Optimal combination of number of participants and number of repeated measurements in longitudinal studies with time-varying exposure

Abstract

In the context of observational longitudinal studies, we explored the values of the number of participants and the number of repeated measurements that maximize the power to detect the hypothesized effect, given the total cost of the study. We considered two different models, one that assumes a transient effect of exposure and one that assumes a cumulative effect. Results were derived for a continuous response variable, whose covariance structure was assumed to be damped exponential, and a binary time-varying exposure. Under certain assumptions, we derived simple formulas for the approximate solution to the problem in the particular case in which the response covariance structure is assumed to be compound symmetry. Results showed the importance of the exposure intraclass correlation in determining the optimal combination of the number of participants and the number of repeated measurements, and therefore the optimized power. Thus, incorrectly assuming a time-invariant exposure leads to inefficient designs. We also analyzed the sensitivity of results to dropout, mis-specification of the response correlation structure, allowing a time-varying exposure prevalence and potential confounding impact. We illustrated some of these results in a real study. In addition, we provide software to perform all the calculations required to explore the combination of the number of participants and the number of repeated measurements.

1. Introduction

Sample size and power formulas in the context of longitudinal studies comparing two groups (e.g., exposed and unexposed) have been widely studied but mostly in the context of experiments. Hence, they assumed that the exposure is assigned by design and, in most cases, they considered that it is time-invariant, as in many clinical trials (e.g. placebo vs. treatment groups) [1–12]. Some studies have considered the case of exposures that vary over time but in a way that is controlled by the investigator, such as in crossover trials [13–15]. However, in observational longitudinal studies, exposure is not controlled by the investigator and often such studies involve time-varying exposures, which can imply both a large number of observed exposure patterns and a high variability in the number of exposed periods per participant. Two recently published papers showed that it is crucial to characterize the within-individual variation of the exposure in order to obtain correct calculations of power and sample size in that context [16, 17].

When neither the number of participants nor the number of repeated measurements are fixed a priori, it can be of interest to find their optimal combination in terms of maximizing the power or minimizing the cost of the study subject to a budget or power constraint, respectively. This is important for the efficient allocation of financial resources. Some authors have studied this problem but they did not consider a time-varying exposure [18–27]. As occurs in the unconstrained design problem, it is expected that the optimal design in the constrained problem will also be sensible to the degree of within-subject variation in exposure.

In this paper, we studied the combination of number of participants and number of repeated measurements in observational longitudinal studies such that the power to detect the hypothesized effect is maximized without exceeding a fixed budget, or the cost of the study is minimized while achieving a certain target power. We considered a multivariate normal response variable with a damped exponential covariance structure and a binary exposure that can be time-varying. We analyzed two response patterns under the alternative hypothesis, one assuming an acute and transient effect and the other a cumulative effect. We explored the influence of several parameters, including the dropout rate, on the design results. In addition, we also explored the potential impact of the presence of confounders in the study. We illustrate the new methodology with data from a study of the respiratory effects of cleaning products [28]. In addition, we provide a publicly available open source R package to perform all the design calculations described in the paper.

2. Parameterization and models

To design a longitudinal study with a fixed follow-up time, we need to find the combination of number of participants (N) and number of repeated measurements per participant (r, i.e., the total number of measurements per participant is r + 1). Without loss of generality, we defined the fixed duration of the study as the time unit. We assumed that all measurements for all participants are taken at the same set of time points, t = 0, 1/r, 2/r, … , 1, where 1/r is the elapsed time between two consecutives measurements. We also assumed a linear form for the mean, $\mathbb{E}\left({\mathrm{Y}}_i\right)={\mathrm{X}}_i\mathrm{B}(i=1,\dots ,N)$, where Y_i and X_i are the multivariate normal response of interest and the covariate matrix for participant i, respectively, and B is the vector of unknown regression parameters; and Var(Y_i|X_i) = Σ(i = 1, … , N), where Σ is the (r + 1) × (r + 1) residual covariance matrix assumed equal for all participants (i.e., we assumed homogeneity across the participants).

The constant mean difference (CMD) response pattern is defined as

$$\mathbb{E}(Y_{\mathit{ij}}\mid E_{\mathit{ij}},t_j)={\beta }_0+{\beta }_1{t}_j+{\beta }_2{E}_{\mathit{ij}},$$ (1)

where Y_ij and E_ij ∈ {0, 1} are the response and the exposure state, respectively, for the i-th participant at the j-th measurement, and t_j = j/r, j = 0, 1, … , r are the time points. The parameter of interest is β₂, which can be interpreted as the expected difference in the mean of the response variable, at any time point, between exposed and non-exposed. The minimum value of r in model (1) is 0, which corresponds to a cross-sectional study.

The linearly divergent difference (LDD) response pattern is defined as

$$\mathbb{E}(Y_{\mathit{ij}}\mid E_{\mathit{ij}},t_j)={\beta }_0+{\beta }_1{E}_{i0}+{\beta }_2{t}_j+{\beta }_3{E}_{\mathit{ij}}^{\star },$$ (2)

where $E_{\mathit{ij}}^{\star }=\frac{1}{r}{\sum }_{k=1}^j{E}_{\mathit{ik}}$ is the cumulative exposure for the i-th individual in the j-th measurement, assuming $E_{i0}^{\star }=0$ for all participants. The parameter of interest is β₃, which can be interpreted as the expected difference in the mean of the response variable at the end of the follow-up between the worst exposure pattern (i.e., those exposed at all measurements) and those not exposed for the entire follow-up. For a time-invariant exposure, model (2) is equivalent to a model with the main effects of time and exposure and their interaction. The minimum value of r in model (2) is 1, given that at least two measurements are needed to estimate the rate of change over time.

To characterize the covariance structure of the response, we considered a damped exponential structure (DEX) [29], whose covariance matrix has diagonal elements σ² and off–diagonal [j, j’] elements, ${\sigma }^2{\rho }^{{\mid \frac{j^{\prime }-j}{r}\mid }^{\theta }}$ where ρ is the correlation between the first and the last response measurements and θ ∈ [0, 1] is the damping parameter. We focused especially on two well-known cases, compound symmetry (CS(σ, ρ)), when θ = 0, and first order autoregressive (AR(1)), when θ = 1.

