = 0 to 1 with an increment of 0.25; the number of patients per group n (= n1 = n2) = 10, 20, 30, and 500 bootstrap samples

λ D	${\widehat{\eta }}_{PAR}$	${\widehat{\eta }}_{BA}$	${\widehat{\eta }}_{OP}$	${\widehat{\eta }}_{OP\_a}$
0	0.929 (1.245)	0.930 (0.609)	0.947 (1.065)	0.939 (1.076)
0.25	0.934 (1.246)	0.814 (0.608)	0.936 (1.077)	0.936 (1.080)
0.5	0.930 (1.247)	0.502 (0.598)	0.917 (1.118)	0.923 (1.108)
0.75	0.929 (1.246)	0.303 (0.609)	0.916 (1.158)	0.921 (1.138)
1	0.927 (1.244)	0.162 (0.610)	0.915 (1.207)	0.920 (1.178)

Note: Each entry is calculated on the basis of 1000 repeated samples.

5 |. AN EXAMPLE

To illustrate the use of our proposed estimators, we consider an example from Senn.⁵ The study was to evaluate the efficacy of two types of bronchodilators that were used for asthma treatment. The two types of treatments are Salbutamol (Sal) and Formoterol (For). A 2×2 crossover design was applied for the study and a total of 12 patients were randomly assigned into two groups. One group used the sequential order of For-then-Sal and another group used the sequential order of Sal-then-For. The outcome measure is peak expiratory flow (PEF). The estimated treatment effect (the difference of For and Sal) using parallel group design (use first period data only) ${\widehat{\eta }}_{PAR}$ (8) is −9.17. If we assume no carry-over effect and apply the most commonly-used unbiased estimator for η_BA, ${\widehat{\eta }}_{BA}$ (2), the estimate is −28.67. When considering the weighted estimator we proposed, the estimated weight, ${\widehat{W}}_0$ is 0.734. This leads to ${\widehat{\eta }}_{OP}$ (18) = −14.35. When we consider the adjusted weight, ${\widehat{W}}_{01}$ (19) = 0.781 and the estimate ${\widehat{\eta }}_{OP\_a}$ (20) is −13.43. Both ${\widehat{\eta }}_{OP}$ and ${\widehat{\eta }}_{OP\_a}$ yield different treatment effect estimates comparing to the estimates of ${\widehat{\eta }}_{PAR}$ and ${\widehat{\eta }}_{BA}$.

6 |. DISCUSSION

From Figure 1 and Figure 2, we notice the derived optimal estimator η_OP yields the smallest MSE if W₀ is known. However, Monte Carlo simulations show that with estimated parameters, the EMSE of ${\widehat{\eta }}_{OP}$ is not uniformly smallest in all scenarios we considered. This is due to the random variations that the estimated optimal weight ${\widehat{W}}_0$ can be smaller than 0 or greater than 1 while the possible range of the W₀ is between 0 and 1. We assigned ${\widehat{W}}_0$ to be 0 (1) when the estimated ${\widehat{W}}_0$ is ≤ 0 (≥ 1). This is the reason that most of the estimated ${\widehat{W}}_0$ ranges from 0.25 to 0.75 while the true value of W₀ ranges from 0 to 1. By observing the behavior of EMSE of ${\widehat{\eta }}_{OP}$ and other estimated parameters such as ${\widehat{\sigma }}^2$ and ${\widehat{\lambda }}_D$, we propose a simple modification to calculate ${\widehat{W}}_0$. Simulation results show that the modified ${\widehat{\eta }}_{OP\_a}$ performs better than the ${\widehat{\eta }}_{OP}$ in most cases. One advantage of our proposed weighted estimator is that there is no need to apply a two-stage test procedure approach as suggested by Grizzle.⁵ We use a weighted estimator and a modified estimator that are based on ${\widehat{\eta }}_{BA}$ and ${\widehat{\eta }}_{PAR}$. Even the simulation results demonstrate that ${\widehat{\eta }}_{OP}$ and ${\widehat{\eta }}_{OP\_a}$ do not perform the best uniformly, it can serve as a safeguard to prevent a huge MSE when the carry-over effect is large.

Simulation results also demonstrate that the coverage probabilities using both ${\widehat{\eta }}_{OP}$ and ${\widehat{\eta }}_{OP\_a}$ are reasonably close to the nominal 0.95 level even with the presence of the carry-over effect. Furthermore, the percentile interval lengths using ${\widehat{\eta }}_{OP}$ and ${\widehat{\eta }}_{OP\_a}$ are shorter than the interval lengths when ${\widehat{\eta }}_{PAR}$ is applied under the situations we considered. Since the coverage probabilities based on the percentile intervals are close to the nominal 95% confidence level, there is an advantage of using ${\widehat{\eta }}_{OP}$ and ${\widehat{\eta }}_{OP\_a}$ rather than conducting a two-stage test procedure which can lead the actual Type I error rate to be higher than the nominal α - level.

In summary, we have developed an optimal estimator that is a weighted average of the two commonly used estimators of AB/BA crossover designs and a modified estimator based on the optimal estimator. Simulation results demonstrate that the overall performance of the optimal estimator and its modified estimator are better than the two commonly used estimators. When sample size is large, the parallel design that uses only the first period can prevent any potential carry-over effect difference between the two considered treatments and it shall be used. The results, findings and discussions should be of use for clinicians and biostatisticians when they employ simple AB/BA crossover designs for their studies.

DATA AVAILABILITY STATEMENT

The data used in the paper was an example from Senn (2002) “Cross-over Trials In Clinical Research.”