Package 'BayesMultMeta'

Summary statistics from a posterior distribution

Description

Given a univariate sample drawn from the posterior distribution, this function computes the posterior mean, the posterior median, the posterior standard deviation, and the limits of the $(1-\alpha)$ probability-symmetric credible interval.

Usage

bayes_inference(x, alp)

Arguments

x	Univariate sample from the posterior distribution of a parameter.
alp	Significance level used in the computation of the credible interval

Value

a matrix with summary statistics

---

Interface for the BayesMultMeta class

Description

The BayesMultMeta package implements two methods of constructing Markov chains to assess the posterior distribution of the model parameters, namely the overall mean vector $\mathbf{\mu}$ and the between-study covariance matrix $\mathbf{\Psi}$, of the generalized marginal multivariate random effects models. The Bayesian inference procedures are performed when the model parameters are endowed with the Berger and Bernardo reference prior (Berger and Bernardo 1992) and the Jeffreys prior (Jeffreys 1946). This is achieved by constructing Markov chains using the Metropolis-Hastings algorithms developed in (Bodnar and Bodnar 2021). The convergence properties of the generated Markov chains are investigated by the rank plots and the split-$\hat{R}$ estimate based on the rank normalization, which are proposed in (Vehtari et al. 2021).

Usage

BayesMultMeta(X, U, N, burn_in, likelihood, prior, algorithm_version, d = NULL)

Arguments

X	A $p \times n$ matrix which contains $n$ observation vectors of dimension $p$
U	A $pn \times pn$ block-diagonal matrix which contains the covariance matrices of observation vectors.
N	Length of the generated Markov chain.
burn_in	Number of burn-in samples
likelihood	Likelihood to use. It currently supports "normal" and "t".
prior	Prior to use. It currently supports "reference" and "jeffrey".
algorithm_version	One of "mu" or "Psi". Both algorithms samples the same quantities.
d	Degrees of freedom for the t-distribution when the "t" option is used for the likelihood.

Value

a BayesMultMeta class which contains simulations from the MCMC inference procedure as well as many of the input parameters. The elements 'psi' and 'mu' in the list contains simulations from the posterior distribution. All other elements are input parameters to the class.

References

Berger JO, Bernardo JM (1992). “On the development of the reference prior method.” Bayesian statistics, 4(4), 35–60.

Bodnar O, Bodnar T (2021). “Objective Bayesian meta-analysis based on generalized multivariate random effects model.” 2104.02105.

Jeffreys H (1946). “An invariant form for the prior probability in estimation problems.” Proceedings of the Royal Society of London Series A, 186(1007), 453-461. doi:10.1098/rspa.1946.0056.

Vehtari A, Gelman A, Simpson D, Carpenter B, Bürkner P (2021). “Rank-normalization, folding, and localization: An improved hatR for assessing convergence of MCMC (with Discussion).” Bayesian analysis, 16(2), 667–718.

Examples

dataREM<-mvmeta::hyp
# Observation matrix X
X<-t(cbind(dataREM$sbp,dataREM$dbp))
p<-nrow(X)  # model dimension
n<-ncol(X)  # sample size
# Matrix U
U<-matrix(0,n*p,n*p)
for (i_n in 1:n) {
  Use<-diag(c(dataREM$sbp_se[i_n],dataREM$dbp_se[i_n]))
  Corr_mat<-matrix(c(1,dataREM$rho[i_n],dataREM$rho[i_n],1),p,p)
  U[(p*(i_n-1)+1):(p*i_n),(p*(i_n-1)+1):(p*i_n)]<- Use%*%Corr_mat%*%Use
}

bmgmr_run <- BayesMultMeta(X, U, 1e2, burn_in = 100,
                   likelihood = "normal", prior="jeffrey",
                   algorithm_version = "mu")
summary(bmgmr_run)
plot(bmgmr_run)

---

Duplication matrix

Description

This function creates the duplication matrix of size $p^2 \times p(p+1)/2$

Usage

duplication_matrix(p)

Arguments

p	Integer which specifies the dimension of the duplication matrix.

