 Research
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Metaanalysis under imbalance in measurement of confounders in cohort studies using only summarylevel data
BMC Medical Research Methodology volume 22, Article number: 143 (2022)
Abstract
Background
Cohort collaborations often require metaanalysis of exposureoutcome association estimates across cohorts as an alternative to pooling individuallevel data that requires a laborious process of data harmonization on individuallevel data. However, it is likely that important confounders are not all measured uniformly across the cohorts due to differences in study protocols. This imbalance in measurement of confounders leads to association estimates that are not comparable across cohorts and impedes the metaanalysis of results.
Methods
In this article, we empirically show some asymptotic relations between fully adjusted and unadjusted exposureoutcome effect estimates, and provide theoretical justification for the same. We leverage these results to obtain fully adjusted estimates for the cohorts with no information on confounders by borrowing information from cohorts with complete measurement on confounders. We implement this novel method in CIMBAL (confounder imbalance), which additionally provides a metaanalyzed estimate that appropriately accounts for the dependence between estimates arising due to borrowing of information across cohorts. We perform extensive simulation experiments to study CIMBAL’s statistical properties. We illustrate CIMBAL using National Children’s Study (NCS) data to estimate association of maternal education and low birth weight in infants, adjusting for maternal age at delivery, race/ethnicity, marital status, and income.
Results
Our simulation studies indicate that estimates of exposureoutcome association from CIMBAL are closer to the truth than those from commonlyused approaches for metaanalyzing cohorts with disparate confounder measurements. CIMBAL is not too sensitive to heterogeneity in underlying joint distributions of exposure, outcome and confounders but is very sensitive to heterogeneity of confounding bias across cohorts. Application of CIMBAL to NCS data for a proofofconcept analysis further illustrates the utility and advantages of CIMBAL.
Conclusions
CIMBAL provides a practical approach for metaanalyzing cohorts with imbalance in measurement of confounders under a weak assumption that the cohorts are independently sampled from populations with the same confounding bias.
Background
In cohort collaborations, such as the Environmental influences on Child Health Outcomes (ECHO), analyses are often done by pooling cohortlevel results using metaanalysis techniques [1]. We use the term ‘collective analysis’ to describe collaborative metaanalysis in which an apriori analytical plan is developed to answer a specific research question, the analytic code is developed centrally, distributed to the individual cohorts, applied to each cohort’s data, and results are collected from each cohort which are then metaanalyzed. This increases uniformity with the idea of reducing bias. Further, due to onerous computational needs of big data– as is common in modern observational epidemiologic studies and genomewide association studies–, divideandconquer approaches are becoming popular. A divideandconquer algorithm involves dividing the data into independent blocks, analyzing each block separately, and combining solutions from each block to get the solution for the full data [2]. If the goal is to estimate model parameters from the full data, then metaanalysis can be considered a divideandconquer approach that avoids computeintensive model fitting on a very large sample size. Several cohort collaborations such as the Chronic Kidney Disease Prognosis Consortium (CKDPC) operate exclusively in this manner [3].
Despite practical advantages of metaanalysis in cohort collaborations, it is likely that important confounders are not necessarily measured across all cohorts since each cohort may have been independently funded with independent study protocols. We refer to this problem as ‘confounder imbalance’. As pointed by Voils et al. [4], there appears to be no consensus on how to synthesize adjusted and unadjusted parameter estimates in a metaanalysis. Therefore, it is an open question of how to deal with this situation in the collective analysis. The simplest option is to combine unadjusted estimates only because it is available from all studies and is easy to interpret. Historically, unadjusted analyses have been emphasized for their interpretability and generalizability [5]. However, unadjusted estimates provide biased inference about the exposureoutcome association. The investigator could also be most conservative by restricting the analysis to cohort studies which have measured all pertinent confounders. This is likely to result in decreased sample size (and hence loss of information) that these cohort collaborations often capitalize upon. Alternatively, as the CKDPC has done previously [6], the cohort collaboration may list deviations such as not having measured particular variables in an appendix. This potentially combines various estimands in which some are adjusted for all possible confounders and others for only a subset of confounders, thus ignoring potential heterogeneity in estimates. Yet another approach is to conduct separate metaanalysis of unadjusted and adjusted estimates, report both estimates, and qualitatively assess the conclusions from each. This may lead to difficulty in interpretation if the two metaanalysis results do not support the same conclusion. These naive approaches have been described with examples elsewhere [4].
A more sophisticated and statistically principled approach is GENMETA, a generalized metaanalysis method that relies on an external reference dataset to provide information on the joint distribution of covariates that are needed for the analysis [7]. The reference data can be of fairly modest size, should be independent of the cohorts under study, should have individuallevel measurements on all possible covariates, and need not be linked to the outcome of interest. Using GENMETA, one can capitalize on this external data source to correct for missing confounders by relying on the joint distribution as a mapping to allow for fully adjusted estimates. However, an external data source representative of the underlying population from which the cohort studies are being drawn may not always be available. Kundu and Chatterjee [8] subsequently relaxed this independence assumption for the external data source in GENMETA.
In this article, using the idea of ‘confounding risk ratio’ (ratio of the unadjusted risk ratio to the adjusted risk ratio) [9, 10], we show how information may be borrowed across cohorts or studies reporting only summarylevel data to result in fully adjusted estimates to be combined in the metaanalysis step. Our approach enables cohorts with incomplete information on confounders to contribute to the collective analysis. Our focus is on metaanalysis in the setting of parametric regression modelling of exposureoutcome association given a set of confounders. Although we describe our approach in the context of cohort studies, it is also relevant for a metaanalysis of randomized controlled trials, where the imbalance in measuring the effect modifiers across trials is prevalent. We first empirically observe that for a large enough cohort the difference of the exposureoutcome effect estimates (fully adjusted vs unadjusted) is independent of the sample size for both linear and logistic regressions. Using a generalized linear model, we theoretically justify that this limiting behavior is not unexpected. We additionally provide exact theoretical limits in the linear regression framework. We then leverage these asymptotic relations to implement our approach in a novel metaanalysis method for confounder imbalance, which we refer to as CIMBAL (implementation available in R, https://github.com/RayDebashree/cimbal). We provide details on how CIMBAL not only imputes the adjusted estimates (effect estimates and their variances) for cohorts with missing confounders but also provides a metaanalyzed estimate that appropriately accounts for the dependence between estimates arising due to borrowing of information across cohorts. We perform extensive simulation experiments to empirically demonstrate the aforementioned asymptotic relations under different generalizations, and to study statistical properties such as bias, type I error and power of CIMBAL. To illustrate the application of CIMBAL, we present a proofofconcept metaanalysis of randomly chosen subsets of the National Children’s Study (NCS) data to estimate association of maternal education and low birth weight in infants, adjusting for maternal age at delivery, race/ethnicity, marital status, and income.
