Operating Characteristics for 2 Sample Design

The oc2S function defines a 2 sample design (priors, sample sizes & decision function) for the calculation of operating characeristics. A function is returned which calculates the calculates the frequency at which the decision function is evaluated to 1 when assuming known parameters.

Usage

oc2S(prior1, prior2, n1, n2, decision, ...)

# S3 method for class 'betaMix'
oc2S(prior1, prior2, n1, n2, decision, eps, ...)

# S3 method for class 'normMix'
oc2S(
  prior1,
  prior2,
  n1,
  n2,
  decision,
  sigma1,
  sigma2,
  eps = 1e-06,
  Ngrid = 10,
  ...
)

# S3 method for class 'gammaMix'
oc2S(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 which when called with two arguments theta1 and theta2 will return the frequencies at which the decision function is evaluated to 1 whenever the data is distributed according to the known parameter values in each sample. Note that the returned function takes vector arguments.

Details

The oc2S function defines a 2 sample design and returns a function which calculates its operating characteristics. This is the frequency with which the decision function is evaluated to 1 under the assumption of a given true distribution of the data defined by the known parameter $\theta_1$ and $\theta_2$. The 2 sample design is defined by the priors, the sample sizes and the decision function, $D(y_1,y_2)$. These uniquely define the decision boundary , see decision2S_boundary.

Calling the oc2S function calculates the decision boundary $D_1(y_2)$ (see decision2S_boundary) and returns a function which can be used to calculate the desired frequency which is evaluated as

$$ \int f_2(y_2|\theta_2) F_1(D_1(y_2)|\theta_1) dy_2. $$

See below for examples and specifics for the supported mixture priors.

Methods (by class)

oc2S(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.
oc2S(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.
oc2S(gammaMix): Applies for the Poisson model with a gamma mixture prior for the rate parameter. The function oc2S 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.

References

Schmidli H, Gsteiger S, Roychoudhury S, O'Hagan A, Spiegelhalter D, Neuenschwander B. Robust meta-analytic-predictive priors in clinical trials with historical control information. Biometrics 2014;70(4):1023-1032.

Examples


# example from Schmidli et al., 2014
dec <- decision2S(0.975, 0, lower.tail=FALSE)

prior_inf <- mixbeta(c(1, 4, 16))
prior_rob <- robustify(prior_inf, weight=0.2, mean=0.5)
prior_uni <- mixbeta(c(1, 1,  1))

N <- 40
N_ctl <- N - 20

# compare designs with different priors
design_uni <- oc2S(prior_uni, prior_uni, N, N_ctl, dec)
design_inf <- oc2S(prior_uni, prior_inf, N, N_ctl, dec)
design_rob <- oc2S(prior_uni, prior_rob, N, N_ctl, dec)

# type I error
curve(design_inf(x,x), 0, 1)
curve(design_uni(x,x), lty=2, add=TRUE)
curve(design_rob(x,x), lty=3, add=TRUE)


# power
curve(design_inf(0.2+x,0.2), 0, 0.5)
curve(design_uni(0.2+x,0.2), lty=2, add=TRUE)
curve(design_rob(0.2+x,0.2), lty=3, add=TRUE)