NPP (version 0.6.0)

BerMNPP_MCMC2: MCMC Sampling for Bernoulli Population of multiple historical data using Normalized Power Prior

Description

Multiple historical data are combined individually. The NPP of multiple historical data is the product of the NPP of each historical data. Conduct posterior sampling for Bernoulli 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 \(p\), Gibbs sampling is used.

Usage

BerMNPP_MCMC2(n0, y0, n, y, prior_p, prior_delta_alpha, prior_delta_beta,
              prop_delta_alpha, prop_delta_beta, delta_ini, prop_delta,
              rw_delta, nsample, burnin, thin)

Value

A list of class "NPP" with three elements:

acceptrate

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

p

posterior of the model parameter \(p\).

delta

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

Arguments

n0

a non-negative integer vector: number of trials in historical data.

y0

a non-negative integer vector: number of successes in historical data.

n

a non-negative integer: number of trials in the current data.

y

a non-negative integer: number of successes in the current data.

prior_p

a vector of the hyperparameters in the prior distribution \(Beta(\alpha, \beta)\) for \(p\).

prior_delta_alpha

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

prior_delta_beta

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

prop_delta_alpha

a vector of the hyperparameter \(\alpha\) in the proposal distribution \(Beta(\alpha, \beta)\) for each \(\delta\).

prop_delta_beta

a vector of the hyperparameter \(\beta\) in the proposal distribution \(Beta(\alpha, \beta)\) for each \(\delta\).

delta_ini

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

prop_delta

the class of proposal distribution for \(\delta\).

rw_delta

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

nsample

specifies the number of posterior samples in the output.

burnin

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

thin

the thinning parameter in MCMC sampling.

Author

Qiang Zhang

Details

The outputs include posteriors of the model parameter(s) and power parameter, acceptance rate in sampling \(\delta\). The normalized power prior distribution is $$\pi_0(\delta)\prod_{k=1}^{K}\frac{\pi_0(\theta)L(\theta|D_{0k})^{\delta_{k}}}{\int \pi_0(\theta)L(\theta|D_{0k})^{\delta_{k}} d\theta}.$$

Here \(\pi_0(\delta)\) and \(\pi_0(\theta)\) are the initial prior distributions of \(\delta\) and \(\theta\), respectively. \(L(\theta|D_{0k})\) is the likelihood function of historical data \(D_{0k}\), and \(\delta_k\) is the corresponding power parameter.

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

BerMNPP_MCMC1; BerOMNPP_MCMC1; BerOMNPP_MCMC2

Examples

BerMNPP_MCMC2(n0 = c(275, 287), y0 = c(92, 125), n = 39, y = 17,
              prior_p=c(1/2,1/2), prior_delta_alpha=c(1/2,1/2),
              prior_delta_beta=c(1/2,1/2), prop_delta_alpha=c(1,1)/2,
              prop_delta_beta=c(1,1)/2, delta_ini=NULL, prop_delta="IND",
              nsample = 2000, burnin = 500, thin = 2)