power.two.grp.random.a0 {BayesPPD}

Power/type I error calculation for two groups (treatment and control group, no covariates) with random a0

Description

Power/type I error calculation using normalized power priors for two groups (treatment and control group, no covariates) with random a_0

Usage

power.two.grp.random.a0(
  data.type,
  n.t,
  n.c,
  historical,
  nullspace.ineq = ">",
  samp.prior.mu.t,
  samp.prior.mu.c,
  samp.prior.var.t = 0,
  samp.prior.var.c = 0,
  prior.mu.t.shape1 = 1,
  prior.mu.t.shape2 = 1,
  prior.mu.c.shape1 = 1,
  prior.mu.c.shape2 = 1,
  prior.a0.shape1 = rep(1, 10),
  prior.a0.shape2 = rep(1, 10),
  lower.limits = rep(0, 10),
  upper.limits = rep(1, 10),
  slice.widths = rep(0.1, 10),
  delta = 0,
  gamma = 0.95,
  nMC = 10000,
  nBI = 250,
  N = 10000
)

Arguments

data.type

Character string specifying the type of response. The options are "Normal", "Bernoulli", "Poisson" and "Exponential".

n.t

Sample size of the treatment group for the simulated datasets.

n.c

Sample size of the control group for the simulated datasets.

historical

Matrix of historical dataset(s). If data.type is "Normal", historical is a matrix with three columns:

• The first column contains the sum of responses for the control group.

• The second column contains the sample size of the control group.

• The third column contains the sample variance of responses for the control group.

For all other data types, historical is a matrix with two columns:

• The first column contains the sum of responses for the control group.

• The second column contains the sample size of the control group.

Each row represents a historical dataset.

nullspace.ineq

Character string specifying the inequality of the null hypothesis. The options are ">" and "<". If ">" is specified, the null hypothesis (for non-exponential data) is H_0: \mu_t - \mu_c \ge \delta. If "<" is specified, the null hypothesis is H_0: \mu_t - \mu_c \le \delta. The default choice is ">".

samp.prior.mu.t

Vector of possible values of \mu_t to sample (with replacement) from. The vector contains realizations from the sampling prior (e.g. normal distribution) for \mu_t.

samp.prior.mu.c

Vector of possible values of \mu_c to sample (with replacement) from. The vector contains realizations from the sampling prior (e.g. normal distribution) for \mu_c.

samp.prior.var.t

Vector of possible values of \sigma^2_t to sample (with replacement) from. Only applies if data.type is "Normal". The vector contains realizations from the sampling prior (e.g. inverse-gamma distribution) for \sigma^2_t.

samp.prior.var.c

Vector of possible values of \sigma^2_c to sample (with replacement) from. Only applies if data.type is "Normal". The vector contains realizations from the sampling prior (e.g. inverse-gamma distribution) for \sigma^2_c

prior.mu.t.shape1

First hyperparameter of the initial prior for \mu_t. The default is 1. Does not apply if data.type is "Normal".

prior.mu.t.shape2

Second hyperparameter of the initial prior for \mu_t. The default is 1. Does not apply if data.type is "Normal".

prior.mu.c.shape1

First hyperparameter of the initial prior for \mu_c. The default is 1. Does not apply if data.type is "Normal".

prior.mu.c.shape2

Second hyperparameter of the initial prior for \mu_c. The default is 1. Does not apply if data.type is "Normal".

prior.a0.shape1

Vector of the first shape parameters of the independent beta priors for a_0. The length of the vector should be equal to the number of historical datasets. The default is a vector of one's.

prior.a0.shape2

Vector of the second shape parameters of the independent beta priors for a_0. The length of the vector should be equal to the number of historical datasets. The default is a vector of one's.

lower.limits

Vector of lower limits for parameters to be used by the slice sampler. The length of the vector should be equal to the number of historical datasets. The default is 0 for all parameters (may not be appropriate for all situations).

upper.limits

Vector of upper limits for parameters to be used by the slice sampler. The length of the vector should be equal to the number of historical datasets. The default is 1 for all parameters (may not be appropriate for all situations).

slice.widths

Vector of initial slice widths used by the slice sampler. The length of the vector should be equal to the number of historical datasets. The default is 0.1 for all parameter (may not be appropriate for all situations).

delta

Prespecified constant that defines the boundary of the null hypothesis. The default is zero.

gamma

Posterior probability threshold for rejecting the null. The null hypothesis is rejected if posterior probability is greater gamma. The default is 0.95.

nMC

Number of iterations (excluding burn-in samples) for the slice sampler or Gibbs sampler. The default is 10,000.

nBI

Number of burn-in samples for the slice sampler or Gibbs sampler. The default is 250.

