Power and sample size calculations for evaluating mediation effects in longitudinal studies

Abstract

Current methods of power and sample size calculations for the design of longitudinal studies to evaluate mediation effects are mostly based on simulation studies and do not provide closed form formulae. A further challenge due to the longitudinal study design is the consideration of missing data, which almost always occur in longitudinal studies due to staggered entry or drop out. In this paper, we consider the product of coefficients as a measure for the longitudinal mediation effect and evaluate three methods for testing the hypothesis on the longitudinal mediation effect: the joint significant test, the normal approximation and the test of b methods. Formulae for power and sample size calculations are provided under each method while taking into account missing data. Performance of the three methods under limited sample size are examined using simulation studies. An example from the Einstein Aging Study (EAS) is provided to illustrate the methods.

1 Introduction

Mediation effects are often of great interest in longitudinal studies. For example, in the Einstein Aging Study (EAS) [1] program project, one primary goal is to investigate the role of clinical cardiovascular disease (CAD) and cerebral vascular reactivity (CVR) with CO2 challenge on the rate of cognitive decline. CVR assessed by transcranial doppler ultrasound (TCD) with CO2 challenge is a measure of cerebromicrovascular function. Decreased reactivity may reflect microvascular damage. One possible pathway of the effect of CAD on cognitive decline is through cerebral microvascular damage. Hence it is hypothesized that the effect of CAD on cognitive decline is partly mediated by CVR.

There are currently limited research on the power analysis for mediation effects on the rate of change in a continuous outcome, i.e., the longitudinal mediation effect. Current work on power estimation for mediation effects is mostly empirical, calculated based on simulations. For example, to evaluate mediation effects in cross-sectional studies, MacKinnon et al. [1] examined empirical power based on simulation studies for some common sample size values, and Fritz and MacKinnon [3] reported sample size needed with 80% power obtained from simulation studies. For more complicated situations including longitudinal studies with repeatedly measured mediator, Thoemmers et al [4] also used simulation studies to calculate power. The empirical approach through simulations is useful for power calculations as it can handle various statistical models and complications such as missing data. However, it can be computationally intensive and is cumbersome to use when sample size estimation is of primary interest.

In this paper, we develop closed form formulae to calculate sample size and power for detecting a mediation effect with consideration of missing data. In particular, a linear mixed effects model is used for evaluating the rate of change, or slope, in the longitudinal outcome. We adopt the product of coefficients as a measure of the longitudinal mediation effect, and evaluate three methods for testing the hypothesis on the longitudinal mediation effect: the joint significance test, the normal approximation and the test of b methods. Section 2 provides the description of the product of coefficients as the measure of the longitudinal mediation effect. In Section 3, three methods of testing the hypothesis are evaluated. The type I error rate and power of each method are examined, and power and sample size calculation formulae are provided. Simulation studies to examine the performance of the three methods under various situations are presented in Section 4. The methods are applied to an example from EAS in Section 5. We conclude the paper in Section 6.

2 Measure of Longitudinal Mediation Effect

2.1 Background and Notation

Consider a longitudinal study of n subjects indexed by i, with k planned visits. Due to staggered entry and drop out, K_i measurements, K_i = 1, …, k, are observed for subject i. Here k is considered fixed so that the asymptotic properties of the estimators are applicable for large n. In presence of missing data, it is assumed that the data are missing at random (MAR) [5]. A monotone missing data pattern is also assumed, i.e., once a subject misses a visit, he/she will not return to the study. This pattern is common in longitudinal studies as missing data are usually caused by staggered entry and drop out. We assume that the observations from different subjects are independent. For subject i, i = 1, …, n, let X_i and M_i denote the risk factor and the mediator, respectively. Both are time-invariant (e.g., baseline measures). Denote Z_i as the set of covariates for subject i. Let Y_ij and t_ij denote the outcome and time, respectively, at the jth measurement (t_ij = 0 corresponds to the baseline), and Y_i = (Y_i1, …, Y_iKi)^T, where superscript T denotes matrix transpose. Here the variables M and Y are both continuous. Although in reality t_ij might vary between subjects, in the design stage it is typical to assume t_ij as fixed with value t_j for all subjects.

Suppose the effect of X on M can be modelled as follows:

$$M_i={\alpha }_0+a{X}_i+{\zeta }^T{\mathbf{Z}}_i+e_i,$$ (1)

where ζ is a vector of coefficients for the vector of covariates Z; e_i is the random error assumed normally distributed with zero mean.

For the longitudinal trajectory of the outcome Y, suppose that, given X, M and Z, the following linear mixed effects model [6] holds:

$$Y_{\mathit{\text{ij}}}={\beta }_0+{\beta }_x^c{X}_i+{\beta }_m^c{M}_i+{\beta }_t{t}_{\mathit{\text{ij}}}+{\beta }_x^l{X}_i{t}_{\mathit{\text{ij}}}+b{M}_i{t}_{\mathit{\text{ij}}}+{\varphi }^T{\mathbf{Z}}_{\mathit{\text{it}}}+e_{\mathit{\text{ij}}},$$ (2)

where Z_it is a vector consisting of Z_i and possibly some interactions with time t_ij, ϕ is the vector of coefficients for Z_it, e_i = (e_i1, …, e_iKi)^T has variance covariance matrix V_i. For example, in a random intercept model, e_ij = u_i+ε_ij, where u_i is the subject-specific random effect which is normally distributed with mean 0 and variance ${\sigma }_u^2$, and ε_ij is the normally distributed random error term with variance ${\sigma }_e^2$ and assumed independent of u_i. The parameter b measures how one unit difference in the mediator M affects the rate of change in the outcome Y adjusting for the risk factor X and the covariate Z, i.e., it is a measure of the longitudinal effect of M on the outcome Y adjusting for X and Z. Similarly, ${\beta }_x^l$ measures the longitudinal effect of X on Y adjusting for M and Z. The parameters ${\beta }_x^c$ and ${\beta }_m^c$ measure the association between X and the outcome at baseline, and between M and the outcome at baseline, respectively, and therefore are referred as the baseline effects.

2.2 Measure of longitudinal mediation effect

According to (2), the expectation of Y_ij given X_i, M_i and Z_i is

$$E(Y_{\mathit{\text{ij}}}|X_i,M_i,{\mathbf{Z}}_i)={\beta }_0+{\beta }_x^c{X}_i+{\beta }_m^c{M}_i+{\beta }_t{t}_{\mathit{\text{ij}}}+{\beta }_x^l{X}_i{t}_{\mathit{\text{ij}}}+b{M}_i{t}_{\mathit{\text{ij}}}+{\varphi }^T{\mathbf{Z}}_{\mathit{\text{it}}}.$$ (3)

From (1), this implies that the expectation of Y_ij given X_i and Z_i is

$$\begin{array}{ll}\hfill & E(Y_{\mathit{\text{ij}}}|X_i,{\mathbf{Z}}_i)\hfill \\ =\hfill & {\beta }_0+{\beta }_x^c{X}_i+{\beta }_m^{c}E(M_i|X_i,{\mathbf{Z}}_i)+{\beta }_t{t}_{\mathit{\text{ij}}}+{\beta }_x^l{X}_i{t}_{\mathit{\text{ij}}}+\mathit{\text{bE}}(M_i|X_i,{\mathbf{Z}}_i)t_{\mathit{\text{ij}}}+{\varphi }^T{\mathbf{Z}}_{\mathit{\text{it}}}\hfill \\ =\hfill & ({\beta }_0+{\alpha }_0{\beta }_m^c)+({\beta }_x^c+a{\beta }_m^c)X_i+({\beta }_t+{\alpha }_{0}b)t_{\mathit{\text{ij}}}+({\beta }_x^l+\mathit{\text{ab}})X_i{t}_{\mathit{\text{ij}}}+{\gamma }^T{\mathbf{Z}}_{\mathit{\text{it}}},\hfill \end{array}$$ (4)

where ${\gamma }^T{\mathbf{Z}}_{\mathit{\text{it}}}={\varphi }^T{\mathbf{Z}}_{\mathit{\text{it}}}+{\beta }_m^c{\zeta }^T{\mathbf{Z}}_i+b{\zeta }^T{\mathbf{Z}}_i{t}_{\mathit{\text{ij}}}$. The difference in the coefficients of X_it_ij between (4) and (3), which measures the effect of X on the rate of change in the outcome, is ab. Hence, the product of a and b, denoted as Δ = ab, measures the difference in the longitudinal effects of the risk factor X on the rate of change in the outcome Y, with and without the presence of the mediator M. In other words, it measures the indirect longitudinal effect, through the mediator M, of the risk factor X on the rate of change in the outcome Y. Thus Δ is a measure of the longitudinal mediation effect. The inference based on Δ = ab is called the product of coefficients method ([7]-[9]).

