Ratio estimators of intervention effects on event rates in cluster randomized trials

Abstract

We consider five asymptotically unbiased estimators of intervention effects on event rates in non-matched and matched-pair cluster randomized trials, including ratio of mean counts (r 1 ), ratio of mean cluster-level event rates (r 2 ), ratio of event rates (r 3 ), double ratio of counts (r 4 ), and double ratio of event rates (r 5 ).In the absence of an indirect effect, they all estimate the direct effect of the intervention. Otherwise, r 1 , r 2 , and r 3 estimate the total effect, which comprises the direct and indirect effects, whereas r 4 and r 5 estimate the direct effect only. We derive the conditions under which each estimator is more precise or powerful than its alternatives. To control bias in studies with a small number of clusters, we propose a set of approximately unbiased estimators. We evaluate their properties by simulation and apply the methods to a trial of seasonal malaria chemoprevention. The approximately unbiased estimators are practically unbiased and their confidence intervals usually have coverage probability close to the nominal level; the asymptotically unbiased estimators perform well when the number of clusters is approximately 32 or more per trial arm. Despite its simplicity, r 1 performs comparably with r 2 and r 3 in trials with a large but realistic number of clusters. When the variability of baseline event rate is large and there is no indirect effect, r 4 and r 5 tend to offer higher power than r 1 , r 2 , and r 3 . We discuss the implications of these findings to the planning and analysis of cluster randomized trials. K E Y W O R D Scluster randomized trial, event rate, incidence rate ratio, ratio estimator, relative incidence INTRODUCTIONThe cluster randomized trial (CRT) is an important study design in medical and health research. 1,2,3 Data on outcome events may be collected by passive surveillance or active surveillance. 4 Passive surveillance methods may or may not provide data at the individual level. That is, they may determine only the number of events in a cluster, without identifying Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not reflect the views of Ministry of

crtrest: A command for ratio estimators of intervention effects on event rates in cluster randomized trials

“…In Ma et al (2022), we showed that Var ( r l ∗ ) = Var ( r l ) for l = 1, 2, 3, 4, and 5. This equality between the variances of the approximately unbiased estimators and asymptotically unbiased estimators also holds for the estimators for matched-pair CRTs.…”

“…The contrast of the target and nontarget groups within each cluster reduces the noise arising from heterogeneity between clusters. If there is no indirect effect or only a trivial indirect effect, r 4 is more precise (smaller standard error) and more powerful than r 2 and r 3 when the heterogeneity between clusters is large (Ma et al 2022). If a nontrivial indirect effect is present, the point estimates are not comparable, but the absolute values of the test statistics, | t ( r l )|, for l = 2, 3, or 4 are still comparable in that they all indicate the probability of rejecting the null hypothesis of the target ratio being 1.…”

“…Asymptotically unbiased estimators for matched-pair CRTs, r l paired ( l = 1, 2, 3, 4, 5), are defined similarly (Ma et al 2022).…”

“…Its degree of bias appeared to depend on not only the number of clusters but also the coefficient of variation of the cluster-level event rate, CV( c ij ) (Bennett et al 2002; Pacheco et al 2009). Previous simulation studies appeared to have limited the degree of heterogeneity between clusters to a range that is too small to be realistic (Ma et al 2022). The number of clusters per trial arm is highly variable in CRTs, ranging from a few to hundreds.…”

“…For matched-pair CRTs, the formulas for each Var ( r l paired ) , l = 1, 2, 3, 4, and 5, involves subtraction of a (scaled) covariance term; see Ma et al (2022). Because the matched-pair design tends to generate positive covariances of the relevant quantities, it tends to provide smaller variances.…”