Methods
Notation, models, and existing approaches
Consider the following measurements from an epidemiologic study: response or outcome Y, exposure X and the full set of q possible confounders \(\boldsymbol {\mathcal {C}}\). Our interest lies in quantifying the true exposureoutcome association. To assess exposureoutcome association, a cohort with no information on any confounder will consider an unadjusted model
and report unadjusted estimate of association (\(\hat {\beta }_{\text {unadj}}\)) and its standard error (SE) (\(\hat {\text {se}}_{\text {unadj}}\)). On the other hand, a cohort that measured all relevant confounders will consider a fully adjusted model
and report adjusted estimate of association (\(\hat {\beta }_{\text {adj}}\)) and its SE (\(\hat {\text {se}}_{\text {adj}}\)). For simplicity, in this article we will focus on cohorts with either no or full confounder information. Let us assume there are K cohorts, of which K_{c} cohorts have complete information on confounders while K_{m} cohorts have no confounder information. Thus, one can gather the unadjusted estimates of exposureoutcome association from all cohorts \(\left \{ \left (\hat {\beta }_{j,\text {unadj}}, \hat {\text {se}}_{j,\text {unadj}}\right),\; j=1,2,...,K \right \}\) and the adjusted estimates from the complete cohorts \(\left \{ \left (\hat {\beta }_{j,\text {adj}}, \hat {\text {se}}_{j,\text {adj}}\right),\; j=1,2,...,K_{c} \right \}\).
Metaanalysis: unadjusted. This is the most common metaanalysis approach for combining parameter estimates from studies with disparate sets of confounders. The fixedeffect inversevariance weighted metaanalysis estimate and its SE are given by \(\hat {\beta }_{\text {unadj}}^{(\text {meta})} = \frac {\sum _{j=1}^{K} w_{j} \hat {\beta }_{j,\text {unadj}}}{\sum _{j=1}^{K} w_{j}} \) and \(\hat {\text {se}}_{\text {unadj}}^{(\text {meta})} = \frac {1}{\sqrt {\sum _{j=1}^{K} w_{j}}}\), where \(w_{j} = 1/\hat {\text {se}}_{j,\text {unadj}}^{2}\) for all j=1,2,...,K.
Metaanalysis: complete only. Since unadjusted estimates are biased and usually artificially large in the presence of unmeasured confounders, this approach only combines the studies with fully adjusted estimates. The metaanalyzed effect estimate and its SE are given by \(\hat {\beta }_{\text {adj}}^{(\text {meta})} = \frac {\sum _{j=1}^{K_{c}} w_{j} \hat {\beta }_{j,\text {adj}}}{\sum _{j=1}^{K_{c}} w_{j}} \) and \(\hat {\text {se}}_{\text {adj}}^{(\text {meta})} = \frac {1}{\sqrt {\sum _{j=1}^{K_{c}} w_{j}}}\), where \(w_{j} = 1/\hat {\text {se}}_{j,\text {adj}}^{2}\) for all j=1,2,...,K_{c}.
Idea of confounding risk ratio
Assuming a binary outcome Y, a binary exposure X and a binary unmeasured confounder C, Cornfield et al. [11] showed the following inequalities must hold in order for the confounder to fully explain the observed exposureoutcome association:
where RR_{XY} is the risk ratio for the association between the outcome and the exposure, RR_{CY} is the risk ratio for the association between the outcome and the confounder, and p_{0} (p_{1}) is the prevalence of the confounder in the unexposed (exposed) group. Later, Flanders and Khoury [9] proposed the idea of confounding risk ratio (coRR) [12] to quantify unmeasured confounding, and determined the bounds of coRR as
where OR_{XC} is the odds ratio for the association of the exposure with the confounder. In an analysis of a single cohort without access to other information about the unmeasured confounder C, an investigator would be required to make estimates of these parameters from some source (e.g., from literature or expert opinion) to define the bounds. However, in a cohort collaboration where one or more cohorts have collected the pertinent set of confounders, this information may be used and applied to the cohorts that did not measure one or more of the confounders. In other words, the cohorts with a complete set of confounders may provide the coRR that may be applied to crude (unadjusted or partially adjusted) estimates of association. We build on this idea to provide fully adjusted estimates of exposureoutcome association for cohorts with no information on confounders using fully adjusted and unadjusted estimates from cohorts with complete information on confounders. We empirically show asymptotic relations between adjusted and unadjusted estimates from both linear (continuous response) and logistic (binary outcome) regressions, and also provide theoretical support for these relations in a generalized linear model setup.
Relations between adjusted and unadjusted estimates and their variances
Linear regression
Using simulations on a continuous response, a binary or a continuous exposure and two binary confounders, we observe that the difference of association estimates \(\hat {\beta }_{\text {unadj}}  \hat {\beta }_{\text {adj}}\) (commonly referred to as ‘omitted variable bias’ in econometrics literature) is stabilizing to a constant as sample size increases (Supplementary S1). This appears to be true regardless of the strength and direction of the true exposureoutcome association.
To theoretically explore this relation, we assume all the variables Y, X and \(\boldsymbol {\mathcal {C}}\) are continuous, and that the following joint distribution of variables holds at the populationlevel:
A cohort, which randomly sampled n individuals from this population and measured Y, X and \(\boldsymbol {\mathcal {C}}\) will consider a fully adjusted model to determine exposureoutcome association using (\(\hat {\beta }_{\text {adj}}, \hat {\text {se}}_{\text {adj}}^{2}\)). It may also consider an unadjusted model without any confounder adjustment and get (\(\hat {\beta }_{\text {unadj}}, \hat {\text {se}}_{\text {unadj}}^{2}\)). Note that in the population (true model) all the variables Y, X and \(\boldsymbol {\mathcal {C}}\) are considered random. In the sample (adjusted or unadjusted model), Y is treated as random while X and \(\boldsymbol {\mathcal {C}}\) are assumed to be fixed. For simplicity of theoretical exposition, we assume α_{unadj}=0=α_{adj}, which is satisfied when the variables in the models are centered around their means. Then, the relation between the unadjusted and the adjusted effect estimates from linear regression is given by the following result.