Suppose a random intercept model is used for model (2), then the residual variance $\text{Var}({\mathbf{Y}}_i|X_i,M_i,{\mathbf{Z}}_i)={\sigma }^2\rho {\mathbf{J}}_{K_i}+{\sigma }^2(1-\rho ){\mathbf{I}}_{K_i}$, where ${\sigma }^2={\sigma }_u^2+{\sigma }_e^2$, $\rho ={\sigma }_u^2/{\sigma }^2$, J_Ki is a K_i × K_i matrix with all elements equal to 1, and I_Ki is the identity matrix with dimension K_i, is compound symmetry with variance σ² and intraclass correlation coefficient ρ. The variance given X and Z is

$$\begin{array}{lll}\text{Var}({\mathbf{Y}}_i|X_i,{\mathbf{Z}}_i)\hfill & =\hfill & E\{\text{Var}({\mathbf{Y}}_i|X_i,M_i,{\mathbf{Z}}_i)|X_i,{\mathbf{Z}}_i\}+\text{Var}\{E({\mathbf{Y}}_i|X_i,M_i,{\mathbf{Z}}_i)|X_i,{\mathbf{Z}}_i\}\hfill \\ \hfill & =\hfill & {\sigma }^2\rho {\mathbf{J}}_{K_i}+{\sigma }^2(1-\rho ){\mathbf{I}}_{K_i}+\mathbf{\Gamma },\hfill \end{array}$$ (5)

where the expectation and variance are taken over the distribution of the mediator M given X and Z. The (j, k)^th entry of the variance-covariance matrix Γ is $({\beta }_m^c+b{t}_{\mathit{\text{ij}}})({\beta }_m^c+b{t}_{\mathit{\text{ik}}}){\sigma }_{m|x,\mathbf{z}}^2$, a quadratic function of time unless b = 0; here ${\sigma }_{m|x,\mathbf{z}}^2=\text{Var}(M|X,\mathbf{Z})$. This shows that in general the structure of the residual variance matrix in the model without the mediator does not have the same structure as that in the model with the mediator. As shown in the above example, when model (2) is a random intercept model, Var(Y_i|X_i, Z_i) is no longer compound symmetry as Var(Y_i|X_i, M_i, Z_i) is. In practice, the difference in coefficients method is sometimes used, in which model (2), and the following linear mixed effects model without M is fit:

$$Y_{\mathit{\text{ij}}}={\beta }_0^{\ast }+{\beta }_x^{c\ast }{X}_i+{\beta }_t^{\ast }{t}_{\mathit{\text{ij}}}+{\beta }_x^{l\ast }{X}_i{t}_{\mathit{\text{ij}}}+{\varphi }^{\ast T}{\mathbf{Z}}_{\mathit{\text{it}}}+e_{\mathit{\text{ij}}}^{\ast }.$$ (6)

The difference in the coefficients of X_it_ij in model (2) and (6), ${\beta }_x^{l\ast }-{\beta }_x^l$, is then used as the measure of the longitudinal the mediation effect. However, model (6) needs to be fit with caution as the residual ${\mathbf{e}}_i^{\ast }={(e_{i1}^{\ast },\dots ,e_{i{K}_i}^{\ast })}^T$ usually has a different and more complicated variance structure compared to model (2). Furthermore, the calculation of the covariance between ${\beta }_x^{l\ast }$ and ${\beta }_x^l$ is not easy. In the cross sectional mediation analysis, the difference in coefficients method does not perform better than the product of coefficients method (e.g. [2]). The product of coefficients measure is thus used and serves as the basis of the hypothesis test.

2.3 Estimation

Denote the estimates of a from model (1) and b from model (2) as â and b̂, respectively. Because â and b̂ are consistent estimate of a and b, the longitudinal mediation effect Δ can be consistently estimated by

$$\hat{\mathrm{\Delta }}=\hat{a}\hat{b}.$$

Similar to [10], it can be shown that â and b̂ are asymptotically independent. The exact variance of Δ̂ can thus be expressed as

$$V_{\mathrm{\Delta }}^e=a^2\text{Var}(\hat{b})+b^2\text{Var}(\hat{a})+\text{Var}(\hat{b})\text{Var}(\hat{a}).$$

However, the distribution of Δ̂ is not normal so the inference on the mediation effect can not be made simply by using the point estimate and its variance. Theoretically, the exact form of the distribution of the product of two random variables has been studied (e.g., [11]-[17]). For example, if a = 0 and b = 0, it is a Bessel function of the second kind [14, 18]. However, the implementation of the distribution is not easy. Although tables of the distribution of the product of two normal variables are available [14, 15, 19], and some algorithms on calculating the distribution function have been developed [16, 17], in general it is not available in commonly used commercial statistical softwares. In practice, the bootstrap technique [20] has been recommended to evaluate the distribution of the product of coefficients and obtain confidence intervals [21]. Although easily applicable at the data analysis stage, this method is not practical at the design stage. In the cross sectional setting, a table of the empirical distribution of the estimate of the product of coefficients based on simulation studies is available [22]. However, this is not particularly useful for longitudinal studies because, in addition to the sample size and other parameters, the variance of b̂ further depends on several factors specific to longitudinal studies including number of repeated measures, time at each measurement, the missing data distribution, and correlations among the repeated measures. A simple and commonly used method is to use a normal distribution to approximate the distribution of Δ̂ [12, 23, 24]. Due to the skewness of the distribution of Δ̂, this method can be problematic, especially when the sample size is not large [2, 21, 25, 26]. However, it has been shown that when a or b is large in magnitude, the product of two normal variables approaches normal distribution [13], in which case the normal approximation method might work well.

In the next section, we examine three methods for testing the mediation effect, some of which do not require assumptions on the distribution of the mediation effect estimate. We also provide formulae for calculating power and sample size under each method.

3 Hypothesis test and power analysis

The hypothesis on the mediation effect, H₀: Δ = ab = 0 versus H₁: Δ = a₀b₀, where a₀≠ 0 and b₀≠ 0, is complicated by the fact that it involves two parameters a and b: values of both parameters have to be specified for either the null or the alternative hypothesis. The null hypothesis H₀: Δ = ab = 0 corresponds to either both a = 0 and b = 0, or one of a and b is zero. We examine the performance of the following three methods for testing the mediation effect.

3.1 Joint significance test

Because ab ≠ 0 is equivalent to a ≠ 0 and b ≠ 0, the hypothesis H₀versus H₁ on the mediation effect Δ can be tested by jointly testing the two hypotheses on a and b. Specifically, the hypothesis on the coefficient a, H_0a : a = 0 versus H_1a : a = a₀, and the hypothesis on the coefficient b, H_0b : b = 0 versus H_1b : b = b₀, are tested jointly, and the null hypothesis on the mediation effect, H₀ : Δ = ab = 0 versus H₁ : Δ = ab, a = a₀, b = b₀, is rejected when both H_0a and H_0b are rejected [2, 27]. This method addresses the problem of testing the mediation effect without dealing with the complicated distribution of Δ̂.

The distributions of the estimates of both components, â and b̂, are asymptotically normal. Without covariate Z, the variance of â can be expressed as

$$\text{Var}(\hat{a})=\frac{{\sigma }_m^2-a^2{\sigma }_x^2}{n{\sigma }_x^2},$$

where ${\sigma }_x^2$ and ${\sigma }_m^2$ are the marginal variance of X and M, respectively. In the presence of covariate Z, using the method of Hsieh (1998) [28], the variance of â can be conservatively approximated by

$$\text{Var}(\hat{a})=\frac{{\sigma }_m^2-a^2{\sigma }_x^2}{n{\sigma }_x^2(1-R_{x|z}^2)},$$

where $R_{x|z}^2$ is the proportion of the variance of X explained by Z.

