bayesmeta (version 2.6)

ess.elir: Expected local-information-ratio ESS

Description

This function computes \(ESS_{ELIR}\), the expected local-information-ratio ESS (effective sample size) of a posterior predictive distribution.

Usage

ess.elir(object, uisd)

Arguments

object

a bayesmeta object.

uisd

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

Value

The expected local-information-ratio ESS (\(ESS_{ELIR}\)).

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 expected local-information-ratio ESS (\(ESS_{ELIR}\)), which was proposed by Neuenschwander et al. (2020), 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 (\(\theta\)). 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 \(2\,\mathrm{cosh}(\theta/2)\) (see also the example below).

References

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

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.

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.

See Also

bayesmeta, uisd.

Examples

# NOT RUN {
# load data set:
data("BaetenEtAl2013")
print(BaetenEtAl2013)

# }
# NOT RUN {
# 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.elir(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.elir(bm, uisd=uisdLogOdds)  # = 43.4 patients
# }