ess {bayesmeta}

Effective sample size (ESS)

Description

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

Usage

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

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

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).

Value

The effective sample size (ESS).

Author(s)

Christian Roever [e-mail]

References

B. Neuenschwander, S. Weber, H. Schmidli, A. O'Hagan. Predictively consistent prior effective sample sizes. Biometrics, 76(2):578-587, 2020. 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. 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. doi:10.1002/jrsm.1475.

See Also

bayesmeta, uisd.

Examples

# 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(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

## End(Not run)