Probability of Success for a 1 Sample Design

The pos1S function defines a 1 sample design (prior, sample size, decision function) for the calculation of the frequency at which the decision is evaluated to 1 when assuming a distribution for the parameter. A function is returned which performs the actual operating characteristics calculations.

Usage

pos1S(prior, n, decision, ...)

# S3 method for class 'betaMix'
pos1S(prior, n, decision, ...)

# S3 method for class 'normMix'
pos1S(prior, n, decision, sigma, eps = 1e-06, ...)

# S3 method for class 'gammaMix'
pos1S(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 a function that takes as single argument mix, which is the mixture distribution of the control parameter. Calling this function with a mixture distribution then calculates the PoS.

Details

The pos1S function defines a 1 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 parameter $\theta$.

Calling the pos1S function calculates the critical value $y_c$ and returns a function which can be used to evaluate the PoS for different predictive distributions and is evaluated as

$$ \int F(y_c|\theta) p(\theta) d\theta, $$

where $F$ is the distribution function of the sampling distribution and $p(\theta)$ specifies the assumed true distribution of the parameter $\theta$. The distribution $p(\theta)$ is a mixture distribution and given as the mix argument to the function.

Methods (by class)

pos1S(betaMix): Applies for binomial model with a mixture beta prior. The calculations use exact expressions.
pos1S(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 pos1S 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 $y$.
pos1S(gammaMix): Applies for the Poisson model with a gamma mixture prior for the rate parameter. The function pos1S 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)

# assume we would like to conduct at an interim analysis
# of PoS after having observed 20 events with a HR of 0.8.
# We first need the posterior at the interim ...
post_ia <- postmix(flat_prior, m=log(0.8), n=20)
#> Using default prior reference scale 2

# dual criterion
decComb <- decision1S(c(1-alpha, 0.5), c(theta_ni, theta_c), lower.tail=TRUE)

# ... and we would like to know the PoS for a successful
# trial at the end when observing 10 more events
pos_ia <- pos1S(post_ia, 10, decComb)
#> Using default prior reference scale 2

# our knowledge at the interim is just the posterior at
# interim such that the PoS is
pos_ia(post_ia)
#> [1] 0.534741