Package 'NPP'

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

n0	A non-negative integer vector representing the number of trials in historical data.
y0	A non-negative integer vector denoting the number of successes in historical data.
n	A non-negative integer indicating the number of trials in the current data.
y	A non-negative integer for the 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 bunrin.
thin	the thinning parameter in MCMC sampling.

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:

acceptrate	Acceptance rate in MCMC sampling for $\delta$ via the Metropolis-Hastings algorithm.
p	Posterior distribution of the model parameter $p$.
delta	Posterior distribution of the power parameter $\delta$.

Author(s)

Qiang Zhang [email protected]

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

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.

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:

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 $\delta$.

Author(s)

Qiang Zhang [email protected]

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

Data.Cur	a non-negative integer vector of two elements: c(number of trials, number of successes) in the current data.
Data.Hist	a non-negative integer vector of two elements: c(number of trials, number of successes) in the historical data.
CompStat	a list of four elements that represents the "compatibility(sufficient) statistics" for $p$. 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 Bernoulli population providing CompStat is equivalent to provide the data summary as in Data.Cur and Data.Cur. n0 is the number of trials in the historical data. y0 is the number of successes in the historical data. n1 is the number of trials in the current data. y1 is the number of successes in the current data.
prior	a list of the hyperparameters in the prior for both $p$ and $\delta$. p.alpha is the hyperparameter $\alpha$ in the prior distribution $Beta(\alpha, \beta)$ for $p$. p.beta is the hyperparameter $\beta$ in the prior distribution $Beta(\alpha, \beta)$ for $p$. delta.alpha is the hyperparameter $\alpha$ in the prior distribution $Beta(\alpha, \beta)$ for $\delta$. delta.beta is 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.

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:

p	posterior of the model parameter $p$.
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.

Author(s)

Zifei Han [email protected]

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

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_gamma	a vector of the hyperparameters in the prior distribution $Dirichlet(\alpha_1, \alpha_2, ... ,\alpha_K)$ for $\gamma$.
prior_p	a vector of the hyperparameters in the prior distribution $Beta(\alpha, \beta)$ for $p$.
gamma_ind_prop	a vector of the hyperparameters in the proposal distribution $Dirichlet(\alpha_1, \alpha_2, ... ,\alpha_K)$ for $\gamma$.
gamma_ini	the initial value of $\gamma$ in MCMC sampling.
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.
adjust	Logical, indicating whether or not to adjust the parameters of the proposal distribution.

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:

acceptrate	the acceptance rate in MCMC sampling for $\gamma$ using Metropolis-Hastings algorithm.
p	posterior of the model parameter $p$.
delta	posterior of the power parameter $\delta$. It is equal to the cumulative sum of $\gamma$.

Author(s)

Qiang Zhang [email protected]

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

n0	a vector of non-negative integers: numbers of trials in historical data.
y0	a vector of non-negative integers: numbers 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_gamma	a vector of the hyperparameters in the prior distribution $Dirichlet(\alpha_1, \alpha_2, ... ,\alpha_K)$ for $\gamma$.
prior_p	a vector of the hyperparameters in the prior distribution $Beta(\alpha, \beta)$ for $p$.
gamma_ind_prop	a vector of the hyperparameters in the proposal distribution $Dirichlet(\alpha_1, \alpha_2, ... ,\alpha_K)$ for $\gamma$.
gamma_ini	the initial value of $\gamma$ in MCMC sampling.
nsample	specifies the number of posterior samples in the output.
burnin	the number of burn-ins. The output will only show MCMC samples after burn-in.
thin	the thinning parameter in MCMC sampling.
adjust	Whether or not to adjust the parameters of the proposal distribution.

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:

acceptrate	the acceptance rate in MCMC sampling for $\gamma$ using Metropolis-Hastings algorithm.
p	posterior of the model parameter $p$.
delta	posterior of the power parameter $\delta$. It is equal to the cumulative sum of $\gamma$

Author(s)

Qiang Zhang [email protected]

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)

---

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

delta	the power parameter between 0 and 1. The function returns $logC(\delta)$
loglikmle	a scalar; the loglikelihood of the historical data evaluated at the maximum likelihood estimates based on the historical data
detHessian	determinant of the Hessian matrix evaluated at the loglikelihood function with respect to the maximum likelihood estimates based on the historical data
ntheta	an positive integer indicating number of parameters in the model

Value

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

Author(s)

Zifei Han [email protected]

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

---

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

D0	a list of $k$ elements representing $k$ historical data, where the $i^{th}$ element corresponds to the $i^{th}$ historical data named as “D0i”.
X	a vector or matrix or data frame of covariate observed in the current data. If more than 1 covariate available, the number of rows is equal to the number of observations.
Y	a vector of individual level of the response y in the current data.
a0	a positive shape parameter for inverse-gamma prior on model parameter $\sigma^2$.
b	a positive scale parameter for inverse-gamma prior on model parameter $\sigma^2$.
mu0	a vector of the mean for prior $\beta|\sigma^2$.
R	a inverse matrix of the covariance matrix for prior $\beta|\sigma^2$.
delta_ini	the initial value of $\delta$ in MCMC sampling.
prop_delta	the class of proposal distribution for $\delta$.
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 $\delta$.
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 bunrin.
thin	the thinning parameter in MCMC sampling.

