MCMC Sampling for Bernoulli Population with Multiple Historical Data using Normalized Power Prior

Description

Incorporate multiple historical data sets for posterior sampling of a Bernoulli population using the normalized power prior. The Metropolis-Hastings algorithm, with either an independence proposal or a random walk proposal on the logit scale, is applied for the power parameter \delta. Gibbs sampling is utilized for the model parameter p.

Usage

  BerMNPP_MCMC1(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)

Arguments

Details

The outputs include posteriors of the model parameter(s) and power parameter, acceptance rate in sampling \delta. The normalized power prior distribution is

\frac{\pi_0(\delta)\pi_0(\theta)\prod_{k=1}^{K}L(\theta|D_{0k})^{\delta_{k}}}{\int \pi_0(\theta)\prod_{k=1}^{K}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.

Value

A list of class "NPP" comprising:

Author(s)

Qiang Zhang zqzjf0408@163.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.

See Also

BerMNPP_MCMC2; BerOMNPP_MCMC1; BerOMNPP_MCMC2

Examples

BerMNPP_MCMC1(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)

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)

Arguments

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.

Value

A list of class "NPP" with three elements:

Author(s)

Qiang Zhang zqzjf0408@163.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.

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)

MCMC Sampling for Bernoulli Population using Normalized Power Prior

Description

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

BerNPP_MCMC(Data.Cur = c(100, 50), Data.Hist = c(100, 50),
            CompStat = list(n0 = NULL, y0 = NULL, n1 = NULL, y1 = NULL),
            prior = list(p.alpha = 1, p.beta = 1, 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

Details

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

Value

A list of class "NPP" with four elements:

Author(s)

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.

See Also

MultinomialNPP_MCMC; NormalNPP_MCMC; PoissonNPP_MCMC

Examples

BerNPP_MCMC(Data.Cur = c(493, 473), Data.Hist = c(680, 669),
            prior = list(p.alpha = 0.5, p.beta = 0.5, delta.alpha = 1, delta.beta = 1),
            MCMCmethod = 'RW', rw.logit.delta = 1, nsample = 5000,
            control.mcmc = list(delta.ini = NULL, burnin = 2000, thin = 5))

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

Description

Multiple ordered historical data are incorporated together. Conduct posterior sampling for Bernoulli population with normalized power prior. For the power parameter \gamma, a Metropolis-Hastings algorithm with independence proposal is used. For the model parameter p, Gibbs sampling is used.

Usage

    BerOMNPP_MCMC1(n0, y0, n, y, prior_gamma, prior_p, gamma_ind_prop,
                   gamma_ini, nsample, burnin, thin, adjust = FALSE)

Arguments

Details

The outputs include posteriors of the model parameter(s) and power parameter, acceptance rate in sampling \gamma. The normalized power prior distribution is given by:

\frac{\pi_0(\gamma)\pi_0(\theta)\prod_{k=1}^{K}L(\theta|D_{0k})^{(\sum_{i=1}^{k}\gamma_i)}}{\int \pi_0(\theta)\prod_{k=1}^{K}L(\theta|D_{0k})^{(\sum_{i=1}^{k}\gamma_i)}d\theta}.

Here, \pi_0(\gamma) and \pi_0(\theta) are the initial prior distributions of \gamma and \theta, respectively. L(\theta|D_{0k}) is the likelihood function of historical data D_{0k}, and \sum_{i=1}^{k}\gamma_i is the corresponding power parameter.

Value

A list of class "NPP" with three elements:

Author(s)

Qiang Zhang zqzjf0408@163.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.

See Also

BerMNPP_MCMC1, BerMNPP_MCMC2, BerOMNPP_MCMC2

Examples

BerOMNPP_MCMC1(n0 = c(275, 287), y0 = c(92, 125), n = 39, y = 17, prior_gamma=c(1,1,1)/3,
               prior_p=c(1/2,1/2), gamma_ind_prop=rep(1,3)/2, gamma_ini=NULL,
               nsample = 2000, burnin = 500, thin = 2, adjust = FALSE)

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

Description

Multiple ordered historical data are combined individually. Conduct posterior sampling for Bernoulli population with normalized power prior. For the power parameter \gamma, a Metropolis-Hastings algorithm with independence proposal is used. For the model parameter p, Gibbs sampling is used.

Usage

BerOMNPP_MCMC2(n0, y0, n, y, prior_gamma, prior_p, gamma_ind_prop, gamma_ini,
               nsample, burnin, thin, adjust = FALSE)

Arguments

Details

The outputs include posteriors of the model parameter(s) and power parameter, acceptance rate in sampling \gamma. The normalized power prior distribution is

\pi_0(\gamma)\prod_{k=1}^{K}\frac{\pi_0(\theta)L(\theta|D_{0k})^{(\sum_{i=1}^{k}\gamma_i)}}{\int \pi_0(\theta)L(\theta|D_{0k})^{(\sum_{i=1}^{k}\gamma_i)} d\theta}.

Here \pi_0(\gamma) and \pi_0(\theta) are the initial prior distributions of \gamma and \theta, respectively. L(\theta|D_{0k}) is the likelihood function of historical data D_{0k}, and \sum_{i=1}^{k}\gamma_i is the corresponding power parameter.

Value

A list of class "NPP" with three elements:

Author(s)

Qiang Zhang zqzjf0408@163.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.

