RBesT (version 1.4-0)

ess: Effective Sample Size for a Conjugate Prior

Description

Calculates the Effective Sample Size (ESS) for a mixture prior. The ESS indicates how many experimental units the prior is roughly equivalent to.

Usage

ess(mix, method = c("elir", "moment", "morita"), ...)

Arguments

mix

Prior (mixture of conjugate distributions).

method

Selects the used method. Can be either elir (default), moment or morita.

...

Optional arguments applicable to specific methods.

Supported Conjugate Prior-Likelihood Pairs

Prior/Posterior	Likelihood	Predictive	Summaries
Beta	Binomial	Beta-Binomial	n, r
Normal	Normal (fixed \(\sigma\))	Normal	n, m, se
Gamma	Poisson	Gamma-Poisson	n, m

Details

The ESS is calculated using either the expected local information ratio (elir) Neuenschwander et al. (submitted), the moments approach or the method by Morita et al. (2008). The moments based method is the default method and provides conservative estimates of the ESS.

The elir approach is the only ESS which fulfills predictive consistency. The predictive consistency of the ESS requires that the ESS of a prior is the same as averaging the posterior ESS after a fixed amount of events over the prior predictive distribution from which the number of forward simulated events is subtracted. The elir approach results in ESS estimates which are neither conservative nor liberal. See the example section for a demonstration of predictive consistency.

For the moments method the mean and standard deviation of the mixture are calculated and then approximated by the conjugate distribution with the same mean and standard deviation. For conjugate distributions, the ESS is well defined. See the examples for a step-wise calculation in the beta mixture case.

The Morita method used here evaluates the mixture prior at the mode instead of the mean as proposed originally by Morita. The method may lead to very optimistic ESS values, especially if the mixture contains many components.

References

Morita S, Thall PF, Mueller P. Determining the effective sample size of a parametric prior. Biometrics 2008;64(2):595-602.

Neuenschwander B, Weber S, Schmidli H, O'Hagen A. Predictively Consistent Prior Effective Sample Sizes. submitted

Examples

# NOT RUN {
# Conjugate Beta example
a <- 5
b <- 15
prior <- mixbeta(c(1, a, b))

ess(prior)
(a+b)

# Beta mixture example
bmix <- mixbeta(rob=c(0.2, 1, 1), inf=c(0.8, 10, 2))

ess(bmix, "elir")

ess(bmix, "moment")
# moments method is equivalent to
# first calculate moments
bmix_sum <- summary(bmix)
# then calculate a and b of a matching beta
ab_matched <- ms2beta(bmix_sum["mean"], bmix_sum["sd"])
# finally take the sum of a and b which are equivalent
# to number of responders/non-responders respectivley
round(sum(ab_matched))

ess(bmix, method="morita")

# Predictive consistency of elir
# }
# NOT RUN {
n_forward <- 1E2
bmixPred <- preddist(bmix, n=n_forward)
pred_samp <- rmix(bmixPred, 1E3)
pred_ess <- sapply(pred_samp, function(r) ess(postmix(bmix, r=r, n=n_forward), "elir") )
ess(bmix, "elir")
mean(pred_ess) - n_forward
# }
# NOT RUN {
# Normal mixture example
nmix <- mixnorm(rob=c(0.5, 0, 2), inf=c(0.5, 3, 4), sigma=10)

ess(nmix, "elir")

ess(nmix, "moment")

## the reference scale determines the ESS
sigma(nmix) <- 20
ess(nmix)

# Gamma mixture example
gmix <- mixgamma(rob=c(0.3, 20, 4), inf=c(0.7, 50, 10))

ess(gmix) ## interpreted as appropriate for a Poisson likelihood (default)

likelihood(gmix) <- "exp"
ess(gmix) ## interpreted as appropriate for an exponential likelihood


# }