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Logo of nihpaAbout Author manuscriptsSubmit a manuscriptHHS Public Access; Author Manuscript; Accepted for publication in peer reviewed journal;
 
Stat Methods Med Res. Author manuscript; available in PMC 2017 September 17.
Published before final editing as:
PMCID: PMC5026912
NIHMSID: NIHMS795985

Sample size determinations for stepped-wedge clinical trials from a three-level data hierarchy perspective

Abstract

Stepped-wedge (SW) designs have been steadily implemented in a variety of trials. A SW design typically assumes a three-level hierarchical data structure where participants are nested within times or periods which are in turn nested within clusters. Therefore, statistical models for analysis of SW trial data need to consider two correlations, the first and second level correlations. Existing power functions and sample size determination formulas had been derived based on statistical models for two-level data structures. Consequently, the second-level correlation has not been incorporated in conventional power analyses. In this paper, we derived a closed-form explicit power function based on a statistical model for three-level continuous outcome data. The power function is based on a pooled overall estimate of stratified cluster-specific estimates of an intervention effect. The sampling distribution of the pooled estimate is derived by applying a fixed-effect meta-analytic approach. Simulation studies verified that the derived power function is unbiased and can be applicable to varying number of participants per period per cluster. In addition, when data structures are assumed to have two levels, we compare three types of power functions by conducting additional simulation studies under a two-level statistical model. In this case, the power function based on a sampling distribution of a marginal, as opposed to pooled, estimate of the intervention effect performed the best. Extensions of power functions to binary outcomes are also suggested.

Keywords: Stepped-wedge design, three level data, statistical power, sample size, design effect, effect size

1 Introduction

Stepped-wedge (SW) clinical trial design is a variation of cluster randomized trials and is a type of crossover design at the cluster level as treatment assignments are designed to be progressively crossed over unilaterally from a control to an experimental arm until all clusters are completely crossed over.1 The random element of a SW design is the assignment of time points of the crossover to the clusters. The main advantage of the SW design is the relaxation of logistical constraints related to human or financial resources for conduct of classical cluster randomized trials,1 although there are challenges in implementing in real-world settings.2,3 The SW design is also useful when clinical equipoise is not met and it is unethical to randomize participants to the control arm for the length of the study. Further detailed discussion on this issue can be found in Prost et al.4 Additional considerations that should be taken into account for conducting SW trials are suggested by Hargreaves et al.5 The SW design has been steadily implemented in a variety of trials,6 and a systematic review concerning characteristics of published SW trials is conducted by Brown and Lilford,7 and more recently and comprehensively by Beard et al.8

As is the case for all types of randomized clinical trials, sample size determination is an indispensible element of the SW design. Hussey and Hughes9 have proposed a widely used closed-form sample size determination formula for the SW design considering a random effects model. Woertman et al.10 converted Hussey and Hughes’ formula to a design effect of the SW design in comparison with a conventional two parallel arm design. Baio et al.11 also suggested a design effect formula under a different setting and statistical model. Hemming et al.12 evaluated the impact of intra-cluster correlations on statistical power or sample size through design effects under various types of SW designs. In addition, simulation studies for power analysis without explicit formula have also been conducted by Biao et al.11 and Van den Heuvel et al.13

In all those derivations above, although the first level correlations (denoted below by ρ1) of outcomes among participants in the same times or periods within the same clusters were taken into account, the second level correlations (denoted below by ρ2) of outcomes among participants between times or periods within the same clusters were not explicitly considered for computing power or determining sample sizes. The latter correlations would need to be modeled in a statistical model for SW design trials because SW designs by definition naturally assume a three-level data hierarchy, as participants are nested within times or periods that are in turn nested within clusters in SW designs. The nomenclatures for units of levels should depend on the study context; for example, depending on research settings, the third level units can be physicians, clinics, hospitals, schools, communities, and districts to name a few. Hereafter, however, we refer to “cluster”, “period”, and “participant” as the third, the second, and the first level data units, respectively, in the SW design.