See Also

BerMNPP_MCMC1; BerMNPP_MCMC2; BerOMNPP_MCMC1

Examples

BerOMNPP_MCMC2(n0 = c(275, 287), y0 = c(92, 125), n = 39, y = 17, prior_gamma=c(1,1,1)/3,
               prior_p=c(1/2,1/2),  gamma_ind_prop=rep(1,3)/2, gamma_ini=NULL,
               nsample = 2000, burnin = 500, thin = 2, adjust = FALSE)

MCMC Sampling for Linear Regression Model of multiple historical data using Normalized Power Prior

Description

Multiple historical data are incorporated together. Conduct posterior sampling for Linear Regression Model 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 parameters (\beta, \sigma^2), Gibbs sampling is used.

Usage

LMMNPP_MCMC1(D0, X, Y, a0, b, mu0, R, delta_ini, prop_delta,
             prior_delta_alpha, prior_delta_beta, prop_delta_alpha,
             prop_delta_beta, rw_delta, nsample, burnin, thin)

Arguments

Details

The outputs include posteriors of the model parameters and power parameter, acceptance rate in sampling \delta. Let \theta=(\beta, \sigma^2), the normalized power prior distribution is

\frac{\pi_0(\delta)\pi_0(\theta)\prod_{k=1}^{K}L(\theta|D_{0k})^{\delta_k}}{\int \pi_0(\theta)\prod_{k=1}^{K}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.

Value

A list of class "NPP" with four elements:

Author(s)

Qiang Zhang zqzjf0408@163.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.

See Also

LMMNPP_MCMC2; LMOMNPP_MCMC1; LMOMNPP_MCMC2

Examples

## Not run: 
set.seed(1234)
sigsq0 = 1

n01 = 100
theta01 = c(0, 1, 1)
X01 = cbind(1, rnorm(n01, mean=0, sd=1), runif(n01, min=-1, max=1))
Y01 = X01%*%as.vector(theta01) + rnorm(n01, mean=0, sd=sqrt(sigsq0))
D01 = cbind(X01, Y01)

n02 = 70
theta02 = c(0, 2, 3)
X02 = cbind(1, rnorm(n02, mean=0, sd=1), runif(n02, min=-1, max=1))
Y02 = X02%*%as.vector(theta02) + rnorm(n02, mean=0, sd=sqrt(sigsq0))
D02 = cbind(X02, Y02)

n03 = 50
theta03 = c(0, 3, 5)
X03 = cbind(1, rnorm(n03, mean=0, sd=1), runif(n03, min=-1, max=1))
Y03 = X03%*%as.vector(theta03) + rnorm(n03, mean=0, sd=sqrt(sigsq0))
D03 = cbind(X03, Y03)

D0 = list(D01, D02, D03)
n0 = c(n01, n02, n03)

n = 100
theta = c(0, 3, 5)
X = cbind(1, rnorm(n, mean=0, sd=1), runif(n, min=-1, max=1))
Y = X%*%as.vector(theta) + rnorm(n, mean=0, sd=sqrt(sigsq0))

LMMNPP_MCMC1(D0=D0, X=X, Y=Y, a0=2, b=2, mu0=c(0,0,0), R=diag(c(1/64,1/64,1/64)),
             delta_ini=NULL, prior_delta_alpha=c(1,1,1), prior_delta_beta=c(1,1,1),
             prop_delta_alpha=c(1,1,1), prop_delta_beta=c(1,1,1),
             prop_delta="RW", rw_delta=0.9, nsample=5000, burnin=1000, thin=3)

## End(Not run)

MCMC Sampling for Linear Regression Model 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 Linear Regression Model 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 parameters (\beta, \sigma^2), Gibbs sampling is used.

Usage

LMMNPP_MCMC2(D0, X, Y, a0, b, mu0, R, delta_ini, prop_delta,
             prior_delta_alpha, prior_delta_beta, prop_delta_alpha,
             prop_delta_beta, rw_delta, nsample, burnin, thin)

Arguments

Details

The outputs include posteriors of the model parameters and power parameter, acceptance rate in sampling \delta. Let \theta=(\beta, \sigma^2), 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.

Value

A list of class "NPP" with four elements:

Author(s)

Qiang Zhang zqzjf0408@163.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.

See Also

LMMNPP_MCMC1; LMOMNPP_MCMC1; LMOMNPP_MCMC2

Examples

## Not run: 
set.seed(1234)
sigsq0 = 1

n01 = 100
theta01 = c(0, 1, 1)
X01 = cbind(1, rnorm(n01, mean=0, sd=1), runif(n01, min=-1, max=1))
Y01 = X01%*%as.vector(theta01) + rnorm(n01, mean=0, sd=sqrt(sigsq0))
D01 = cbind(X01, Y01)

n02 = 70
theta02 = c(0, 2, 3)
X02 = cbind(1, rnorm(n02, mean=0, sd=1), runif(n02, min=-1, max=1))
Y02 = X02%*%as.vector(theta02) + rnorm(n02, mean=0, sd=sqrt(sigsq0))
D02 = cbind(X02, Y02)

n03 = 50
theta03 = c(0, 3, 5)
X03 = cbind(1, rnorm(n03, mean=0, sd=1), runif(n03, min=-1, max=1))
Y03 = X03%*%as.vector(theta03) + rnorm(n03, mean=0, sd=sqrt(sigsq0))
D03 = cbind(X03, Y03)

D0 = list(D01, D02, D03)
n0 = c(n01, n02, n03)

n = 100
theta = c(0, 3, 5)
X = cbind(1, rnorm(n, mean=0, sd=1), runif(n, min=-1, max=1))
Y = X%*%as.vector(theta) + rnorm(n, mean=0, sd=sqrt(sigsq0))

LMMNPP_MCMC2(D0=D0, X=X, Y=Y, a0=2, b=2, mu0=c(0,0,0), R=diag(c(1/64,1/64,1/64)),
             delta_ini=NULL, prior_delta_alpha=c(1,1,1), prior_delta_beta=c(1,1,1),
             prop_delta_alpha=c(1,1,1), prop_delta_beta=c(1,1,1),
             prop_delta="RW", rw_delta=0.9, nsample=5000, burnin=1000, thin=5)

## End(Not run)

MCMC Sampling for Normal Linear Model using Normalized Power Prior

Description

Conduct posterior sampling for normal linear model 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 regression parameter \beta and \sigma^2, Gibbs sampling is used.

