Description

Compute statistics relating to heterogeneity (whole model) and pooling (for each group) given a baggr meta-analysis model. The statistics are the pooling metric by Gelman & Pardoe (2006) or its complement, the I-squared statistic.

Usage

Arguments

Details

Pooling statistic describes the extent to which group-level estimates of treatment effect are "pooled" (or pulled!) toward average treatment effect in the meta-analysis model. If pooling = "none" or "full" in baggr, then the returned values are always 0 or 1, respectively. If pooling = "partial", the value is somewhere between 0 and 1.

Formulae for the calculations below are provided in main package vignette. See vignette("baggr").

#' Estimate of pooling in a group: this is the calculation done by pooling() if type = "groups" (default).

In a partial pooling model (see baggr), group k (e.g. study) has a treatment effect estimate, with some SE around the real treatment effect (TE). Each TE itself is distributed with mean and variance.

The quantity of interest is ratio of variability in τ to total variability. By convention, we subtract it from 1, to obtain a pooling metric p.

p = 1 - (σ(τ)^2 / (σ_(τ)^2 + se_k^2))

• If p < 0.5, that means the variation across studies is higher than variation within studies.

• Values close to 1 indicate nearly full pooling. Variation across studies dominates.

• Values close to 0 – no pooling. Variation within studies dominates.

Note that, since σ_{τ}^2 is a Bayesian parameter (rather than a single fixed value) p is also a parameter. It is typical for p to have very high dispersion, as in many cases we cannot precisely estimate σ_{τ}. To obtain the whole distribution of_p_ (rather than summarised values), set summary=FALSE.

Overall pooling (in the model)

Typically it is a single measure of heterogeneity that is of interest to researchers. This is calculated by setting type = "total" or simply writing heterogeneity(mymodel)

In many contexts, i.e. medical statistics, it is typical to report 1-P, called I^2 (see Higgins et al, 2003). Higher values of I-squared indicate higher heterogeneity. Von Hippel (2015) provides useful details for I-squared calculations.

Same as for group-specific estimates, P is a Bayesian parameter and its dispersion can be high.

Relationship to R-squared statistic

See Gelman & Pardoe (2006) Section 1.1 for a short explanation of how R^2 statistic relates to the pooling metric.

Value

Matrix with mean and intervals for chosen pooling metric, each row corresponding to one meta-analysis group.

References

Gelman, Andrew, and Iain Pardoe. "Bayesian Measures of Explained Variance and Pooling in Multilevel (Hierarchical) Models." Technometrics 48, no. 2 (May 2006): 241-51. https://doi.org/10.1198/004017005000000517.

Higgins, Julian P T, Simon G Thompson, Jonathan J Deeks, and Douglas G Altman. "Measuring Inconsistency in Meta-Analyses." British Medical Journal 327, no. 7414 (September 6, 2003): 557-60.

Hippel, Paul T von. "The Heterogeneity Statistic I2 Can Be Biased in Small Meta-Analyses." BMC Medical Research Methodology 15 (April 14, 2015). https://doi.org/10.1186/s12874-015-0024-z.