We considered missing completely at random data by allowing a monotone dropout pattern, i.e. that losing one individual measurement implies losing all the subsequent measurements of that individual. We assumed that there is no missing data at the first measurement and that each subject that has not dropped out of the study at a given measurement time has a probability π_m of dropout at the subsequent measurement, assuming a missing completely at random (MCAR) pattern. Thus, there are r + 1 possible dropout patterns. By representing the dropout patterns by a 0/1 string of length r + 1, the g-th pattern is ($(1,\stackrel{\left(g\right)}{\dots ,}1,0,\stackrel{(r+1-g)}{\dots },0)$) with probability

$${\pi }_g=\{\begin{array}{cc}{\pi }_m{(1-{\pi }_m)}^{g-1}\hfill & ,g=1,2,\dots ,r\hfill \\ {(1-{\pi }_m)}^r\hfill & ,g=r+1\hfill \end{array}\phantom{\}}.$$ (3)

If r = 0, there is only one measurement and π_m = 0.

For investigators performing study design calculations, it may be easier to provide a value for π_M, the proportion of subjects lost at the end of follow-up. Its relation to π_m is expressed as

$${\pi }_M=1-{(1-{\pi }_m)}^r.$$

The variance of the estimate of the regression coefficient of interest, β₂ in model (1) and β₃ in model (2), and hereinafter generically identified as β, is needed. We used the generalized least squares (GLS) estimator which takes into account the within-participant correlation. Since the covariate matrix X_i is unknown a priori, we used, following Whittemore [30] and Shieh [31], the asymptotic variance of the GLS estimate of B which is $\frac{1}{N}{\Sigma }_B$, where, after adapting for dropout following Yi [10],

$${\Sigma }_B={\left({\mathbb{E}}_X\left[\sum _{g=1}^{r+1}{\mathrm{X}}_{\mathit{ig}}^{\prime }{\Sigma }_g^{-1}{\mathrm{X}}_{\mathit{ig}}{\pi }_g\right]\right)}^{-1},$$ (4)

π_g is defined in (3), and $\frac{1}{N}{\Sigma }_B$ can be fully specified by the first- and second-order moments of the covariate distribution because Σ is assumed independent of the covariates in our context [32].

We are interested in the [m, m]–th element of Σ_B, where m is the position of β in the vector of regression coefficients B or, equivalently, the position of the column associated to the β parameter of interest in the design matrix, X_i,

$${\tilde{\sigma }}^2={\Sigma }_{B[m,m]}.$$ (5)

For some covariance structures of Σ, such as CS, and under assumptions above, there is a simple formula for (5), while for others it must be computed numerically.

The prevalence vector of exposures p_e = (p_e0, … , p_er) is needed. We allow the prevalence to vary linearly from p_e0 at the first measurement to p_er at the end of the follow-up. This linear trend of the exposure prevalence can thus be parameterized as

$$p_{\mathit{ej}}=\frac{1+\gamma j∕r}{1+\gamma ∕2}{\stackrel{‒}{p}}_e,j=0,1,\dots ,r,$$

where ${\stackrel{‒}{p}}_e=\frac{p_{e0}+p_{\mathit{er}}}{2}$ is the mean of the exposure prevalence along the r + 1 measurements and $\gamma =\frac{p_{\mathit{er}}-p_{e0}}{p_{e0}}$ is the relative change in the exposure prevalence between the first and the last measurements. When the exposure prevalence is constant, γ = 0.

In order to perform design calculations under models (1) and (2), the covariance of the exposure process, Σ_E, must also be characterized. Once the intraclass correlation of exposure is known, the exact value of ${\tilde{\sigma }}^2$ if the response covariance structure is CS and the response pattern is CMD is determined exactly (web-based supporting document, section A) or to good approximation otherwise [16, 17] without characterization of the full matrix Σ_E. The intraclass correlation of exposure is defined as

$${\rho }_e=\frac{\mathrm{sum}\left({\Sigma }_E\right)-\mathrm{Tr}\left({\Sigma }_E\right)}{r\mathrm{Tr}\left({\Sigma }_E\right)},$$

where sum() and Tr() denote the sum of the elements and the trace of a matrix respectively [33]. ρ_e can be interpreted as a measure of within-participant variation of exposure. When ρ_e takes its maximum value of 1, each of the participants are either exposed or unexposed for the entire follow-up (i.e., the exposure is time-invariant). Conversely, when ρ_e takes its minimum value, the within-participant variation of exposure is greatest [33]. The upper bound of ρ_e is lower than 1 when the exposure prevalence is time-varying [16] and 1 otherwise. For binary variables, as here, the lower bound of ρ_e is

$$-\frac{1}{r}+\frac{\text{frac}\left(\right(r+1\left){\stackrel{‒}{p}}_e\right)[1-\text{frac}((r+1){\stackrel{‒}{p}}_e\left)\right]}{r(r+1){\stackrel{‒}{p}}_e(1-{\stackrel{‒}{p}}_e)}$$

where frac(x) denotes the fractional (non-integer) part of x [34]. When the covariance structure of the exposure, Σ_E, is CS and the exposure prevalence is constant, ρ_e becomes the common off-diagonal term of the exposure correlation matrix.

As a tool for deciding an appropriate value for ρ_e at the study design phase, it can be useful to explore the distribution of the number of exposed periods per participant, once the values of ρ_e, ${\stackrel{‒}{p}}_e$, and r have been fixed and Σ_E is assumed to follow CS. For instance, Figure 1 shows this distribution for the case in which r = 3 and ${\stackrel{‒}{p}}_e=1∕4$, for ρ_e = −1/3, 0, 1/2 and 1. Our R package can be used to build this kind of figures.

Figure 1.

Distribution of the number of exposed periods, E_i. for r = 3, ${\stackrel{‒}{p}}_e=1∕4$ and different values of ρ_e, assuming exposure covariance structure CS with correlation parameter ρ_e, and no dropout.

All parameters that need to be provided by the investigator to compute the optimal combination of number of participants and number of repeated measurements are described in Table 1.

Table 1.

Parameters needed to compute (r_opt;N_opt).

