Accounting for small-study effects using a bivariate trim and fill meta-analysis procedure

In meta-analyses, small-study effects (SSE) refer to the phenomenon that smaller studies show different, often larger, treatment effects than larger studies, which may lead to incorrect, commonly optimistic estimates of treatment effects. Visualization tools such as funnel plots have been widely used to investigate the SSE in univariate meta-analyses. The trim and fill procedure is a non-parametric method to identify and adjust for SSE and is widely used in practice due to its simplicity. However, most visualization tools and SSE bias correction methods have been focusing on univariate outcomes. For a meta-analysis with multiple outcomes, the estimated number of trimmed studies by trim and fill for different outcomes may be different, leading to inconsistent conclusions. In this paper, we propose a bivariate trim and fill procedure to account for SSE in a bivariate meta-analysis. Based on a recently developed visualization tool of bivariate meta-analysis, known as the galaxy plot, we develop a sensible data-driven imputation algorithm for SSE bias correction. The method relies on the symmetry of the galaxy plot and assumes that some studies are suppressed based on a linear combination of outcomes. The studies are projected along a particular direction and the univariate trim and fill method is used to estimate the number of trimmed studies. Compared to the univariate method, the proposed method yields consistent conclusion about SSE and trimmed studies. The proposed approach is validated using simulated data and is applied to a meta-analysis of efficacy and safety of antidepressant drugs.