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Calculate Bayesian Posterior Probability or Bayesian Posterior Predictive Probability for a Clinical Trial When Outcome is Continuous

Source: R/pPPsinglecontinuous.R
pPPsinglecontinuous.Rd

This function computes Bayesian posterior probability or posterior predictive probability for continuous outcome clinical trials. The function supports controlled, uncontrolled, and external control designs with Normal-Inverse-Chi-squared or vague priors, using three calculation methods: numerical integration, Monte Carlo simulation, and Welch-Satterthwaite approximation. For external control designs, power priors are incorporated using exact conjugate representation as Normal-Inverse-Chi-squared distributions, enabling closed-form computation without MCMC sampling.

Usage

pPPsinglecontinuous(
  prob = "posterior",
  design = "controlled",
  prior = "vague",
  CalcMethod = "NI",
  theta0,
  nMC = NULL,
  n1,
  n2,
  m1 = NULL,
  m2 = NULL,
  kappa01 = NULL,
  kappa02 = NULL,
  nu01 = NULL,
  nu02 = NULL,
  mu01 = NULL,
  mu02 = NULL,
  sigma01 = NULL,
  sigma02 = NULL,
  bar.y1,
  bar.y2,
  s1,
  s2,
  r = NULL,
  ne1 = NULL,
  ne2 = NULL,
  alpha01 = NULL,
  alpha02 = NULL,
  bar.ye1 = NULL,
  bar.ye2 = NULL,
  se1 = NULL,
  se2 = NULL,
  lower.tail = TRUE
)

Arguments

prob

A character string specifying the type of probability to use (prob = 'posterior' or prob = 'predictive').

design

A character string specifying the type of trial design (design = 'controlled', design = 'uncontrolled', or design = 'external').

prior

A character string specifying the prior distribution (prior = 'N-Inv-Chisq' or prior = 'vague').

CalcMethod

A character string specifying the calculation method (CalcMethod = 'NI' for numerical integration, CalcMethod = 'MC' for Monte Carlo method, or CalcMethod = 'WS' for Welch-Satterthwaite approximation).

theta0

A numeric value representing the pre-specified threshold value.

nMC

A positive integer representing the number of iterations for Monte Carlo simulation (required if CalcMethod = 'MC').

n1

A positive integer representing the number of patients in group 1 for the PoC trial.

n2

A positive integer representing the number of patients in group 2 for the PoC trial.

m1

A positive integer representing the number of patients in group 1 for the future trial data.

m2

A positive integer representing the number of patients in group 2 for the future trial data.

kappa01

A positive numeric value representing the prior precision parameter related to the mean for conjugate prior of Normal-Inverse-Chi-squared in group 1.

kappa02

A positive numeric value representing the prior precision parameter related to the mean for conjugate prior of Normal-Inverse-Chi-squared in group 2.

nu01

A positive numeric value representing the prior degrees of freedom related to the variance for conjugate prior of Normal-Inverse-Chi-squared in group 1.

nu02

A positive numeric value representing the prior degrees of freedom related to the variance for conjugate prior of Normal-Inverse-Chi-squared in group 2.

mu01

A numeric value representing the prior mean value of outcomes in group 1 for the PoC trial.

mu02

A numeric value representing the prior mean value of outcomes in group 2 for the PoC trial.

sigma01

A positive numeric value representing the prior standard deviation of the outcomes for group 1.

sigma02

A positive numeric value representing the prior standard deviation of the outcomes for group 2.

bar.y1

A numeric value representing the sample mean of group 1.

bar.y2

A numeric value representing the sample mean of group 2.

s1

A positive numeric value representing the sample standard deviation of group 1.

s2

A positive numeric value representing the sample standard deviation of group 2.

r

A positive numeric value representing the ratio of the hypothesized values for the null hypotheses (required if design = 'uncontrolled').

ne1

A positive integer representing the number of patients in group 1 for the external data (required if external treatment data available).

ne2

A positive integer representing the number of patients in group 2 for the external data (required if external control data available).

alpha01

A numeric value in [0, 1]: representing the scale parameter (power prior) for group 1 (required if external treatment data available). Controls the degree of borrowing: 0 = no borrowing, 1 = full borrowing.

alpha02

A numeric value in [0, 1] representing the scale parameter (power prior) for group 2 (required if external control data available). Controls the degree of borrowing: 0 = no borrowing, 1 = full borrowing.

bar.ye1

A numeric value representing the external sample mean of group 1 (required if external treatment data available).

bar.ye2

A numeric value representing the external sample mean of group 2 (required if external control data available).

se1

A positive numeric value representing the external sample standard deviation of group 1 (required if external treatment data available).

se2

A positive numeric value representing the external sample standard deviation of group 2 (required if external control data available).

lower.tail

logical; if TRUE (default), probabilities are P(theta <= theta0), otherwise, P(theta > theta0)

Value

A numeric vector representing the Bayesian posterior probability or Bayesian posterior predictive probability. The function can handle vectorized inputs.