Result 1
Under the probability law (true model) assumed above,
where \(\hat {\beta }_{\text {unadj}}\) and \(\hat {\beta }_{\text {adj}}\) are obtained from linear regression models of Y on X unadjusted and adjusted for confounders respectively, and \(\overset {P}{\longrightarrow }\) denotes convergence in probability.
In other words, for a large enough cohort, the difference of the unadjusted and the adjusted effect estimates from a linear regression model are independent of the true exposureoutcome effect size (β_{adj}) and also of the sample size (n).
Result 2
Under the true model assumed above, the effect estimates from a linear regression model are nonnegatively correlated, i.e., \(\text {Cov}(\hat {\beta }_{\text {unadj}}, \hat {\beta }_{\text {adj}})\geq 0\).
Proofs of these results are outlined in Supplementary S2. We also provide empirical proof of Result 1 by simulating data from the same data generating model as the populationlevel true model assumed above (Supplementary S2). For more general settings, such as nonnormal distributions for the exposure and the confounders, our empirical evidence supports that \(\hat {\beta }_{\text {unadj}}  \hat {\beta }_{\text {adj}}\) is asymptotically independent of true exposureoutcome association and the sample size (Supplementary S1). It is quite possible that these results are not new since many researchers across different quantitative fields have done theoretical work on linear models for several years [13, 14]. Notwithstanding this possibility, we state them here for completeness as these relations are leveraged by our novel metaanalysis approach.
Logistic regression
Using simulations, the above asymptotic sample size invariance property appears to hold for a logistic regression with binary outcome Y, any exposure X and any confounders \(\boldsymbol {\mathcal {C}}\) (Supplementary S3). In other words, for a large enough cohort, \(\hat {\beta }_{\text {unadj}}  \hat {\beta }_{\text {adj}} \stackrel {\text {\tiny def}}{=} \log (\hat {\text {OR}}_{\text {unadj}})  \log (\hat {\text {OR}}_{\text {adj}})\) does not appear to depend on the sample size. Unlike the linear regression scenario, this relation does not appear to be independent of the true odds ratio (\(\phantom {\dot {i}\!}\text {OR}_{\text {adj}} = e^{\beta _{\text {adj}}}\)) in logistic regression. In fact, under certain assumptions– including binary exposure X, independence of X and \(\boldsymbol {\mathcal {C}}\) as in a randomized trial, and weak effects γ of covariates \(\boldsymbol {\mathcal {C}}\)–, Gail et al. [15] derived the asymptotic approximate bias \(\hat {\beta }_{\text {unadj}}  \hat {\beta }_{\text {adj}}\) for nonlinear models using second order Taylor series. In particular for the logistic model, they found this bias to depend on the true exposureoutcome association in their numerical studies: \(\hat {\beta }_{\text {unadj}}\) is negatively biased when β_{adj}>0 and positively biased for β_{adj}<0. Note, this conclusion about direction of bias was strictly based on small values of \(\frac {1}{4}\boldsymbol \gamma '\boldsymbol {\Omega }\boldsymbol \gamma \), where γ consists of covariate effects in the fully adjusted model and Ω is the covariate variancecovariance matrix [15].
Generalized linear model
The following result generalizes the above asymptotic sample size invariance property for generalized linear models using asymptotic normality of maximum likelihood estimates.
Result 3
For a large enough cohort, the difference of the unadjusted and the adjusted effect estimates, \(\hat {\beta }_{\text {unadj}}  \hat {\beta }_{\text {adj}}\), from a generalized linear model is asymptotically constant (independent of the sample size).
The proof is outlined in Supplementary S4.
CIMBAL: proposed approach to adjust for confounder imbalance
Assumptions
For the ease of exposition, we will first consider only two cohorts and later generalize our approach for multiple cohorts. CIMBAL relies on asymptotic Result 3 and depends on a weak assumption that the cohorts are drawn independently from populations with the same confounding bias regardless of other types of heterogeneity (e.g., different distributions of confounders across cohorts). As will become evident in the following sections, CIMBAL does not depend on strong assumptions such as cohorts drawn from the same underlying population or homogeneity of joint distributions \([Y,X,\boldsymbol {\mathcal {C}}]\) underlying each cohort.
Imputed adjusted effect estimate
If there are two independent cohorts– cohort 1 with no information on any confounder and is able to report only \(\hat {\beta }_{\text {unadj}}\), and cohort 2 with complete information to be able to report both \(\hat {\beta }_{\text {adj}}\) and \(\hat {\beta }_{\text {unadj}}\)– the investigator can impute the adjusted association estimate for the cohort with missing confounder information leveraging Result 3:
However, for metaanalysis, having the adjusted effect estimates is not enough. The SE of the effect estimate from the fully adjusted model is required from all the cohorts for inverse variance weighting and for obtaining the SE and the 95% confidence interval (CI) of the metaanalyzed effect.
Imputed adjusted standard error estimate
From Eq. 1, the adjusted variance estimate for cohort 1 may be obtained as
We call our proposed correction approach for confounder imbalance CIMBAL.
Metaanalysis using imputed estimates under confounder imbalance
To explain metaanalysis in this context, we continue discussion with cohort 1, which has the imputed adjusted estimates (\(\tilde \beta _{1,\text {adj}}, \tilde {se}^{2}_{1,\text {adj}}\)) and cohort 2, which reported the fully adjusted estimates (\(\hat {\beta }_{2,\text {adj}}, \hat {\text {se}}^{2}_{2,\text {adj}}\)). Inversevariance weighted fixedeffect metaanalysis is a popular approach for pooling estimates from independent cohorts. Although we assume cohort 1 to be independent of cohort 2 (i.e., no sharing of samples between cohorts), Eq. 1 indicates that the estimates \(\tilde \beta _{1,\text {adj}}\) and \(\hat {\beta }_{2,\text {adj}}\) are no longer uncorrelated:
where Cov_{b}(.) denotes betweencohort covariance, to differentiate from Cov(.) that captures withincohort covariability. Consequently, the inversevariance weights are not optimal in terms of statistical efficiency of the metaanalyzed estimate [16, 17]. The following result gives the linear combination of exposureoutcome association estimates with the smallest asymptotic variance among all linear estimators.
