bayes_sim_unknownvar: Bayesian Simulation with Composite Sampling

In jpan928/bayesassurance_rpackage: Bayesian Assurance Computation

Bayesian Simulation with Composite Sampling

Description

Approximates the Bayesian assurance of a one-sided hypothesis test through Monte Carlo sampling with the added assumption that the variance is unknown.

Usage

bayes_sim_unknownvar(
  n,
  p = NULL,
  u,
  C,
  R,
  Xn = NULL,
  Vn,
  Vbeta_d,
  Vbeta_a_inv,
  mu_beta_d,
  mu_beta_a,
  a_sig_a,
  b_sig_a,
  a_sig_d,
  b_sig_d,
  alt = "greater",
  alpha,
  mc_iter
)

Arguments

n	sample size (either vector or scalar)
p	column dimension of design matrix Xn. If Xn = NULL, p must be specified to denote the column dimension of the default design matrix generated by the function.
u	a scalar or vector to evaluate u'β > C, where β is an unknown parameter that is to be estimated.
C	constant value to be compared to when evaluating u'β > C
R	number of iterations we want to pass through to check for satisfaction of the analysis stage objective. The proportion of those iterations meeting the analysis objective corresponds to the approximated Bayesian assurance.
Xn	design matrix that characterizes where the data is to be generated from. This is specifically given by the normal linear regression model yn = Xnβ + ε, ε ~ N(0, σ^2 Vn), where σ^2 is unknown in this setting. Note that Xn must have column dimension p.
Vn	an n by n correlation matrix for the marginal distribution of the sample data yn. Takes on an identity matrix when set to NULL.
Vbeta_d	correlation matrix that helps describe the prior information on β in the design stage
Vbeta_a_inv	inverse-correlation matrix that helps describe the prior information on β in the analysis stage
mu_beta_d	design stage mean
mu_beta_a	analysis stage mean
a_sig_a, b_sig_a	shape and scale parameters of the inverse gamma distribution where variance σ^2 is sampled from in the analysis stage
a_sig_d, b_sig_d	shape and scale parameters of the inverse gamma distribution where variance σ^2 is sampled from in the design stage
alt	specifies alternative test case, where alt = "greater" tests if u'β > C, alt = "less" tests if u'β < C, and alt = "two.sided" performs a two-sided test. By default, alt = "greater".
alpha	significance level
mc_iter	number of MC samples evaluated under the analysis objective

Value

a list of objects corresponding to the assurance approximations

• assurance_table: table of sample size and corresponding assurance values

• assur_plot: plot of assurance values

• mc_samples: number of Monte Carlo samples that were generated and evaluated

See Also

bayes_sim for the Bayesian assurance function for known variance.

Examples

## O'Hagan and Stevens (2001) include a series of examples with 
## pre-specified parameters that we will be using to replicate their 
## results through our Bayesian assurance simulation.
## The inputs are as follows:

n <- 285
p <- 4 ## includes two parameter measures (cost and efficacy) for each of 
       ## the two treatments, for a total of p = 4 parameters. 
K <- 20000
C <- 0
u <- as.matrix(c(-K, 1, K, -1))

## Set up correlation matrices
Vbeta_a_inv <- matrix(rep(0, p^2), nrow = p, ncol = p)

sigsq <- 4.04^2
tau1 <- tau2 <- 8700
sig <- sqrt(sigsq)
Vn <- matrix(0, nrow = n*p, ncol = n*p)
Vn[1:n, 1:n] <- diag(n)
Vn[(2*n - (n-1)):(2*n), (2*n - (n-1)):(2*n)] <- (tau1 / sig)^2 * diag(n)
Vn[(3*n - (n-1)):(3*n), (3*n - (n-1)):(3*n)] <- diag(n)
Vn[(4*n - (n-1)):(4*n), (4*n - (n-1)):(4*n)] <- (tau2 / sig)^2 * diag(n)

Vbeta_d <- (1 / sigsq) * 
matrix(c(4, 0, 3, 0, 0, 10^7, 0, 0, 3, 0, 4, 0, 0, 0, 0, 10^7),
nrow = 4, ncol = 4)
mu_beta_d <- as.matrix(c(5, 6000, 6.5, 7200))
mu_beta_a <- as.matrix(rep(0, p))
alpha <- 0.05
epsilon <- 10e-7
a_sig_d <- (sigsq / epsilon) + 2
b_sig_d <- sigsq * (a_sig_d - 1)
a_sig_a <- -p / 2
b_sig_a <- 0

bayesassurance::bayes_sim_unknownvar(n = n, p = 4, 
u = as.matrix(c(-K, 1, K, -1)), C = 0, R = 40,
Xn = NULL, Vn = Vn, Vbeta_d = Vbeta_d, 
Vbeta_a_inv = Vbeta_a_inv, mu_beta_d = mu_beta_d,
mu_beta_a = mu_beta_a, a_sig_a = a_sig_a, b_sig_a = b_sig_a, 
a_sig_d = a_sig_d, b_sig_d = b_sig_d, alt = "two.sided", alpha = 0.05, 
mc_iter = 500)

jpan928/bayesassurance_rpackage documentation built on June 25, 2022, 5:24 a.m.