engleski

Systematically missing data in meta-analysis: a graphical model approach

Systematically missing data in meta-analysis arise when studies measure different sets of adjusting variables. Common approaches to this problem can cause bias (using only variables measured in all studies) or loss of precision (using only studies with all variables ; complete-case approach). A previous study showed that exploiting correlation between partially and fully adjusted regression coefficient estimates in a bivariate meta-analysis increases precision (The Fibrinogen Studies Collaboration 2009, Stat Med 28:1218-37). We propose a graphical model that uses these correlations and, additionally, hypothesized conditional independence between differently adjusted estimates to calculate fully adjusted estimate. Unlike previously proposed approaches, full data is not required ; only estimated coefficients with standard errors. Moreover, even studies with only one reported coefficient can be included in the analysis. The proposed graphical model is illustrated in Fig.1. Node X_i in a graph corresponds to regression coefficient adjusted for the set of variables S_i, and edge from X_i to X_j indicates that S_j ⊂ S_i. Joint density is derived from graph structure (Fig.1) and nodes’ distributions, which we specify as X_i ∼ N(beta_i, sigma_i^2 + tau_i^2) if X_i has no parents in the graph, or X_i | X_j ~ N(beta_i + rho_ij * sqrt((sigma_i^2 + tau_i^2)/ (sigma_j^2 + tau_j^2)) * (X_j – beta_j), (sigma_i^2 + tau_i^2)*(1-rho_ij^2)) if its parent is X_j. Here sigma_i^2 and tau_i^2 are within- and between-study variances, and rho_ij correlation coefficient between X_i and X_j. Parameters are then estimated either by Markov Chain Monte Carlo sampling or maximum likelihood, both in a standard way. The method was tested using data for 11 countries from the Survey of Health, Ageing and Retirement in Europe. As our goal was solely to develop methodology, we repeat the analysis by Siegrist et al. (Siegrist et al. 2007, Eur J Public Health 17:62-8). The analysis aimed to estimate the effect of effort-reward imbalance on intended early retirement, adjusted for 9 covariates representing sex, age, socio-economic status, physical and mental health parameters, and work conditions (3 binary, 3 3-level, and 3 continuous variables ; Fig.1). We considered two missing data patterns, one monotone (Fig.1 a) and the other non-monotone (Fig.1 b). For each pattern, countries were randomly assigned to missing data categories for 3 times. The proposed graphical model (G-MA) was compared to meta-analysis of all 11 fully-adjusted estimates (“gold standard”) and complete-case meta-analysis with normal distribution-based confidence intervals (CC-MA ; conventional approach). Graphical model estimates were based on 50 000 iterations from Markov Chain Monte Carlo analysis, with uninformative priors for coefficients’ means (beta_i) and between-study variances. Meta-analysis of all 11 fully-adjusted estimates yielded an odds ratio estimate of 1.87, indicating that higher effort-reward imbalance increases odds of retiring early. In both missing pattern settings (Fig 1.a), G-MA showed less bias than CC-MA (Table 1). In one example with simple missing pattern 95% G-MA credible interval had the same width as the 95% CC-MA confidence interval, while in all other cases CC-MA intervals were wider, by 16% to 55%. None of the G-MA models indicated any lack of convergence. To conclude, the proposed approach showed less bias and more precision than the conventional approach in our examples, while at the same time requiring less data than alternative approaches. It is easily extended to more complex, non-monotone missing data settings, all types of regression models with any kind of variables, and fixed-effects meta-analysis, while also allowing for incorporation of prior knowledge. All of this makes it suitable for a variety of applications.

meta-analysis; systematically missing data; graphical models

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