Usage

LMNPP_MCMC(y.Cur, y.Hist, x.Cur = NULL, x.Hist = NULL,
           prior = list(a = 1.5, b = 0, mu0 = 0,
                   Rinv = matrix(1, nrow = 1), 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

Details

If b = 1, prior for (\beta, \sigma) is (1/\sigma^2)^a * N(mu0, \sigma^2 R^{-1}), which includes the g-prior. If b = 0, prior for (\beta, \sigma) is (1/\sigma^2)^a. The outputs include posteriors of the model parameter(s) and power parameter, acceptance rate when sampling \delta, and the deviance information criteria.

Value

A list of class "NPP" with five elements:

Author(s)

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.

Berger, J.O. and Bernardo, J.M. (1992). On the development of reference priors. Bayesian Statistics 4: Proceedings of the Fourth Valencia International Meeting, Bernardo, J.M, Berger, J.O., Dawid, A.P. and Smith, A.F.M. eds., 35-60, Clarendon Press:Oxford.

Jeffreys, H. (1946). An Invariant Form for the Prior Probability in Estimation Problems. Proceedings of the Royal Statistical Society of London, Series A 186:453-461.

See Also

BerNPP_MCMC; MultinomialNPP_MCMC; PoissonNPP_MCMC; NormalNPP_MCMC

Examples

set.seed(123)
x1 = runif(100, min = 0, max = 10)
x0 = runif(100, min = 0, max = 1)
y1 = 10+ 2*x1 + rnorm(100, mean = 0, sd = 1)
y0 = 10+ 1.5*x0 + rnorm(100, mean = 0, sd = 1)

RegPost = LMNPP_MCMC(y.Cur = y1, y.Hist = y0, x.Cur = x1, x.Hist = x0,
                     prior = list(a = 1.5, b = 0, mu0 = c(0, 0),
                                  Rinv = diag(100, nrow = 2),
                     delta.alpha = 1, delta.beta = 1), MCMCmethod = 'IND',
                     ind.delta.alpha= 1, ind.delta.beta= 1, nsample = 5000,
                     control.mcmc = list(delta.ini = NULL,
                                         burnin = 2000, thin = 2))

MCMC Sampling for Linear Regression Model of multiple historical data using Ordered Normalized Power Prior

Description

Multiple historical data are incorporated together. Conduct posterior sampling for Linear Regression Model with ordered normalized power prior. For the power parameter \gamma, a Metropolis-Hastings algorithm with independence proposal is used. For the model parameters (\beta, \sigma^2), Gibbs sampling is used.

Usage

LMOMNPP_MCMC1(D0, X, Y, a0, b, mu0, R, gamma_ini, prior_gamma,
              gamma_ind_prop, nsample, burnin, thin, adjust)

Arguments

Details

The outputs include posteriors of the model parameters and power parameter, acceptance rate in sampling \gamma. Let \theta=(\beta, \sigma^2), the normalized power prior distribution is

\frac{\pi_0(\gamma)\pi_0(\theta)\prod_{k=1}^{K}L(\theta|D_{0k})^(\sum_{i=1}^{k}\gamma_i)}{\int \pi_0(\theta)\prod_{k=1}^{K}L(\theta|D_{0k})^(\sum_{i=1}^{k}\gamma_i)\,d\theta}.

Here \pi_0(\gamma) and \pi_0(\theta) are the initial prior distributions of \gamma and \theta, respectively. L(\theta|D_{0k}) is the likelihood function of historical data D_{0k}, and \sum_{i=1}^{k}\gamma_i is the corresponding power parameter.

Value

A list of class "NPP" with four elements:

Author(s)

Qiang Zhang zqzjf0408@163.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.

See Also

LMMNPP_MCMC1; LMMNPP_MCMC2; LMOMNPP_MCMC2

Examples

## Not run: 
set.seed(1234)
sigsq0 = 1

n01 = 100
theta01 = c(0, 1, 1)
X01 = cbind(1, rnorm(n01, mean=0, sd=1), runif(n01, min=-1, max=1))
Y01 = X01%*%as.vector(theta01) + rnorm(n01, mean=0, sd=sqrt(sigsq0))
D01 = cbind(X01, Y01)

n02 = 70
theta02 = c(0, 2, 3)
X02 = cbind(1, rnorm(n02, mean=0, sd=1), runif(n02, min=-1, max=1))
Y02 = X02%*%as.vector(theta02) + rnorm(n02, mean=0, sd=sqrt(sigsq0))
D02 = cbind(X02, Y02)

n03 = 50
theta03 = c(0, 3, 5)
X03 = cbind(1, rnorm(n03, mean=0, sd=1), runif(n03, min=-1, max=1))
Y03 = X03%*%as.vector(theta03) + rnorm(n03, mean=0, sd=sqrt(sigsq0))
D03 = cbind(X03, Y03)