Details

The function can obtain:

  • Bayesian posterior probability

  • Bayesian posterior predictive probability

Prior distribution of mean and variance of outcomes for each treatment group (k=1,2) can be either (1) Normal-Inverse-Chi-squared or (2) Vague. The posterior distribution or posterior predictive distribution of outcome for each treatment group follows a t-distribution.

For controlled and uncontrolled designs, three calculation methods are available:

  • NI: Numerical integration method for exact computation

  • MC: Monte Carlo simulation for flexible approximation

  • WS: Welch-Satterthwaite approximation for computational efficiency

For external control designs, power priors are incorporated using exact conjugate representation:

  • Power priors for normal data are mathematically equivalent to Normal-Inverse-Chi-squared distributions

  • This enables closed-form posterior computation without MCMC sampling

  • Alpha parameters control the degree of borrowing (0 = no borrowing, 1 = full borrowing)

  • The method preserves complete Bayesian rigor with no approximation

The external design supports:

  • External control data only (ne2, alpha02, bar.ye2, se2)

  • External treatment data only (ne1, alpha01, bar.ye1, se1)

  • Both external control and treatment data

Examples

# Example 1: Numerical Integration (NI) method with N-Inv-Chisq prior
pPPsinglecontinuous(
  prob = 'posterior', design = 'controlled', prior = 'N-Inv-Chisq', CalcMethod = 'NI',
  theta0 = 2, n1 = 12, n2 = 12, kappa01 = 5, kappa02 = 5, nu01 = 5, nu02 = 5,
  mu01 = 5, mu02 = 5, sigma01 = sqrt(5), sigma02 = sqrt(5),
  bar.y1 = 2, bar.y2 = 0, s1 = 1, s2 = 1, lower.tail = FALSE
)
#> [1] 0.4167132

# Example 2: Monte Carlo (MC) method with vague prior
pPPsinglecontinuous(
  prob = 'posterior', design = 'controlled', prior = 'vague', CalcMethod = 'MC',
  theta0 = 1, nMC = 10000, n1 = 12, n2 = 12,
  bar.y1 = 3, bar.y2 = 1, s1 = 1.5, s2 = 1.2, lower.tail = FALSE
)
#> [1] 0.9501

# Example 3: Welch-Satterthwaite (WS) approximation with N-Inv-Chisq prior
pPPsinglecontinuous(
  prob = 'predictive', design = 'controlled', prior = 'N-Inv-Chisq', CalcMethod = 'WS',
  theta0 = 0.5, n1 = 15, n2 = 15, m1 = 100, m2 = 100,
  kappa01 = 3, kappa02 = 3, nu01 = 4, nu02 = 4, mu01 = 2, mu02 = 2,
  sigma01 = 2, sigma02 = 2, bar.y1 = 2.5, bar.y2 = 1.8, s1 = 1.8, s2 = 1.6, lower.tail = FALSE
)
#> [1] 0.5429566

# Example 4: External control design with power prior (NI method)
pPPsinglecontinuous(
  prob = 'posterior', design = 'external', prior = 'vague', CalcMethod = 'NI',
  theta0 = 1.5, n1 = 12, n2 = 12, bar.y1 = 4, bar.y2 = 2, s1 = 1.2, s2 = 1.1,
  ne2 = 20, alpha02 = 0.5, bar.ye2 = 1.8, se2 = 1.0, lower.tail = FALSE
)
#> [1] 0.9116773

# Example 5: External design with both treatment and control data
pPPsinglecontinuous(
  prob = 'posterior', design = 'external', prior = 'N-Inv-Chisq', CalcMethod = 'WS',
  theta0 = 1.0, n1 = 15, n2 = 15, bar.y1 = 3.5, bar.y2 = 2.0, s1 = 1.3, s2 = 1.1,
  kappa01 = 2, kappa02 = 2, nu01 = 3, nu02 = 3, mu01 = 3, mu02 = 2,
  sigma01 = 1.5, sigma02 = 1.5,
  ne1 = 25, ne2 = 25, alpha01 = 0.7, alpha02 = 0.7,
  bar.ye1 = 3.2, bar.ye2 = 1.9, se1 = 1.4, se2 = 1.2, lower.tail = FALSE
)
#> [1] 0.8868462