Optimal Sample Size Calculation in Cost-Effectiveness Longitudinal Cluster Randomized Trials

Abstract

Longitudinal cluster randomized trials (L-CRTs) are increasingly used to evaluate the cost-effectiveness of healthcare interventions across multiple assessment periods, yet design methods for powering these trials remain underdeveloped. Existing methods for cost-effectiveness analyses in cluster settings are limited to simple parallel-arm cluster randomized trials with a single follow-up assessment period. These methods cannot accommodate the complex correlation structures in L-CRTs conducted over multiple periods, which require differentiation between within-period and between-period correlations for both clinical and cost outcomes, as well as between-outcome correlations. Moreover, while substantial methodological advances have been made for the design of L-CRTs with univariate outcomes, none specifically address cost-effectiveness objectives where clinical and cost outcomes must be jointly modeled. We provide a design-stage framework for powering cost-effectiveness L-CRTs across three design variants: parallel-arm, crossover, and stepped wedge designs. We derive closed-form variance expressions for the generalized least squares estimator of the average incremental net monetary benefit under a bivariate linear mixed model. We propose a standardized ceiling ratio that adjusts willingness-to-pay for relative outcome variability to inform optimal design. We then develop local optimal designs that maximize statistical power under known correlation parameters and MaxiMin designs that ensure robust performance across parameter uncertainty for all three design variants. Through a real stepped wedge trial data example, we demonstrate the sample size calculation for testing intervention cost-effectiveness under local optimal and MaxiMin designs.

Figures (8)

• Figure 1: Schematic illustration of L-CRT designs with $J = 6$ periods. Purple cells indicate intervention periods and white cells indicate control periods. Panel (a) shows a CRXO trial, where sequences $q \in \{1, 3\}$ receive intervention in periods $j \in \{1, 3, 5\}$ and control in periods $j \in \{2, 4, 6\}$, while sequences $q \in \{2, 4\}$ receive control in periods $j \in \{1, 3, 5\}$ and intervention in periods $j \in \{2, 4, 6\}$. Panel (b) displays a PA-LCRT, where sequences $q \in \{1, 2\}$ are randomized to intervention and sequences $q \in \{3, 4\}$ to control for all periods $j \in \{1, \ldots, 6\}$. Panel (c) presents an SW-CRT, where sequence $q$ transitions from control to intervention in period $j = q + 1$, with all clusters under control in period $j = 1$ and all under intervention in period $j \in \{5, 6\}$.
• Figure 2: Graphical representation of the data structure within the $i$-th cluster of a cost-effectiveness L-CRT, where $j \neq j'$ and $k \neq k'$. Each dashed oval represents an individual observed in a period (dashed square) nested in the cluster, with both a clinical outcome ($E$) and cost ($C$) measured per individual. Arrows depict the ICCs corresponding to within-period correlations ($\rho_0^E$, $\rho_0^C$), between-period correlations ($\rho_1^E$, $\rho_1^C$), and three types of between-outcome correlations ($\rho_0^{EC}$, $\rho_1^{EC}$, $\rho_2^{EC}$). The width of each arrow indicates the relative strength of correlation under the following constraints: (i) $\rho_1^E \leq \rho_0^E$; (ii) $\rho_1^C \leq \rho_0^C$; (iii) $\rho_0^{EC} \leq \min(\rho_0^E, \rho_0^C)$; (iv) $\rho_1^{EC} \leq \min(\rho_1^E, \rho_1^C)$; and (v) $\rho_1^{EC} \leq \rho_0^{EC} \leq \rho_2^{EC}$.
• Figure 3: LODs for CRXO trials, PA-LCRTs, and SW-CRTs with varying CAC. Panels (a), (c), and (e) present $I_{\text{LOD}}$ and $K_{\text{LOD}}$ as functions of CAC; panels (b), (d), and (f) display the corresponding power of the LOD. Both integer (solid lines) and decimal (dotted lines) estimates are shown. We consider $J = 4$ periods, $B = \$300{,}000$, $c_1 = \$3{,}000$, and $c_2 = \$250$. For CRXO trials, clusters are equally allocated to two treatment sequences (ICIC/CICI); for PA-LCRTs, clusters are equally allocated to treatment or control for the entire study duration; for SW-CRTs, the number of sequences is $L = 3$. For ICC parameters, we fix $\rho_0^E = 0.1$, $\rho_0^C/\rho_0^E = 1$, $\rho_0^{EC} = 0.04$, $\rho_2^{EC} = 0.5$, with CAC $= \rho_1^E/\rho_0^E = \rho_1^C/\rho_0^C = \rho_1^{EC}/\rho_0^{EC} \in [0.1, 0.8]$. Additional parameters include $\beta_1 = 4{,}000$, $\lambda = 20{,}000$, and $r = \sigma_E/\sigma_C = 1/3000$.
• Figure 4: MMDs for CRXO trials, PA-LCRTs, and SW-CRTs with varying maximum between-period effect ICC. Panels (a), (c), and (e) present $I_{\text{MMD}}$ and $K_{\text{MMD}}$ as functions of $\rho_{1,\max}^E$, and panel (b), (d), and (f) display the corresponding RE. Both integer (solid lines) and decimal (dotted lines) estimates are shown. The MMDs are computed for CRXO trials with $J = 4$ periods and balanced design $\pi = 0.5$ under a fixed budget constraint of $B = \$300{,}000$, with $c_1 = \$3{,}000$ and $c_2 = \$250$. The ICC parameter ranges are specified as $\rho_0^E \in [0.05, 0.10]$, $\rho_0^C \in [0.04, 0.08]$, $\rho_0^{EC} \in [0.01, 0.02]$, $\rho_1^{EC} \in [0.005, 0.01]$, $\rho_2^{EC} \in [0.5, 0.8]$. We set $\rho_1^E \in [0.025, \rho_{1,\max}^E]$ and $\rho_1^C \in [0.02, 0.8\rho_{1,\max}^E]$ where $\rho_{1,\max}^E \in (0.025, 0.045]$. Additional parameters include $\beta_1 = 4{,}000$, $\lambda = 20{,}000$, and $r = \sigma_E/\sigma_C = 1/3000$.
• Figure 5: First row of the R Shiny application interface. The left panel displays the bivariate linear mixed model specification for jointly modeling clinical outcomes ($E_{ijk}$) and costs ($C_{ijk}$), including the variance-covariance matrices $\boldsymbol{\Sigma}_b$, $\boldsymbol{\Sigma}_s$, and $\boldsymbol{\Sigma}_e$ for the cluster-level, cluster-period-level, and individual-level random effects, respectively, and the INMB definition. The right panel displays a graphical representation of the data structure within the $i$-th cluster of a cost-effectiveness L-CRT, illustrating the seven intracluster correlation coefficients: within-period ICCs ($\rho_0^E$, $\rho_0^C$), between-period ICCs ($\rho_1^E$, $\rho_1^C$), and three between-outcome ICCs ($\rho_0^{EC}$, $\rho_1^{EC}$, $\rho_2^{EC}$), along with their formal definitions and ordering constraints.

Theorems & Definitions (11)

• Theorem 1
• Theorem 2
• Lemma 1
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Theorem 7
• Theorem 8
• Theorem 9