D0 = list(D01, D02, D03)
n0 = c(n01, n02, n03)

n = 100
theta = c(0, 3, 5)
X = cbind(1, rnorm(n, mean=0, sd=1), runif(n, min=-1, max=1))
Y = X%*%as.vector(theta) + rnorm(n, mean=0, sd=sqrt(sigsq0))

LMOMNPP_MCMC1(D0=D0, X=X, Y=Y, a0=2, b=2, mu0=c(0,0,0), R=diag(c(1/64,1/64,1/64)),
              gamma_ini=NULL, prior_gamma=rep(1/4,4), gamma_ind_prop=rep(1/4,4),
              nsample=5000, burnin=1000, thin=5, adjust=FALSE)

## End(Not run)

MCMC Sampling for Linear Regression Model of multiple historical data using Ordered 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 Linear Regression Model with ordered normalized power prior. For the power parameter \gamma, a Metropolis-Hastings algorithm with independence proposal is used. For the model parameters (\beta, \sigma^2), Gibbs sampling is used.

Usage

LMOMNPP_MCMC2(D0, X, Y, a0, b, mu0, R, gamma_ini, prior_gamma,
              gamma_ind_prop, nsample, burnin, thin, adjust)

Arguments

Details

The outputs include posteriors of the model parameters and power parameter, acceptance rate in sampling \gamma. Let \theta=(\beta, \sigma^2), the normalized power prior distribution is

\pi_0(\gamma)\prod_{k=1}^{K}\frac{\pi_0(\theta)L(\theta|D_{0k})^(\sum_{i=1}^{k}\gamma_i)}{\int \pi_0(\theta)L(\theta|D_{0k})^(\sum_{i=1}^{k}\gamma_i)\,d\theta}.

Here \pi_0(\gamma) and \pi_0(\theta) are the initial prior distributions of \gamma and \theta, respectively. L(\theta|D_{0k}) is the likelihood function of historical data D_{0k}, and \sum_{i=1}^{k}\gamma_i is the corresponding power parameter.

Value

A list of class "NPP" with four elements:

Author(s)

Qiang Zhang zqzjf0408@163.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.

See Also

LMMNPP_MCMC1; LMMNPP_MCMC2; LMOMNPP_MCMC1

Examples

## Not run: 
set.seed(1234)
sigsq0 = 1

n01 = 100
theta01 = c(0, 1, 1)
X01 = cbind(1, rnorm(n01, mean=0, sd=1), runif(n01, min=-1, max=1))
Y01 = X01%*%as.vector(theta01) + rnorm(n01, mean=0, sd=sqrt(sigsq0))
D01 = cbind(X01, Y01)

n02 = 70
theta02 = c(0, 2, 3)
X02 = cbind(1, rnorm(n02, mean=0, sd=1), runif(n02, min=-1, max=1))
Y02 = X02%*%as.vector(theta02) + rnorm(n02, mean=0, sd=sqrt(sigsq0))
D02 = cbind(X02, Y02)

n03 = 50
theta03 = c(0, 3, 5)
X03 = cbind(1, rnorm(n03, mean=0, sd=1), runif(n03, min=-1, max=1))
Y03 = X03%*%as.vector(theta03) + rnorm(n03, mean=0, sd=sqrt(sigsq0))
D03 = cbind(X03, Y03)

D0 = list(D01, D02, D03)
n0 = c(n01, n02, n03)

n = 100
theta = c(0, 3, 5)
X = cbind(1, rnorm(n, mean=0, sd=1), runif(n, min=-1, max=1))
Y = X%*%as.vector(theta) + rnorm(n, mean=0, sd=sqrt(sigsq0))

LMOMNPP_MCMC1(D0=D0, X=X, Y=Y, a0=2, b=2, mu0=c(0,0,0), R=diag(c(1/64,1/64,1/64)),
              gamma_ini=NULL, prior_gamma=rep(1/4,4), gamma_ind_prop=rep(1/4,4),
              nsample=5000, burnin=1000, thin=5, adjust=FALSE)

## End(Not run)

A Function to Calculate logC(\delta) Based on Laplace Approximation

Description

The function assumes that the prior of the model parameters is very flat that had very minor impact on the shape of the power prior (posterior based on the D0).

Usage

LaplacelogC(delta, loglikmle, detHessian, ntheta)

Arguments

Value

logC(\delta) based on the Laplace approximation. Can be used for the posterior sampling in the normalized power prior.

Author(s)

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.

See Also

logCknot

Calculate Posterior Mode of the Power Parameter in Normalized Power Prior with Grid Search, Bernoulli Population

Description

The function returns the posterior mode of the power parameter \delta in Bernoulli population. It calculates the log of the posterior density (up to a normalizing constant), and conduct a grid search to find the approximate mode.

Usage

ModeDeltaBerNPP(Data.Cur, Data.Hist,
                CompStat = list(n0 = NULL, y0 = NULL, n1 = NULL, y1 = NULL),
                npoints = 1000,
                prior = list(p.alpha = 1, p.beta = 1,
                             delta.alpha = 1, delta.beta = 1))

Arguments

Details

See example.

Value

A numeric value between 0 and 1.

Author(s)

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.

