Decision Boundary for 1 Sample Designs

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 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 $10^{-6}$.

Value

Returns the critical value $y_c$.

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).

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 $y$.

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)
#> Using default prior reference scale 2
#> [1] 0.1357511

# ... is limited by the statistical significance ...
decision1S_boundary(flat_prior, nL, dec1)
#> Using default prior reference scale 2
#> [1] 0.1844494

# ... or the indecision point (whatever is smaller)
decision1S_boundary(flat_prior, nL, dec2)
#> Using default prior reference scale 2
#> [1] 0.1357511