power.two.grp.fixed.a0: Power/type I error calculation for data with two groups...

Description Usage Arguments Details Value References See Also Examples

View source: R/main_func.R

Description

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

Usage

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power.two.grp.fixed.a0(
  data.type,
  n.t,
  n.c,
  historical = matrix(0, 1, 4),
  samp.prior.mu.t,
  samp.prior.mu.c,
  samp.prior.var.t,
  samp.prior.var.c,
  prior.mu.t.shape1 = 1,
  prior.mu.t.shape2 = 1,
  prior.mu.c.shape1 = 1,
  prior.mu.c.shape2 = 1,
  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

(Optional) matrix of historical dataset(s). If data.type is "Normal", historical is a matrix with four 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.

  • The fourth column contains the discounting parameter value a_0 (between 0 and 1).

For all other data types, 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 discounting parameter value a_0 (between 0 and 1).

Each row represents a historical dataset.

samp.prior.mu.t

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

samp.prior.mu.c

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

samp.prior.var.t

Vector of possible values of σ^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 σ^2_t.

samp.prior.var.c

Vector of possible values of σ^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 σ^2_c

prior.mu.t.shape1

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

prior.mu.t.shape2

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

prior.mu.c.shape1

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

prior.mu.c.shape2

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

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(μ_t), Pois(μ_t) or Exp(rate=μ_t), respectively, where μ_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(μ_t, τ^{-1}) where τ 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 μ_t is beta(prior.mu.t.shape1, prior.mu.t.shape2). If data.type is "Poisson", the initial prior for μ_t is Gamma(prior.mu.t.shape1, rate=prior.mu.t.shape2). If data.type is "Exponential", the initial prior for μ_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", each historical dataset D_{0k} is assumed to have a different precision parameter τ_k. The initial prior for τ is the Jeffery's prior, τ^{-1}, and the initial prior for τ_k is τ_k^{-1}. The initial prior for the μ_c is the uniform improper prior.

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.

If data.type is "Normal", Gibbs sampling is used for model fitting. For all other data types, numerical integration is used for modeling fitting.

Value

Power or type I error is returned, depending on the sampling prior used. If data.type is "Normal", average posterior means of μ_c, τ and τ_k's (if historical data is given) are also returned.

References

Yixuan Qiu, Sreekumar Balan, Matt Beall, Mark Sauder, Naoaki Okazaki and Thomas Hahn (2019). RcppNumerical: 'Rcpp' Integration for Numerical Computing Libraries. R package version 0.4-0. https://CRAN.R-project.org/package=RcppNumerical

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

See Also

two.grp.fixed.a0

Examples

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

# Simulate three historical datasets
historical <- matrix(0, ncol=3, nrow=3)
historical[1,] <- c(70, 100, 0.3)
historical[2,] <- c(60, 100, 0.5)
historical[3,] <- c(50, 100, 0.7)

# 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 <- 1000 # N should be larger in practice
result <- power.two.grp.fixed.a0(data.type=data.type, n.t=n.t, n.c=n.t, historical=historical,
                                 samp.prior.mu.t=samp.prior.mu.t, samp.prior.mu.c=samp.prior.mu.c,
                                 delta=0, N=N)

BayesPPD documentation built on Sept. 8, 2021, 5:06 p.m.