The primary aim of this paper is to derive explicit closed-form power functions which consider also the second level correlations by formulating a three-level model accounting for the two types of correlations. To this end, in section 2, we introduce a SW design with design parameter notations. In section 3, we specify the three-level model and formulate the two types of correlations. In section 4, we estimate an overall treatment effect by pooling cluster-specific effect estimates since the number of periods exposed to the experimental condition is not identical across all clusters. A power function is derived based on a sampling distribution of the pooled estimate of the overall treatment effect. In section 5, as a secondary aim, we compare performances of three power functions including that of Hussey and Hughes9 under a two-level model when ρ2 is assumed to be 0 as has previously been implicitly assumed. In section 6, simulation studies compare validities of all power functions under both two- and three-level models. Discussion follows in section 7.

2 Stepped wedge design parameters

Here we consider the SW design depicted in Figure 1, similar to that which was considered in Woertman et al.,10 to illustrate design parameters. The total number of steps is represented by S (≥1); the number of clusters in each step is represented by c (≥1); the number of periods for each “depth” of a step under an experimental condition (gray areas) is represented by p (≥1); and each cluster has b (≥0) number of “baseline” periods under a control condition (blank areas). The clusters are indexed by i = 1, 2, …, I = cS, this being the total number of clusters. Let us further denote by S(i) the depths of steps for the ith cluster under the experimental condition: e.g., S(i) = 1 for i = 1, 2; S(i) = 2 for i = 3, 4; and so on. The periods nested within clusters are indexed by j = 1, 2, …, J = b + pS, this being the total number of periods per cluster. Study participants nested within each period are indexed by k = 1, 2, …, K, this being referred to as “cell” size or the number of participants for each cell in Figure 1. Let us assume that the participants belong to only one cell without cross-over to other clusters or periods. Then the total number of participants N will be N = IJK = cS(b + pS)K. The parameter values for the SW design depicted in Figure 1 can be found in its footnote. The sets of indices indicating observations from the experimental and control arms will be denoted by E and C, respectively.

Figure 1
A stepped wedge design with design parameters. Note: Gray areas represent periods under an experimental condition whereas blanks areas represent those under a control condition; S = total number of steps (=5); c = number of clusters per step (=2); p = ...

3 Statistical model for three level data structure

A statistical model for testing an experimental intervention/treatment effect under a SW design can be formulated as follows.

Yijk=β0+δXijk+ui+uj(i)+eijk
(1)

The study outcome is denoted by Yijk (i = 1, 2, …, I; j = 1, 2, …, J; k = 1, 2…, K) and the experimental arm indicator is denoted by Xijk = 1 for experimental arm, and=0 otherwise. Likewise, the control arm indicator is denoted by Wijk=1 – Xijk=1 for control arm, and =0 otherwise. Then, XijkWi′j′k′= 0 if i = i′, j = j′, and k = k′; otherwise, the product is either 0 or 1 depending on the configurations of the indices. The two sets, E and C, are defined as E = {i, j, k | Xijk = 1} = {i, j, k | Wijk = 0} and C = {i, j, k | Wijk = 1} = {i, j, k | Xijk = 0}.

The fixed-effect overall intercept is denoted by β0, and the fixed experimental intervention effect by δ in model (1). The distribution of the cluster-level random intercepts ui is assumed to be normal as ui~N(0,σ32) and so is that of the period-level random intercepts uj(i) as. uj(i)~N(0,σ22). The distribution of the errors eijk is also assumed to be normal as eijk~N(σe2). Among these random components, it is further assumed that ui[perpendicular]uj(i)[perpendicular]eijk, i.e., these three random components are mutually independent. In addition, conditional independence is assumed for all uj(i) and eijk, whereas the ui are unconditionally independent. That is, uj(i) are independent over j conditional on ui, and eijk are independent over k conditional on ui, and uj(i).