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:

acceptrate	the acceptance rate in MCMC sampling for $\delta$ using Metropolis-Hastings algorithm.
beta	posterior of the model parameter $\beta$ in vector or matrix form.
sigma	posterior of the model parameter $\sigma^2$.
delta	posterior of the power parameter $\delta$.

Author(s)

Qiang Zhang [email protected]

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

D0	a list of $k$ elements representing $k$ historical data, where the $i^{th}$ element corresponds to the $i^{th}$ historical data named as “D0i”.
X	a vector or matrix or data frame of covariate observed in the current data. If more than 1 covariate available, the number of rows is equal to the number of observations.
Y	a vector of individual level of the response y in the current data.
a0	a positive shape parameter for inverse-gamma prior on model parameter $\sigma^2$.
b	a positive scale parameter for inverse-gamma prior on model parameter $\sigma^2$.
mu0	a vector of the mean for prior $\beta|\sigma^2$.
R	a inverse matrix of the covariance matrix for prior $\beta|\sigma^2$.
delta_ini	the initial value of $\delta$ in MCMC sampling.
prop_delta	the class of proposal distribution for $\delta$.
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 $\delta$.
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 bunrin.
thin	the thinning parameter in MCMC sampling.

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:

acceptrate	the acceptance rate in MCMC sampling for $\delta$ using Metropolis-Hastings algorithm.
beta	posterior of the model parameter $\beta$ in vector or matrix form.
sigma	posterior of the model parameter $\sigma^2$.
delta	posterior of the power parameter $\delta$.

Author(s)

Qiang Zhang [email protected]

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

y.Cur	a vector of individual level of the response y in current data.
y.Hist	a vector of individual level of the response y in historical data.
x.Cur	a vector or matrix or data frame of covariate observed in the current data. If more than 1 covariate available, the number of rows is equal to the number of observations.
x.Hist	a vector or matrix or data frame of covariate observed in the historical data. If more than 1 covariate available, the number of rows is equal to the number of observations.
prior	a list of the hyperparameters in the prior for model parameters $(\beta, \sigma^2)$ and $\delta$. The form of the prior for model parameter $(\beta, \sigma^2)$ is in the section "Details". a a positive hyperparameter for prior on model parameters. It is the power $a$ in formula $(1/\sigma^2)^a$; See details. b equals 0 if a flat prior is used for $\beta$. Equals 1 if a normal prior is used for $\beta$; See details. mu0 a vector of the mean for prior $\beta|\sigma^2$. Only applicable if b = 1. Rinv inverse of the matrix $R$. The covariance matrix of the prior for $\beta|\sigma^2$ is $\sigma^2 R^{-1}$. delta.alpha is the hyperparameter $\alpha$ in the prior distribution $Beta(\alpha, \beta)$ for $\delta$. delta.beta is 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.

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:

beta	posterior of the model parameter $\beta$ in vector or matrix form.
sigmasq	posterior of the model parameter $\sigma^2$.
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.

Author(s)

Zifei Han [email protected]

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

D0	a list of $k$ elements representing $k$ historical data, where the $i^{th}$ element corresponds to the $i^{th}$ historical data named as “D0i”.
X	a vector or matrix or data frame of covariate observed in the current data. If more than 1 covariate available, the number of rows is equal to the number of observations.
Y	a vector of individual level of the response y in the current data.
a0	a positive shape parameter for inverse-gamma prior on model parameter $\sigma^2$.
b	a positive scale parameter for inverse-gamma prior on model parameter $\sigma^2$.
mu0	a vector of the mean for prior $\beta|\sigma^2$.
R	a inverse matrix of the covariance matrix for prior $\beta|\sigma^2$.
gamma_ini	the initial value of $\gamma$ in MCMC sampling.
prior_gamma	a vector of the hyperparameters in the prior distribution $Dirichlet(\alpha_1, \alpha_2, ... ,\alpha_K)$ for $\gamma$.
gamma_ind_prop	a vector of the hyperparameters in the proposal distribution $Dirichlet(\alpha_1, \alpha_2, ... ,\alpha_K)$ for $\gamma$.
nsample	specifies the number of posterior samples in the output.
burnin	the number of burn-ins. The output will only show MCMC samples after bunrin.
thin	the thinning parameter in MCMC sampling.
adjust	Whether or not to adjust the parameters of the proposal distribution.

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:

acceptrate	the acceptance rate in MCMC sampling for $\gamma$ using Metropolis-Hastings algorithm.
beta	posterior of the model parameter $\beta$ in vector or matrix form.
sigma	posterior of the model parameter $\sigma^2$.
delta	posterior of the power parameter $\delta$.