Value

a matrix of size $p^2 \times p(p+1)/2$

---

Computes the ranks within the pooled draws of Markov chains

Description

The function computes the ranks within the pooled draws of Markov chains. Average ranks are used for ties.

Usage

MC_ranks(MC)

Arguments

MC	An $N \times M$ matrix with N draws in each of M constructed Markov chains.

Value

a matrix with the ranks from the MCMC procedure

Examples

dataREM<-mvmeta::hyp
# Observation matrix X
X<-t(cbind(dataREM$sbp,dataREM$dbp))
p<-nrow(X) # model dimension
n<-ncol(X) # sample size
# Matrix U
U<-matrix(0,n*p,n*p)
for (i_n in 1:n) {
  Use<-diag(c(dataREM$sbp_se[i_n],dataREM$dbp_se[i_n]))
  Corr_mat<-matrix(c(1,dataREM$rho[i_n],dataREM$rho[i_n],1),p,p)
  U[(p*(i_n-1)+1):(p*i_n),(p*(i_n-1)+1):(p*i_n)]<- Use%*%Corr_mat%*%Use
}
# Generating M Markov chains for mu_1
M<-4 # number of chains
MC <-NULL
for (i in 1:M) {
chain <-  BayesMultMeta(X, U, 1e2, burn_in = 1e2,
                          likelihood = "t", prior="jeffrey",
                          algorithm_version = "mu",d=3)
  MC<- cbind(MC,chain$mu[1,])
}
ranks<-MC_ranks(MC)
id_chain <- 1
hist(ranks[,id_chain],breaks=25,prob=TRUE, labels = FALSE, border = "dark blue",
  col = "light blue", main = expression("Chain 1,"~mu[1]), xlab = expression(),
  ylab = expression(),cex.axis=1.2,cex.main=1.7,font=2)

---

Plot a BayesMultMeta object

Description

This function produces the trace plots of the constructed Markov chains.

Usage

## S3 method for class 'BayesMultMeta'
plot(x, ...)

Arguments

x	a BayesMultMeta object
...	additional arguments

Value

No return value, produces trace plots

---

Metropolis-Hastings algorithm for the normal distribution and the Jeffreys prior, where $\mathbf{\mu}$ is generated from the marginal posterior.

Description

This function implements Metropolis-Hastings algorithm for drawing samples from the posterior distribution of $\mathbf{\mu}$ and $\mathbf{\Psi}$ under the assumption of the normal distribution when the Jeffreys prior is employed. At each step, the algorithm starts with generating a draw from the marginal distribution of $\mathbf{\mu}$.

Usage

sample_post_nor_jef_marg_mu(X, U, Np)

Arguments

X	A $p \times n$ matrix which contains $n$ observation vectors of dimension $p$.
U	A $p n \times p n$ block-diagonal matrix which contains the covariance matrices of observation vectors.
Np	Length of the generated Markov chain.

Value

List with the generated samples from the joint posterior distribution of $\mathbf{\mu}$ and $\mathbf{\Psi}$, where the values of $\mathbf{\Psi}$ are presented by using the vec operator.

---

Metropolis-Hastings algorithm for the normal distribution and the Jeffreys prior, where $\mathbf{\Psi}$ is generated from the marginal posterior.

Description

This function implements Metropolis-Hastings algorithm for drawing samples from the posterior distribution of $\mathbf{\mu}$ and $\mathbf{\Psi}$ under the assumption of the normal distribution when the Jeffreys prior is employed. At each step, the algorithm starts with generating a draw from the marginal distribution of $\mathbf{\Psi}$.

Usage

sample_post_nor_jef_marg_Psi(X, U, Np)

Arguments

X	A $p \times n$ matrix which contains $n$ observation vectors of dimension $p$.
U	A $p n \times p n$ block-diagonal matrix which contains the covariance matrices of observation vectors.
Np	Length of the generated Markov chain.

Value

List with the generated samples from the joint posterior distribution of $\mathbf{\mu}$ and $\mathbf{\Psi}$, where the values of $\mathbf{\Psi}$ are presented by using the vec operator.