Result 4
The exposureoutcome association estimate from metaanalyzing CIMBALimputed adjusted estimate from one cohort and the fully adjusted estimate from another cohort is given by
where the optimal weights maximizing statistical efficiency are
and the corresponding metaanalyzed adjusted SE estimate is
The proof is outlined in Supplementary S4.
The metaanalyzed SE estimate as well as the weights depend on \(\text {Cov}\left (\hat {\beta }_{2,\text {unadj}},\hat {\beta }_{2,\text {adj}}\right)\) which does not have a closed form for logistic regression. We have theoretically shown that \(\text {Cov}\left (\hat {\beta }_{2,\text {unadj}},\hat {\beta }_{2,\text {adj}}\right) \geqslant \ 0\) for linear regression (Result 2), and empirically shown that it holds for logistic regression for a large enough number of cohorts (Supplementary S3). If there are multiple cohorts with full confounder information, we can use the estimated covariance between their adjusted and unadjusted estimates as an estimate for \(\text {Cov}\left (\hat {\beta }_{2,\text {unadj}},\hat {\beta }_{2,\text {adj}}\right)\). If there are insufficient number of cohorts to estimate this covariance, we suggest ignoring the covariance (i.e., assume \(\text {Cov}\left (\hat {\beta }_{2,\text {unadj}},\hat {\beta }_{2,\text {adj}}\right)=0\)). This will only lead to overestimation of the metaanalyzed SE or a larger CI and hence a less efficient estimate of the exposureoutcome association. The following corollary states this special case of CIMBAL.
Corollary 1
Ignoring \(\text {Cov}(\hat {\beta }_{2,\text {unadj}},\hat {\beta }_{2,\text {adj}})\) in Result 4 is equivalent to taking \(\hat {w}_{1}=0\), thus ignoring the study reporting only unadjusted estimates. Consequently, metaanalysis using only the adjusted estimates available from complete cohorts is a special case of CIMBAL.
Generalization of CIMBAL to multiple cohorts with and without confounder measurements
The proposed correction approach can be easily extended to >2 independent cohorts. If there are K_{c} cohorts with complete confounder information, we first metaanalyze these cohorts (using fixedeffect inversevariance weighted metaanalysis) to obtain the metaanalyzed adjusted estimates and the metaanalyzed unadjusted estimates and their corresponding SE estimates: \(\left (\hat {\beta }_{2,\text {adj}}^{(\text {meta})}, \hat {\text {se}}_{2,\text {adj}}^{(\text {meta})}\right)\) and \(\left (\hat {\beta }_{2,\text {unadj}}^{(\text {meta})}, \hat {\text {se}}_{2,\text {unadj}}^{(\text {meta})}\right)\). Similarly, if there are K_{m} cohorts with no confounder information, we metaanalyze these cohorts to obtain the metaanalyzed unadjusted estimate and the corresponding SE estimate: \(\left (\hat {\beta }_{1,\text {unadj}}^{(\text {meta})}, \hat {\text {se}}_{1,\text {unadj}}^{(\text {meta})}\right)\). Now we can apply the imputation approach using Eqs. 1 and 2 to obtain CIMBALimputed adjusted estimates for the pooled noconfounder cohort \(\left (\tilde \beta _{1,\text {adj}}^{(\text {meta})}, \tilde {\text {se}}_{1,\text {adj}}^{(\text {meta})}\right)\). For the final metaanalysis, we first estimate \(\text {Cov}\left (\hat {\beta }_{2,\text {unadj}},\hat {\beta }_{2,\text {adj}}\right)\) from the K_{c} complete cohorts and then use formulae from Result 4. We have implemented our imputation approach and the subsequent metaanalysis in a R [18] program at https://github.com/RayDebashree/cimbal.
Simulation design
To demonstrate pitfalls of metaanalysis in cohort collaborations in the presence of unbalanced measurement of confounders, and to study the performance of our proposed approach, CIMBAL, we conduct extensive simulation experiments using a binary outcome Y, a binary exposure X, and two binary confounding variables C_{1},C_{2}. Note, preliminary simulation analysis in support of asymptotic relations claimed in “Linear regression” and “Logistic regression” sections are presented in Supplementary S1 and Supplementary S3.
For a given individual, we use the model logit(P(Y=1))=γ_{0}+γ_{1}C_{1}+γ_{2}C_{2}+βX to generate the outcome, where C_{1} and C_{2} are Bernoulli variables with success probabilities 0.1 and 0.6 respectively, and the exposure X is generated using the logistic model logit(P(X=1))=η_{0}+η_{1}C_{1}+η_{2}C_{2}. The choices of parameters γ_{0},γ_{1},γ_{2},β,η_{0},η_{1},η_{2} are provided in Table 1. Setting I of parameter choices involve equal confounder effects in the same direction for both exposure and outcome (η_{1}=η_{2}=γ_{1}=γ_{2}=2). Keeping the confounder effect on exposure same, Setting II involves one confounder having equal and same effect and the other confounder having equal but opposite effect on the outcome (η_{1}=η_{2}=γ_{1}=2,γ_{2}=−2). Setting III considers equal and opposite effect of both confounders for the outcome (η_{1}=η_{2}=2,γ_{1}=γ_{2}=−2). For simplicity, we assume 60 independent cohorts of equal sample size (n=150) are available. We simulate 3 scenarios: (1) fewer cohorts or (2) equal number of cohorts or (3) more cohorts with no confounder information than with complete confounder information. For each scenario, we simulate 2,500 replicates of 60 cohorts. We are interested in estimating the association between the outcome and the exposure by using cohortlevel summary statistics.
