View source: R/pooling_metrics.R
pooling  R Documentation 
Compute statistics relating to
pooling
in a given baggr metaanalysis model returns statistics, for
either the entire model or individual groups, such as
pooling statistic by Gelman & Pardoe (2006), Isquared, Hsquared, or study weights;
heterogeneity
is a shorthand for pooling(type = "total")
weights
is shorthand for pooling(metric = "weights")
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, ...)
bg 
a baggr model 
metric 

type 
In 
summary 
logical; if 
object 
baggr model for which to calculate group (study) weights 
... 
Unused, please ignore. 
Pooling statistic (Gelman & Pardoe, 2006) describes the extent to which
grouplevel estimates of treatment
effect are "pooled" toward average treatment effect in the metaanalysis 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 1P, called I^2 (see Higgins and Thompson, 2002; sometimes another statistic, H^2 = 1 / P, is used). Higher values of Isquared indicate higher heterogeneity; Von Hippel (2015) provides useful details for Isquared 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.
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
hyperSD 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
.
Contributions of each group (e.g. each study) to the mean metaanalysis estimate can be calculated by calculating for each study w_k the inverse of sum of groupspecific SE squared and betweenstudy 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 betweenstudy variance is a model parameter, hence the weights themselves are also random variables.
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 withinstudy variance term to total variance.
To obtain such single estimate we need to substitute average variability of groupspecific 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 groupspecific estimates, P is a Bayesian parameter and its dispersion can be high.
Matrix with mean and intervals for chosen pooling metric, each row corresponding to one metaanalysis group.
Gelman, Andrew, and Iain Pardoe. "Bayesian Measures of Explained Variance and Pooling in Multilevel (Hierarchical) Models." Technometrics 48, no. 2 (May 2006): 24151.
Higgins, Julian P. T., and Simon G. Thompson. "Quantifying Heterogeneity in a MetaAnalysis." Statistics in Medicine, vol. 21, no. 11, June 2002, pp. 153958.
Hippel, Paul T von. "The Heterogeneity Statistic I2 Can Be Biased in Small MetaAnalyses." BMC Medical Research Methodology 15 (April 14, 2015).
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