See Also

ModeDeltaNormalNPP; ModeDeltaPoisNPP; ModeDeltaMultinomialNPP

Examples

ModeDeltaBerNPP(Data.Cur = c(100, 40), Data.Hist = c(100, 40), npoints = 1000,
                prior = list(p.alpha = 1, p.beta = 1, delta.alpha = 1, delta.beta = 1))

ModeDeltaBerNPP(Data.Cur = c(100, 40), Data.Hist = c(100, 35), npoints = 1000,
                prior = list(p.alpha = 1, p.beta = 1, delta.alpha = 1, delta.beta = 1))

ModeDeltaBerNPP(Data.Cur = c(100, 40), Data.Hist = c(100, 50), npoints = 1000,
                prior = list(p.alpha = 1, p.beta = 1, delta.alpha = 1, delta.beta = 1))

Calculate Posterior Mode of the Power Parameter in Normalized Power Prior with Grid Search, Normal Linear Model

Description

The function returns the posterior mode of the power parameter \delta in normal linear model. It calculates the log of the posterior density (up to a normalizing constant), and conduct a grid search to find the approximate mode.

Usage

ModeDeltaLMNPP(y.Cur, y.Hist, x.Cur = NULL, x.Hist = NULL, npoints = 1000,
               prior = list(a = 1.5, b = 0, mu0 = 0, Rinv = matrix(1, nrow = 1),
                            delta.alpha = 1, delta.beta = 1))

Arguments

Details

If b = 1, prior for (\beta, \sigma) is (1/\sigma^2)^a * N(mu0, \sigma^2 R^{-1}), which includes the g-prior. If b = 0, prior for (\beta, \sigma) is (1/\sigma^2)^a. The outputs include posteriors of the model parameter(s) and power parameter, acceptance rate when sampling \delta, and the deviance information criteria.

Author(s)

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.

Berger, J.O. and Bernardo, J.M. (1992). On the development of reference priors. Bayesian Statistics 4: Proceedings of the Fourth Valencia International Meeting, Bernardo, J.M, Berger, J.O., Dawid, A.P. and Smith, A.F.M. eds., 35-60, Clarendon Press:Oxford.

Jeffreys, H. (1946). An Invariant Form for the Prior Probability in Estimation Problems. Proceedings of the Royal Statistical Society of London, Series A 186:453-461.

See Also

ModeDeltaBerNPP; ModeDeltaNormalNPP; ModeDeltaMultinomialNPP; ModeDeltaNormalNPP

Calculate Posterior Mode of the Power Parameter in Normalized Power Prior with Grid Search, Multinomial Population

Description

The function returns the posterior mode of the power parameter \delta in multinomial population. It calculates the log of the posterior density (up to a normalizing constant), and conduct a grid search to find the approximate mode.

Usage

ModeDeltaMultinomialNPP(Data.Cur, Data.Hist, CompStat = list(n0 = NULL, n1 = NULL),
                        npoints = 1000, prior = list(theta.dir.alpha = c(0.5, 0.5, 0.5),
                        delta.alpha = 1, delta.beta = 1))

Arguments

Details

See example.

Value

A numeric value between 0 and 1.

Author(s)

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.

See Also

ModeDeltaBerNPP; ModeDeltaNormalNPP; ModeDeltaPoisNPP

Examples

ModeDeltaMultinomialNPP(CompStat = list(n0 = c(25,25,25,25), n1 = c(25,25,25,25)),
                        prior = list(theta.dir.alpha = c(0.5, 0.5, 0.5, 0.5),
                                     delta.alpha = 1, delta.beta = 1))

ModeDeltaMultinomialNPP(CompStat = list(n0 = c(22,25,28,25), n1 = c(25,22,25,28)),
                        prior = list(theta.dir.alpha = c(0.5, 0.5, 0.5, 0.5),
                                     delta.alpha = 1, delta.beta = 1))

ModeDeltaMultinomialNPP(CompStat = list(n0 = c(15,25,30,30), n1 = c(25,25,25,25)),
                        prior = list(theta.dir.alpha = c(0.5, 0.5, 0.5, 0.5),
                                     delta.alpha = 1, delta.beta = 1))

Calculate Posterior Mode of the Power Parameter in Normalized Power Prior with Grid Search, Normal Population

Description

The function returns the posterior mode of the power parameter \delta in multinomial population. It calculates the log of the posterior density (up to a normalizing constant), and conduct a grid search to find the approximate mode.

Usage

ModeDeltaNormalNPP(Data.Cur, Data.Hist,
                   CompStat = list(n0 = NULL, mean0 = NULL, var0 = NULL,
                                   n1 = NULL, mean1 = NULL, var1 = NULL),
                   npoints = 1000,
                   prior = list(a = 1.5, delta.alpha = 1, delta.beta = 1))

Arguments

Details

See example.

Value

A numeric value between 0 and 1.

Author(s)

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.

Berger, J.O. and Bernardo, J.M. (1992). On the development of reference priors. Bayesian Statistics 4: Proceedings of the Fourth Valencia International Meeting, Bernardo, J.M, Berger, J.O., Dawid, A.P. and Smith, A.F.M. eds., 35-60, Clarendon Press:Oxford.

Jeffreys, H. (1946). An Invariant Form for the Prior Probability in Estimation Problems. Proceedings of the Royal Statistical Society of London, Series A 186:453-461.

See Also

ModeDeltaBerNPP; ModeDeltaMultinomialNPP; ModeDeltaPoisNPP

Examples

ModeDeltaNormalNPP(CompStat = list(n0 = 50, mean0 = 0, var0 = 1,
                                   n1 = 50, mean1 = 0, var1 = 1), npoints = 1000,
                   prior = list(a = 1.5, delta.alpha = 1, delta.beta = 1))

ModeDeltaNormalNPP(CompStat = list(n0 = 50, mean0 = 0, var0 = 1,
                                   n1 = 40, mean1 = 0.2, var1 = 1), npoints = 1000,
                   prior = list(a = 1.5, delta.alpha = 1, delta.beta = 1))

ModeDeltaNormalNPP(CompStat = list(n0 = 50, mean0 = 0, var0 = 1,
                                   n1 = 40, mean1 = 0.6, var1 = 1), npoints = 1000,
                   prior = list(a = 1.5, delta.alpha = 1, delta.beta = 1))

Calculate Posterior Mode of the Power Parameter in Normalized Power Prior with Grid Search, Poisson Population

Description

The function returns the posterior mode of the power parameter \delta in multinomial population. It calculates the log of the posterior density (up to a normalizing constant), and conduct a grid search to find the approximate mode.