The variance of the estimate of the vector of parameters from model (2) is

$${\{\mathrm{E}({\mathbf{W}}_i^T{\mathbf{V}}_i^{-1}{\mathbf{W}}_i)\}}^{-1}/n,$$ (7)

where W_i is the design matrix for the fixed effects, and V_i = Var(Y_i|X_i, M_i, Z_i) is the residual variance. When V_i is compound symmetry, the closed form of the variance of b̂ without missing data is well known (e.g. [29]). It has also been studied extensively in the presence of missing data ([30] - [38]). For a general structure of V_i, denote ϕ_lj as the (l, j)^th element of V_i⁻¹, l, j = 1, …, K_i, t^(K_i) = (1, …, t_Ki)^T, similar to [38], it can be shown that the variance of b̂ can be expressed as

$$\text{Var}(\hat{b})=\frac{1}{n{A}_t{\sigma }_m^2(1-R_{m|x,z}^2)},$$ (8)

where A_t = E(c_i) − {E(b_i)}²/E(a_i), $a_i={\mathbf{1}}_{K_i}^T{{\mathbf{V}}_{\mathbf{i}}}^{-1}{\mathbf{1}}_{K_i}={\sum }_{l=1}^{K_i}{\sum }_{j=1}^{K_i}{\varphi }_{\mathit{\text{lj}}}$, $b_i={\mathbf{1}}_{K_i}^T{{\mathbf{V}}_{\mathbf{i}}}^{-1}{\mathbf{t}}^{({\mathbf{K}}_{\mathbf{i}})}={\sum }_{l=1}^{K_i}{\sum }_{j=1}^{K_i}({\varphi }_{\mathit{\text{lj}}}{t}_l)$, $c_i={\mathbf{t}}^{{({\mathbf{K}}_{\mathbf{i}})}^T}{{\mathbf{V}}_{\mathbf{i}}}^{-1}{\mathbf{t}}^{({\mathbf{K}}_{\mathbf{i}})}={\sum }_{l=1}^{K_i}{\sum }_{j=1}^{K_i}({\varphi }_{\mathit{\text{lj}}}{t}_l{t}_j)$, and $R_{m|x,z}^2$ is the proportion of the variance of M explained by X and Z. If there is no covariate Z included in the model, then $R_{m|x,z}^2$ is replaced by $R_{m|x}^2=a^2{\sigma }_x^2/{\sigma }_m^2$.

If V_i is compound symmetry with common correlation coefficient ρ and variance σ², then $A_t=\frac{{\mathrm{E}}_{(S_{K_i}^2)+(1-\rho )\delta }}{(1-\rho ){\sigma }^2}$, where $S_{K_i}^2={\sum }_{j=1}^{K_i}{t}_j^2-{({\sum }_{j=1}^{K_i}{t}_j)}^2/K_i$, $\delta =\mathrm{E}\{\frac{{({\sum }_{j=1}^{K_i}{t}_j)}^2}{K_i(1-\rho +K_i\rho )}\}-{\{\mathrm{E}(\frac{{\sum }_{j=1}^{K_i}{t}_j}{1-\rho +K_i\rho })\}}^2/\mathrm{E}(\frac{K_i}{1-\rho +K_i\rho })$. Note that A_t increases as ρ or the planned number of measurements k increases, and decreases as the drop out rate increases. When there is no missing data, i.e., K_i ≡ k, then δ = 0 and $\mathrm{E}(S_{K_i}^2)=S_k^2$, thus $A_t=\frac{S_k^2}{(1-\rho ){\sigma }^2}$, and (3.1) reduces to the well-known full data formula

$$\text{Var}(\hat{b})=\frac{(1-\rho ){\sigma }^2}{n{S}_k^2{\sigma }_m^2(1-R_{m|x,z}^2)},$$

which is the basis for formula (2.4.1) in chapter 2 of [29].

If the structure of V_i is first order autoregressive (AR1), i.e., the (l, j)^th element of V_i is σ²ρ^|j-l|, in which the correlation weakens as the time lag between observations increases, then it can be shown that the components a_i, b_i and c_i of A_t in the formula can be expressed as follows: $a_i=\frac{K_i-K_i\rho +2\rho }{(1+\rho ){\sigma }^2}$, $b_i=\frac{t_1}{{\sigma }^2}$ if K_i = 1, $\frac{t_1+t_2}{(1+\rho ){\sigma }^2}$ if K_i = 2, and $\frac{t_1+t_{K_i}+(1-\rho ){\sum }_{j=2}^{K_i-1}{t}_j}{(1+\rho ){\sigma }^2}$ if K_i ≥ 3; and $c_i=\frac{t_1^2}{{\sigma }^2}$ if K_i = 1, $\frac{t_1^2+t_2^2-2\rho t_1{t}_2}{(1-{\rho }^2){\sigma }^2}$ if K_i= 2, and $\frac{t_1^2+t_{K_i}^2+(1+{\rho }^2){\sum }_{j=2}^{K_i-1}{t}_j^2-2\rho {\sum }_{j=1}^{K_i-1}{t}_j{t}_j+1}{(1-{\rho }^2){\sigma }^2}$ if K_i ≥ 3. Similar as in the compound symmetry case, A_t increases as ρ increases or as the drop out rate decreases. When k > 2, due to the reduced correlation coefficient as time lag increases in the AR1 structure, with the same parameter ρ, using an AR1 residual variance matrix will result in a lower power than a compound symmetry one. It is therefore important to use the correct variance structure for the longitudinal outcome in power calculations. Information of the variance structure can usually be obtained from preliminary data.

Although a monotone missing data pattern is assumed in the formula for Var(b̂), when the missing data pattern is not monotone, Var(b̂) can be obtained using the general formula (7). To approximate Var(b̂) under a non-monotone missing data pattern using the closed-form formula obtained from monotone missing data patterns, data from the subsequent visits after a subject missed a visit are either discarded or moved backward in time as if they were from a monotone missing data pattern. Since a subject with measurements at later time points contributes more information to the estimation of slopes, the monotone approximation is conservative. If the proportion of missed visits is low, it will be similar to the exact value calculated using the general formula (7).

Suppose significance levels α₁ and α₂ are used for the tests of a and b, respectively. Denote Z_p as the p^th percentile of the standard normal distribution. To test the hypothesis on a, we use the test statistics $T_a=\frac{\hat{a}}{\sqrt{\text{Var}(\hat{a})}}$, and reject H_0a if |T_a| > Z_1−α1/2. Similarly, to test the hypothesis on b, we use the test statistics $T_b=\frac{\hat{b}}{\sqrt{\text{Var}(\hat{b})}}$, and reject H_0b if |T_b| > Z_1−α2/2. The null hypothesis H₀ on the mediation effect is rejected when both H_0a and H_0b are rejected.

We examine the type I error rate and power of the joint significance test. Denote P_a and P_b as the powers for detecting the values of a₀ and b₀, respectively, under the alternative hypothesis for testing a and b, respectively. There are three possible situations under H₀: (1) a = 0, b = 0; (2) a = a₀, b = 0; and (3) a = 0, b = b₀. The type I error rate, α_J, of the joint significance test under each situation can be expressed as follows because of the asymptotic independence of â and b̂. Under situation (1),

$${\alpha }_J\doteq P(\text{reject}{H}_{0a}|H_{0a}|\text{is true})P(\text{reject}{H}_{0b}|H_{0b}\text{is true})={\alpha }_1{\alpha }_2;$$

under situation (2),

$${\alpha }_J\doteq P(\text{reject}{H}_{0a}|H_{1a}\text{is true})P(\text{reject}{H}_{0b}|H_{0b}\text{is true})=P_a{\alpha }_2;$$

and under situation (3),

$${\alpha }_J\doteq P(\text{reject}{H}_{0a}|H_{0a}\text{is true})P(\text{reject}{H}_{0b}|H_{1b}\text{is true})={\alpha }_1{P}_b,$$

where ≐ denotes asymptotic equivalence. Suppose the target significance level for the test of H₀ is α. If the tests of a and b are both set at the significance level α, i.e., α₁ = α₂ = α, then in any of the above situations α_J ≤ α. Hence, the joint significance test is conservative in the sense that the type I error rate is no more than the target level. Under situation (1) of the null hypothesis H₀ in which a = 0 and b = 0, the joint significance test is especially conservative as α_J = α². Under situations (2) and (3) of the null hypothesis H₀ in which only one of a or b equals zero, the actual type I error rate is the product of the power for detecting the non-zero coefficient and the significance level for the test of the zero coefficient. It is close to the target value α if the power for testing the non-zero coefficient is close to 1. In order to yield the target significance level α, one needs to inflate either one or both significance levels for the tests of a and b. For example, under situation (1), α₁ and α₂ can be set at value $\sqrt{\alpha }$. Under situation (2), if α₁ = α and the power P_a can be determined, then the value of α/P_a can be chosen for α₂. But in reality the true value of a and b are unknown, it is common to set both α₁ and α₂ at α because it guarantees a type I error rate no more than α. We adopt this setting of α₁ = α₂ = α here.