Parameter	Description
Pattern	Assumed pattern of the response Y (CMD or LDD)
σ 2	Diagonal element of the DEX covariance structure matrix of the response Y
ρ	Correlation between the first and the last measurements of the response Y
θ	Damping parameter of the DEX covariance structure matrix of the response Y
ρ e	Intraclass correlation of the exposure E
p e0	Prevalence of the exposure E at the first measurement (at time 0)
per	Prevalence of the exposure E at the last measurement (at the end of the follow-up)
π M	Proportion of subjects lost by the end of follow-up
κ	Ratio between the cost of the first measurement and each of the subsequent ones
c 1	Cost of the first measurement (including recruitment) for one participant
Budget a	Total budget for the study, including recruitment and follow-up
Power b	Target power of the study
α b	Significance level
${\tilde{\beta }}^b$ b	Effect size under the alternative hypothesis

^aIf interested in maximizing the power of the study without exceeding a fixed budget.

^bIf interested in minimizing the total cost of the study while achieving a certain target power.

3. Optimal allocation

Let us call c₁ the monetary cost of the first measurement of each participant and suppose it is κ times more expensive than each of the subsequent ones, i.e

$$\kappa =\frac{\text{Cost of}1\text{st measurement}\left(c_1\right)}{\text{Cost of subsequent measurements}}\ge 1.$$

This is justified by the incorporation of recruitment costs into the first measurement. Then, the total cost of recruitment and follow-up of N participants for a total of r + 1 measurements is

$$\text{Cost}(N,r)={\mathit{Nc}}_1\left(1+\frac{r}{\kappa }\right).$$ (6)

Asymptotically, the power of a study testing the significance of an individual regression coefficient, β, through a Wald test can be expressed as

$$\text{Power}=\Phi \left(\frac{\mid \tilde{\beta }\mid \sqrt{N}}{\tilde{\sigma }}-z_{1-\alpha ∕2}\right),$$ (7)

where $\tilde{\beta }$ is the value of β under the alternative hyphotesis, $\tilde{\sigma }∕\sqrt{N}$ is the standard error of $\tilde{\beta }$, α is the significance level and z_q and Φ(·) are the q–th quantile and the cumulative density of the standard normal distribution, respectively.

Our aim is to find the combination of N and r (N_opt, r_opt) that maximizes the power of the study without exceeding a fixed budget, i.e.

$$\begin{array}{cc}\hfill {\text{max}}_{N,r}\hfill & \text{Power}\hfill \\ \hfill \text{Subject to}\hfill & \text{Cost}\le \text{Budget}\hfill \end{array}.$$

It can be easily shown that the value of r that solves this problem is the same as the solution to the problem of minimizing the cost of a study while achieving a certain target power, i.e.

$$\begin{array}{cc}\hfill {\text{min}}_{N,r}\hfill & \text{Cost}\hfill \\ \hfill \text{Subject to}\hfill & \text{Power}\ge {\text{Power}}_{\text{target}}\hfill \end{array}.$$

With the cost definition in (6), both problems are equivalent to solving the following unconstrained optimization problem,

$$\underset{r\in \mathbb{N}}{\min }(\kappa +r){\tilde{\sigma }}^2,$$ (8)

whose solution provides the value of r_opt. Then, the value of N_opt can be obtained from the constraint in the optimization problem as detailed below.

In the presence of missing data, the expression for the total cost of the study changes. The cost function (6) can be adapted by computing the expected cost over the r + 1 missing patterns (see web-based supporting document, section B), which results in

$$\text{Cost}={\mathit{Nc}}_1\left[1+\frac{(1-{\pi }_m)[1-{(1-{\pi }_m)}^r]}{{\pi }_m\kappa }\right].$$ (9)

The function to minimize becomes then

$$\underset{r\in \mathbb{N}}{\min }\left\{\kappa +\frac{1-{\pi }_m}{{\pi }_m}[1-{(1-{\pi }_m)}^r]\right\}{\left\{{\mathbb{E}}_X\left[\sum _{g=1}^{r+1}{\mathrm{X}}_{\mathit{ig}}^{\prime }{\Sigma }_g^{-1}{\mathrm{X}}_{\mathit{ig}}{\pi }_g\right]\right\}}_{[m,m]}^{-1},$$ (10)

which provides a solution for r_opt.

The optimal number of participants N_opt, is obtained as follows. If the investigator is interested in fixing the maximum acceptable budget in order to maximize the power, then the total budget and c₁ must be specified and N_opt is obtained from expression (9) as

$$N_{\mathrm{opt}}=\lfloor \tilde{N}\rfloor ,\text{where}\tilde{N}=\{\begin{array}{cccc}\frac{\text{Budget}}{c_1}{\left(1+\frac{r_{\mathrm{opt}}}{\kappa }\right)}^{-1}\hfill & ,\hfill & \text{if}\hfill & {\pi }_M=0\hfill \\ \frac{\text{Budget}}{c_1}{\left[1+\frac{{\pi }_M}{\kappa \left({(1-{\pi }_M)}^{-\frac{1}{r_{\mathrm{opt}}}}-1\right)}\right]}^{-1}\hfill & ,\hfill & \text{if}\hfill & {\pi }_M>0\hfill \end{array}\phantom{\}}$$ (11)

and the floor function ⌊x⌋ denotes the greatest integer not greater than x.

If the investigator is interested in achieving a certain power while minimizing the cost, then the power and the expected effect under the alternative hypothesis, $\tilde{\beta }$, are needed and N_opt is obtained from expression (7) as

$$N_{\mathrm{opt}}=\lceil \tilde{N}\rceil ,\text{where}\tilde{N}=\frac{{\tilde{\sigma }}_{\mathrm{opt}}^2}{{\mid \tilde{\beta }\mid }^2}{[z_{1-\alpha ∕2}+{\Phi }^{-1}(\text{Power}\left)\right]}^2$$ (12)

where ${\tilde{\sigma }}_{\mathrm{opt}}^2$ is the result of computing expression (5) for r = r_opt and the ceiling function ⌈x⌉ denotes the smallest integer not smaller than x.

