Probability of Success for 2 Sample Design

The pos2S function defines a 2 sample design (priors, sample sizes & decision function) for the calculation of the probability of success. A function is returned which calculates the calculates the frequency at which the decision function is evaluated to 1 when parameters are distributed according to the given distributions.

Usage

pos2S(prior1, prior2, n1, n2, decision, ...)

# S3 method for class 'betaMix'
pos2S(prior1, prior2, n1, n2, decision, eps, ...)

# S3 method for class 'normMix'
pos2S(
  prior1,
  prior2,
  n1,
  n2,
  decision,
  sigma1,
  sigma2,
  eps = 1e-06,
  Ngrid = 10,
  ...
)

# S3 method for class 'gammaMix'
pos2S(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 mix1 and mix2 will return the frequencies at which the decision function is evaluated to 1. Each argument is expected to be a mixture distribution representing the assumed true distribution of the parameter in each group.

Details

The pos2S function defines a 2 sample design and returns a function which calculates its probability of success. The probability of success is the frequency with which the decision function is evaluated to 1 under the assumption of a given true distribution of the data implied by a distirbution of the parameters $\theta_1$ and $\theta_2$.

The calculation is analogous to the operating characeristics oc2S with the difference that instead of assuming known (point-wise) true parameter values a distribution is specified for each parameter.

Calling the pos2S function calculates the decision boundary $D_1(y_2)$ and returns a function which can be used to evaluate the PoS for different predictive distributions. It is evaluated as

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

where $F$ is the distribution function of the sampling distribution and $p(\theta_1)$ and $p(\theta_2)$ specifies the assumed true distribution of the parameters $\theta_1$ and $\theta_2$, respectively. Each distribution $p(\theta_1)$ and $p(\theta_2)$ is a mixture distribution and given as the mix1 and mix2 argument to the function.

For example, in the binary case an integration of the predictive distribution, the BetaBinomial, instead of the binomial distribution will be performed over the data space wherever the decision function is evaluated to 1. All other aspects of the calculation are as for the 2-sample operating characteristics, see oc2S.

Methods (by class)

pos2S(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.
pos2S(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.
pos2S(gammaMix): Applies for the Poisson model with a gamma mixture prior for the rate parameter. The function pos2S 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)

# example interim outcome
postP_interim <- postmix(priorP, n=10, m=-50)
#> Using default prior reference scale 88
postT_interim <- postmix(priorT, n=20, m=-80)
#> Using default prior reference scale 88

# assume that mean -50 / -80 were observed at the interim for
# placebo control(n=10) / active treatment(n=20) which gives
# the posteriors
postP_interim
#> Univariate normal mixture
#> Reference scale: 88
#> Mixture Components:
#>   comp1    
#> w   1.00000
#> m -49.33333
#> s  16.06653
postT_interim
#> Univariate normal mixture
#> Reference scale: 88
#> Mixture Components:
#>   comp1    
#> w   1.00000
#> m -79.99600
#> s  19.67691

# then the PoS to succeed after another 20/30 patients is
pos_final <- pos2S(postP_interim, postT_interim, 20, 30, successCrit)
#> Using default prior 1 reference scale 88
#> Using default prior 2 reference scale 88

pos_final(postP_interim, postT_interim)
#> [1] 0.145567