We perform a few additional experiments to see how sensitive CIMBAL is compared to other approaches when underlying assumptions are not satisfied. In particular, for the Sensitivity I scenario, we generate all the cohorts without confounder information from the data generating model described above, and the remaining cohorts (those with complete confounder information) from another distribution so that the underlying joint distributions [Y,X,C_{1},C_{2}] are not the same between complete and incomplete cohorts. The second data generating model is assumed to have different success probabilities of confounders C_{1} and C_{2} (0.2 and 0.7 respectively). This changes the mean vector as well as the variancecovariance matrix of the confounder distribution and the joint distribution [Y,X,C_{1},C_{2}]. We keep the true exposureoutcome association fixed at 0 for both populations. For the Sensitivity II scenario, we consider a variation of the Sensitivity I scenario by keeping the joint distribution same across cohorts but changing the strength of confounding on X and Y. Briefly, the cohorts with no confounder information are drawn from the data generating model with strong confounding as described before, while the remaining cohorts are drawn from a data generating model with much weaker confounding effects. For the Sensitivity III scenario, we consider a third confounder C_{3}∼N(0,1) in the data generating model for Y and X, where we assume γ_{3}=2 and η_{3}=2 are parameters corresponding to the association of C_{3} with Y and X respectively. For the analysis models however, we assume C_{3} is an unmeasured confounder so that the models fit by each cohort is either adjusted for the first two confounders (C_{1} and C_{2}) or adjusted for none.
For each simulation setting and scenario, we obtain the logodds estimate and its SE for the combined cohort using CIMBAL and compare it against two metaanalysis approaches: metaanalysis of only the available adjusted estimates, and the oracle (gold standard) metaanalysis of adjusted estimates from all cohorts. Note, we do not include metaanalysis of the unadjusted estimates from all cohorts in this comparison since it is wellestablished to be a biased estimate in the presence of confounders. We visually compare all three approaches by plotting the distribution of estimated logodds and its SE. Further, we use the following metrics for comparison: mean squared error (MSE), mean width of 95% CI, and type I error (only when data are generated under β= log(OR)=0). For a given method, we estimate MSE as mean of the squared difference between estimated logodds and the true logodds; mean width as mean of the difference between the upper and the lower 95% confidence limits; and type I error as the proportion of times the null hypothesis that β=0 is rejected at 5% significance level. For all these metrics, average or proportion is calculated over 2,500 independent replicates. We conduct all statistical analyses in R and create plots using R package ggplot2 [19].
Application to NCS data on low birth weight of infants
We use NCS [20, 21] data on 5,604 children enrolled between 2009 and 2013 to evaluate all the metaanalysis approaches including CIMBAL. For this proofofconcept analysis, we study if and how maternal education influences infant birth weightforgestationalage (BWforGA). We assume the NCS dataset is our population and we randomly sample 40 independent cohorts of equal sample size from this population.
We define a dichotomized version of BWforGA zscore as our outcome. In particular, we extract child sex, birth weight, and gestational age from the medical records and calculate child BWforGA zscore according to the 2017 US reference [22]. After excluding children with missing sex, birth weight and/or gestational age at birth, and those with gestational age at birth <22 or >42 weeks, we calculate BWforGA zscore for 4,658 children. ‘Small’ for gestational age is typically defined as BWforGA zscore <10th percentile. However, this definition would lead to an outcome with low prevalence in our study, and combined with small sample size in each of the 40 cohorts may lead to unstable estimates. Hence, in this proofofconcept analysis we define ‘small’ for gestational age as BWforGA zscore <25th percentile to mimic a relatively common outcome.
For our exposure, we consider two categories of maternal education. The reference category is ‘Some college or below’ while the other category is ‘Bachelor’s degree or above’. We consider 4 key maternal covariates (confounders) that may influence the exposureoutcome association: maternal age at delivery, race/ethnicity, marital status, and annual household income. While maternal age is a continuous variable, the others are categorical. We consider 4 categories for race/ethnicity: Hispanic, nonHispanic White, nonHispanic Black, and nonHispanic other; 3 categories for marital status: married or living together with a partner, never been married, and divorced, separated, or widowed; 4 categories for annual household income: <$30,000, $30,000$49,999, $50,000$99,999, and >$100,000. After removing observations with missing confounders, we have 4,089 motherinfant pairs in our final analytical dataset.
We draw a random sample of 4,080 dyads without replacement and split them into 40 cohorts of equal sample size (n=102). Of the 4,080 dyads, 22.6% (n=921) of the infants had BWforGA zscore <25th percentile. The prevalence of our outcome in the 40 independent cohorts ranged from 12.7% (n=13) to 31.4% (n=32). For all cohorts, we obtain unadjusted as well as adjusted estimates of exposureoutcome association along with their SE estimates using Stata [23]. We also combine all the cohorts and conduct a pooled analysis without and with confounder adjustment, giving us unadjusted as well as adjusted estimates for the combined cohort. To evaluate CIMBAL in comparison to the other metaanalysis approaches described before, we assume that either 10, 20 or 30 cohorts have no information on any confounders, while the rest have complete information on all confounders. We randomly select the cohorts assumed to have no confounder information. Before metaanalyzing all the cohorts, we exclude any outlying cohorts. Specifically, if logodds estimate or SE estimate (unadjusted and/or adjusted) from a cohort falls outside the 3 times interquartile range, then we exclude that cohort. We remove one such outlier, leaving 39 independent cohorts from the NCS population.
Results
Simulated data analysis
Under confounder imbalance, distributions of estimates from CIMBAL are closer to oracle than when restricted to cohorts with complete data
Figure 1 shows the distribution of the estimate of exposureoutcome association (\(\hat {\beta }\)) and its SE (\(\hat {\text {se}}\)) across different simulation scenarios. Although metaanalysis using adjusted estimates from cohorts with complete information provides unbiased estimates, it has high variability due to small effective sample size, leading to wide CIs. CIMBAL, on the other hand, provides not only unbiased estimates but also smaller variability than completeonly metaanalysis, leading to point estimates and CIs that are closest to the oracle.
Note that the current simulation model assumes strong confounder effects only, and that there is no association between the exposure and the outcome (β= log(1)). Therefore, we additionally simulate Setting I with weak confounder effects (Figure S7), and Setting I with strong positive (β= log(3)) or negative (β= log(1/3)) exposureoutcome association (Figure S8). We observe that relative behavior of the methods is the same regardless of the true association, the strength of the confounding effects, and whether we have fewer, equal or more cohorts with missing confounder information (Fig. 1, S7 and S8). The variability of the exposureoutcome estimate from each metaanalysis method increases as the number of cohorts with missing confounders increases. Similar behavior is observed for other simulation settings involving varying directions of confounder effects (Fig. 1). In other words, metaanalysis using CIMBAL seems to provide estimates that are closest to what one would have obtained if fully adjusted estimates were available from all cohorts.