It should be made clear that the combination of N and r that solves problem (10) can be considered optimal in terms of maximizing the power or minimizing the cost, and under the assumptions implicity made in the problem as, for instance, the asymptotic approximation in both (4) and (7), the cost function and dropout structures assumed. In the next section, we first solve problem (10) for some basic scenarios in which we derived simple formulas for the solution, and subsequently explore numerically other more complex scenarios.

4. Results for the optimal number of repeated measurements

4.1. Basic scenarios results

In addition to assuming asymptotic regime, the basic scenarios for which we have obtained formulas for r_opt are characterized by the following assumptions: 1) No dropout, i.e. π_m = 0; 2) the correlation structure of the response is CS(σ, ρ), i.e. DEX with θ = 0; and 3) the exposure prevalence is constant, i.e. p_ej = p_e, ∀j = 0, … , r. In addition, we make a fourth assumption when response pattern is LDD and the exposure is time-varying; in this case, the covariance structure of the exposure is assumed to be CS with correlation parameter ρ_e. Expressions for ${\tilde{\sigma }}^2$ for these basic scenarios are given in Table 2 (web-based supporting document, section A). We then derived the expressions for the optimal total number of repeated measurements, r_opt, and proved that r_opt depends only on ρ_e, ρ and κ (web-based supporting document, sections C and D).

Table 2.

Expressions for ${\tilde{\sigma }}^2$ under CMD and LDD patterns for the response assuming CS(σ, ρ) for the response covariance structure, no dropout (μ_m = 0) and a constant exposure prevalence (p_ej = p_e; ∀_j = 1, … , r + 1). Results for CMD are derived for any correlation structure for the exposure while results for LDD are derived under the assumption of CS correlation structure for the exposure.

Response pattern	${\tilde{\sigma }}^2$
CMDa	$\frac{{\sigma }^2(1-\rho )}{p_e(1-p_e)}\cdot \frac{\rho r+1}{(r+1)\left[\rho \right(1-{\rho }_e)r+1-\rho ]}$
LDDb	$\frac{12{\sigma }^2(1-\rho )}{p_e(1-p_e)}\cdot \frac{(\rho r+1)r}{(r+1)\{\rho {\rho }_e{r}^2+[2\rho +{\rho }_e+3(1-\rho ){\rho }_e(1-{\rho }_e)]r+2[1+(1-{\rho }_e)(2-\rho )\left]\right\}}$
If time–invariant exposure (ρe = 1)
CMD	$\frac{{\sigma }^2}{p_e(1-p_e)}\cdot \frac{\rho r+1}{r+1}$
LDD	$\frac{12{\sigma }^2(1-\rho )}{p_e(1-p_e)}\cdot \frac{r}{(r+1)(r+2)}$

^aFor any correlation structure of exposure.

^bAssuming compound symmetry for the correlation structure of exposure.

For CMD under the basic scenario, the solution to the unconstrained problem (8) is

$$r_{\mathrm{opt}}=\{\begin{array}{cc}\hfill 0,& \text{if}({\rho }_e\ge \rho \text{and}\kappa =1)\text{or}({\rho }_e>\rho \text{and}\kappa \le {\kappa }_0)\hfill \\ \hfill \frac{\kappa -1-({\kappa }_c-1)\rho +\sqrt{(1-\rho )({\kappa }_c-1)(\kappa -1)[1-\rho +\rho ({\kappa }_c-\kappa \left)\right]}}{\rho ({\kappa }_c-\kappa )},& \text{if}{\rho }_e>\rho \text{and}\kappa \in ({\kappa }_0,{\kappa }_c)\hfill \\ \hfill \infty ,& \text{for other combinations of}(\kappa ,\rho ,{\rho }_e)\hfill \end{array}\phantom{\}}$$ (13)

where

$${\kappa }_0≔\frac{1-\rho }{1-{\rho }_e+{(1-\rho )}^2}\text{and}{\kappa }_c≔\frac{{\rho }_e(1-\rho )}{\rho (1-{\rho }_e)}.$$

For a time-invariant exposure (ρ_e = 1), expression (13) becomes

$$r_{\mathrm{opt}}=\{\begin{array}{ccc}\hfill 0\hfill & \hfill \text{if}\hfill & \hfill \kappa \le \frac{1}{1-\rho }\hfill \\ \hfill \sqrt{\frac{1-\rho }{\rho }(\kappa -1)}-1\hfill & \hfill \text{if}\hfill & \hfill \kappa >\frac{1}{1-\rho }\hfill \end{array}\phantom{\}}.$$ (14)

Result (14) has been given, in the particular context of cluster randomized trials, by Cochran [18] and Raudenbush [22]. According to (14), the optimal under CMD is to take only one measurement (i.e., no repeated measurements) if κ is not greater than the threshold $\frac{1}{1-\rho }$, which increases as ρ increases. Otherwise, the optimal is to take a number of repeated measurements that increases as κ increases or ρ decreases. Overall, in the time-invariant case, values of κ and ρ close to 1 favor taking only one measurement, while the high cost of the first measurement relative to the following ones and the low correlation of the response become factors that favor more repeated measurements. When the solution is generalized to the context of a time-varying exposure (ρ_e < 1), the relationship between the optimal number of repeated measurements and the parameters ρ_e, ρ and κ is complex, as given by (13). Figure 2 illustrates more clearly this relationship. The intraclass correlation of exposure ρ_e strongly affects the optimal number of measurements. When comparing the value of r_opt for a given combination of κ and ρ across different values of ρ_e in Figure 2, one can readily see that, in most cases, more repeated measurements are advisable as the within-subject variation in exposure increases (i.e. ρ_e decreases). In many instances, the optimal is to take as many repeated measurements as possible (corresponding to points without a label in the plot), a situation that becomes more common as κ or ρ increase. This behavior is pronounced as ρ_e decreases, to the point that if ρ_e = 0.1 then r_opt = ∞ for any combination of κ ≥ 2 and ρ ≥ 0.1.

Figure 2.

Each number in the plot area indicates the optimal number of repeated measurements, r_opt, under the CMD response pattern in the basic scenario, which assumes covariance structure CS(σ, θ) for the response (θ = 0), no dropout (π_m = 0) and constant exposure prevalence (p_ej = p_e, ∀j = 0, … , r). Points without label correspond to those cases in which we should make as many measurements as possible (mathematically, infinite). Results correspond to values of the ratio between the economic cost of the first measurement and one of the following ones, κ = 1, 2, … , 20, values of ρ = 0.05, 0.10, … 0.95 and values of the intraclass correlation ρ_e = 0.1, 0.5, 0.9 and the time-invariant exposure case, ρ_e = 1.