Under model (1), and the elements of the covariance matrix are

Cov(Yijk,Yijk)=1(i=i&j=j&k=k)σe2+1(i=i&j=j)σ22+1(i=i)σ32
(2)

where 1(.) is an indicator function. It follows that Var(Yijk) = Cov(Yijk, Yijk) = σ2, where σ2=σe2+σ22+σ32. Therefore, the correlations among the level two data (i.e., among outcomes from different periods but the same cluster) can be expressed for jj′ as follows:

ρ2=σ32/σ2
(3)

The correlations among the level one data (i.e., among outcomes measured from different participants in the same period within the same cluster) can be expressed for kk′ as

ρ1=(σ22+σ32)/σ2
(4)

As a result, ρ1 is greater than or equal to ρ2, that is, ρ1ρ2.

4 Estimate of intervention effect and power function

To estimate the overall intervention effect δ, we consider each cluster as a stratum because outcome observations between periods within clusters are correlated and the numbers of periods exposed to control and experimental conditions are not necessarily identical across the clusters. This means that the variances of the cluster-specific effect estimates are not necessarily identical. In our approach, we first estimate an intervention effect for each cluster/stratum in a cluster-specific fashion, and then pool the cluster-specific estimates into an overall estimate of δ in model (1) by applying a fixed-effect meta-analytic approach14 as δ is assumed to be fixed and homogenous across clusters.

The intervention effect for the ith cluster is denoted by δi. A moment of estimate, δi, of δi can then be obtained as δi=θi,Eθi,C, where θi,E=jJkKXijkYijk/Ni,E and θi,C=jJkKWijkYijk/Ni,C are the means of outcome Y for the participants in the experimental and control arms in the ith cluster, respectively; Ni,E and Ni,C represent the number of participants in the experimental and control arms in the ith cluster, respectively. That is, Ni,E=jJkKXijk=Ji,EK and Ni,C=jJkKWijk=Ji,CK, where Ji,E= #{j|Xajk =1, a= i} = pS(i) and Ji,C = #{j|Wajk = 1, a = i} = b + p(SS(i)) are the numbers of periods under the experimental and control conditions in the ith cluster; #{.} denotes the number of elements in the set {.}. It follows that Ji,E + Ji,C = J, Ni,E = pS(i)K and Ni,C = bK + p(SS(i))K. Under this setting, the variances of θi,E and θi,C and the covariance between them can be derived as follows:

Var(θi,E)=σ2Ji,EK{1+(K1)ρ1+K(Ji,E1)ρ2},Var(θi,C)=σ2Ji,CK{1+(K1)ρ1+K(Ji,C1)ρ2}

and Cov(θi,C,θi,E)=σ32=σ2ρ2. It follows that the variance of δi can be expressed as below:

Var(δi)=Var(θi,E)+Var(θi,C)2Cov(θi,E,θi,C)=fJσ2Ji,EJi,CK

where

f=1+(K1)ρ1Kρ2
(5)

which is the design effect for three-level trials that randomly assign treatments at the second level within clusters.15,16 This design effect f is an increasing function of ρ1 and a decreasing function of ρ2.

An estimate δ of overall intervention effect can now be obtained as a pooled estimate of δi ‘s weighted by their corresponding inverse variances as follows:

δ=i=1Iωiδi/i=1Iωi
(6)

where ωi=1/Var(δi). This pooled estimate is a weighted mean of δis. It follows that

Var(δ)=1/i=1Iωi=fJσ2Ki=1IJi,EJi,C
(7)

Under the setting depicted in Figure 1, the following equation is straightforward:

i=1IJi,EJi,C=cm=0S1(pm+b)(Sm)p=cpS(S+1){p(S1)+3b6}

This equation enables a power function to be expressed in terms of design parameters as follows:

φ=Φ(|δ|/Var(δ)z1α/2)=Φ(|Δ|cpKS(S+1)(pSp+3b)6f(b+pS)z1α/2)
(8)

where Φ (.) is the cumulative density function of a standard normal distribution, z1−α/2 = Φ−1(1 − α/2), Φ−1(.) is the inverse of Φ(.), and Δ = δ/σ which is known as standardized effect size or Cohen’s d.17 The statistical power increases with increasing Δ, b, c, S (or I), p, K, α, and ρ2, all of which decrease the variance (7). However, the statistical power decreases with increasing ρ1 which increases f and thus increases the variance (7).

