NPP (version 0.1.0)

MultinomialNPP_MCMC: MCMC Sampling for Multinomial Population using Normalized Power Prior

Description

Conduct posterior sampling for multinomial population with normalized power prior. For the power parameter \(\delta\), a Metropolis-Hastings algorithm with either independence proposal, or a random walk proposal on its logit scale is used. For the model parameter vector \(\theta\), Gibbs sampling is used. Assume the prior for model parameter \(\theta\) comes from a Dirichlet distribution.

Usage

MultinomialNPP_MCMC(Data.Cur = c(10, 10, 10), Data.Hist = c(10, 10, 10),
                    CompStat = list(n0 = NULL, n1 = NULL),
                    prior = list(theta.dir = c(0.5, 0.5, 0.5),
                                 delta.alpha = 1, delta.beta = 1),
                    MCMCmethod = 'IND', rw.logit.delta = 0.1,
                    ind.delta.alpha = 1, ind.delta.beta = 1, nsample = 5000,
                    control.mcmc = list(delta.ini = NULL, burnin = 0, thin = 1))

Arguments

Data.Cur

a non-negative integer vector of \(K\) elements: c(number of success in group 1, number of success in group 2, ..., number of success in group \(K\)) in the current data.

Data.Hist

a non-negative integer vector of \(K\) elements: c(number of success in group 1, number of success in group 2, ..., number of success in group \(K\)) in the historical data.

CompStat

a list of two elements that represents the "compatibility(sufficient) statistics" for \(\theta\). Default is NULL so the fitting will be based on the data. If the CompStat is provided then the inputs in Data.Cur and Data.Hist will be ignored. Note: in multinomial case providing CompStat is equivalent to provide the data summary as in Data.Cur and Data.Cur.

n0 is a non-negative integer vector of \(K\) elements for compatible statistics in historical data: c(number of success in group 1, number of success in group 2, ..., number of success in group \(K\)).

n1 is a non-negative integer vector of \(K\) elements for compatible statistics in current data: c(number of success in group 1, number of success in group 2, ..., number of success in group \(K\)).

prior

a list of the hyperparameters in the prior for both \(p\) and \(\delta\).

theta.dir is a vector of \(K\) elements of the hyperparameter \(\alpha\) in the prior distribution \(Dir(\alpha[1],\alpha[2],...,\alpha[K])\) for \(\theta\).

delta.alpha a scalar, the hyperparameter \(\alpha\) in the prior distribution \(Beta(\alpha, \beta)\) for \(\delta\).

delta.beta a scalar, the hyperparameter \(\beta\) in the prior distribution \(Beta(\alpha, \beta)\) for \(\delta\).

MCMCmethod

sampling method for \(\delta\) in MCMC. It can be either 'IND' for independence proposal; or 'RW' for random walk proposal on logit scale.

rw.logit.delta

the stepsize(variance of the normal distribution) for the random walk proposal of logit \(\delta\). Only applicable if MCMCmethod = 'RW'.

ind.delta.alpha

specifies the first parameter \(\alpha\) when independent proposal \(Beta(\alpha, \beta)\) for \(\delta\) is used. Only applicable if MCMCmethod = 'IND'

ind.delta.beta

specifies the first parameter \(\beta\) when independent proposal \(Beta(\alpha, \beta)\) for \(\delta\) is used. Only applicable if MCMCmethod = 'IND'

nsample

specifies the number of posterior samples in the output.

control.mcmc

a list of three elements used in posterior sampling.

delta.ini is the initial value of \(\delta\) in MCMC sampling.

burnin is the number of burn-ins. The output will only show MCMC samples after bunrin.

thin is the thinning parameter in MCMC sampling.

Value

A list of class "NPP" with four elements:

p

posterior of the model parameter \(\theta\).

delta

posterior of the power parameter \(\delta\).

acceptance

the acceptance rate in MCMC sampling for \(\delta\) using Metropolis-Hastings algorithm.

DIC

the deviance information criteria for model diagnostics.

Details

The outputs include posteriors of the model parameter(s) and power parameter, acceptance rate in sampling \(\delta\), and the deviance information criteria.

References

Ibrahim, J.G., Chen, M.-H., Gwon, Y. and Chen, F. (2015). The Power Prior: Theory and Applications. Statistics in Medicine 34:3724-3749.

Duan, Y., Ye, K. and Smith, E.P. (2006). Evaluating Water Quality: Using Power Priors to Incorporate Historical Information. Environmetrics 17:95-106.

See Also

BerNPP_MCMC; NormalNPP_MCMC; PoissonNPP_MCMC

Examples

# NOT RUN {
MultinomialNPP_MCMC(Data.Cur = c(3,11,3,669), Data.Hist = c(9,20,9,473),
                    prior = list(theta.dir = c(1,1,1,1),
                    delta.alpha = 1, delta.beta = 1),
                    MCMCmethod = 'IND', rw.logit.delta = 0.1,
                    ind.delta.alpha = 1, ind.delta.beta = 1, nsample = 10000,
                    control.mcmc = list(delta.ini = NULL,
                    burnin = 2000, thin = 5))
# }