For LDD in the basic scenario, the optimal is to take only two measurements (i.e., only one repeated measurement) if the ratio of costs of the first to the subsequent measurements, κ, is not greater than ${\kappa }^{\star }≔5+\frac{6(1-{\rho }_e)[2+(1-\rho \left){\rho }_e\right]}{(1+\rho ){\rho }_e}$ and as many measurements as possible (mathematically, infinity) otherwise. The threshold κ*, which equals 5 for time-invariant exposure, increases as ρ_e or ρ decrease, as shown in Figure 3, where each number in the plot area represents the value of κ* for a specific combination of ρ_e and ρ. Thus, if κ is not greater than 5, the optimal number of repeated measurements is 1 independently of the values of ρ_e and ρ. Compared to the time-invariant case, when the exposure is time-varying it is harder to justify taking more than one repeated measurement, since the ratio of costs that justifies doing so, κ*, is higher.

Figure 3.

Threshold of the ratio of costs of the first measurement over the subsequent ones (κ*) above which it is advisable to take as many repeated measurements as possible in the LDD case under the basic scenario. Otherwise, if the ratio of costs is less than κ*, the optimal is to take r_opt = 1. The basic scenario for the LDD response pattern assumes covariance structure CS(σ, ρ) for the response (θ = 0), no dropout (π_m = 0), constant exposure prevalence (p_ej = p_e, ∀j = 0, … , r) and CS(ρ_e) exposure correlation structure. The threshold takes the form ${\kappa }^{\star }=5+\frac{6(1-{\rho }_e)[2+(1-\rho \left){\rho }_e\right]}{(1+\rho ){\rho }_e}$. The column with constant value of κ* = 5 corresponds to a time-invariant exposure (ρ_e = 1).

Expressions (13) and (14) provide, in general, a non integer value for r_opt. The effective value of r_opt is that which, in combination with the value for N_opt given by (11) or (12), provides the maximum power or the minimum cost, respectively. The R package that we provide performs this correction.

4.2. Effect of varying the covariance stucture of the response

The results for the basic scenarios in the previous section were obtained assuming a CS(σ, ρ) covariance structure of the response, which is the particular case of DEX covariance structure with damping parameter θ = 0. There are many longitudinal situations where CS is not plausible and thus, in this section, we examine how the results change for θ > 0, in which case the minimization problem must be solved numerically. It can be easily shown that r_opt does not depend on σ. Thus, one can evaluate the objective function for a specific combination of the parameters θ, ρ_e, ρ and κ and a range of values r = 0, 1, … , r_max and then find the minimum subject to r_opt ≤ r_max. The value of r_max was fixed to 20 and 30 for CMD and LDD response pattern respectively. The maximum explored value of κ was 20.

For CMD response pattern, when varying θ from 0 (CS) to 1 (AR(1)), if the exposure is time-invariant (ρ_e = 1), differences in the result for r_opt appear only for low values of ρ or high cost of the first measurement (κ ≥ 10). These differences were not greater than 3 units. For instance, for AR(1) covariance structure of the response, the optimal total number of repeated measurements varies only between 0 and 2 for κ ≤ 20 while for CS covariance structure of the response, it varies between 0 and 4. If the exposure is time-varying (ρ_e < 1), the effect of θ on r_opt complexly depend on the combination of the values of the parameters ρ_e, ρ and κ.

For LDD response pattern, the exploration, constrained to r_max = 30 and κ ≤ 20, showed that positive values of θ break the dichotomy 1 vs. ∞ in the results for the optimal number of repeated measurements that was observed for CS (θ = 0). Still, the value 1 was obtained for most of the combinations of the parameters θ, ρ_e, ρ and κ, but higher values were obtained in a small portion of cases. Specifically, if θ ≥ 0.5, the optimal number of repeated measurements was 1 regardless of the value of the remaining parameters. The optimal number of repeated measurements was also 1 for any κ ≤ 9. Increasing ρ_e or κ, or decreasing ρ tends to increase r_opt. As an illustration, Table 3 shows the approximate minimum value of κ, κ*, for which the optimal number of repeated measurements is greater than 1 and the corresponding maximum value of the optimal number of repeated measurements obtained in the exploration over ρ ∈ (0.05, 0.95). The exploration was restricted to κ ≤ 20 and the maximum value of r_opt was reached systematically for the maximum explored value of κ and the minimum explored value of ρ.

Table 3.

Effect of θ in determining r_opt for the LDD response pattern assuming no dropout, a constant exposure prevalence, CS(ρ_e) exposure covariance structure and DEX(θ, ρ) response covariance structure. Exploration was constrained to κ ≤ 20 and ρ ∈ (0:05; 0:95). Each pair represents (max(r_opt); κ_min) where κ_min is an approximation to the minimum value of κ for which ropt is greater than 1 (over the range of ρ) and max(r_opt) is the maximum value of r_opt, which was always reached at the maximun explored value of κ (20) and the minimum explored value of ρ (0.05).

Intraclass correlation of exposure (ρe)
θ	0.1	0.5	0.6	0.7	0.8	0.9	1
0 (CS)	(∞, 60:5)	(∞, 11:2)	(∞, 9:2)	(∞, 7:7)	(∞, 6:6)	(∞, 5:7)	(∞, 5)
0.10	*	*	(18, 19)	(21, 14)	(23, 12)	(26, 10)	(30, 8)
0.20	*	*	*	*	(14, 16)	(15, 13)	(17, 10)
0.25	*	*	*	*	(11, 19)	(12, 15)	(14, 11)
0.30	*	*	*	*	*	(10, 17)	(12, 13)
0.40	*	*	*	*	*	*	( 8, 16)
≥ 0:50	*	*	*	*	*	*	*

^*r_opt = 1 for any combination of κ ≤ 20, ρ_e ≥ 0:1 and ρ ∈ (0:05; 0:95).