5 Power under model for two-level data

If ρ2=σ32/σ2(3) is assumed to be 0, then this assumption is equivalent to assuming σ32=0, and reduces model (1) to a model

Yijk=β0+δXijk+uj(i)+eijk
(9)

for a two-level data structure. Subsequently, Cov(Yijk, Yi′j′k′) in equation (2) reduces to

Cov(Yijk,Yijk)=1(i=i&j=j&k=k)σe2+1(i=i&j=j)σ22

and likewise σ2σe2+σ22 and

ρρ1=σ22/σ2
(10)

The statistical power expressed in equation (7) in Hussey and Hughes,9 can be re-expressed, denoted here by [var phi]HH, utilizing the equations in the supplements of Woertman et al.10 as follows in terms of the design parameters depicted in Figure 1

φHH=Φ(|Δ|cpKS(S1/S){1+ρ(pKS/2+bK1)}6(1ρ){1+ρ(pKS+bK1)}z1α/2)
(11)

This function is not a monotone increasing or decreasing function of ρ. Furthermore, [var phi]HH cannot be defined if ρ = 1 although this is unrealistic to occur.

In addition, the statistical power [var phi] (8) for the three-level model can straightforwardly be reduced to

φ0=Φ(|Δ|cpKS(S+1)(pSp+3b)6f0(b+pS)z1α/2)
(12)

where

f0=1+(K1)ρ
(13)

which is the same as f (5) with ρ2 (3) and ρ1 (4) replaced by 0 and ρ (10), respectively; f0 (13) is the design effect for level-two data structure.18 The statistical power [var phi]0 is in fact based on pooling of cluster-specific effect estimates weighted by the inverses of the cluster-specific variances of the estimates, and is a monotone decreasing function of ρ as is the case for [var phi] (8) that decreases with ρ1.

We note that the clusters, however, became nominal without any influence on statistical inference, since σ32=0 is assumed. That is to say that the periods are no longer assumed to be nested within clusters although individual observations Y are still assumed to be nested with periods. Therefore, statistical power [var phi](2) below can be based on a sampling distribution of a marginal estimate of δ in model (9) for two-level data as follows:

δM=i=1Ij=1Jk=1KXijkYijk/i=1Ij=1Jk=1KXijki=1Ij=1Jk=1KWijkYijk/i=1Ij=1Jk=1KWijk
(14)

It follows that the power [var phi](2) can be obtained as follows19

φ(2)=Φ(|Δ|Kf0(1/NE(2)+1/NC(2))z1α/2)
(15)

where NE(2)=#{i,j|Xijk=1}=pcS(S+1)/2 and NC(2)=#{i,j|Wijk=1}=cS(b+p(S1)/2) are the numbers of total periods for the experimental and control arms, respectively (the numbers of gray-colored and blank “cells” in Figure 1, respectively). It can be seen that the power function [var phi](2) is also a monotone decreasing function of ρ (10).

6 Simulation study

We conducted simulations using the SAS v9.3 PROC MIXED routine with a restricted maximum likelihood fitting option to (1) validate the power function [var phi] (8) derived under the three-level model (1); and (2) compare three power functions [var phi]HH (11), [var phi]0 (12), and [var phi](2) (15) under the two-level model (9). We note that it is possible to theoretically derive closed-form power functions with varying Kij, the number of observations per period per cluster. However, it will be cumbersome not only to express exact formulas but also to compute power functions. Therefore, to assess applicability of the power functions under varying Kij, we randomly drew Kij from uniform distributions Kij ~ U(a, b) with a = K − floor(3 K/4) and b = K + floor(3 K/4) so that a > 0 and E{U(a, b)} = (a + b)/2 = K, where floor(x) returns the greatest integer smaller than or equal to x.

