pooling {baggr} | R Documentation |

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.

pooling(bg, type = c("groups", "total"), summary = TRUE) heterogeneity(bg, summary = TRUE)

`bg` |
output of a baggr() function |

`type` |
In |

`summary` |
logical; if |

Pooling statistic describes the extent to which group-level estimates of treatment
effect are "pooled" (or pulled!) closer to average treatment effect in the meta-analysis model.
If `pooling = "none"`

or "full" in baggr, then the 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.**

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

This is the calculation done by `pooling()`

if `type = "groups"`

(default).
See `vignette("baggr")`

for more details on pooling calculations.

In a partial pooling model (see baggr), group *k* (e.g. study) has
standard error of treatment effect estimate, *se_k*.
The treatment effect (across *k* groups) is variable across groups, with
hyper-SD parameter *σ_(τ)*.

The quantity of interest is ratio of variation in treatment effects to the
total variation.
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*, 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 indicate 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`

.

Typically researchers want to report a single measure from the model,
relating to heterogeneity across groups.
This is calculated by either `pooling(mymodel, type = "total")`

or simply `heterogeneity(mymodel)`

In many contexts, i.e. medical statistics, it is typical to report *1-P*, called *I^2*
(see Higgins and Thompson, 2002; sometimes another statistic, *H^2 = 1 / P*,
is used).
Higher values of *I-squared* indicate higher heterogeneity;
Von Hippel (2015) provides useful details for *I-squared* calculations.

To obtain such single estimate we need to substitute average variability of group-specific
treatment effects and then calculate the same way we would calculate *p*.
By default we use the mean across *k* *se_k^2* values. Typically, implementations of
*I^2* in statistical packages use a different calculation for this quantity,
which may make *I*'s not comparable when different studies have different SE's.

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.

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

Higgins, Julian P. T., and Simon G. Thompson.
“Quantifying Heterogeneity in a Meta-Analysis.”
*Statistics in Medicine, vol. 21, no. 11, June 2002, pp. 1539–58*.

Hippel, Paul T von. "The Heterogeneity Statistic I2 Can Be Biased in Small Meta-Analyses."
*BMC Medical Research Methodology 15 (April 14, 2015).*

[Package *baggr* version 0.6.4 Index]