4.3. Effect of dropout

We analyzed the effect of dropout on the results for the basic scenarios. To do so, we explored a grid of values for the proportion of losses at the end of follow-up, π_M, between 0 (which corresponds to the basic scenarios) and 0.65. The exploration was constrained to r ≤ 30 with θ fixed at 0 (CS covariance structure of the response). For both the CMD and LDD response patterns, the results for r_opt were the same as those obtained when π_M = 0, except for a small percentage of combinations of values for ρ_e, κ and ρ. Specifically, for the CMD response pattern, increases of more than 3 units in r_opt were observed in only 5% of explored scenarios corresponding to high values of π_M (≥ 0.5). In few scenarios, characterized by a high value of r_opt (24 to 30) and κ = 1, an slight decrease of r_opt (1 to 3 units) was detected for high values of π_M (0.6). For the LDD response pattern, an increase in π_M favors r_opt = 1 for high values of π_M for a time-invariant exposure, while it favors increasing r_opt for lower values of ρ and high values of κ for a time-varying exposure. In addition, very few changes were observed when θ was fixed to 1 (AR(1) covariance structure of the response).

According to (9), dropout can affect the value of N_opt even in those cases in which no changes were observed in r_opt. For instance, N_opt increases as π_M increases for a fixed value of r_opt.

In addition, we performed a simulation study in order to explore the accuracy of our formulas and the potential impact of the violation of the MCAR structure assumed for the missing data in our methodology. Thus, we performed a simulation study (web-based supporting document, section E) assuming a missing at random (MAR) structure. In particular, we assumed that the probability of dropout of participant i at measurement j was π₁ if the response at the measurement j – 1 was lower than its third quartile or π₂ otherwise. We explored 9 scenarios, obtained from all combinations of the overall dropout fraction (0.1, 0.3 and 0.6) and the ratio π₁/π₂ (0.5, 0.8 and 1, the latter corresponding to MCAR), and fixing the values for the remaining parameters. In these scenarios, no significant differences were found when comparing the expected power according to our formulas (which was set at 0.8) with the empirical power obtained by simulation. Regarding the dropout pattern, these results provide some robustness to our formulas to departures from MCAR. However, the number of explored scenarios was very limited.

4.4. Effect of a time-varying exposure prevalence

To explore the effect of a time-varying exposure prevalence, we considered a grid of values for the average exposure prevalence, ${\stackrel{‒}{p}}_e$, between 0.05 and 0.95 and for γ between −0.95 (i.e. p_er = p_e0/20) and 20 (i.e. p_er = 21p_e0). The exploration was constrained to r ≤ 30, θ = 0 and κ ≤ 20.

For the CMD response pattern, our exploration revealed that increasing the difference between the prevalance at the beginning and the end of follow-up (i.e. the higher ∣γ∣) tends to increases the value of r_opt. These changes, almost negligible in most cases, become meaningful for extreme changes in the exposure prevalence and high values of the mean prevalence (web-based supporting document, section F, Table F.1).

For the LDD response pattern, a time-varying exposure prevalence disrupts the dichotomy 1 vs. ∞ for r_opt. As under the CMD response pattern, differences appeared only for large changes in the exposure prevalence between the first and the last measurements. For greatly time-increasing exposure prevalence, almost no changes were detected for values of κ ≤ 5 while for κ ≥ 8 and mean prevalence not higher than 0.3, the effect was to change r_opt from ∞ to 1. For greatly time-decreasing exposure prevalence, essentially r_opt tended to increase a few units (from 1 to 4) in those cases in which r_opt was 1 under constant exposure prevalence (web-based supporting document, section F, Table F.2).

4.5. Confounding

In the context of observational studies, the confounding problem is unavoidable. The potential effect of confounding can not be addressed in a systematic way due to the wide range of possible scenarios, and some of the previously proposed methods are limited [35]. We performed a simulation study to explore the effect of confounding by assessing bounds for power, in line with Haneuse et al. [35]. We considered up to 6 confounders and 5 scenarios for strength of confounding, which were explored for different combinations of the values of ρ_e, N and r, resulting in a total of 840 scenarios with 1000 simulations per scenario (more details on the simulation design are available at the web-based supporting document, section G). In each scenario, we compared the empirical power, calculated after including the confounders in the model, with the expected power using our formulas, which do not take into account the effect of confounders. The scenarios were set so that the expected power was 0.9. Table 4 shows results under the CMD response pattern. As expected, it can be seen that, in general, the empirical power decays with both the strength of confounding and the number of confounders. The value of ρ_e seems to play a role in the confounding impact. Indeed, Table 4 shows that, in longitudinal studies with high values of ρ_e (0.8 in our explorations), the power after accounting for confounding was no lower than 0.8 (or 0.72 in scenarios with r = 1) when the target power was 0.9. For low values of ρ_e, occasional values lower than 0.8 were obtained for scenarios with several confounders with moderate to strong effect. Under the LDD response pattern, almost no impact of confounders was observed except in scenarios with ρ_e = 0.2 and r = 2. In these cases, the empirical power dropped to a minimum value of 0.82. The effect of confounding could be lower in the LDD case because when building the cumulative exposure variable the confounding effect simulated for (non-cumulative) exposure is diluted.

Table 4.

Estimated bounds of confounding impact on power under the CMD response pattern. In each confounding scenario and for any number of confounders from 0 to 6, the median, (minimum, maximum) values of the empirical power (%) were calculated across scenarios with common value of ρ_e and all combinations of the values of the parameters N = 50, 200, 500 and 2000; r = 1, 8 and 20. In all scenarios, the expected power according to our formulas, disregarding the potential confounding effect, was 0.9. The empirical power was calculated using 1000 simulations per scenario and including the confounders in the model. Further details on this simulation study are available in the web-based supporting document, section G.