The magnitudes of all of theoretical power functions are compared with those of empirical power estimated from the simulations. To compute simulation-based empirical power, which we consider as the “reference” power, we fit models (1) and (9) with unknown variances which are usually assumed in practice, although all the power functions are derived under known variance components. We generated 1000 simulated data sets for each combination of pre-specified design parameters and estimated the empirical power as follows:

φ=s=110001{ps(δ)<α}/1000
(16)

where ps(δ) is the p value for the sth simulated data set (s = 1, 2, …, 1000). The p values were computed based on critical values of Wald t-distributions under the null hypotheses with degrees of freedom determined by the method of Kenward and Roger.20 SAS simulation codes are provided as Supplementary materials.

Three-level model

The pre-specified design parameters can be found in Table 1. The results show that the theoretical power [var phi] (8) and the simulation-based empirical power φ(16) are very close to each other regardless of whether K is fixed or varying: mean(φ)mean(φ)=0.012 for fixed K and = 0.001 for varying Kij; and range(φφ)=(0.049,0.010)and=(0.041,0.036) respectively. The power function is proven to be an increasing function of all design parameters except ρ2 (3). The effects of ρ1 and ρ2 on the statistical power under three-level model parameters are graphically depicted in Figure 2.

Figure 2
Relationship of ρ1 and ρ2 with statistical power [var phi] (8) for a three-level model: Δ = 0.3, b = 2, c = 2, p = 2, S = 5, K = 5, and α = 0.05. Note: rho_1 = ρ1 and rho_2 = ρ2.
Table 1
Comparison of theoretical and empirical power with the following design parameters held fixed: c = 2, S = 5 (or I = 10), p = 2, and σ2 = 1.

Two-level model

The pre-specified design parameters can be found in Table 2, in which ρ2 is considered 0. The results show that the performances of the three theoretical power functions [var phi]HH (11), [var phi]0 (12), and [var phi](2) (15) are quite different in comparison with the reference empirical power under both fixed K and varying Kij: mean(φHH)mean(φ)=0.129 for fixed K and = 0.158 for varying Kij; mean(φ0)mean(φ)=0.127 and = −0.098, respectively; and mean(φ(2))mean(φ)=0.001 and = 0.030, respectively. With respect to the ranges of biases: range (φHHφ)=(0.000,0.355) for fixed K and = (0.018, 0.385) for varying Kij, range(φ0φ)=(0.183,0.064)and(0.147,0.034), respectively; and range(φ(2)φ)=(0.014,0.021)and(0.002,0.067), respectively. Overall, [var phi](2) is least biased and very close to the empirical power. In contrast, [var phi]HH and [var phi]0 overestimated and underestimated the empirical power, respectively. Therefore, if a sample size were determined based on [var phi]HH or [var phi]0, a study would be under-powered or over-powered, respectively. Furthermore, unlike the other two power functions, [var phi]HH is seen to be increasing with increasing ρ (10) for the values considered for the simulations. For this reason, [var phi]HH could more severely overestimate true power and thus underestimate sample sizes for larger values of ρ. The effect of ρ on the statistical power under a two-level model parameters are graphically depicted in Figure 3.

Figure 3
Relationship of ρ with statistical power [var phi]HH (11), [var phi]0 (12), and [var phi](2) (15) for a two-level model: Δ = 0.3, b = 2, c = 2, p = 2, S = 5, K = 5, and α = 0.05. Note: power_HH = [var phi]HH, power_0 = [var phi] ...
Table 2
Comparison of theoretical and empirical power with the following design parameters held fixed: c = 2, S = 5 (or I = 10), p = 2, and σ2 = 1.

