MultinomialNPP() R function from [NPP]

MCMC Sampling for Multinomial Population using Normalized Power Prior

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.


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.

Returns

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.

Examples


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))

Author(s)

Tianyu Bai tianyu.bai24@gmail.com

Zifei Han hanzifei1@gmail.com

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.

MultinomialNPP function