ρe = 0:2
Confounding scenario	OR between Zm and E, ϕm	Effect of Zm on Y, βZm †	Number of confounders
			0	1	2	3	4	5	6
Constant strength
1. Weak	1.5	0.5	90 (87, 91)	88 (88, 89)	89 (86, 91)	88 (85, 89)	88 (86, 90)	87 (85, 88)	86 (84, 88)
2. Moderate	2.0	1.0	90 (88, 92)	87 (84, 90)	87 (85, 88)	84 (82, 86)	84 (81, 86)	81 (78, 83)	79 (74, 83)
3. Strong	2.5	1.5	90 (88, 91)	86 (84, 88)	85 (83, 87)	80 (76, 81)	78 (74, 80)	75 (70, 76)	73 (70, 77)
Diminish strength
4. Moderate	3.0 → 1.5	2.0 → 0.5	90 (89, 92)	84 (81, 86)	82 (80, 85)	78 (75, 80)	79 (77, 83)	75 (72, 77)	76 (73, 78)
5. Strong	4:0 → 1:5	3:0 → 0:5	90 (88, 91)	79 (77, 82)	78 (75, 82)	72 (68, 75)	70 (64, 72)	69 (66, 72)	68 (64, 72)

ρe = 0:8
Confounding scenario	OR between Zm and E, ϕm	Effect of Zm on Y, βZm†	Number of confounders
			0	1	2	3	4	5	6
Constant strength
1. Weak	1.5	0.5	90 (89, 92)	90 (88, 92)	89 (87, 91)	89 (88, 91)	89 (86, 91)	89 (87, 91)	88 (86, 90)
2. Moderate	2.0	1.0	90 (89, 92)	89 (87, 90)	89 (87, 91)	87 (84, 90)	87 (84, 90)	86 (81, 89)	85 (80, 88)
3. Strong	2.5	1.5	91 (90, 92)	88 (87, 90)	88 (84, 90)	86 (81, 89)	85 (80, 89)	83 (77, 88)	83 (77, 88)
Diminish strength
4. Moderate	3:0 → 1:5	2:0 → 0:5	90 (88, 92)	88 (84, 90)	86 (83, 89)	85 (78, 87)	85 (80, 89)	83 (78, 87)	84 (76, 88)
5. Strong	4:0 → 1:5	3:0 → 0:5	90 (89, 92)	86 (80, 89)	84 (78, 89)	82 (73, 86)	81 (75, 85)	80 (72, 86)	80 (72, 86)

^†: In units of the β of interest.

In addition, we performed another simulation study (see web-based supporting document, section G) in order to explore how confounding affected the optimal combination of N and r. Under the CMD response pattern, discrepancies in the study design were observed exclusively for high values of ρ_e and κ. Similar results were obtained under the LDD response pattern.

Even taking into account that this exploration is not exhaustive, it appears that one can use conservative values of the target power (e.g. 0.9 when 0.8 is intended) to account for the effect of confounding when using our formulas. More detailed results on the simulations are presented in the web-based supporting document, section G.

5. Illustrative example

In this section, we used data from a study on cleaners and respiratory health to provide a design for a new hypothetical study on the same topic. Briefly, Medina-Ramón et al. [28] followed a group of N = 43 female domestic cleaners during r + 1 = 15 days. Each day, they provided measures of pulmonary function and annotated in a diary whether they performed certain cleaning tasks or used certain cleaning products. The study was observational and therefore the exposures were not assigned by design, rather, the cleaners performed the tasks and used the products that their work day required. All exposures showed day-to-day variations within-subjects. Here, we focus on the two exposures that had the highest and lowest value of ρ_e, namely vacuum cleaning and using air freshener sprays. The first one had ρ_e = 0.13 and an average prevalence of ${\stackrel{‒}{p}}_e=0.37$ while the second had ρ_e = 0.60 and ${\stackrel{‒}{p}}_e$ = 0.17. As expected, the prevalence of the exposures showed no trend so we assumed a constant prevalence of the exposures. Thirty-one participants in the original study provided complete data, so we set π_M = 0.28. The residual variance and the response covariance damping parameter were taken from the study and set to σ² = 0.43 and θ = 0.12, respectively. We used low (0.3) and high (0.7) values for ρ. Regarding the hypothesized effect, we fixed it at a difference of 10% in the expected mean value of the response between exposed and non exposed assuming the CMD response pattern. This results in $\tilde{\beta }$ = −0.39. The objective was to minimize the total cost of the study fixing a minimum required power of 0.9. The first measurement was assumed to be 2 times more expensive than each of the subsequent ones (i.e., κ = 2). We constrained the maximum number of repeated measurements to 20. All calculations were performed fixing a significance level α = 0.05.

Results, shown in Table 5, revealed a notable discrepancy in both the optimal number of repeated measurements and optimal number of participants between the assumptions of a time-invariant exposure (i.e., ρ_e = 1) and a time-varying exposure (using the observed value of ρ_e). When using the observed value of ρ_e, the optimal design was to take as many measurements as possible for both exposures, while assuming ρ_e = 1, the optimal was to perform a cross-sectional study (if ρ = 0.7) or a longitudinal study with only two measurements (if ρ = 0.3). As can be seen in Table 5, incorrectly using the design formulas for ρ_e = 1 when the exposure is actually time-varying can lead to discrepancies not only in r_opt and N_opt, but also in the final cost of the study. For example, using the time-invariant exposure formulas leads to designs with an increase in cost of between 140% and 480% for using air freshener sprays, and of between 30% to 190% for vacuum cleaning. In some cases, the slope of the cost as a function of r attenuates as r increases and thus, the investigator could be interested in increasing the number of participants in exchange for reducing the number of repeated measurements without a significant increase of the cost (web-based supporting document, section H).

Table 5.

Optimal combination of number of repeated measurements and number of participants to achieve at least 90% power, and resulting cost of the study. The problem was constrained to r_opt ≤ 20. Based on the data of the example, the following values for the parameters were used: For vacuum cleaning, ρ_e = 0:13 and ${\stackrel{‒}{p}}_e$ = 0:37, and for using air freshener sprays ρ_e = 0:60 and ${\stackrel{‒}{p}}_e$ = 0:17. Constant exposure prevalence was assumed. For all calculations, σ² = 0:43, θ = 0:12, π_M = 0:28, c₁ = 1, κ = 2, α = 0:05. Calculations were performed to detect a difference of 10% between exposed and non-exposed under the CMD response pattern.