7 Discussion

Our results suggest that the second level correlations ρ2 must be accounted for determining sample size when designing a SW assuming a three-level model. However, no SW trials have so far reported an estimate of ρ2, although a couple of SW trial studies21,22 reported only ρ1 based on the recent review of Davey et al.23 As observed in this paper (Figure 2), the effects of both ρ1 and ρ2 on the power are substantial when a three-level model is considered. Therefore, it would be valuable to report estimates of ρ2 from conducted SW trials for aiding designs of future SW trials. For two-level models, many studies addressed impacts of ρ (e.g. see literature12,18,24,25) as reflected in Figure 3. However, relationship between ρ and [var phi]HH is hardly predictable and mostly contradictory to that between ρ and [var phi]0 and that between ρ and [var phi](2) as well.

The derived power function [var phi] (8) is proven to be unbiased and valid for that purpose of accounting for both the first and second level correlations. This finding suggests that the pooled estimate δ(6) may indeed be a maximum likelihood estimate of δ in model (1). Although it was derived under a special case depicted in Figure 1, the power function [var phi] is also proven to be applicable to SW designs with varying Kij, the cell size. Therefore, the pooling estimation approach (6) based on the cluster-specific moment estimates can also be extended to the general cases where c varies over steps, and p or J varies over clusters. Nevertheless, even if it could be possible to derive, a closed-form expression of a power function under those situations would be much more complex and much less tractable for calculations.

When ρ2 does not need to be considered in a two-level model, the power function [var phi](2) (15) based on the marginal estimate δM(14) performs the best with the ignorable biases regardless of whether the cell sizes are fixed or varying. In contrast, the power function [var phi]0 (12) that is reduced from [var phi] (8) by plugging 0 into ρ2 in [var phi] underestimates the reference power estimated by simulations. This may be because when stratification is unnecessary, pooled estimates can have an unduly inflated variance and thus lose efficiency compared to the marginal approach. On the contrary, the widely used power function [var phi]HH (11) overestimates the reference empirical power and thus underestimates sample sizes under the values of ρ in Table 2. We suspect that the Hussey and Hughes’ approach might unduly over or underestimate the variance of the estimate of δ depending heavily on the values of ρ (Figure 3).

Both models (1) and (9) assume that participants are different across the periods within clusters, let alone between clusters. However, when participants are followed up longitudinally over the periods within the same clusters and crossed over from control to experimental arm, another level of data structure should be modeled by expanding the three-level model (1) to a four-level model that additionally incorporates correlations of outcomes over periods within the same participants. In addition, the random intercepts could be correlated each other violating the independence assumption we took in this paper. Derivations of power functions under these situations design would be a worthy contribution to power literature.

Although only continuous outcome is considered in this paper, categorical or non-normal outcomes such as proportions, incidence rates, ordinal, and survival outcomes are more often of interest in many SW trials.8 Extension of sample size determinations for such SW trials would be of great interest. The extension might be possible by modeling those outcomes with generalized estimating equations or non-linear mixed-effects models although derivations of closed forms could be intractable. For this reason, sample size determinations based on simulation approaches might be preferable as attempted by Baio et al.11 Nonetheless, we suspect that simulation of non-normal data with multi-level data hierarchy for a specified correlation structure would be challenging particularly because correlations may well vary with means on which variances depend unlike normal distributions. Therefore, it would also be interesting to examine if extensions based on normal approximations would be comparable. For example, although it has not been verified by simulation studies for a binary outcome, a simple replacement of Δ by Δp=(p1p0)/p¯(1p¯) in [var phi] or [var phi](2) might be a good approximation owing to a central limit theorem, where p0 and p1 are the “success” probabilities under the null and the alternative hypotheses, respectively, and p¯=(p1+p0)/2.

In conclusion, the power functions [var phi] (8) and [var phi](2) (15) should be used for sample size determinations for designing SW trials depending on whether the second level correlations ρ2 is assumed to be 0 or not. Both are applicable when cell sizes vary.

Supplementary Material

SMMR_SUPP

Acknowledgments

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This study was in part supported by the following NIH Grants: UL1RR025750, R01HS023608, and R01DK097096.

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.

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