# pooling: Pooling metrics and related statistics for baggr In baggr: Bayesian Aggregate Treatment Effects

 pooling R Documentation

## Pooling metrics and related statistics for baggr

### Description

Compute statistics relating to `pooling` in a given baggr meta-analysis model returns statistics, for either the entire model or individual groups, such as pooling statistic by Gelman & Pardoe (2006), I-squared, H-squared, or study weights; `heterogeneity` is a shorthand for `pooling(type = "total")` `weights` is shorthand for `pooling(metric = "weights")`

### Usage

```pooling(
bg,
metric = c("pooling", "isq", "hsq", "weights"),
type = c("groups", "total"),
summary = TRUE
)

heterogeneity(
bg,
metric = c("pooling", "isq", "hsq", "weights"),
summary = TRUE
)

## S3 method for class 'baggr'
weights(object, ...)
```

### Arguments

 `bg` a baggr model `metric` `"pooling"` for Gelman & Pardoe statistic P, `"isq"` for I-squared statistic (1-P, Higgins & Thompson, 2002) `"hsq"` for H squared statistic (1/P, ibid.); `"weights"` for study weights; also see Details `type` In `pooling` calculation is done for each of the `"groups"` (default) or for `"total"` hypereffect(s). `summary` logical; if `FALSE` a whole vector of pooling values is returned, otherwise only the means and intervals `object` baggr model for which to calculate group (study) weights `...` Unused, please ignore.

### Details

Pooling statistic (Gelman & Pardoe, 2006) describes the extent to which group-level estimates of treatment effect are "pooled" toward average treatment effect in the meta-analysis model. If `pooling = "none"` or `"full"` (which you specify when calling baggr), then the values are always 0 or 1, respectively. If `pooling = "partial"`, the value is somewhere between 0 and 1. We can distinguish between pooling of individual groups and overall pooling in the model.

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 (and some issues related to it, especially in frequentist models). See Gelman & Pardoe (2006) Section 1.1 for a short explanation of how R^2 statistic relates to the pooling metric.

### Group pooling

This is the calculation done by `pooling()` if `type = "groups"` (default). In a partial pooling model (see baggr and above), 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 samples from the distribution of P (rather than summarised values), set `summary=FALSE`.

### Study weights

Contributions of each group (e.g. each study) to the mean meta-analysis estimate can be calculated by calculating for each study w_k the inverse of sum of group-specific SE squared and between-study variation. To obtain weights, this vector (across all studies) has to be normalised to 1, i.e. w_k/sum(w_k) for each k.

SE is typically treated as a fixed quantity (and usually reported on the reported point estimate), but between-study variance is a model parameter, hence the weights themselves are also random variables.

### Overall pooling in the model

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)`

Formulae for the calculations below are provided in main package vignette and almost analogous to the group calculation above, but using mean variance across all studies. In other words, pooling P is simply ratio of the expected within-study variance term to total variance.

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.

### 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.

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).

baggr documentation built on March 18, 2022, 7:02 p.m.