Metaanalysis using CIMBAL is closest to oracle across multiple statistical metrics
Table 2 shows MSE (along with relative MSE compared to the oracle metaanalysis), mean width of 95% CI, and type I error performance of each method across different simulation scenarios. Across all parameter settings and scenarios, MSE of CIMBAL is closest to the MSE of the oracle. Completeonly metaanalysis has the largest mean width of 95% CI that increases with increasing number of cohorts without complete confounder information, as expected. Mean width of CIMBAL’s CI is only slightly larger than the oracle. These observations continue to hold for simulations involving weaker confounder effects (Table S5). As for the type I error metric that one can evaluate only under the null (i.e., when the underlying data have no exposureoutcome association), both CIMBAL and completeonly metaanalysis maintains appropriate type I error at 5% level. CIMBAL’s type I error performance resembles that of the oracle across most scenarios (Table 2), and this continues to be true regardless of strength of confounder effects (Table S5).
Metaanalysis using CIMBAL coincides with completeonly metaanalysis when there are too few cohorts with complete confounder information
Metaanalysis of CIMBALimputed adjusted estimates from incomplete cohorts and adjusted estimates from complete cohorts requires an estimate of \(\text {Cov}\left (\hat {\beta }_{\text {unadj}}, \hat {\beta }_{\text {adj}}\right)\). This estimated covariance not only depends on the sample sizes of the complete cohorts but also the number of such cohorts used to estimate the covariance, the strengths and directions of confounder effects and exposureoutcome association (Figure S4). We suggest that at least 20 cohorts be used for appropriately estimating this covariance. If the estimate turns out to be negative, our program cimbal automatically assumes 0 covariance (the theoretical lower limit we found for linear regression case and the empirical asymptotic lower limit we found for logistic regression). Negative covariance estimate occurs when the correlation estimate is negative, which may arise due to reasons such as insufficient number of cohorts used in the estimation process, insufficient percohort sample size relative to confounder adjustment, or skewed sample distribution of categorical confounders between cases and controls leading to model fit issues (Figures S5 and S6). Consistent with Corollary 1, the CIMBAL metaanalyzed estimate boils down to the estimate from completeonly metaanalysis when 0 covariance is assumed (Figure S9).
Sensitivity analysis
When there exists heterogeneity in the underlying joint distributions– here, complete cohorts are from one population, the remaining are from another with a different joint distribution (Sensitivity I scenario)–, CIMBAL’s estimates may be slightly biased (Figure S10) but still closer to the oracle in terms of MSE than completeonly metaanalysis (Table S4). This appears to be true regardless of whether fewer or more cohorts have no information on any confounder. When the underlying joint distributions are homogenous but the confounding bias is not the same across cohorts– here, counfounding effects are weak in complete cohorts but very strong in the others (Sensitivity II scenario)–, we see massive increase in bias (Figure S11), MSE and type I error rates (Table S4) for CIMBAL. Thus, CIMBAL is very sensitive to heterogeneity of confounding bias but not as sensitive to heterogeneity in underlying joint distributions. When there is an unmeasured confounder (Sensitivity III scenario), all the metaanalysis approaches, including the oracle, are biased as expected; however, metaanalysis using CIMBAL is again closest to the oracle (Figure S12).
NCS data analysis
Figure 2 shows not only the reported logodds estimate and its 95% CI from each cohort but also the different metaanalyzed estimates for the pooled cohort. For most cohorts with complete confounder information, we see appreciable difference between their adjusted and unadjusted estimates. As expected, metaanalysis using only the unadjusted estimates from all the cohorts leads to considerably biased logodds estimate with a narrow CI around them. On the other hand, metaanalysis only using the adjusted estimates from the complete cohorts leads to less bias but larger CIs compared to the oracle. When only 10 or 20 out of 39 cohorts (1 cohort removed for being an outlier) have no confounder information, we find CIMBAL has similar bias as the completeonly metaanalysis but narrower CI. When 30 cohorts have no confounder information, we do not have enough complete cohorts to estimate the weight CIMBAL should put on incomplete cohorts and consequently we assign all the weight to complete cohorts. We see that CIMBAL and completeonly metaanalysis estimates and CIs coincide as expected.
Discussion
In this article, we develop a novel and practical approach for metaanalysis of exposureoutcome association estimates from cohorts with disparate confounder information. Currently, there is no consensus on how unadjusted and adjusted estimates from cohorts may be meaningfully combined without sacrificing on sample size or unbiasedness of estimate. Our approach, CIMBAL, mitigates this issue by borrowing information from cohorts with complete confounder information to impute adjusted estimates for cohorts without counfounder information. This borrowing of information is grounded in asymptotic relations between adjusted and unadjusted estimates that we have justified theoretically for generalized linear models. We additionally derive the metaanalysis weights for the reported and the imputed adjusted estimates that minimizes the variance of the metaanalyzed estimate. CIMBAL is a practical approach for data integration if only summary statistics are available from cohorts. It is a generalization of metaanalysis of available adjusted estimates only, which ignores contribution from cohorts without full confounder information. When compared to popular metaanalysis approaches in the presence of confounder imbalance among cohorts, we find the CIMBAL metaanalyzed estimate to be closer to the gold standard (metaanalysis of fully adjusted estimates from all cohorts, when available) across statistical metrics such as type I error rate (for null data only), MSE, and meanwidth of 95% CI. As proof of principle, we apply CIMBAL and other metaanalysis approaches to estimate association of maternal education with low birth weight of infants from NCS data after adjusting for four key maternal variables (confounders). Despite having access to individuallevel data, we randomly split the NCS data into multiple cohorts, some with the necessary confounder information, others with all confounder information removed. Our real data analysis results conform with our findings from simulation experiments.
An alternative approach to circumvent missing data on confounders will be to use individuallevel data from the cohorts with complete information to derive \([\boldsymbol {\mathcal {C}}Y,X]\), the conditional distribution of the confounders given the outcome and the exposure of interest. Standard methods of multiple imputation can be applied on the cohorts with incomplete confounder information before obtaining fully adjusted estimates from them [24]. Not only does this require access to individuallevel data from the cohorts with complete information but also the confounders are likely to be of large dimension making the imputation models challenging. Yet another approach is GENMETA, which too requires individuallevel reference data representative of the underlying population [7, 8]. Imputing fully adjusted estimates using CIMBAL is not subjected to those requirements, and thus is a practical metaanalysis approach.
