Find optimal sample size and progression criteria

Given a null and alternative hypothesis, this function finds the lowest sample size such that a design with optimal progression criteria (as determined by the function opt_pc) satisfies upper constraints on three operating characteristics.

Usage

tout_design(
  rho_0,
  rho_1,
  alpha_nom,
  beta_nom,
  gamma_nom = 1,
  eta_0 = 0.5,
  eta_1 = eta_0,
  tau = c(0, 0),
  max_n = NULL,
  n = NULL,
  x = NULL,
  sigma = NULL
)

Arguments

rho_0: null hypothesis.
rho_1: alternative hypothesis.
alpha_nom: nominal upper constraint on alpha.
beta_nom: nominal upper constraint on beta.
gamma_nom: nominal upper constraint on gamma. Defaults to 1.
eta_0: probability of an incorrect decision under the null hypothesis after an intermediate result. Defaults to 0.5.
eta_1: probability of an incorrect decision under the alternative hypothesis after an intermediate result. Defaults to eta_0.
tau: two element vector denoting lower and upper limits of the effect of adjustment.
max_n: optional upper limit to use in search over sample sizes.
n: optional sample size (optimised if left unspecified).
x: optional vector of decision thresholds (optimised if left unspecified).
sigma: standard deviation of outcome. If left unspecified, a binary outcome is assumed.

Value

An object of class tout, which is a list containing the following components:

valid: boolean indicating if the nominal constraints are met.
n: sample size.
thesholds: numeric vector of the two decision thresholds.
alpha: attained value of operating characteristic alpha.
beta: attained value of operating characteristic beta.
gamma: attained value of operating characteristic gamma.

Examples

rho_0 <- 0.5
rho_1 <- 0.7
alpha_nom <- 0.05
beta_nom <- 0.2

tout_design(rho_0, rho_1, alpha_nom, beta_nom)
#> Three-outcome design
#> 
#> Sample size: 37 
#> Decision thresholds: 23 23 
#> 
#> alpha = 0.04943587 
#> beta = 0.1929043 
#> gamma = 1 
#> 
#> Hypotheses: 0.5 (null), 0.7 (alternative)
#> Modification effect range: 0 0 
#> Error probability following an intermediate result: 0.5 0.5 

# Allowing for adjustment effects:

tout_design(rho_0, rho_1, alpha_nom, beta_nom, tau = c(0.08, 0.12))
#> Three-outcome design
#> 
#> Sample size: 145 
#> Decision thresholds: 69 82 
#> 
#> alpha = 0.04819701 
#> beta = 0.199825 
#> gamma = 0.3574197 
#> 
#> Hypotheses: 0.5 (null), 0.7 (alternative)
#> Modification effect range: 0.08 0.12 
#> Error probability following an intermediate result: 0.5 0.5 

# Allowing for different error probabilities following a pause decision

tout_design(rho_0, rho_1, alpha_nom, beta_nom, eta_0 = 0.3)
#> Three-outcome design
#> 
#> Sample size: 32 
#> Decision thresholds: 19 21 
#> 
#> alpha = 0.04983493 
#> beta = 0.1995952 
#> gamma = 0.7427826 
#> 
#> Hypotheses: 0.5 (null), 0.7 (alternative)
#> Modification effect range: 0 0 
#> Error probability following an intermediate result: 0.3 0.3 

# Designs for continuous outcomes:

tout_design(rho_0 = 0, rho_1 = 0.4, alpha_nom, beta_nom, sigma = 1)
#> Three-outcome design
#> 
#> Sample size: 39 
#> Decision thresholds: 1.55615 1.748788 
#> 
#> alpha = 0.05 
#> beta = 0.2 
#> gamma = 0.9292477 
#> 
#> Hypotheses: 0 (null), 0.4 (alternative)
#> Standard deviation: 1 
#> Modification effect range: 0 0 
#> Error probability following an intermediate result: 0.5 0.5