N

Number of simulated datasets to generate. The default is 10,000.

Details

If data.type is "Bernoulli", "Poisson" or "Exponential", a single response from the treatment group is assumed to follow Bern(\mu_t), Pois(\mu_t) or Exp(rate=\mu_t), respectively, where \mu_t is the mean of responses for the treatment group. If data.type is "Normal", a single response from the treatment group is assumed to follow N(\mu_t, \tau^{-1}) where \tau is the precision parameter. The distributional assumptions for the control group data are analogous.

samp.prior.mu.t and samp.prior.mu.c can be generated using the sampling priors (see example).

If data.type is "Bernoulli", the initial prior for \mu_t is beta(prior.mu.t.shape1, prior.mu.t.shape2). If data.type is "Poisson", the initial prior for \mu_t is Gamma(prior.mu.t.shape1, rate=prior.mu.t.shape2). If data.type is "Exponential", the initial prior for \mu_t is Gamma(prior.mu.t.shape1, rate=prior.mu.t.shape2). The initial priors used for the control group data are analogous.

If data.type is "Normal", historical datasets are assumed to have the same precision parameter as the current dataset for computational simplicity. The initial prior for \tau is the Jeffery's prior, \tau^{-1}. The initial prior for the \mu_c is the uniform improper prior. Posterior samples of \mu_c and \tau are obtained through Gibbs sampling.

Independent beta(prior.a0.shape1,prior.a0.shape1) priors are used for a_0. Posterior samples of a_0 are obtained through slice sampling. The default lower limits for the parameters are 0. The default upper limits for the parameters are 1. The default slice widths for the parameters are 0.1. The defaults may not be appropriate for all situations, and the user can specify the appropriate limits and slice width for each parameter.

If a sampling prior with support in the null space is used, the value returned is a Bayesian type I error rate. If a sampling prior with support in the alternative space is used, the value returned is a Bayesian power.

Value

The function returns a S3 object with a summary method. Power or type I error is returned, depending on the sampling prior used. The posterior probabilities of the alternative hypothesis are returned. Average posterior means of \mu_t and \mu_c and their corresponding biases are returned. The average posterior mean of a_0 is returned. If data.type is "Normal", the average posterior mean of \tau is also returned.

References

Chen, Ming-Hui, et al. "Bayesian design of noninferiority trials for medical devices using historical data." Biometrics 67.3 (2011): 1163-1170.

Neal, Radford M. Slice sampling. Ann. Statist. 31 (2003), no. 3, 705–767.

See Also

two.grp.random.a0

Examples

data.type <- "Bernoulli"
n.t <- 100
n.c <- 100

# Simulate three historical datasets
historical <- matrix(0, ncol=2, nrow=3)
historical[1,] <- c(70, 100)
historical[2,] <- c(60, 100)
historical[3,] <- c(50, 100)

# Generate sampling priors
set.seed(1)
b_st1 <- b_st2 <- 1
b_sc1 <- b_sc2 <- 1
samp.prior.mu.t <- rbeta(50000, b_st1, b_st2)
samp.prior.mu.c <- rbeta(50000, b_st1, b_st2)
# The null hypothesis here is H0: mu_t - mu_c >= 0. To calculate power,
# we can provide samples of mu.t and mu.c such that the mass of mu_t - mu_c < 0.
# To calculate type I error, we can provide samples of mu.t and mu.c such that
# the mass of mu_t - mu_c >= 0.
sub_ind <- which(samp.prior.mu.t < samp.prior.mu.c)
# Here, mass is put on the alternative region, so power is calculated.
samp.prior.mu.t <- samp.prior.mu.t[sub_ind]
samp.prior.mu.c <- samp.prior.mu.c[sub_ind]

N <- 10 # N should be larger in practice
result <- power.two.grp.random.a0(data.type=data.type, n.t=n.t, n.c=n.c, historical=historical,
                                  samp.prior.mu.t=samp.prior.mu.t, samp.prior.mu.c=samp.prior.mu.c,
                                  delta=0, nMC=10000, nBI=250, N=N)
summary(result)