Usage

ModeDeltaPoisNPP(Data.Cur, Data.Hist,
                 CompStat = list(n0 = NULL, mean0 = NULL, n1 = NULL, mean1 = NULL),
                 npoints = 1000, prior = list(lambda.shape = 1/2,
                 lambda.scale = 100, delta.alpha = 1, delta.beta = 1))

Arguments

Details

See example.

Value

A numeric value between 0 and 1.

Author(s)

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.

See Also

ModeDeltaBerNPP; ModeDeltaNormalNPP; ModeDeltaMultinomialNPP

Examples

ModeDeltaPoisNPP(CompStat = list(n0 = 50, mean0 = 10, n1 = 50, mean1 = 10), npoints = 1000,
                 prior = list(lambda.shape = 1/2, lambda.scale = 100,
                              delta.alpha = 1, delta.beta = 1))

ModeDeltaPoisNPP(CompStat = list(n0 = 50, mean0 = 10, n1 = 50, mean1 = 9.5), npoints = 1000,
                 prior = list(lambda.shape = 1/2, lambda.scale = 100,
                              delta.alpha = 1, delta.beta = 1))

ModeDeltaPoisNPP(CompStat = list(n0 = 50, mean0 = 10, n1 = 50, mean1 = 9), npoints = 1000,
                 prior = list(lambda.shape = 1/2, lambda.scale = 100,
                              delta.alpha = 1, delta.beta = 1))

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

Details

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

Value

A list of class "NPP" with four elements:

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.

See Also

BerNPP_MCMC; NormalNPP_MCMC; PoissonNPP_MCMC

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))

MCMC Sampling for Normal Population using Normalized Power Prior

Description

Conduct posterior sampling for normal population with normalized power prior. The initial prior \pi(\mu|\sigma^2) is a flat 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 \mu and \sigma^2, Gibbs sampling is used.

Usage

NormalNPP_MCMC(Data.Cur, Data.Hist,
               CompStat = list(n0 = NULL, mean0 = NULL, var0 = NULL,
                               n1 = NULL, mean1 = NULL, var1 = NULL),
               prior = list(a = 1.5, 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

Details

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

Value

A list of class "NPP" with five elements:

Author(s)

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.

Berger, J.O. and Bernardo, J.M. (1992). On the development of reference priors. Bayesian Statistics 4: Proceedings of the Fourth Valencia International Meeting, Bernardo, J.M, Berger, J.O., Dawid, A.P. and Smith, A.F.M. eds., 35-60, Clarendon Press:Oxford.

Jeffreys, H. (1946). An Invariant Form for the Prior Probability in Estimation Problems. Proceedings of the Royal Statistical Society of London, Series A 186:453-461.

See Also

BerNPP_MCMC; MultinomialNPP_MCMC; PoissonNPP_MCMC;

Examples

set.seed(1234)
NormalData0 <- rnorm(n = 100, mean= 20, sd = 1)

set.seed(12345)
NormalData1 <- rnorm(n = 50, mean= 30, sd = 1)

NormalNPP_MCMC(Data.Cur = NormalData1, Data.Hist = NormalData0,
               CompStat = list(n0 = 100, mean0 = 10, var0 = 1,
               n1 = 100, mean1 = 10, var1 = 1),
               prior = list(a = 1.5, delta.alpha = 1, delta.beta = 1),
               MCMCmethod = 'RW', rw.logit.delta = 1,
               ind.delta.alpha= 1, ind.delta.beta= 1, nsample = 10000,
               control.mcmc = list(delta.ini = NULL, burnin = 0, thin = 1))

PH Data on four sites in Virginia

Description

The dataset is used to assess if there is site impairment. The site impairment is defined as whether the pH values at a site indicate that the site violates a (lower) standard of 6.0 more than 10% of the time.

Usage

data("PHData")

Format

A data frame with 325 observations on the following 3 variables.

Station

the site number, labeled as 1 to 4

Data.Time

indicator of historical data (coded as 0) or current data (coded as 1)

PH

value of PH on the site

Examples

data(PHData)

MCMC Sampling for Poisson Population using Normalized Power Prior with Multiple Historical Data

Description

This function incorporates multiple sets of historical data for posterior sampling in a Poisson population using a normalized power prior. The power parameter \delta uses a Metropolis-Hastings algorithm, which can be either an independence proposal or a random walk proposal on its logit scale. For the model parameter \lambda, Gibbs sampling is employed.

Usage

PoiMNPP_MCMC1(n0, n, prior_lambda, prop_delta, prior_delta_alpha,
              prior_delta_beta, rw_delta, delta_ini, nsample, burnin, thin)

Arguments

Details

The function returns posteriors for both the model and power parameters, as well as the acceptance rate for sampling \delta. The normalized power prior distribution is given by:

\frac{\pi_0(\delta)\pi_0(\lambda)\prod_{k=1}^{K}L(\lambda|D_{0k})^{\delta_{k}}}{\int \pi_0(\lambda)\prod_{k=1}^{K}L(\lambda|D_{0k})^{\delta_{k}} d\lambda}.