Given a sample size n, the power of detecting a₀ for the test of a is

$$P_a=\mathrm{\Phi }\left(\sqrt{\frac{a_0^2}{\text{Var}(\hat{a})}}-Z_{1-\alpha /2}\right)=\mathrm{\Phi }\left(\sqrt{\frac{n{a}_0^2{\sigma }_x^2(1-R_{x|z}^2)}{{\sigma }_m^2-a_0^2{\sigma }_x^2}}-Z_{1-\alpha /2}\right),$$ (9)

where Φ is the cumulative distribution function of a standard normal random variable, and the power of detecting b₀ for the test of b is

$$P_b=\mathrm{\Phi }\left(\sqrt{\frac{b_0^2}{\text{Var}(\hat{b})}}-Z_{1-\alpha /2}\right)=\mathrm{\Phi }\left(\sqrt{n{b}_0^2{A}_t{\sigma }_m^2(1-R_{m|x,z}^2)}-Z_{1-\alpha /2}\right),$$ (10)

The power from the joint significance test is thus

$$P_J=P_a{P}_b=\mathrm{\Phi }\left(\sqrt{\frac{n{a}_0^2{\sigma }_x^2(1-R_{x|z}^2)}{{\sigma }_m^2-a_0^2{\sigma }_x^2}}-Z_{1-\alpha /2}\right)\mathrm{\Phi }\left(\sqrt{n{b}_0^2{A}_t{\sigma }_m^2(1-R_{m|x,z}^2)}-Z_{1-\alpha /2}\right),$$ (11)

which reduces to

$$P_J=\mathrm{\Phi }\left(\sqrt{\frac{n{a}_0^2{\sigma }_x^2}{{\sigma }_m^2-a_0^2{\sigma }_x^2}}-Z_{1-\alpha /2}\right)\mathrm{\Phi }\left(\sqrt{n{b}_0^2{A}_t({\sigma }_m^2-a_0^2{\sigma }_x^2)}-Z_{1-\alpha /2}\right)$$

when there is no covariate Z.

It shows clearly from (11) that the power from the joint significance test increases as the sample size n or the value b₀ increases. From the formulae of A_t, the power also increases as the drop out rate decreases, or as σ² decreases in the case of common residual variance at each visit, or as the correlation among repeated measures increases for the compound symmetry or AR1 residual variance structure. However, it is not monotone with a₀, $\begin{array}{l}{\sigma }_x^2\end{array}$ and ${\sigma }_m^2$.

For a given power level P for testing H₀: Δ = 0 versus H₁: Δ = ab,a = a₀, b = b₀, the sample size needed, n_J, is the solution of n to the equation P_J = P. A closed form formula for the sample size n is not directly available, but it can be easily obtained through numerical iteration.

3.2 Normal approximation method

Since â and b̂ are the maximum likelihood estimates (MLE) of a and b, respectively, based on asymptotic theory, the asymptotic distribution of Δ̂ is normal with mean ab and variance $V_{\mathrm{\Delta }}^a$. Sobel ([23]) derived the asymptotic variance of Δ̂ using the multivariate delta method ([39]) based on a first order Taylor series approximation, which can be expressed as

$$V_{\mathrm{\Delta }}^a=a^2\text{Var}(\hat{b})+b^2\text{Var}(\hat{a}).$$

Combining with the expression of Var(â) and Var(b̂), we have

$$V_{\mathrm{\Delta }}^a=\frac{a^2}{n{A}_t{\sigma }_m^2(1-R_{m|x,z}^2)}+\frac{b^2({\sigma }_m^2-a^2{\sigma }_x^2)}{n{\sigma }_x^2(1-R_{x|z}^2)}.$$

If a second order Taylor series approximation is used, the asymptotic variance expression is the same as the exact variance $V_{\mathrm{\Delta }}^e$ ([12]). The difference between $V_{\mathrm{\Delta }}^a$ and $V_{\mathrm{\Delta }}^e$ is usually very small because the additional component, Var(b̂) Var(â), in $V_{\mathrm{\Delta }}^e$ is usually ignorable relative to $V_{\mathrm{\Delta }}^a$. Other variance formulae are also available (e.g., [24]), but they are all close to $V_{\mathrm{\Delta }}^a$. In practice, $V_{\mathrm{\Delta }}^a$ is the most frequently used.

The (1 – α) confidence interval based on the normal approximation can be obtained as

$$\hat{\mathrm{\Delta }}\pm Z_{1-\alpha /2}\sqrt{V_{\mathrm{\Delta }}^a}.$$

To test the hypothesis on the mediation effect Δ at significance level α based on the asymptotic normal approximation, we use the test statistic

$$T_{\mathrm{\Delta }}=\frac{\hat{\mathrm{\Delta }}}{\sqrt{V_{\mathrm{\Delta }}}},$$

in which the null hypothesis that Δ = 0 is rejected if |T_Δ| > Z_1−α/2. This test is sometimes referred as Sobel's test.

However, the actual type I error rate is at or below the target level α. This can be seen by comparing it with the joint significance test. Since |T_Δ| > Z_1−α/2 is equivalent to $\frac{1}{T_{\mathrm{\Delta }}^2}<\frac{1}{Z_{1-\alpha /2}^2}$, and

$$\frac{1}{T_{\mathrm{\Delta }}^2}=\frac{{\hat{a}}^2\text{Var}(\hat{b})+{\hat{b}}^2\text{Var}(\hat{a})}{{\hat{a}}^2{\hat{b}}^2}=\frac{\text{Var}(\hat{a})}{{\hat{a}}^2}+\frac{\text{Var}(\hat{b})}{{\hat{b}}^2}=\frac{1}{T_a^2}+\frac{1}{T_b^2},$$

it follows that

$$P(|T_{\mathrm{\Delta }}|>Z_{1-\alpha /2})=P(\frac{1}{T_a^2}+\frac{1}{T_b^2}<\frac{1}{Z_{1-\alpha /2}^2}).$$

Because the set $\{\frac{1}{T_a^2}+\frac{1}{T_b^2}<\frac{1}{Z_{1-\alpha /2}^2}\}$ is a subset of $\{\frac{1}{T_a^2}<\frac{1}{Z_{1-\alpha /2}^2},\frac{1}{T_b^2}<\frac{1}{Z_{1-\alpha /2}^2}\}$, we have

$$P(|T_{\mathrm{\Delta }}|>Z_{1-\alpha /2})\le P(|T_a|>Z_{1-\alpha /2})P(|T_b|>Z_{1-\alpha /2}).$$

The left hand of the above inequality is the probability of rejecting H₀ using the normal approximation method, while the right hand side is the probability of rejecting H₀ using the joint significant test method. The equality would hold when either |T_a| > Z_1−α/2 or |T_b| > Z_1−α/2 occurs with probability 1. This means that the null hypothesis on the mediation effect is less likely to be rejected in the normal approximation approach than the joint significance test approach, hence the type I error rate and power from the normal approximation approach are lower than or equal to that from the joint significance approach. As the type I error rate from the joint significance approach is already no more than α, the normal approximation approach also has a type I error rate lower than or equal to the target level α.

Consistent with our theoretical finding, in the cross-sectional setting, the joint significance test has been shown to be more powerful and has better type I error rate than the normal approximation method through simulation [2, 27].

The power from the normal approximation method for a given sample size n can be expressed as

$$\begin{array}{lll}{P}_S\hfill & =\hfill & \mathrm{\Phi }\left(\sqrt{\frac{a_0^2{b}_0^2}{V_{\mathrm{\Delta }}}}-Z_{1-\alpha /2}\right)\hfill \\ \hfill & =\hfill & \mathrm{\Phi }\left(\sqrt{\frac{n}{\frac{1}{b_0^2{A}_t{\sigma }_m^2(1-R_{m|x,z}^2)}+\frac{{\sigma }_m^2-a_0^2{\sigma }_x^2}{a_0^2{\sigma }_x^2(1-R_{x|z}^2)}}}-Z_{1-\alpha /2}\right),\hfill \end{array}$$

which reduces to

$$P_S=\mathrm{\Phi }\left(\sqrt{\frac{n}{\frac{1}{b_0^2{A}_t({\sigma }_m^2-a_0^2{\sigma }_x^2)}+\frac{{\sigma }_m^2-a_0^2{\sigma }_x^2}{a_0^2{\sigma }_x^2}}}-Z_{1-\alpha /2}\right)$$

when there is no covariate Z.

To calculate the sample size n needed to achieve power P for testing H₀: Δ = 0 versus H₁: Δ = ab, with a = a₀, b = b₀, where a₀ ≠ 0 and b₀ ≠ 0, the equation (a₀b₀)² = V_Δ(Z_1−α/2 + Z_P)² is solved, where Z_p is the p^th percentile of the standard normal distribution. The sample size needed can then be expressed as

$$n_S=\{\frac{1}{b_0^2{A}_t{\sigma }_m^2(1-R_{m|x,z}^2)}+\frac{{\sigma }_m^2-a_0^2{\sigma }_x^2}{a_0^2{\sigma }_x^2(1-R_{x|z}^2)}\}{(Z_{1-\alpha /2}+Z_P)}^2.$$ (12)

When there is no covariate, it is simplified as

$$n_S=\{\frac{1}{b_0^2{A}_t({\sigma }_m^2-a_0^2{\sigma }_x^2)}+\frac{{\sigma }_m^2-a_0^2{\sigma }_x^2}{a_0^2{\sigma }_x^2}\}{(Z_{1-\alpha /2}+Z_P)}^2.$$

Note that the power or sample size estimates using the normal approximation method can be misleading if the true distribution of Δ̂ deviates severely from the normal distribution.

