bayesmeta (version 3.4)

ess: Effective sample size (ESS)

Description

This function computes the effective sample size (ESS) of a posterior predictive distribution.

Usage

ess(object, ...)
  # S3 method for bayesmeta
ess(object, uisd, method=c("elir", "vr", "pr", "mtm.pt"), ...)

Value

The effective sample size (ESS).

Arguments

object

a bayesmeta object.

uisd

the unit infomation standard deviation (a single numerical value, or a function of the parameter (\(\mu\))).

method

a character string specifying the method to be used for ESS computation. By default, the expected local-information-ratio ESS (\(ESS_{ELIR}\)) is returned.

...

additional arguments

Author

Christian Roever

Details

The information conveyed by a prior distribution may often be quantified in terms of an effective sample size (ESS). Meta-analyses are commonly utilized to summarize “historical” information in order to inform a future study, leading to a meta-analytic-predictive (MAP) prior (Schmidli et al., 2014). In the context of the normal-normal hierarchical model (NNHM), the MAP prior results as the (posterior) predictive distribution for a “new” study mean \(\theta_{k+1}\). This function computes the ESS for the posterior predictive distribution based on a bayesmeta object.

Within the NNHM, the notion of an effective sample size requires the specification of a unit information standard deviation (UISD) (Roever et al., 2020); see also the ‘uisd()’ function's help page. The UISD \(\sigma_\mathrm{u}\) here determines the Fisher information for one information unit, effectively assuming that a study's sample size \(n_i\) and standard error \(\sigma_i\) are related simply as $$\sigma_i=\frac{\sigma_\mathrm{u}}{\sqrt{n_i}},$$ i.e., the squared standard error is inversely proportional to the sample size. For the (possibly hypothetical) case of a sample size of \(n_i=1\), the standard error then is equal to the UISD \(\sigma_\mathrm{u}\).

Specifying the UISD as a constant is often an approximation, sometimes it is also possible to specify the UISD as a function of the parameter (\(\mu\)). For example, in case the outcome in the meta-analyses are log-odds, then the UISD varies with the (log-) odds and is given by \(2\,\mathrm{cosh}(\mu/2)\) (see also the example below).

The ESS may be computed or approximated in several ways. Possible choices here are:

• "elir": the expected local-information-ratio (ELIR) method (the default),

• "vr": the variance ratio (VR) method,

• "pr": the precision ratio (PR) method,

• "mtm.pt": the Morita-Thall-Mueller / Pennello-Thompson (MTM.PM) method.

For more details on these see also Neuenschwander et al. (2020).

References

B. Neuenschwander, S. Weber, H. Schmidli, A. O'Hagan. Predictively consistent prior effective sample sizes. Biometrics, 76(2):578-587, 2020. tools:::Rd_expr_doi("10.1111/biom.13252").

H. Schmidli, S. Gsteiger, S. Roychoudhury, A. O'Hagan, D. Spiegelhalter, B. Neuenschwander. Robust meta-analytic-predictive priors in clinical trials with historical control information. Biometrics, 70(4):1023-1032, 2014. tools:::Rd_expr_doi("10.1111/biom.12242").

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. Research Synthesis Methods, 12(4):448-474, 2021. tools:::Rd_expr_doi("10.1002/jrsm.1475").

See Also

bayesmeta, uisd.

Examples

# load data set:
data("BaetenEtAl2013")
print(BaetenEtAl2013)

if (FALSE) {
# compute effect sizes (logarithmic odds) from the count data:
as <- escalc(xi=events, ni=total, slab=study,
             measure="PLO", data=BaetenEtAl2013)

# estimate the unit information standard deviation (UISD):
uisd(as, individual=TRUE)
uisd(as)  # = 2.35

# perform meta-analysis
# (using uniform priors for effect and heterogeneity):
bm <- bayesmeta(as)

# show forest plot:
forestplot(bm, zero=NA, xlab="log-odds")

# compute ESS_ELIR (based on fixed UISD):
ess(bm, uisd=2.35)  # = 45.7 patients

# compute ESS_ELIR based on UISD as a function of the log-odds:
uisdLogOdds <- function(logodds)
{
  return(2 * cosh(logodds / 2))
}

# Note: in the present example, probabilities are
# at approximately 0.25, corresponding to odds of 1:3.
uisdLogOdds(log(1/3))
# The UISD value of 2.31 roughly matches the above empirical figure.

ess(bm, uisd=uisdLogOdds)  # = 43.4 patients
}