Author(s)

Qiang Zhang [email protected]

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

D0	a list of $k$ elements representing $k$ historical data, where the $i^{th}$ element corresponds to the $i^{th}$ historical data named as “D0i”.
X	a vector or matrix or data frame of covariate observed in the current data. If more than 1 covariate available, the number of rows is equal to the number of observations.
Y	a vector of individual level of the response y in the current data.
a0	a positive shape parameter for inverse-gamma prior on model parameter $\sigma^2$.
b	a positive scale parameter for inverse-gamma prior on model parameter $\sigma^2$.
mu0	a vector of the mean for prior $\beta|\sigma^2$.
R	a inverse matrix of the covariance matrix for prior $\beta|\sigma^2$.
gamma_ini	the initial value of $\gamma$ in MCMC sampling.
prior_gamma	a vector of the hyperparameters in the prior distribution $Dirichlet(\alpha_1, \alpha_2, ... ,\alpha_K)$ for $\gamma$.
gamma_ind_prop	a vector of the hyperparameters in the proposal distribution $Dirichlet(\alpha_1, \alpha_2, ... ,\alpha_K)$ for $\gamma$.
nsample	specifies the number of posterior samples in the output.
burnin	the number of burn-ins. The output will only show MCMC samples after bunrin.
thin	the thinning parameter in MCMC sampling.
adjust	Whether or not to adjust the parameters of the proposal distribution.

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:

acceptrate	the acceptance rate in MCMC sampling for $\gamma$ using Metropolis-Hastings algorithm.
beta	posterior of the model parameter $\beta$ in vector or matrix form.
sigma	posterior of the model parameter $\sigma^2$.
delta	posterior of the power parameter $\delta$.

Author(s)

Qiang Zhang [email protected]

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 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

delta	a scalar of the input value of $\delta$.
deltaknot	a vector of the knots for $\delta$. It should be selected before conduct the sampling.
lCknot	a vector of the values $logC(\delta)$ on selected knots, coming from the function logCknot.

Value

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

Author(s)

Zifei Han [email protected]

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

deltaknot	a vector of the knots for $\delta$. It should be selected before conduct the sampling.
llikf0	a matrix of the log-likelihoods of class "npp".

Value

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

Author(s)

Zifei Han [email protected]

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

D0	a vector of each observation(binary) in historical data.
thetalist	a list of parameter values. The number of elements is equal to ntheta. Each element is a matrix. The sample should come from the posterior of the powered likelihood for historical data, with each column corresponds to a distinct value of the power parameter $\delta$ (the corresponding power parameter increases from left to right). The number of rows is the number of Monte Carlo samples for each $\delta$ fixed. The number of columns is the number of selected knots (number of distinct $\delta$).
ntheta	a positive integer indicating number of parameters to be estimated in the model. Default is 1 for Bernoulli.

Value

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

Author(s)

Zifei Han [email protected]

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

D0	a vector of each observation in historical data.
thetalist	a list of parameter values. The number of elements is equal to ntheta. Each element is a matrix. The sample should come from the posterior of the powered likelihood for historical data, with each column corresponds to a distinct value of the power parameter $\delta$ (the corresponding power parameter increases from left to right). The number of rows is the number of Monte Carlo samples for each $\delta$ fixed. The number of columns is the number of selected knots (number of distinct $\delta$).
ntheta	a positive integer indicating number of parameters to be estimated in the model.

Value

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

Author(s)

Zifei Han [email protected]

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

---

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

Data.Cur	a non-negative integer vector of two elements: c(number of success, number of failure) in the current data.
Data.Hist	a non-negative integer vector of two elements: c(number of success, number of failure) in the historical data.
CompStat	a list of four elements that represents the "compatibility(sufficient) statistics" for $p$. 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 Bernoulli population providing CompStat is equivalent to provide the data summary as in Data.Cur and Data.Cur. n0 is the number of trials in the historical data. y0 is the number of successes in the historical data. n1 is the number of trials in the current data. y1 is the number of successes in the current data.
npoints	is a non-negative integer scalar indicating number of points on a regular spaced grid between [0, 1], where we calculate the log of the posterior and search for the mode.
prior	a list of the hyperparameters in the prior for both $p$ and $\delta$. p.alpha is the hyperparameter $\alpha$ in the prior distribution $Beta(\alpha, \beta)$ for $p$. p.beta is the hyperparameter $\beta$ in the prior distribution $Beta(\alpha, \beta)$ for $p$. delta.alpha is the hyperparameter $\alpha$ in the prior distribution $Beta(\alpha, \beta)$ for $\delta$. delta.beta is the hyperparameter $\beta$ in the prior distribution $Beta(\alpha, \beta)$ for $\delta$.

Details

See example.

Value

A numeric value between 0 and 1.

Author(s)

Zifei Han [email protected]

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))

---