We describe CIMBAL using a fixedeffect metaanalysis framework. One could alternatively consider a randomeffect metaanalysis framework, where heterogeneity among effect estimates is modeled as a variance component reflecting betweencohort variance. Specifically, one can metaanalyze unadjusted estimates from K_{m} cohorts with no confounder information using the inversevariance weights for randomeffect metaanalysis: \(\omega _{j} = \frac {1}{\hat {\text {se}}_{j,\text {unadj}}^{2}+\hat \tau ^{2}}\) for j=1,2,...,K_{m}, where \(\hat \tau ^{2}\) is an estimate of betweencohort variance in \(\left \{ \hat {\beta }_{j,\text {unadj}} \right \}_{j=1}^{K_{m}}\) obtained using, for instance, DerSimonian and Laird method of moments approach [25]. Alternative approaches, such as restricted maximum likelihood or nonparametric DerSimonian and Laird methods, may be used to obtain \(\hat \tau ^{2}\) when method of moments gives biased estimate, as suggested by recent comparative studies [26, 27]. Similarly, the betweencohort variances in \(\left \{ \hat {\beta }_{j,\text {unadj}} \right \}_{j=1}^{K_{c}}\) and \(\left \{ \hat {\beta }_{j,\text {adj}} \right \}_{j=1}^{K_{c}}\) may be estimated separately, and subsequently used to obtain the inversevariance weights for randomeffect metaanalysis of the unadjusted and the adjusted estimates from K_{c} complete cohorts. This set of randomeffect metaanalyzed estimates may then be used to get the CIMBALimputed adjusted estimates for the pooled noconfounder cohort. The final metaanalysis step (Result 4) in CIMBAL, however, needs to use a fixedeffect framework since the two estimates corresponding to two groups of cohorts are correlated and the betweengroup variance cannot be reliably estimated. A caveat of this randomeffect framework is potentially allowing for heterogeneity of confounding bias between the complete cohorts and the incomplete ones due to differences in aspects of the underlying populations or the study designs, and CIMBAL is very sensitive to this heterogeneity.
While we demonstrate CIMBAL on cohorts with either full or no confounder information, it may also be applied to cohorts with either full or partial confounder information. Among the q possible confounders, suppose all cohorts have the same p(<q) confounders measured. For instance, sex and race/ethnicity information are commonly collected in epidemiologic studies. In such a scenario, one may provide the cimbal program with partially adjusted estimates instead of the unadjusted estimates. Under the hood, cimbal program first metaanalyzes all the complete cohorts to obtain \(\left (\hat {\beta }_{2,\text {adj}}^{(\text {meta})}, \hat {\text {se}}_{2,\text {adj}}^{(\text {meta})}\right)\) and \(\left (\hat {\beta }_{2,\text {padj}}^{(\text {meta})}, \hat {\text {se}}_{2,\text {padj}}^{(\text {meta})}\right)\), where suffix ‘padj’ denotes partial confounder adjustment. All the cohorts with partial confounder information are metaanalyzed to obtain \(\left (\hat {\beta }_{1,\text {padj}}^{(\text {meta})}, \hat {\text {se}}_{1,\text {padj}}^{(\text {meta})}\right)\). Then, CIMBALimputed adjusted estimates for the pooled partialconfounder cohort are obtained as \(\left (\tilde \beta _{1,\text {adj}}^{(\text {meta})}, \tilde {\text {se}}_{1,\text {adj}}^{(\text {meta})}\right)\), and metaanalyzed with the available fully adjusted estimates using Result 4.
CIMBAL is not without limitations. We make the simplifying assumption that cohorts either have full confounder information or have the same partial confounder information. By ‘full confounder information’, we mean measurements on the minimally sufficient confounder set are available to complete cohorts. However, multiple minimally sufficient adjustment sets may exist and each can be used to obtain an unbiased estimate of the exposureoutcome association. While each minimally sufficient adjustment set can, in principle, adjust for confounding, the estimands are fundamentally different. Consequently, noncollapsible association estimator like the odds ratio from a logistic regression can show substantial heterogeneity across minimally sufficient confounder sets, and metaanalyzing cohorts using different sets may result in an unreliable estimate [28]. We also assume there is no model misspecification when fitting the fully adjusted model. While heterogeneity is inevitable in metaanalysis, CIMBAL does not have a diagnostic test for heterogeneity of fully adjusted estimates from all cohorts. However, we find CIMBAL is not too sensitive to heterogeneity in underlying joint distributions across cohorts; instead it is extremely sensitive to heterogeneity in confounding bias. For instance, age and sex or gender distributions can often be different across studies, and that is not expected to strongly influence CIMBAL as long as the confounding bias is the same across studies. Currently, CIMBAL cannot handle a combination of unadjusted, partially adjusted, and fully adjusted estimates from different cohorts. If many cohorts report partially adjusted estimates with different subsets of confounders, for practical purposes we recommend that investigators choose the most commonly occurring subset of confounder, and regard the corresponding cohorts as ‘complete cohorts’ and the remaining cohorts as those with no confounder information. We suggest that this choice be influenced by not just the number of cohorts reporting a particular set of confounders but also by the sample sizes of such cohorts. Once we have the two groups of cohorts, CIMBAL can be used to impute the ‘fully adjusted’ estimates for the incomplete cohorts as described earlier. We acknowledge this recommendation is suboptimal: it results in loss of information and statistical efficiency of the metaanalyzed estimate when many cohorts report partially adjusted estimates. Future work will generalize CIMBAL in this aspect.
Conclusions
Our novel method CIMBAL provides a practical yet valid approach for metaanalyzing independently sampled cohorts with imbalance in measurement of confounders. It is particularly useful when investigators have access to only summarylevel data from each cohort. As long as the confounding bias is the same across cohorts, CIMBAL is not too sensitive to heterogeneity in underlying joint distributions of exposure, outcome and confounders. Although we describe CIMBAL in the context of cohort studies, it is also relevant for a metaanalysis of randomized controlled trials.
Availability of data and materials
The NICHD Data and Specimen Hub (DASH) contains the public version of the NCS data (https://dash.nichd.nih.gov/study/228954). The CIMBAL method is implemented in a R program, which is publicly available at https://github.com/RayDebashree/cimbal.
