pos1S: Probability of Success for a 1 Sample Design

Description

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

Arguments

prior

Prior for analysis.

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.

Methods (by class)

betaMix: Applies for binomial model with a mixture beta prior. The calculations use exact expressions.
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 $y$.
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 $y$.

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.

Examples

Run this code

# NOT RUN {
# 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)

# 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)

# our knowledge at the interim is just the posterior at
# interim such that the PoS is
pos_ia(post_ia)


# }

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