---

Metropolis-Hastings algorithm for the normal distribution and the Berger and Bernardo reference prior, where $\mathbf{\mu}$ is generated from the marginal posterior.

Description

This function implements Metropolis-Hastings algorithm for drawing samples from the posterior distribution of $\mathbf{\mu}$ and $\mathbf{\Psi}$ under the assumption of the normal distribution when the Berger and Bernardo reference prior is employed. At each step, the algorithm starts with generating a draw from the marginal distribution of $\mathbf{\mu}$.

Usage

sample_post_nor_ref_marg_mu(X, U, Np)

Arguments

X	A $p \times n$ matrix which contains $n$ observation vectors of dimension $p$.
U	A $p n \times p n$ block-diagonal matrix which contains the covariance matrices of observation vectors.
Np	Length of the generated Markov chain.

Value

List with the generated samples from the joint posterior distribution of $\mathbf{\mu}$ and $\mathbf{\Psi}$, where the values of $\mathbf{\Psi}$ are presented by using the vec operator.

---

Metropolis-Hastings algorithm for the normal distribution and the Berger and Bernardo reference prior, where $\mathbf{\Psi}$ is generated from the marginal posterior.

Description

This function implements Metropolis-Hastings algorithm for drawing samples from the posterior distribution of $\mathbf{\mu}$ and $\mathbf{\Psi}$ under the assumption of the normal distribution when the Berger and Bernardo reference prior is employed. At each step, the algorithm starts with generating a draw from the marginal distribution of $\mathbf{\Psi}$.

Usage

sample_post_nor_ref_marg_Psi(X, U, Np)

Arguments

X	A $p \times n$ matrix which contains $n$ observation vectors of dimension $p$.
U	A $p n \times p n$ block-diagonal matrix which contains the covariance matrices of observation vectors.
Np	Length of the generated Markov chain.

Value

List with the generated samples from the joint posterior distribution of $\mathbf{\mu}$ and $\mathbf{\Psi}$, where the values of $\mathbf{\Psi}$ are presented by using the vec operator.

---

Metropolis-Hastings algorithm for the t-distribution and the Jeffreys prior, where $\mathbf{\mu}$ is generated from the marginal posterior.

Description

This function implements Metropolis-Hastings algorithm for drawing samples from the posterior distribution of $\mathbf{\mu}$ and $\mathbf{\Psi}$ under the assumption of the t-distribution when the Jeffreys prior is employed. At each step, the algorithm starts with generating a draw from the marginal distribution of $\mathbf{\mu}$.

Usage

sample_post_t_jef_marg_mu(X, U, d, Np)

Arguments

X	A $p \times n$ matrix which contains $n$ observation vectors of dimension $p$.
U	A $p n \times p n$ block-diagonal matrix which contains the covariance matrices of observation vectors.
d	Degrees of freedom for the t-distribution
Np	Length of the generated Markov chain.

Value

List with the generated samples from the joint posterior distribution of $\mathbf{\mu}$ and $\mathbf{\Psi}$, where the values of $\mathbf{\Psi}$ are presented by using the vec operator.

---

Metropolis-Hastings algorithm for the t-distribution and the Jeffreys prior, where $\mathbf{\Psi}$ is generated from the marginal posterior.

Description

This function implements Metropolis-Hastings algorithm for drawing samples from the posterior distribution of $\mathbf{\mu}$ and $\mathbf{\Psi}$ under the assumption of the t-distribution when the Jeffreys prior is employed. At each step, the algorithm starts with generating a draw from the marginal distribution of $\mathbf{\Psi}$.

Usage

sample_post_t_jef_marg_Psi(X, U, d, Np)

Arguments

X	A $p \times n$ matrix which contains $n$ observation vectors of dimension $p$.
U	A $p n \times p n$ block-diagonal matrix which contains the covariance matrices of observation vectors.
d	Degrees of freedom for the t-distribution
Np	Length of the generated Markov chain.

Value

List with the generated samples from the joint posterior distribution of $\mathbf{\mu}$ and $\mathbf{\Psi}$, where the values of $\mathbf{\Psi}$ are presented by using the vec operator.

