# uisd: Unit information standard deviation In bayesmeta: Bayesian Random-Effects Meta-Analysis

## Description

This function estimates the unit information standard deviation (UISD) from a given set of standard errors and associated sample sizes.

## Usage

 ```1 2 3 4 5``` ``` uisd(n, ...) ## Default S3 method: uisd(n, sigma, sigma2=sigma^2, labels=NULL, individual=FALSE, ...) ## S3 method for class 'escalc' uisd(n, ...) ```

## Arguments

 `n` vector of sample sizes or an `escalc` object. `sigma` vector of standard errors associated with `n`. `sigma2` vector of squared standard errors (variances) associated with `n`. `labels` (optional) a vector of labels corresponding to `n` and `sigma`. `individual` a `logical` flag indicating whether individual (study-specific) UISDs are to be returned. `...` other `uisd` arguments.

## Details

The unit information standard deviation (UISD) reflects the “within-study” variability, which, depending on the effect measure considered, sometimes is a somewhat heuristic notion (Roever et al., 2020). For a single study, presuming that standard errors result as

sigma[i] = sigma[u] / sqrt(n[i]),

where sigma[u] is the within-study (population) standard deviation, the UISD simply results as

sigma[u] = sqrt(n[i] * sigma[i]^2).

This is often appropriate when assuming an (approximately) normal likelihood.

Assuming a constant sigma[u] value across studies, this figure then may be estimated by

s[u] = sqrt(mean(n) * hmean(sigma^2)) = sqrt(sum(n)/sum(sigma^-2)),

where mean(n) is the average (arithmetic mean) of the studies' sample sizes, and hmean(sigma^2) is the harmonic mean of the squared standard errors (variances).

The estimator s[u] is motivated via meta-analysis using the normal-normal hierarchical model (NNHM). In the special case of homogeneity (zero heterogeneity, tau=0), the overall mean estimate has standard error

sqrt(1/sum(sigma^(-2))).

Since this estimate corresponds to complete pooling, the standard error may also be expressed via the UISD as

sigma[u] / sqrt(sum(n)).

Equating both above standard error expressions yields s[u] as an estimator of the UISD sigma[u] (Roever et al, 2020).

## Value

Either a (single) estimate of the UISD, or, if `individual` was set to ‘`TRUE`’, a (potentially named) vector of UISDs for each individual study.

## Author(s)

Christian Roever christian.roever@med.uni-goettingen.de

## References

C. Roever, R. Bender, S. Dias, C.H. Schmid, H. Schmidli, S. Sturtz, S. Weber, T. Friede. On weakly informative prior distributions for the heterogeneity parameter in Bayesian random-effects meta-analysis. arXiv preprint 2007.08352 (submitted for publication), 2020.

`escalc`.
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19``` ```# load data set: data("CrinsEtAl2014") # compute logarithmic odds ratios (log-ORs): CrinsAR <- escalc(measure="OR", ai=exp.AR.events, n1i=exp.total, ci=cont.AR.events, n2i=cont.total, slab=publication, data=CrinsEtAl2014) # estimate the UISD: uisd(n = CrinsAR\$exp.total + CrinsAR\$cont.total, sigma = sqrt(CrinsAR\$vi), label = CrinsAR\$publication) # for an "escalc" object, one may also apply the function directly: uisd(CrinsAR) # compute study-specific UISDs: uisd(CrinsAR, individual=TRUE) ```