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.
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, ...)
prior |
Prior for analysis. |
n |
Sample size for the experiment. |
decision |
One-sample decision function to use; see |
... |
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 |
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
θ.
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|θ) p(θ) dθ,
where F is the distribution function of the sampling
distribution and p(θ) specifies the assumed true
distribution of the parameter θ. The distribution
p(θ) is a mixture distribution and given as the
mix argument to the function.
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.
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 σ 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.
Other design1S: decision1S_boundary,
decision1S, oc1S
# 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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