3.3 Test of b

In practice, there might be situations that the investigators are quite certain that one of the component of Δ is not zero. In this case, test of the other parameter would be almost the same as that of the test of the mediation effect. Since the estimation of a only requires cross-sectional data of X and M while the estimation of b requires the collection of all data on X, M and the longitudinal outcome Y, most likely the investigators have more knowledge of a than b at the design stage. If there are evidences that the mediator has some effect on the outcome, usually this effect is established without adjustment of X. Furthermore, it is straightforward to test a with existing softwares. Therefore, we focus on developing power and sample size estimates for the test of b. Using the test of b approach in the power and sample size calculations for evaluating mediation effect has been proposed in other settings [40].

The power for testing b under H_1b: b = b₀, is P_b as expressed in (10). And the sample size needed for achieving power P from this method is

$$n_b=\frac{{(Z_{1-\alpha /2}+Z_P)}^2}{b_0^2{A}_t{\sigma }_m^2(1-R_{m|x,z}^2)},$$ (13)

which reduces to

$$n_b=\frac{{(Z_{1-\alpha /2}+Z_P)}^2}{b_0^2{A}_t({\sigma }_m^2-a_0^2{\sigma }_x^2)}$$

when there is no covariate Z.

However, in general, the test of b and the test of Δ is not the same. The test of Δ can be approximated by the test of b only when P_a, the power for testing a, is close to 1. From the joint significance test, the power for testing the mediation effect P_J is the product of P_a and P_b, so P_b ≥ P_J, and the two are close when P_a ≃ 1. For example, suppose α₁ = 0.05 and there is no covariates, and the value of a₀ and the sample size n are large enough so that $\frac{n}{1/r^2-1}\ge 18.373$, where $r^2=\frac{a_0^2{\sigma }_x^2}{{\sigma }_m^2}$ is the correlation coefficient between X and M, then P_a ≥ 0.99 so P_J is no more than 1% less than P_b. For n = 100, 200, 300, 400, 500, this occurs when r ≥ 0.394, 0.290, 0.240, 0.210, 0.188, respectively.

The fact that the test of b is more liberal than the test of Δ can be also seen from the comparison with the normal approximation approach. In the normal approximation approach, the test statistics, T_Δ, can be expressed in the forms of T_a and T_b, the test statistics for testing a and b, respectively, as follows

$$T_{\mathrm{\Delta }}^2=\frac{a^2{b}^2}{a^2\text{Var}(\hat{b})+b^2\text{Var}(\hat{a})}=\frac{1}{\frac{\text{Var}(\hat{b})}{b^2}+\frac{\text{Var}(\hat{a})}{a^2}}=\frac{1}{\frac{1}{T_b^2}+\frac{1}{T_a^2}}.$$

Hence |T_Δ| ≤ |T_b and |T_Δ| ≤ |T_a. This also shows that the test of the mediation effect has lower power than the test of either a or b. The test of mediation and the test of b are equivalent when |T_a| is infinitely large. When all the other conditions are fixed, this occurs when |a| and n are large.

A comparison of the three methods discussed in this section is summarized in Table 1.

Table 1. Comparison of the three hypothesis testing methods.

	Test of b*	Joint Significance test	Normal Approximation
Power	Pb	PaPb	$\mathrm{\Phi }(\sqrt{\frac{a_0^2{b}_0^2}{V_{\mathrm{\Delta }}^a}-{{\rm Z}}_{1-\alpha /2})}$
Type I error
a = b = 0	NA	α2	≤α2
a = 0, b = b0	NA	αPb	≤αPb
a = a0, b = 0	α	αPa	≤αPa

Note

^*Applicable only when P_a = 1

4 Simulation studies

Simulation studies were performed to examine the performance of the three methods for testing the longitudinal mediation effect, in particular, their Type I error rate and power, under limited sample size. We considered longitudinal studies with 2 and 5 visits, representing short and moderate length of follow-up, respectively. For k = 2, the planned follow-up times t = (0, 1), representing baseline and follow-up visits; for k = 5, t = (0, 1, 2, 3, 4), representing baseline and 4 follow-up visits with equal time intervals (e.g, annual follow-up). Values of 100, 200, …, 1000 were considered for the sample size n. The risk factor X was generated from Bernoulli distribution with P(X = 1) = 0.5 for the case of k = 2, and standard normal for the case of k = 5. The mediator M was generated as a normal random variable with marginal variance ${\sigma }_m^2=1$ and depends on X through model (1), with α₀ = 0, and a took values 0.25, 0.5 and 0.75 for the case of k = 2, and 0.10, 0.20 and 0.30 for the case of k = 5. Here a is the correlation coefficient between X and M for continuous X (k = 5), and the expected difference in M in SD unit between X = 1 and X = 0 for the binary X case (k = 2). The repeatedly measured outcome Y was generated from a random intercept model for model (2), with ${\beta }_0={\beta }_x^c={\beta }_m^c={\beta }_t={\beta }_x^l=0$, σ² = 1, ${\sigma }_u^2=\rho {\sigma }^2$, ${\sigma }_e^2=(1-\rho ){\sigma }^2$, and b was set as 0.1, 0.2 and 0.3 for k = 2, and 0.02, 0.03, 0.04 and 0.05 for the case of k = 5. Here b is the expected difference in the change in the outcome at follow up from baseline for k = 2, and the expected difference in the slope for k = 5, corresponding to 1 SD unit difference in M adjusting for X. We considered values of 0.25, 0.50 and 0.75 for the correlation coefficient ρ, representing low, medium and high levels of correlation, respectively. The first (or baseline) measure Y₁ was observed for every subject. For the missing data distribution, a constant dropout rate p_d, defined as the probability of missing Y_ij given Y_ij−1 was observed, j = 2, …, k, was considered. Values of 0, 0.1, 0.2 and 0.3 were considered for the dropout rate p_d. All simulations were repeated 2000 times. The significance level for all tests was set at α = 0.05.

To examine the empirical type I error rates of the methods, the data were generated with either a = b = 0, or one of a or b was zero, and the proportions of rejecting the null hypothesis H₀ were reported. To examine the performance of the power calculation formulae, the data were generated from non-zero values of a and b. The asymptotic power estimates (Asym) and empirical power estimates from 2000 simulations (Emp) were obtained for the separate tests of a and b, the joint significance test (Joint Sig) and the normal approximation (Norm Appr) methods for testing the mediation effect. The bias of the point estimate of the longitudinal mediation effect (Bias) was calculated as the difference between the average of the estimates of Δ from 2000 simulations and the true value. The 95% confidence intervals were obtained from the normal approximation method and the proportion that the true values of Δ were covered out of 2000 simulations (Cover) were reported.

Results on the empirical type I error rates for dropout rates of p_d=0, 0.3, ρ=0.25, 0.75 and sample size n=100, 200, 300, 500 and 1000 were reported in Table 2a and Table 2b for k = 2 and k = 5, respectively. In the case of b = 0, only the cases with ρ = 0.25 were reported. As expected, when both a = 0 and b = 0, the empirical Type I error rates of the joint significance test were close to the expected level of α² = 0.0025; when one of a or b is not zero, the empirical type I error rate of the join significance test is approximately 0.05 multiplied by the proportion of rejecting the hypothesis on the non-zero component, and hence is lower than 0.05, but is close to 0.05 when the power for testing the non-zero parameter is close to 1. Also consistent with the theoretical results, the empirical type I error rate of the normal approximation method is lower than that of the joint significance test method, and the differences are smaller when either a or b is not zero and the power for detecting the non-zero component is close to 1.

Table 2a. Empirical Type 1 error rates for 2-visit studies (ρ = 0.25) under various values for a₀ (true value of a), b₀ (true value of b), sample size n and drop out rate, with joint significance test (Joint Sig) and normal approximation (Norm Appr) methods used for the test of Δ = ab.

