Meta-regression to explain shrinkage and heterogeneity in large-scale replication projects

Abstract

Recent large-scale replication projects (RPs) have estimated concerningly low reproducibility rates. Further, they reported substantial degrees of shrinkage of effect size, where the replication effect size was found to be, on average, much smaller than the original effect size. Within these RPs, the included original-replication study-pairs can vary with respect to aspects of study design, outcome measures, and descriptive features of both original and replication study population and study team. This often results in between-study-pair heterogeneity, i.e., variation in effect size differences across study-pairs that goes beyond expected statistical variation. When broader claims about the reproducibility of an entire field are based on such heterogeneous data, it becomes imperative to conduct a rigorous analysis of the amount and sources of shrinkage and heterogeneity within and between included study-pairs. Methodology from the meta-analysis literature provides an approach for quantifying the heterogeneity present in RPs with an additive or multiplicative parameter. Meta-regression methodology further allows for an investigation into the sources of shrinkage and heterogeneity. We propose the use of location-scale meta-regressions as a means to directly relate the identified characteristics with shrinkage (represented by the location) and heterogeneity (represented by the scale). This provides valuable insights into drivers and factors associated with high or low reproducibility rates and therefore contextualises results of RPs. The proposed methodology is illustrated using publicly available data from the Replication Project Psychology and the Replication Project Experimental Economics. All analysis scripts and data are available online.

Fig 1. Ordered differences between effect estimates on Fisher-z scale Δ^i=θ^oi−θ^ri for all included study-pairs (n = 87) in both replication projects with their 95% confidence interval.

The dashed horizontal line indicates no difference in effect size.

Fig 2. Left: The distribution of the observed standardized difference δ^i of the original-replication-study-pair (n = 87) compared to the standard normal distribution.

Right: The p-values from the Q-test for heterogeneity within original-replication study-pairs, as well as the p-value for the overall test for heterogeneity between all study-pairs, included in both replication projects.

Fig 3. The AIC for the multiplicative and the additive models with best performance (min AIC) for each possible number of covariates included.

The residual multiplicative heterogeneity parameter $\tilde{\phi }$ and additive heterogeneity variance ${\tilde{\tau }}^2$ of the respective models are also shown. At least one covariate, namely the original standard error, is included. The minimum AIC value is highlighted.