Here, \pi_0(\delta) and \pi_0(\lambda) are the initial prior distributions for \delta and \lambda, respectively. L(\lambda|D_{0k}) is the likelihood function based on historical data D_{0k}, with \delta_k being its corresponding power parameter.

Value

A list of class "NPP" comprising:

Author(s)

Qiang Zhang zqzjf0408@163.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.

See Also

PoiMNPP_MCMC2, PoiOMNPP_MCMC1, PoiOMNPP_MCMC2

Examples

PoiMNPP_MCMC1(n0 = c(0, 3, 5), n = 3, prior_lambda = c(1, 1/10), prop_delta = "IND",
              prior_delta_alpha = c(1, 1, 1), prior_delta_beta = c(1, 1, 1),
              rw_delta = 0.1, delta_ini = NULL, nsample = 2000, burnin = 500, thin = 2)

MCMC Sampling for Poisson Population of multiple historical data using Normalized Power Prior

Description

Multiple historical data are combined individually. Conduct posterior sampling for Poisson 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 \lambda, Gibbs sampling is used.

Usage

PoiMNPP_MCMC2(n0,n,prior_lambda,prop_delta,prior_delta_alpha,
                  prior_delta_beta,rw_delta, delta_ini,nsample,burnin,thin)

Arguments

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(\lambda)L(\lambda|D_{0k})^{\delta_{k}}}{\int \pi_0(\lambda)L(\lambda|D_{0k})^{\delta_{k}} d\lambda}.

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

Value

A list of class "NPP" with three elements:

Author(s)

Qiang Zhang zqzjf0408@163.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.

See Also

PoiMNPP_MCMC1; PoiOMNPP_MCMC1; PoiOMNPP_MCMC2

Examples

PoiMNPP_MCMC2(n0=c(0,3,5),n=3,prior_lambda=c(1,1/10),prop_delta="IND",
              prior_delta_alpha=c(1,1,1), prior_delta_beta=c(1,1,1), rw_delta=0.1,
              delta_ini=NULL, nsample = 2000, burnin = 500, thin = 2)

MCMC Sampling for Poisson Population of multiple ordered historical data using Normalized Power Prior

Description

Multiple ordered historical data are incorporated together. Conduct posterior sampling for Poisson population with normalized power prior. For the power parameter \gamma, a Metropolis-Hastings algorithm with independence proposal is used. For the model parameter \lambda, Gibbs sampling is used.

Usage

PoiOMNPP_MCMC1(n0,n,prior_gamma,prior_lambda, gamma_ind_prop,
               gamma_ini,nsample,burnin,thin)

Arguments

Details

The outputs include posteriors of the model parameter(s) and power parameter, acceptance rate in sampling \gamma. The normalized power prior distribution is

\frac{\pi_0(\gamma)\pi_0(\lambda)\prod_{k=1}^{K}L(\lambda|D_{0k})^{(\sum_{i=1}^{k}\gamma_i)}}{\int \pi_0(\lambda)\prod_{k=1}^{K}L(\lambda|D_{0k})^{(\sum_{i=1}^{k}\gamma_i)}d\lambda }.

Here \pi_0(\gamma) and \pi_0(\lambda) are the initial prior distributions of \gamma and \lambda, respectively. L(\lambda|D_{0k}) is the likelihood function of historical data D_{0k}, and \sum_{i=1}^{k}\gamma_i is the corresponding power parameter.

Value

A list of class "NPP" with three elements:

Author(s)

Qiang Zhang zqzjf0408@163.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.

See Also

PoiMNPP_MCMC1; PoiMNPP_MCMC2; PoiOMNPP_MCMC2

Examples

PoiOMNPP_MCMC1(n0=c(0,3,5),n=3,prior_gamma=c(1/2,1/2,1/2,1/2), prior_lambda=c(1,1/10),
               gamma_ind_prop=rep(1,4),gamma_ini=NULL, nsample = 2000, burnin = 500, thin = 2)

MCMC Sampling for Poisson Population of multiple ordered historical data using Normalized Power Prior

Description

Multiple ordered historical data are combined individually. Conduct posterior sampling for Poisson population with normalized power prior. For the power parameter \gamma, a Metropolis-Hastings algorithm with independence proposal is used. For the model parameter \lambda, Gibbs sampling is used.

Usage

PoiOMNPP_MCMC2(n0,n,prior_gamma,prior_lambda, gamma_ind_prop,gamma_ini,
               nsample, burnin, thin)

Arguments

Details

The outputs include posteriors of the model parameter(s) and power parameter, acceptance rate in sampling \gamma. The normalized power prior distribution is

\pi_0(\gamma)\prod_{k=1}^{K}\frac{\pi_0(\lambda)L(\lambda|D_{0k})^{(\sum_{i=1}^{k}\gamma_i)}}{\int \pi_0(\lambda)L(\lambda|D_{0k})^{(\sum_{i=1}^{k}\gamma_i)} d\lambda}.

Here \pi_0(\gamma) and \pi_0(\lambda) are the initial prior distributions of \gamma and \lambda, respectively. L(\lambda|D_{0k}) is the likelihood function of historical data D_{0k}, and \sum_{i=1}^{k}\gamma_i is the corresponding power parameter.

Value

A list of class "NPP" with three elements:

Author(s)

Qiang Zhang zqzjf0408@163.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.

See Also

PoiMNPP_MCMC1; PoiMNPP_MCMC2; PoiOMNPP_MCMC1

Examples

PoiOMNPP_MCMC2(n0=c(0,3,5),n=3,prior_gamma=c(1/2,1/2,1/2,1/2),
               prior_lambda=c(1,1/10), gamma_ind_prop=rep(1,4),
               gamma_ini=NULL, nsample = 2000, burnin = 500, thin = 2)

MCMC Sampling for Bernoulli Population using Normalized Power Prior

Description

Conduct posterior sampling for Poisson 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 \lambda, Gibbs sampling is used.