Abbreviations
 BWforGA:

Birth Weight for Gestational Age
 CIMBAL:

metaanalysis method for Confounder IMBALance
 CKDPC:

Chronic Kidney Disease Prognosis Consortium
 CI:

Confidence Interval
 coRR:

confounding Risk Ratio
 ECHO:

Environmental influences on Child Health Outcomes
 GENMETA:

GENeralized METAanalysis
 MSE:

Mean Squared Error
 NCS:

National Children’s Study
 SE:

Standard Error
References
Jacobson LP, Lau B, Catellier D, et al. An Environmental influences on Child Health Outcomes viewpoint of data analysis centers for collaborative study designs. Curr Opin Pediatr. 2018; 30:269–75.
Wang C, Chen M, Schifano E., et al.Statistical methods and computing for big data. Stat Interface. 2016; 9(4):399–414.
Matsushita K, Coresh J, Sang Y., et al. Estimated glomerular filtration rate and albuminuria for prediction of cardiovascular outcomes: a collaborative metaanalysis of individual participant data. Lancet Diabetes Endocrinol. 2015; 3(7):514–25.
Voils CI, Crandell JL, Chang Y, et al. Combining adjusted and unadjusted findings in mixed research synthesis. J Eval Clin Pract. 2011; 17(3):429–34.
Ciolino JD, Martin RH, Zhao W, et al. Covariate imbalance and adjustment for logistic regression analysis of clinical trial data. J Biopharm Stat. 2013; 23(6):1383–402.
Chronic Kidney Disease Prognosis Consortium. Association of estimated glomerular filtration rate and albuminuria with allcause and cardiovascular mortality in general population cohorts: a collaborative metaanalysis. Lancet. 2010; 375(9731):2073–81.
Kundu P, Tang R, Chatterjee N. Generalized metaanalysis for multiple regression models across studies with disparate covariate information. Biometrika. 2019; 106(3):567–85.
Kundu P, Chatterjee N. Analysis of twophase studies using generalized method of moments. arXiv. 2019. arXiv:1910.11991.
Flanders WD, Khoury MJ. Indirect assessment of confounding: graphic description and limits on effect of adjusting for covariates. Epidemiology. 1990; 1(3):239–46.
Lash TL, Fox MP, Fink AK: Springer Science & Business Media; 2011.
Cornfield J, Haenszel W, Hammond EC, et al. Smoking and Lung Cancer: Recent Evidence and a Discussion of Some Questions. J Natl Cancer Inst. 1959; 22(1):173–203.
Miettinen OS. Components of the crude risk ratio. Am J Epidemiol. 1972; 96(2):168–72.
Lehmann EL. New York, NY: Springer; 1999.
DasGupta A. New York: Springer; 2008.
Gail MH, Wieand S, Piantadosi S. Biased estimates of treatment effect in randomized experiments with nonlinear regressions and omitted covariates. Biometrika. 1984; 71(3):431–44.
Wei LJ, Lin DY, Weissfeld L. Regression analysis of multivariate incomplete failure time data by modeling marginal distributions. J Am Stat Assoc. 1989; 84(408):1065–73.
Lin DY, Sullivan PF. Metaanalysis of genomewide association studies with overlapping subjects. Am J Hum Genet. 2009; 85(6):862–72.
R Core Team. Vienna, Austria: R Foundation for Statistical Computing; 2018. https://www.Rproject.org/.
Wickham H, Vol. ISBN 9783319242774. New York: SpringerVerlag; 2016. http://ggplot2.org.
Hudak ML, Park CH, Annett RD, et al. The National Children’s Study: an introduction and historical overview. Pediatrics. 2016; 137(Supplement 4):S213–8.
Hirschfeld S. Introduction and goals for the National Children’s Study. Front Pediatr. 2018; 5:240.
Aris IM, Kleinman KP, Belfort MB, et al. A 2017 US reference for singleton birth weight percentiles using obstetric estimates of gestation. Pediatrics. 2019; 144(1):e20190076.
StataCorp. TX: StataCorp LLC, College Station; 2017.
Little RJ, Rubin DB: John Wiley & Sons; 2019. ISBN 9780470526798.
DerSimonian R, Laird N. Metaanalysis in clinical trials. Control Clin Trials. 1986; 7(3):177–88.
Petropoulou M, Mavridis D. A comparison of 20 heterogeneity variance estimators in statistical synthesis of results from studies: a simulation study. Stat Med. 2017; 36(27):4266–80.
Langan D, Higgins JP, Jackson D, et al. A comparison of heterogeneity variance estimators in simulated randomeffects metaanalyses. Res Synth Methods. 2019; 10(1):83–98.
Hamra GB, Lesko CR, Buckley JP, et al. Combining effect estimates across cohorts and sufficient adjustment sets for collaborative research: a simulation study. Epidemiology. 2021; 32(3):421–4.
Acknowledgements
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Funding
This research was supported by the NIH for the Environmental influences of Child Health Outcomes Data Analysis Center (U24OD023382). BL was additionally supported by funds from the JHU CFAR NIH/NIAID (P30AI094189) and the NIH/NIAAA (U01AA020793).
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Conceptualization: AM, LPJ, BL. Data curation: DR, MZ, XL. Formal analysis: DR, MZ. Funding acquisition: LPJ. Investigation: DR. Methodology: DR, AM, NC, BL. Project administration: DR, AM, BL. Software: DR. Supervision: AM, BL. Validation: DR. Visualization: DR. Writing – original draft: DR. Writing – review & editing: DR, AM, MZ, XL, NC, LPJ, BL. The authors read and approved the final manuscript.
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This article uses real data from only NCS for the purposes of methodologic development. The NCS data was approved for methodologic development by the Institutional Review Board (IRB) at the Johns Hopkins Bloomberg School of Public Health (IRB No. 11542/CR389). The data analyses in this article are based on a repository of deidentified NCS data under this IRB. Informed consent was obtained for the original study from all participants. All methods were carried out in accordance with relevant guidelines and regulations.
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Ray, D., Muñoz, A., Zhang, M. et al. Metaanalysis under imbalance in measurement of confounders in cohort studies using only summarylevel data. BMC Med Res Methodol 22, 143 (2022). https://doi.org/10.1186/s12874022016149
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DOI: https://doi.org/10.1186/s12874022016149