Exposure	Response correlation	Exposure covariance	Optimum		Costa
			r opt	N opt
Vacuuming	ρ = 0.3	CS(ρe)	18	6	51.6
		t.i.e.b	1	92	125.1
	ρ = 0.7	CS(ρe)	15	3	22.0
		t.i.e.b	0	128	128.0
Air freshener sprays	ρ = 0:3	CS(ρe)	20	17	160.7
		t.i.e.b	1	152	206.7
	ρ = 0:7	CS(ρe)	19	8	72.2
		t.i.e.b	0	211	211.0

^aIn units of the cost of the first measurement.

^bUsing formulas assuming a time-invariant exposure (i.e. ρ_e = 1).

6. Discussion

In this work, we developed methods to explore, under certain assumptions, the optimal combination of number of participants and number of repeated measurements when there are cost or power constraints. We examined a variety of situations typical for observational longitudinal studies in which the exposure can vary over time within participants in a manner not controlled by the investigator. The degree of within-subject variation in exposure, as measured by the intraclass correlation of exposure, proved to be a key parameter for the optimal design. We provide an R package so that the optimal design can be calculated for specific studies, and also examined general patterns. In models studying acute exposure effects (CMD), results have already been presented for a time-invariant exposure, suggesting that when the repeated measurements are not much cheaper than the first, the optimal is to take one or just a few repeated measurements [18, 22]. If the exposure is time-varying, however, more repeated measurements may be advisable, and when the within-subject variation is high, the optimal design may be to take as many repeated measurements as possible. In models studying cumulative exposure effects (LDD), if the exposure is time-invariant, the optimal is to take just one repeated measure if the ratio of costs of the first measurement vs. the subsequent ones is less than 5, and to take as many measurements as possible otherwise. When the exposure is time-varying, the pattern is the same but the ratio of the costs threshold is higher than 5, thus increasing the number of situations where the optimal is to take just one repeated measure.

Our results extend the previous work on optimal allocation in longitudinal studies [18–26] to the case of a time-varying exposure, as often occurs in observational research. Previous work considered somewhat sligthly different settings, including different covariance structures, cost functions or missing data structure. The methodology described here can be easily adapted to those other settings. The general results described in the previous paragraph were found for a basic scenario in which the response had CS covariance, there was no dropout and the exposure prevalence was constant. We explored how the results changed when some of these assumptions were relaxed. Importantly, our explorations were performed by changing one assumption at a time, precluding general interpretations. If needed, the R package provided could be used to explore other combinations.

In terms of the covariance of the response, we explored variations following a DEX covariance structure. Others have studied covariances that resulted from mixed models with a random intercept and slope [8, 16], or covariances that mix a random intercept with autoregressive errors [26]. When the response follows DEX covariance instead of CS, under the acute exposure effect model, the optimal number of repeated measurements increases as the damping parameter θ increases. This amplifies the differences with the time-invariant exposure case. Under the cumulative exposure effects model, increasing the damping parameter breaks the dichotomy 1 vs. ∞ in the optimal number of repeated measurements observed when the response follows CS covariance. In general, high values of θ favor a smaller number of repeated measurements compared to the CS case.

Monotone dropout had almost no influence on r_opt, except in a few combinations of the values of the study design parameters. Galbraith [25] also investigated the effect of missing data on the optimal combination (N, r) under the LDD response pattern in the context of a time-invariant exposure. They examined several dropout patterns and also found that the optimal allocation was in general not affected by the extent or kind of dropout. They suggested computing the sample size for 90 % power when 80 % power is intended, if the expected overall dropout is no more than 30%. In our case, the amount of dropout is directly used as an input parameter for the design calculation.

The exposure prevalence is a parameter that does not affect r_opt when the exposure is time-invariant. However, it does play a role when the exposure is time-varying, and therefore we explored the effects of p_e on r_opt. Under the acute exposure effect model, the higher the variability of this prevalence the higher the optimal number of measurements. Under the cumulative exposure effect model, the effect depends on the sign of the time trend of the exposure prevalence. If the exposure prevalence is time-increasing, the optimal number of repeated measurements tends to decrease (changing always from ∞ to 1), while if the exposure prevalence is time-decreasing, the dichotomy 1 vs. ∞ is disrupted with no clear patterns with respect to the remaining parameters involved in the study design.

We performed a simulation study to explore the effect of confounding by assessing bounds for power, in line with Haneuse et al. We considered a variety of confounding scenarios. Results showed that the value of ρ_e plays an important role on the impact of confounding. In both CMD and LDD scenarios, we showed that in most cases one can account for the effect of confounding by using a conservative value of 0.9 when 0.8 power is intended.

Additional issues could affect our results. For example, it has been shown that the F-test using restricted maximum likelihood (REML) with Satterthwaite approximation for the degrees of freedom was found to provide better results than the Wald test for small samples [36]. In a simulation study, we found no significant differences in empirical power between the two tests in studies with more than about 30 participants while the Wald test resulted in overestimation of power for smaller values of N (web-based supporting document, section I). Another issue is that we assumed fixed measuring times but in real studies there might be participant variation around that schedule. Results from a simulation study suggest that this had minimal impact on our results (web-based supporting document, J).

Our methodology is easily adaptable for departures from several assumptions considered. For instance, different cost functions or more complex response covariance structures (such as those arising in mixed models with random intercept and slope). Also a continuous exposure can be considered. In this case, the first- and second-order moments of the exposure would be needed.

It should be noted that this study has limitations due to a number of approximations and assumptions. For instance, a linear time effect is assumed. This assumption made possible that the optimal is to take only 2 measurements in several scenarios. If an acceleration or higher order effect is expected or wants to be detected, such a design would not be appropriate. Results presented in this paper were obtained under asymptotic approximations, so one should be careful interpreting results for small samples. The scenarios explored in this paper are just a very small subset of simple cases, and much more complicated scenarios can be found in real studies. Furthermore, reasons other than power and cost can also favor other designs. For example, there might be design strategies that maximize the commitment of study participants. Thus, different aspects of the problem can be addressed in order to refine the methodology presented.

In summary, we have generalized, under certain assumptions, an algorithm for the search for the optimal combination of number of participants and number of repeated measurements in observational longitudinal studies to the case where the exposure is time-varying. We applied our methodology to an illustrative example whose results showed that assuming incorrectly a time-invariant exposure may produce an inefficient design. We provide an R package to perform all calculations. This package is available at http://www.creal.cat/xbasagana/software.html.