# decision1S_boundary: Decision Boundary for 1 Sample Designs In RBesT: R Bayesian Evidence Synthesis Tools

## Description

Calculates the decision boundary for a 1 sample design. This is the critical value at which the decision function will change from 0 (failure) to 1 (success).

## Usage

  1 2 3 4 5 6 7 8 9 10 decision1S_boundary(prior, n, decision, ...) ## S3 method for class 'betaMix' decision1S_boundary(prior, n, decision, ...) ## S3 method for class 'normMix' decision1S_boundary(prior, n, decision, sigma, eps = 1e-06, ...) ## S3 method for class 'gammaMix' decision1S_boundary(prior, n, decision, eps = 1e-06, ...) 

## Arguments

 prior Prior for analysis. n Sample size for the experiment. decision One-sample decision function to use; see decision1S. ... Optional arguments. sigma The fixed reference scale. If left unspecified, the default reference scale of the prior is assumed. eps Support of random variables are determined as the interval covering 1-eps probability mass. Defaults to 10^{-6}.

## Details

The specification of the 1 sample design (prior, sample size and decision function, D(y)), uniquely defines the decision boundary

y_c = max_{y}{D(y) = 1},

which is the maximal value of y whenever the decision D(y) function changes its value from 1 to 0 for a decision function with lower.tail=TRUE (otherwise the definition is y_c = max_{y}{D(y) = 0}). The decision function may change at most at a single critical value as only one-sided decision functions are supported. Here, y is defined for binary and Poisson endpoints as the sufficient statistic y = ∑_{i=1}^{n} y_i and for the normal case as the mean \bar{y} = 1/n ∑_{i=1}^n y_i.

The convention for the critical value y_c depends on whether a left (lower.tail=TRUE) or right-sided decision function (lower.tail=FALSE) is used. For lower.tail=TRUE the critical value y_c is the largest value for which the decision is 1, D(y ≤q y_c) = 1, while for lower.tail=FALSE then D(y > y_c) = 1 holds. This is aligned with the cumulative density function definition within R (see for example pbinom).

## Value

Returns the critical value y_c.

## Methods (by class)

• betaMix: Applies for binomial model with a mixture beta prior. The calculations use exact expressions.

• normMix: Applies for the normal model with known standard deviation σ and a normal mixture prior for the mean. As a consequence from the assumption of a known standard deviation, the calculation discards sampling uncertainty of the second moment. The function decision1S_boundary has an extra argument eps (defaults to 10^{-6}). The critical value y_c is searched in the region of probability mass 1-eps for y.

• gammaMix: Applies for the Poisson model with a gamma mixture prior for the rate parameter. The function decision1S_boundary takes an extra argument eps (defaults to 10^{-6}) which determines the region of probability mass 1-eps where the boundary is searched for y.

Other design1S: decision1S(), oc1S(), pos1S()
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 # non-inferiority example using normal approximation of log-hazard # ratio, see ?decision1S for all details s <- 2 flat_prior <- mixnorm(c(1,0,100), sigma=s) nL <- 233 theta_ni <- 0.4 theta_a <- 0 alpha <- 0.05 beta <- 0.2 za <- qnorm(1-alpha) zb <- qnorm(1-beta) n1 <- round( (s * (za + zb)/(theta_ni - theta_a))^2 ) theta_c <- theta_ni - za * s / sqrt(n1) # double criterion design # statistical significance (like NI design) dec1 <- decision1S(1-alpha, theta_ni, lower.tail=TRUE) # require mean to be at least as good as theta_c dec2 <- decision1S(0.5, theta_c, lower.tail=TRUE) # combination decComb <- decision1S(c(1-alpha, 0.5), c(theta_ni, theta_c), lower.tail=TRUE) # critical value of double criterion design decision1S_boundary(flat_prior, nL, decComb) # ... is limited by the statistical significance ... decision1S_boundary(flat_prior, nL, dec1) # ... or the indecision point (whatever is smaller) decision1S_boundary(flat_prior, nL, dec2)