LMOMNPP_MCMC1() R function from [NPP]

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

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.


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.

Returns

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 $\delta$ .

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)

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.

LMOMNPP_MCMC1 function