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.

Keywords: intraclass correlation; longitudinal study; optimal design; sample size.

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.

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.

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).