Small‐sample inference for cluster‐based outcome‐dependent sampling schemes in resource‐limited settings: Investigating low birthweight in Rwanda

Abstract

The neonatal mortality rate in Rwanda remains above the United Nations Sustainable Development Goal 3 target of 12 deaths per 1,000 live births. As part of a larger effort to reduce preventable neonatal deaths in the country, we conducted a study to examine risk factors for low birthweight. The data was collected via a cost-efficient cluster-based outcome-dependent sampling scheme wherein clusters of individuals (health centers) were selected on the basis of, in part, the outcome rate of the individuals. For a given dataset collected via a cluster-based outcomedependent sampling scheme, estimation for a marginal model may proceed via inverse-probability-weighted generalized estimating equations, where the cluster-specific weights are the inverse probability of the health center's inclusion in the sample. In this paper, we provide a detailed treatment of the asymptotic properties of this estimator, together with an explicit expression for the asymptotic variance and a corresponding estimator. Furthermore, motivated by the study we conducted in Rwanda, we propose a number of small-sample bias corrections to both the point estimates and the standard error estimates. Through simulation, we show that applying these corrections when the number of clusters is small generally reduces the bias in the point estimates, and results in closer to nominal coverage. The proposed methods are applied to data from 18 health centers and 1 district hospital in Rwanda.

Optimal allocation in stratified cluster‐based outcome‐dependent sampling designs

“…Letting R k denote the N k × 1 vector with all entries equal to R k , R is the N × 1 vector obtained by concatenating the R k together. Following arguments similar to those presented by Xie and Yang 26 in the complete data setting, Sauer et al 27 showed that under regularity conditions,̂w, the solution to (3), is consistent for 0 , the true value of , and is asymptotically multivariate normal, with the asymptotic variance given by…”

“…Letting denote the N k × 1 vector with all entries equal to R k , is the N × 1 vector obtained by concatenating the together. Following arguments similar to those presented by Xie and Yang 26 in the complete data setting, Sauer et al 27 showed that under regularity conditions, , the solution to (), is consistent for , the true value of , and is asymptotically multivariate normal, with the asymptotic variance given by where H ( β ) = − E [∂𝒰( β )], and 𝒰( β ) = U T 1 N ×1 ; see Appendix B of the Supporting Information for details.…”

“…For each sample taken, we computed the point estimates by solving expression () and estimated the standard errors using an estimator of (), the form of which is given in Sauer et al 27 Due to the potential downward bias of the robust sandwich variance estimator in finite samples, we computed four estimates of the standard errors: (i) unadjusted standard errors, (ii) degrees‐of‐freedom adjusted standard errors, in which the variance‐covariance matrix is multiplied by the factor K s /( K s − p ), 27,31 (iii) a Mancl and DeRouen‐type bias correction 32 that is adapted to accommodate the weights needed to account for the sampling design, 27 and (iv) a Kauermann and Carroll‐type 33 bias correction that is similarly adapted 27 . For each design, across the 10 000 iterations, we computed the mean point estimates and the relative uncertainty of each design for the estimation of each parameter, defined as the ratio: where R = 10000 is the number of simulation iterations.…”

Practical strategies for operationalizing optimal allocation in stratified cluster‐based outcome‐dependent sampling designs

“…Given data that has been collected through a single‐stage cluster‐stratified ODS design, estimation and inference can proceed via IPW‐GEE 1,2 . In particular, the regression parameters $$$ \boldsymbol{\beta} $$$ can be estimated as the solution to the following: where $$$ {R}_k $$$ is equal to 1 if cluster $$$ k $$$ is sampled and is equal to 0 otherwise; $$$ {\pi}_k $$$ is the probability of cluster $$$ k $$$ being sampled; $$$ {\boldsymbol{\epsilon}}_k $$$ = $$$ \left({\boldsymbol{Y}}_k-{\boldsymbol{\mu}}_k\right) $$$, with $$$ {\boldsymbol{Y}}_k $$$ = $$$ \left({Y}_{k1},\dots, {Y}_{k{N}_k}\right) $$$ and …”

“…One way to operationalize a cluster‐based ODS design is to stratify the clusters based on pieces of information known at the design stage, and to then sample a certain number of clusters from each of the strata. Data that has been collected in such a way can be analyzed using inverse‐probability weighted generalized estimating equations (IPW‐GEE), 1,2 where the weights are the inverse of the cluster‐specific probabilities of selection. The number of clusters sampled from each of the defined strata can influence the efficiency gain or loss for a particular parameter in the analysis model, and Sauer et al 3 derived formulae for the optimal allocation of the (cluster‐level) sample size across the strata, when the intended analysis is IPW‐GEE.…”