---

Metropolis-Hastings algorithm for the t-distribution and Berger and Bernardo reference prior, where $\mathbf{\mu}$ is generated from the marginal posterior.

Description

This function implements Metropolis-Hastings algorithm for drawing samples from the posterior distribution of $\mathbf{\mu}$ and $\mathbf{\Psi}$ under the assumption of the t-distribution when the Berger and Bernardo prior is employed. At each step, the algorithm starts with generating a draw from the marginal distribution of $\mathbf{\mu}$.

Usage

sample_post_t_ref_marg_mu(X, U, d, Np)

Arguments

X	A $p \times n$ matrix which contains $n$ observation vectors of dimension $p$.
U	A $p n \times p n$ block-diagonal matrix which contains the covariance matrices of observation vectors.
d	Degrees of freedom for the t-distribution
Np	Length of the generated Markov chain.

Value

List with the generated samples from the joint posterior distribution of $\mathbf{\mu}$ and $\mathbf{\Psi}$, where the values of $\mathbf{\Psi}$ are presented by using the vec operator.

---

Metropolis-Hastings algorithm for the t-distribution and Berger and Bernardo reference prior, where $\mathbf{\Psi}$ is generated from the marginal posterior.

Description

This function implements Metropolis-Hastings algorithm for drawing samples from the posterior distribution of $\mathbf{\mu}$ and $\mathbf{\Psi}$ under the assumption of the t-distribution when the Berger and Bernardo prior is employed. At each step, the algorithm starts with generating a draw from the marginal distribution of $\mathbf{\Psi}$.

Usage

sample_post_t_ref_marg_Psi(X, U, d, Np)

Arguments

X	A $p \times n$ matrix which contains $n$ observation vectors of dimension $p$.
U	A $p n \times p n$ block-diagonal matrix which contains the covariance matrices of observation vectors.
d	Degrees of freedom for the t-distribution
Np	Length of the generated Markov chain.

Value

List with the generated samples from the joint posterior distribution of $\mathbf{\mu}$ and $\mathbf{\Psi}$, where the values of $\mathbf{\Psi}$ are presented by using the vec operator.

---

Computes the split-$\hat{R}$ estimate based on the rank normalization

Description

The function computes the split-$\hat{R}$ estimate based on the rank normalization.

Usage

split_rank_hatR(MC)

Arguments

MC	An $N \times M$ matrix with N draws in each of M constructed Markov chains.

Value

a value with the the split-$\hat{R}$ estimate based on the rank normalization

Examples

dataREM<-mvmeta::hyp
# Observation matrix X
X<-t(cbind(dataREM$sbp,dataREM$dbp))
p<-nrow(X) # model dimension
n<-ncol(X) # sample size
# Matrix U
U<-matrix(0,n*p,n*p)
for (i_n in 1:n) {
  Use<-diag(c(dataREM$sbp_se[i_n],dataREM$dbp_se[i_n]))
  Corr_mat<-matrix(c(1,dataREM$rho[i_n],dataREM$rho[i_n],1),p,p)
  U[(p*(i_n-1)+1):(p*i_n),(p*(i_n-1)+1):(p*i_n)]<- Use%*%Corr_mat%*%Use
}
# Generating M Markov chains for mu_1
M<-4 # number of chains
MC <-NULL
for (i in 1:M) {
  chain <-  BayesMultMeta(X, U, 1e2, burn_in = 1e2,
                          likelihood = "t", prior="jeffrey",
                          algorithm_version = "mu",d=3)
  MC<- cbind(MC,chain$mu[1,])
}
split_rank_hatR(MC)

---

Summary statistics from the posterior of a BayesMultMeta class

Description

Summary statistics from the posterior of a BayesMultMeta class

Usage

## S3 method for class 'BayesMultMeta'
summary(object, alpha = 0.95, ...)

Arguments

object	BayesMultMeta class
alpha	Significance level used in the computation of the credible interval.
...	not used

Value

a list with summary statistics

---