Decision Boundary for 2 Sample Designs

The decision2S_boundary function defines a 2 sample design (priors, sample sizes, decision function) for the calculation of the decision boundary. A function is returned which calculates the critical value of the first sample $y_{1,c}$ as a function of the outcome in the second sample $y_2$. At the decision boundary, the decision function will change between 0 (failure) and 1 (success) for the respective outcomes.

Usage

decision2S_boundary(prior1, prior2, n1, n2, decision, ...)

# S3 method for class 'betaMix'
decision2S_boundary(prior1, prior2, n1, n2, decision, eps, ...)

# S3 method for class 'normMix'
decision2S_boundary(
  prior1,
  prior2,
  n1,
  n2,
  decision,
  sigma1,
  sigma2,
  eps = 1e-06,
  Ngrid = 10,
  ...
)

# S3 method for class 'gammaMix'
decision2S_boundary(prior1, prior2, n1, n2, decision, eps = 1e-06, ...)

Arguments

prior1: Prior for sample 1.
prior2: Prior for sample 2.
n1, n2: Sample size of the respective samples. Sample size n1 must be greater than 0 while sample size n2 must be greater or equal to 0.
decision: Two-sample decision function to use; see decision2S.
...: Optional arguments.
eps: Support of random variables are determined as the interval covering 1-eps probability mass. Defaults to $10^{-6}$.
sigma1: The fixed reference scale of sample 1. If left unspecified, the default reference scale of the prior 1 is assumed.
sigma2: The fixed reference scale of sample 2. If left unspecified, the default reference scale of the prior 2 is assumed.
Ngrid: Determines density of discretization grid on which decision function is evaluated (see below for more details).

Value

Returns a function with a single argument. This function calculates in dependence of the outcome $y_2$ in sample 2 the critical value $y_{1,c}$ for which the defined design will change the decision from 0 to 1 (or vice versa, depending on the decision function).

Details

For a 2 sample design the specification of the priors, the sample sizes and the decision function, $D(y_1,y_2)$, uniquely defines the decision boundary

$$D_1(y_2) = \max_{y_1}\{D(y_1,y_2) = 1\},$$

which is the critical value of $y_{1,c}$ conditional on the value of $y_2$ whenever the decision $D(y_1,y_2)$ function changes its value from 0 to 1 for a decision function with lower.tail=TRUE (otherwise the definition is $D_1(y_2) = \max_{y_1}\{D(y_1,y_2) = 0\}$). The decision function may change at most at a single critical value for given $y_{2}$ as only one-sided decision functions are supported. Here, $y_2$ is defined for binary and Poisson endpoints as the sufficient statistic $y_2 = \sum_{i=1}^{n_2} y_{2,i}$ and for the normal case as the mean $\bar{y}_2 = 1/n_2 \sum_{i=1}^{n_2} y_{2,i}$.

Methods (by class)

decision2S_boundary(betaMix): Applies for binomial model with a mixture beta prior. The calculations use exact expressions. If the optional argument eps is defined, then an approximate method is used which limits the search for the decision boundary to the region of 1-eps probability mass. This is useful for designs with large sample sizes where an exact approach is very costly to calculate.
decision2S_boundary(normMix): Applies for the normal model with known standard deviation $\sigma$ and normal mixture priors for the means. As a consequence from the assumption of a known standard deviation, the calculation discards sampling uncertainty of the second moment. The function has two extra arguments (with defaults): eps ($10^{-6}$) and Ngrid (10). The decision boundary is searched in the region of probability mass 1-eps, respectively for $y_1$ and $y_2$. The continuous decision function is evaluated at a discrete grid, which is determined by a spacing with $\delta_2 = \sigma_2/\sqrt{N_{grid}}$. Once the decision boundary is evaluated at the discrete steps, a spline is used to inter-polate the decision boundary at intermediate points.
decision2S_boundary(gammaMix): Applies for the Poisson model with a gamma mixture prior for the rate parameter. The function decision2S_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 $y_1$ and $y_2$, respectively.

Examples


# see ?decision2S for details of example
priorT <- mixnorm(c(1,   0, 0.001), sigma=88, param="mn")
priorP <- mixnorm(c(1, -49, 20   ), sigma=88, param="mn")
# the success criteria is for delta which are larger than some
# threshold value which is why we set lower.tail=FALSE
successCrit  <- decision2S(c(0.95, 0.5), c(0, 50), FALSE)
# the futility criterion acts in the opposite direction
futilityCrit <- decision2S(c(0.90)     , c(40),    TRUE)

# success criterion boundary
successBoundary <- decision2S_boundary(priorP, priorT, 10, 20, successCrit)
#> Using default prior 1 reference scale 88
#> Using default prior 2 reference scale 88

# futility criterion boundary
futilityBoundary <- decision2S_boundary(priorP, priorT, 10, 20, futilityCrit)
#> Using default prior 1 reference scale 88
#> Using default prior 2 reference scale 88

curve(successBoundary(x), -25:25 - 49, xlab="y2", ylab="critical y1")
curve(futilityBoundary(x), lty=2, add=TRUE)


# hence, for mean in sample 2 of 10, the critical value for y1 is
y1c <- futilityBoundary(-10)

# around the critical value the decision for futility changes
futilityCrit(postmix(priorP, m=y1c+1E-3, n=10), postmix(priorT, m=-10, n=20))
#> Using default prior reference scale 88
#> Using default prior reference scale 88
#> [1] 0
futilityCrit(postmix(priorP, m=y1c-1E-3, n=10), postmix(priorT, m=-10, n=20))
#> Using default prior reference scale 88
#> Using default prior reference scale 88
#> [1] 1