			no drop out				30% drop out
			Test of a	Test of b	Test of Δ = ab		Test of a	Test of b	Test of Δ = ab
a0	b0	n			Joint Sig	Norm App			Joint Sig	Norm Appr
0	0	100	0.0595	0.0505	0.0030	0.0000	0.0455	0.0540	0.0015	0.0000
		200	0.0540	0.0600	0.0050	0.0005	0.0500	0.0525	0.0025	0.0010
		300	0.0490	0.0510	0.0025	0.0000	0.0540	0.0475	0.0015	0.0000
		500	0.0535	0.0455	0.0035	0.0000	0.0545	0.0415	0.0015	0.0000
		1000	0.0490	0.0480	0.0025	0.0000	0.0540	0.0595	0.0030	0.0000
0	0.10	100	0.0520	0.1295	0.0070	0.0005	0.0500	0.1240	0.0050	0.0000
		200	0.0515	0.2200	0.0110	0.0020	0.0610	0.1840	0.0090	0.0000
		300	0.0570	0.2985	0.0160	0.0020	0.0465	0.2505	0.0095	0.0000
		500	0.0510	0.4315	0.0240	0.0025	0.0535	0.3845	0.0215	0.0030
		1000	0.0520	0.7295	0.0380	0.0085	0.0490	0.6380	0.0315	0.0050
	0.20	100	0.0495	0.3820	0.0185	0.0055	0.0495	0.3065	0.0155	0.0010
		200	0.0485	0.6325	0.0290	0.0050	0.0470	0.5045	0.0180	0.0035
		300	0.0515	0.7840	0.0395	0.0065	0.0510	0.6880	0.0315	0.0080
		500	0.0485	0.9505	0.0440	0.0185	0.0535	0.9010	0.0485	0.0140
		1000	0.0515	0.9995	0.0515	0.0305	0.0500	0.9970	0.0500	0.0265
	0.30	100	0.0455	0.6745	0.0300	0.0075	0.0535	0.5720	0.0260	0.0020
		200	0.0525	0.9260	0.0475	0.0170	0.0550	0.8505	0.0450	0.0120
		300	0.0430	0.9845	0.0430	0.0205	0.0475	0.9620	0.0460	0.0150
		500	0.0505	1.0000	0.0505	0.0335	0.0420	0.9975	0.0420	0.0260
0.25	0	100	0.2490	0.0495	0.0145	0.0025	0.2480	0.0515	0.0135	0.0015
		200	0.4330	0.0520	0.0170	0.0005	0.4370	0.0445	0.0200	0.0035
		300	0.6030	0.0545	0.0310	0.0040	0.5675	0.0525	0.0335	0.0035
		500	0.8055	0.0530	0.0435	0.0140	0.8110	0.0515	0.0435	0.0095
		1000	0.9795	0.0470	0.0460	0.0190	0.9805	0.0485	0.0470	0.0225
0.50	0	100	0.7355	0.0390	0.0240	0.0045	0.7430	0.0605	0.0505	0.0135
		200	0.9455	0.0440	0.0410	0.0160	0.9595	0.0555	0.0540	0.0205
		300	0.9930	0.0495	0.0490	0.0255	0.9930	0.0470	0.0470	0.0250
		500	1.0000	0.0560	0.0560	0.0420	0.9995	0.0490	0.0490	0.0370
0.75	0	100	0.9790	0.0480	0.0470	0.0205	0.9820	0.0570	0.0560	0.0270
		200	1.0000	0.0575	0.0575	0.0380	1.0000	0.0495	0.0495	0.0365
		300	1.0000	0.0500	0.0500	0.0375	1.0000	0.0555	0.0555	0.0435
		500	1.0000	0.0495	0.0495	0.0420	1.0000	0.0545	0.0545	0.0465

Table 2b. Empirical Type 1 error rates for 5-visit studies (ρ = 0.25) under various values for a₀ (true value of a), b₀ (true value of b), sample size n and drop out rate, with joint significance test (Joint Sig) and normal approximation (Norm Appr) methods used for the test of Δ = ab.

			no drop out				30% drop out
			Test of a	Test of b	Test of Δ = ab		Test of a	Test of b	Test of Δ = ab
a0	b0	n			Joint Sig	Norm App			Joint Sig	Norm Appr
0	0	100	0.0540	0.0435	0.0025	0.0000	0.0610	0.0525	0.0010	0.0000
		200	0.0530	0.0425	0.0010	0.0000	0.0535	0.0515	0.0030	0.0000
		300	0.0575	0.0525	0.0025	0.0000	0.0555	0.0450	0.0035	0.0000
		500	0.0490	0.0615	0.0025	0.0000	0.0490	0.0485	0.0020	0.0000
		1000	0.0450	0.0490	0.0015	0.0000	0.0505	0.0420	0.0020	0.0000
0	0.02	100	0.0495	0.1260	0.0065	0.0000	0.0505	0.0760	0.0045	0.0010
		200	0.0445	0.1710	0.0080	0.0005	0.0500	0.1060	0.0055	0.0005
		300	0.0500	0.2315	0.0110	0.0005	0.0555	0.1380	0.0065	0.0000
		500	0.0535	0.3720	0.0195	0.0025	0.0500	0.1960	0.0115	0.0010
		1000	0.0500	0.6215	0.0320	0.0075	0.0425	0.3245	0.0155	0.0015
	0.03	100	0.0550	0.1840	0.0065	0.0005	0.0550	0.1120	0.0060	0.0000
		200	0.0520	0.3195	0.0180	0.0020	0.0465	0.1635	0.0100	0.0005
		300	0.0550	0.4755	0.0245	0.0045	0.0555	0.2320	0.0155	0.0010
		500	0.0480	0.6885	0.0340	0.0035	0.0395	0.3540	0.0140	0.0020
		1000	0.0485	0.9350	0.0445	0.0180	0.0505	0.5990	0.0330	0.0040
	0.05	100	0.0545	0.4415	0.0210	0.0025	0.0495	0.2035	0.0080	0.0010
		200	0.0540	0.7415	0.0395	0.0055	0.0465	0.3830	0.0170	0.0015
		300	0.0540	0.8775	0.0470	0.0130	0.0530	0.5130	0.0270	0.0040
		500	0.0440	0.9795	0.0425	0.0205	0.0440	0.7455	0.0305	0.0055
		1000	0.0465	1.0000	0.0465	0.0325	0.0570	0.9540	0.0555	0.0240
0.10	0	100	0.1625	0.0590	0.0125	0.0005	0.1615	0.0480	0.0080	0.0005
		200	0.2830	0.0475	0.0155	0.0000	0.2890	0.0395	0.0125	0.0015
		300	0.4100	0.0590	0.0195	0.0020	0.3900	0.0505	0.0210	0.0005
		500	0.6215	0.0500	0.0355	0.0045	0.5905	0.0545	0.0325	0.0075
		1000	0.8900	0.0495	0.0445	0.0095	0.8860	0.0525	0.0460	0.0120
0.20	0	100	0.5400	0.0525	0.0285	0.0050	0.5250	0.0500	0.0265	0.0070
		200	0.8225	0.0495	0.0425	0.0120	0.8150	0.0585	0.0490	0.0120
		300	0.9345	0.0515	0.0470	0.0140	0.9430	0.0570	0.0560	0.0225
		500	0.9970	0.0470	0.0465	0.0280	0.9950	0.0505	0.0500	0.0290
		1000	1.0000	0.0455	0.0455	0.0385	1.0000	0.0485	0.0485	0.0345
0.30	0	100	0.8745	0.0500	0.0425	0.0120	0.8805	0.0435	0.0380	0.0170
		200	0.9915	0.0500	0.0495	0.0235	0.9930	0.0485	0.0485	0.0225
		300	0.9995	0.0515	0.0515	0.0355	0.9995	0.0500	0.0500	0.0300
		500	1.0000	0.0400	0.0400	0.0330	1.0000	0.0520	0.0520	0.0430

Results on the empirical power for k = 2, with a₀ = 0.25, 0.75 and b₀ = 0.1, 0.3, and for k = 5, with a₀ = 0.1, 0.3 and b₀ = 0.02, 0.05, were shown in Table 3a and Table 3b, respectively, for sample size n=100, 200, 500 and 1000, ρ=0.25, 0.75, and dropout rates p_d=0, 0.1 and 0.3. In all scenarios the bias of the estimate of the mediation effect is ignorable, showing that Δ̂ is a consistent estimate. The empirical power estimates from the joint significance test are close to the asymptotic power estimates in all situations. The test of b method has empirical power similar to that of the joint significance test only when the power for detecting a is close to 1. The asymptotic power for testing b is close to the empirical power in all situations considered, which increases as ρ increases or the dropout rate decreases. The asymptotic power estimate of the normal approximation method deviates from its empirical power estimate unless the sample size or at least one of the values of a and b is large. The coverage proportion of the 95% confidence interval from the normal approximation method is either within or below the nominal level.

Table 3a.

Asymptotic (Asym) and empirical (Emp) powers for 2-visit studies from 2000 simulations under various values for a₀ (true value of a), b₀ (true value of b), sample size n and drop out rate p_d, with joint significance test (Joint Sig) and normal approximation (Norm Appr) methods used for the test of Δ = ab. For normal approximation method, percentage that 95% confidence interval covers the true value (Cover) is also presented. Bias is the simulation bias of Δ̂ = âb̂.

