RBesT (version 1.8-1)

decision1S_boundary: Decision Boundary for 1 Sample Designs

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

decision1S_boundary(prior, n, decision, ...)
# S3 method for betaMix
decision1S_boundary(prior, n, decision, ...)
# S3 method for normMix
decision1S_boundary(prior, n, decision, sigma, eps = 1e-06, ...)
# S3 method for gammaMix
decision1S_boundary(prior, n, decision, eps = 1e-06, ...)

Value

Returns the critical value \(y_c\).

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 \(10^{-6}\).

Methods (by class)

• decision1S_boundary(betaMix): Applies for binomial model with a mixture beta prior. The calculations use exact expressions.

• decision1S_boundary(normMix): Applies for the normal model with known standard deviation \(\sigma\) 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 \(y\).

• decision1S_boundary(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 \(y\).

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 \(y = \sum_{i=1}^{n} y_i\) and for the normal case as the mean \(\bar{y} = 1/n \sum_{i=1}^n y_i\).

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, \(D(y \leq y_c) = 1\), 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).

See Also

Other design1S: decision1S(), oc1S(), pos1S()

Examples

# 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)