Usage

PoissonNPP_MCMC(Data.Cur, Data.Hist,
                CompStat = list(n0 = NULL, mean0 = NULL, n1 = NULL, mean1 = NULL),
                prior = list(lambda.shape = 1/2, lambda.scale = 100,
                             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

Details

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

Value

A list of class "NPP" with four elements:

Author(s)

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.

See Also

MultinomialNPP_MCMC; NormalNPP_MCMC; BerNPP_MCMC;

Examples

set.seed(1234)
DataHist <- rpois(n = 100, lambda = 49)
set.seed(12345)
DataCur <- rpois(n = 100, lambda = 49)

PoissonNPP_MCMC(Data.Cur = DataCur, Data.Hist = DataHist,
                CompStat = list(n0 = 20, mean0 = 10, n1 = 30, mean1 = 11),
                prior = list(lambda.shape = 1/2, lambda.scale = 100,
                             delta.alpha = 1, delta.beta = 1),
                MCMCmethod = 'RW', rw.logit.delta = 1,
                ind.delta.alpha= 1, ind.delta.beta= 1,nsample = 10000,
                control.mcmc = list(delta.ini = NULL, burnin = 2000, thin = 1))

Dataset for Diagnostic Test (PartoSure Test, Medical Device) Evaluation for Spontaneous Preterm Delivery

Description

The diagnostic test was developed to aid in rapidly assess the risk of spontaneouspreterm delivery within 7 days from the time of diagnosis in pre-pregnant women with signs and symptoms. The same diagnostic test was used for two populations in US and EU respectively. The number of counts in the four cells (True positive, false positive, false negative, true negative) was recorded.

Usage

data("SPDData")

Format

A data frame with 2 observations on the following 5 variables.

Data.Region

region where the diagnostic test was conducted

TPDP

number of subjects with tested positive and the disease status positive (true positive)

TPDN

number of subjects with tested positive but the disease status negative (false positive)

TNDP

number of subjects with tested negative and the disease status positive (false negative)

TNDN

number of subjects with tested negative and the disease status negative (true negative)

Source

https://www.accessdata.fda.gov/cdrh_docs/pdf16/P160052C.pdf

Examples

data(SPDData)

Dataset of a Vaccine Trial for RotaTeq and Multiple Historical Trials for Control Group

Description

The study was designed to investigate the concomitant use of RotaTeq(Test Vaccine) and some routine pediatric vaccines between 2001-2005. The dataset includes four historical control trials. The purpose of the study is to borrow the historical controls for the non-inferiority trial. The interest is in the response rate to the routine vaccines.

Usage

data("VaccineData")

Format

A data frame with 6 observations on the following 7 variables.

Data.Time

indicator of historical data (coded as 0) or current data (coded as 1).

StudyID

character to distinguish different studies.

Group

indicator of control group (coded as 0) or treatment group (coded as 1).

Start.Year

start year of the trial

End.Year

end year of the trial

N

total number of patients enrolled and dosed in the group

y

total number of patients respond to the vaccine

References

Liu, G.F. (2018). A Dynamic Power Prior for Borrowing Historical Data in Noninferiority Trials with Binary Endpoint. Pharmaceutical Statistics 17:61-73.

Examples

data(VaccineData)

A Function to Interpolate logC(\delta) Based on Its Values on Selected Knots

Description

The function returns the interpolated value (a scalar) of logC(\delta) based on its results on selected knots, given input vector of \delta.

Usage

logCdelta(delta, deltaknot, lCknot)

Arguments

Value

A sequence of the values, logC(\delta) on selected knots.

Author(s)

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.

See Also

loglikNormD0; loglikBerD0; logCknot

A Function to Calculate logC(\delta) on Selected Knots

Description

The function returns a sequence of the values, logC(\delta) on selected knots, given input vector of \delta.

Usage

logCknot(deltaknot, llikf0)

Arguments

Value

A sequence of the values, logC(\delta) on selected knots.

Author(s)

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.

See Also

loglikNormD0; loglikBerD0; logCdelta

A Function to Calculate Log-likelihood of the Historical Data, Given Matrix-valued Parameters, for Bernoulli Population

Description

The function returns a matrix of class "npp", each element is a log-likelihood of the historical data. It is an intermediate step to calculate the "normalizing constant" C(\delta) in the normalized power prior, for the purpose of providing a flexible implementation. Users can specify their own likelihood function of the same class following this structure.

Usage

loglikBerD0(D0, thetalist, ntheta = 1)

Arguments

Value

A numeric matrix of log-likelihood, for the historical data given the matrix(or array)-valued parameters.

Author(s)

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.

See Also

loglikNormD0; logCknot; logCdelta

A Function to Calculate Log-likelihood of the Historical Data, Given Array-valued Parameters, for Normal Population

Description

The function returns a matrix of class "npp", each element is a log-likelihood of the historical data. It is an intermediate step to calculate the "normalizing constant" C(\delta) in the normalized power prior, for the purpose of providing a flexible implementation. Users can specify their own likelihood function of the same class following this structure.

Usage

loglikNormD0(D0, thetalist, ntheta = 2)

Arguments

Value

A numeric matrix of log-likelihood, for the historical data given the matrix(or array)-valued parameters.

Author(s)

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.

See Also

loglikBerD0; logCknot; logCdelta