						Test of a		Test of b		Test of Δ = ab
										Joint Sig		Norm Appr
a	b	n	p	pd	Bias Δ̂	Asym	Emp	Asym	Emp	Asym	Emp	Cover	Asym	Emp
0.25	0.10	100	0.25	0	−0.0004	0.242	0.231	0.125	0.125	0.030	0.026	0.951	0.101	0.004
				0.10	0.0005	0.242	0.247	0.120	0.136	0.029	0.030	0.945	0.098	0.003
				0.30	0.0024	0.242	0.245	0.107	0.139	0.026	0.036	0.961	0.091	0.007
			0.75	0	0.0001	0.242	0.234	0.289	0.283	0.070	0.077	0.919	0.153	0.013
				0.10	−0.0003	0.242	0.239	0.268	0.279	0.065	0.071	0.915	0.149	0.013
				0.30	−0.0007	0.242	0.258	0.223	0.212	0.054	0.050	0.934	0.137	0.007
		200	0.25	0	0.0008	0.429	0.426	0.208	0.214	0.089	0.095	0.923	0.160	0.021
				0.10	−0.0003	0.429	0.424	0.197	0.193	0.085	0.084	0.919	0.154	0.021
				0.30	0.0005	0.429	0.429	0.173	0.178	0.074	0.081	0.928	0.141	0.014
			0.75	0	0.0002	0.429	0.442	0.510	0.498	0.219	0.222	0.911	0.263	0.075
				0.10	0.0000	0.429	0.432	0.474	0.462	0.203	0.197	0.912	0.254	0.063
				0.30	−0.0005	0.429	0.417	0.395	0.381	0.169	0.151	0.901	0.232	0.038
		500	0.25	0	0.0002	0.804	0.798	0.441	0.442	0.355	0.351	0.924	0.331	0.164
				0.10	−0.0002	0.804	0.800	0.418	0.409	0.336	0.332	0.918	0.318	0.144
				0.30	0.0005	0.804	0.794	0.363	0.385	0.292	0.301	0.928	0.287	0.130
		1000	0.25	0	−0.0002	0.979	0.979	0.726	0.714	0.711	0.699	0.943	0.577	0.550
				0.10	0.0000	0.979	0.982	0.698	0.702	0.683	0.691	0.935	0.557	0.525
				0.30	−0.0001	0.979	0.981	0.624	0.613	0.610	0.603	0.935	0.506	0.459
	0.30	100	0.25	0	0.0012	0.242	0.250	0.681	0.674	0.165	0.166	0.926	0.200	0.048
				0.10	0.0011	0.242	0.245	0.652	0.624	0.158	0.150	0.913	0.198	0.047
				0.30	0.0000	0.242	0.240	0.579	0.564	0.140	0.136	0.903	0.192	0.045
		200	0.25	0	0.0009	0.429	0.434	0.930	0.920	0.399	0.400	0.934	0.353	0.224
				0.10	0.0014	0.429	0.442	0.914	0.911	0.392	0.401	0.940	0.348	0.214
				0.30	0.0013	0.429	0.426	0.863	0.838	0.370	0.359	0.924	0.337	0.185
		500	0.25	0	0.0005	0.804	0.808	1.000	1.000	0.804	0.807	0.936	0.706	0.759
				0.10	0.0007	0.804	0.800	1.000	1.000	0.804	0.800	0.930	0.700	0.748
				0.30	0.0004	0.804	0.808	0.998	0.998	0.803	0.806	0.935	0.682	0.728
0.75	0.10	100	0.25	0	−0.0028	0.982	0.983	0.115	0.112	0.112	0.110	0.964	0.112	0.056
				0.10	−0.0015	0.982	0.975	0.110	0.117	0.108	0.114	0.961	0.108	0.060
				0.30	0.0009	0.982	0.977	0.099	0.106	0.097	0.103	0.965	0.097	0.056
		200	0.25	0	−0.0022	1.000	1.000	0.187	0.166	0.187	0.166	0.959	0.182	0.139
				0.10	−0.0006	1.000	1.000	0.178	0.170	0.178	0.170	0.947	0.173	0.137
				0.30	0.0020	1.000	1.000	0.156	0.162	0.156	0.162	0.964	0.153	0.127
		500	0.25	0	0.0001	1.000	1.000	0.395	0.400	0.395	0.400	0.952	0.384	0.381
				0.10	−0.0004	1.000	1.000	0.373	0.368	0.373	0.368	0.946	0.363	0.347
				0.30	−0.0003	1.000	1.000	0.324	0.321	0.324	0.321	0.951	0.317	0.304
		1000	0.25	0	−0.0008	1.000	1.000	0.668	0.665	0.668	0.665	0.952	0.653	0.654
				0.10	0.0006	1.000	1.000	0.639	0.633	0.639	0.633	0.958	0.625	0.627
				0.30	0.0007	1.000	1.000	0.566	0.572	0.566	0.572	0.953	0.555	0.561
	0.30	100	0.25	0	−0.0003	0.982	0.985	0.622	0.617	0.611	0.607	0.937	0.508	0.453
				0.10	0.0034	0.982	0.980	0.593	0.597	0.582	0.583	0.937	0.488	0.434
				0.30	0.0012	0.982	0.982	0.523	0.517	0.513	0.507	0.948	0.438	0.365
		200	0.25	0	−0.0013	1.000	1.000	0.895	0.880	0.895	0.880	0.945	0.800	0.851
				0.10	0.0041	1.000	1.000	0.874	0.873	0.874	0.873	0.949	0.779	0.842
				0.30	−0.0027	1.000	1.000	0.814	0.792	0.814	0.792	0.934	0.723	0.745

Table 3b.

Asymptotic (Asym) and empirical (Emp) powers for 5-visit studies from 2000 simulations under various values for a₀ (true value of a), b₀ (true value of b), sample size n and drop out rate p_d, with joint significance test (Joint Sig) and normal approximation (Norm Appr) methods used for the test of Δ = ab. For normal approximation method, percentage that 95% confidence interval covers the true value (Cover) is also presented. Bias is the simulation bias of Δ̂ = âb̂.

						Test of a		Test of b		Test of Δ = ab
										Joint Sig		Norm Appr
a	b	n	p	pd	Bias Δ̂	Asym	Emp	Asym	Emp	Asym	Emp	Cover	Asym	Emp
0.10	0.02	100	0.25	0	0.0001	0.170	0.174	0.109	0.124	0.019	0.022	0.970	0.085	0.002
				0.10	−0.0001	0.170	0.169	0.093	0.099	0.016	0.019	0.973	0.078	0.002
				0.30	0.0001	0.170	0.181	0.067	0.075	0.011	0.018	0.982	0.062	0.003
			0.75	0	0.0001	0.170	0.176	0.242	0.244	0.041	0.045	0.938	0.120	0.006
				0.10	0.0000	0.170	0.171	0.188	0.180	0.032	0.029	0.942	0.110	0.003
				0.30	0.0000	0.170	0.172	0.112	0.111	0.019	0.017	0.970	0.087	0.002
		200	0.25	0	−0.0001	0.295	0.292	0.176	0.173	0.052	0.048	0.928	0.130	0.009
				0.10	0.0001	0.295	0.310	0.145	0.164	0.043	0.049	0.940	0.115	0.010
				0.30	0.0001	0.295	0.304	0.096	0.105	0.028	0.026	0.971	0.086	0.003
			0.75	0	0.0001	0.295	0.315	0.429	0.439	0.127	0.126	0.909	0.198	0.031
				0.10	0.0000	0.295	0.300	0.329	0.324	0.097	0.094	0.907	0.178	0.017
				0.30	0.0001	0.295	0.287	0.182	0.203	0.054	0.059	0.925	0.132	0.008
		500	0.25	0	0.0001	0.613	0.607	0.369	0.382	0.226	0.225	0.919	0.260	0.074
				0.10	0.0000	0.613	0.620	0.296	0.330	0.181	0.204	0.911	0.225	0.057
				0.30	0.0001	0.613	0.627	0.178	0.184	0.109	0.115	0.932	0.154	0.025
			0.75	0	0.0000	0.613	0.608	0.804	0.808	0.493	0.492	0.921	0.419	0.268
				0.10	0.0000	0.613	0.620	0.670	0.657	0.411	0.405	0.915	0.375	0.194
				0.30	0.0000	0.613	0.606	0.383	0.381	0.235	0.230	0.908	0.266	0.072
		1000	0.25	0	0.0000	0.888	0.897	0.632	0.660	0.562	0.589	0.927	0.461	0.359
				0.10	0.0000	0.888	0.901	0.521	0.530	0.463	0.479	0.916	0.398	0.263
				0.30	0.0000	0.888	0.890	0.311	0.317	0.276	0.286	0.934	0.265	0.141
			0.75	0	0.0000	0.888	0.882	0.978	0.980	0.869	0.865	0.923	0.700	0.762
				0.10	0.0000	0.888	0.886	0.924	0.915	0.821	0.808	0.929	0.641	0.652
				0.30	−0.0001	0.888	0.890	0.652	0.637	0.579	0.565	0.922	0.472	0.338
	0.05	100	0.25	0	0.0001	0.170	0.171	0.443	0.430	0.075	0.078	0.925	0.140	0.012
				0.10	−0.0002	0.170	0.160	0.356	0.349	0.061	0.054	0.936	0.134	0.004
				0.30	−0.0001	0.170	0.158	0.212	0.225	0.036	0.032	0.939	0.115	0.006
		200	0.25	0	0.0000	0.295	0.289	0.729	0.738	0.215	0.209	0.927	0.237	0.080
				0.10	0.0000	0.295	0.297	0.615	0.619	0.181	0.182	0.912	0.224	0.056
				0.30	−0.0001	0.295	0.282	0.374	0.364	0.110	0.100	0.916	0.188	0.021
		500	0.25	0	0.0000	0.613	0.618	0.982	0.982	0.602	0.607	0.933	0.503	0.454
				0.10	0.0000	0.613	0.614	0.945	0.942	0.580	0.576	0.936	0.476	0.390
				0.30	0.0001	0.613	0.624	0.736	0.727	0.451	0.451	0.915	0.397	0.237
		1000	0.25	0	0.0000	0.888	0.890	1.000	1.000	0.888	0.890	0.950	0.794	0.854
				0.10	−0.0001	0.888	0.879	0.999	1.000	0.888	0.879	0.932	0.767	0.826
				0.30	0.0000	0.888	0.886	0.956	0.965	0.849	0.857	0.936	0.670	0.716
0.30	0.02	100	0.25	0	−0.0004	0.882	0.869	0.103	0.097	0.091	0.081	0.974	0.100	0.030
				0.10	−0.0005	0.882	0.873	0.089	0.096	0.078	0.084	0.975	0.087	0.024
				0.30	−0.0002	0.882	0.877	0.065	0.079	0.057	0.069	0.978	0.064	0.027
		200	0.25	0	−0.0002	0.994	0.993	0.165	0.152	0.164	0.151	0.958	0.159	0.099
				0.10	0.0000	0.994	0.990	0.136	0.147	0.136	0.145	0.954	0.133	0.093
				0.30	−0.0002	0.994	0.988	0.092	0.088	0.091	0.088	0.967	0.091	0.060
		500	0.25	0	0.0000	1.000	1.000	0.344	0.337	0.344	0.337	0.958	0.330	0.310
				0.10	−0.0002	1.000	1.000	0.276	0.271	0.276	0.271	0.951	0.268	0.241
				0.30	0.0000	1.000	1.000	0.167	0.166	0.167	0.166	0.955	0.165	0.145
		1000	0.25	0	0.0000	1.000	1.000	0.596	0.597	0.596	0.597	0.959	0.576	0.579
				0.10	0.0000	1.000	1.000	0.488	0.506	0.488	0.506	0.950	0.474	0.489
				0.30	0.0000	1.000	1.000	0.290	0.292	0.290	0.292	0.955	0.285	0.277
	0.05	100	0.25	0	−0.0001	0.882	0.871	0.414	0.407	0.365	0.354	0.924	0.331	0.182
				0.10	−0.0007	0.882	0.861	0.332	0.300	0.293	0.251	0.927	0.279	0.114
				0.30	−0.0003	0.882	0.874	0.198	0.191	0.175	0.167	0.947	0.181	0.066
		200	0.25	0	0.0001	0.994	0.993	0.693	0.693	0.688	0.687	0.938	0.577	0.571
				0.10	0.0002	0.994	0.992	0.579	0.586	0.575	0.583	0.938	0.493	0.465
				0.30	−0.0001	0.994	0.993	0.349	0.342	0.347	0.340	0.945	0.316	0.241

5 Example

We applied the power analysis methods to an example from the Einstein Aging Study (EAS) [1]. EAS is a prospective cohort study aimed to identify risk factors for dementia which was initiated in 1994 and last renewed in 2011. One aim in this study is to examine whether baseline cerebral vascular reactivity (CVR) from Transcranial Doppler ultrasonography (TCD) with carbon dioxide (CO2) challenge, denoted by M, will mediate the effect of baseline clinical cardiovascular disease (CAD) status, denoted by X, on the rate of decline in cognitive performance (denoted by Y) measured by the free and cued reminding test (FCSRT) [41]. The rationale behind this hypothesis is that cerebral microvascular damage might be a possible pathway for the effect of CAD on cognitive decline, where CVR is a measure of cerebromicrovascular function and decreased reactivity may reflect microvascular damage. There are 400 subjects available for annual follow-up for up to 5 years. The sample consists of a mix of current EAS participants and new subjects that will be enrolled. It is projected that 30%, 8%, 24%, 19% and 19% of the sample will have 1, 2, 3, 4 and 5 repeated measures of FCSRT.

Based on results from previous studies, the residual variance is compound symmetry, with a correlation coefficient ρ conservatively estimated as 0.50 and the residual standard deviation σ approximately 6.4. The prevalence of clinical CAD at baseline is p_x = 14% so that ${\sigma }_x^2=p_x(1-p_x)=0.12$. Suppose the CVR measure M is standardized, i.e., ${\sigma }_m^2=1$, and the difference in CVR between subjects with and without CAD, the value of a₀, is 0.5. The value of b₀, i.e., the change in the rate of decline in FCSRT corresponding to 1 SD unit increase in CVR adjusting for CAD, is expected to be at least 0.6. The joint significance test method is proposed to test the hypothesis on the mediation effect, and the estimate of power to detect the mediation effect Δ = a₀b₀ = 0.30 is 94%. If the normal approximation method is used, the power estimate is 81%. As expected, it underestimates power as compared to the joint significance test. If the test of b method is used, the power estimate is 99.8%. Because the power for detecting a₀ is 94%, compared to the joint significance test, it overestimates power by 6%. The difference is not big here because the power for detecting a₀ is not too far from 1. The empirical power estimates from 2000 simulations based on the joint significance test, the normal approximation and the test of b methods are 94.1%, 90.2% and 99.95%, respectively; similar to their corresponding asymptotic estimates except for the normal approximation method whose asymptotic power estimate is lower than the empirical power estimate.

6 Discussion

We have shown that the product of coefficients a, the parameter measures the effect of the risk factor on the mediator, and b, the parameter measures the independent effect of the mediator on the change in outcome adjusting for the risk factor, is an appropriate measure of the mediation effect. Currently available methods for estimating power and sample size for detecting a mediation effect mostly rely on simulation studies and do not incorporate the presence of missing data. We have evaluated three methods to test the presence of longitudinal mediation effect: the joint significance test, the normal approximation and the test of b methods. Closed form formulae for power and sample size estimation are developed while allowing the existence of missing data, which is commonly seen in longitudinal studies. Such closed form not only allows easy computation for power and/or sample size, it clearly demonstrates how each parameter affects the power for testing the mediation effect so that it can provide guidance on how to design an optimal longitudinal study, taking into consideration of the extent of missing data, number of possible repeated measurements and level of correlations among repeated measured outcome, etc.

Simulation studies show that the joint significance test has better empirical type I error rate and empirical power than the normal approximation method. The asymptotic power estimate from the joint significance test is close to the empirical power estimate under limited sample sizes. The joint significance test is conservative because the type I error rate is lower than the target value, but it is close to the target value when the null hypothesis corresponds to only one zero component and the power for detecting the nonzero component is close to 1. For the normal approximation method, the asymptotic power estimate may deviate from the empirical power unless the sample size or at least one of values of a and b is large. It can provide confidence intervals for the mediation effect, but the coverage proportion can be lower than expected. The test of b is not the same as the test of the mediation effect unless the null hypothesis of the mediation effect corresponds to nonzero a and the power for detecting it is close to 1, in which case it is a reasonable shortcut for testing the mediation effect. In general, the joint significance test performs the best for testing mediation effects and is recommended. Although the joint significance test is only for hypothesis testing and can not provide confidence intervals of the mediation effects, it is sufficient for the power analysis purpose at the design stage. Since the bootstrap method can provide a reliable estimate of the distribution of the mediation effect estimate and thus can serve for both the purposes of hypothesis testing and estimation of confidence intervals, it is usually recommended at the data analysis stage. However, it is hard to implement at the design stage without actual data unless simulation data are used. Nevertheless, values calculated from our formulae can serve as a starting point for the bootstrap or other computation intensive power analysis approaches.

When the mediator is binary, the linear model (1) is not appropriate for the association between the risk factor and the mediator, so the product of coefficients ab is no longer the difference in the coefficients of X_it_ij in model (2) and (6). Approaches that either model the mediator directly or treat it as the indicator of a latent continuous variable can be applied [42, 43, 44]. Recent years also saw growing interests in the mediation model that includes the interaction between the risk factor and the mediator on the outcome (e.g., [45, 46, 47]). In our case, this means that the interaction term between X and M, X_iM_i, measures the interaction effect at baseline, as well as its product with time, X_iM_it_ij, measures the interaction effect in the rate of change in the outcome, will be added to model (2). The general formula (7) can be used to calculate the variance of the parameter estimates for the model with interactions between X and M. Power analysis under these situations will be considered in future research.