The function sets up a 1 sample one-sided decision function with an arbitrary number of conditions.
Usage
decision1S(pc = 0.975, qc = 0, lower.tail = TRUE)
oc1Sdecision(pc = 0.975, qc = 0, lower.tail = TRUE)Arguments
- pc
Vector of critical cumulative probabilities.
- qc
Vector of respective critical values. Must match the length of
pc.- lower.tail
Logical; if
TRUE(default), probabilities are \(P(X \leq x)\), otherwise, \(P(X > x)\).
Value
The function returns a decision function which takes two arguments. The first argument is expected to be a mixture (posterior) distribution which is tested if the specified conditions are met. The logical second argument determines if the function acts as an indicator function or if the function returns the distance from the decision boundary for each condition in log-space, i.e. the distance is 0 at the decision boundary, negative for a 0 decision and positive for a 1 decision.
Details
The function creates a one-sided decision function which
takes two arguments. The first argument is expected to be a mixture
(posterior) distribution. This distribution is tested whether it
fulfills all the required threshold conditions specified with the
pc and qc arguments and returns 1 if all conditions
are met and 0 otherwise. Hence, for lower.tail=TRUE
condition \(i\) is equivalent to
$$P(\theta \leq q_{c,i}) > p_{c,i}$$
and the decision function is implemented as indicator function on the basis of the heavy-side step function \(H(x)\) which is \(0\) for \(x \leq 0\) and \(1\) for \(x > 0\). As all conditions must be met, the final indicator function returns
$$\Pi_i H_i(P(\theta \leq q_{c,i}) - p_{c,i} ).$$
When the second argument is set to TRUE a distance metric is
returned component-wise per defined condition as
$$ D_i = \log(P(\theta < q_{c,i})) - \log(p_{c,i}) .$$
These indicator functions can be used as input for 1-sample
boundary, OC or PoS calculations using oc1S or
pos1S .
References
Neuenschwander B, Rouyrre N, Hollaender H, Zuber E, Branson M. A proof of concept phase II non-inferiority criterion. Stat. in Med.. 2011, 30:1618-1627
See also
Other design1S:
decision1S_boundary(),
oc1S(),
pos1S()
Examples
# see Neuenschwander et al., 2011
# example is for a time-to-event trial evaluating non-inferiority
# using a normal approximation for the log-hazard ratio
# reference scale
s <- 2
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 ) # n for which design was intended
nL <- 233
c1 <- theta_ni - za * s / sqrt(n1)
# flat prior
flat_prior <- mixnorm(c(1,0,100), sigma=s)
# standard NI design
decA <- decision1S(1 - alpha, theta_ni, lower.tail=TRUE)
# for double criterion with indecision point (mean estimate must be
# lower than this)
theta_c <- c1
# double criterion design
# statistical significance (like NI design)
dec1 <- decision1S(1-alpha, theta_ni, lower.tail=TRUE)
# require mean to be at least as good as theta_c
dec2 <- decision1S(0.5, theta_c, lower.tail=TRUE)
# combination
decComb <- decision1S(c(1-alpha, 0.5), c(theta_ni, theta_c), lower.tail=TRUE)
theta_eval <- c(theta_a, theta_c, theta_ni)
# we can display the decision function definition
decComb
#> 1 sample decision function
#> Conditions for acceptance:
#> P(theta <= 0.4) > 0.95
#> P(theta <= 0.13576435472344) > 0.5
# and use it to decide if a given distribution fulfills all
# criterions defined
# for the prior
decComb(flat_prior)
#> [1] 0
# or for a possible outcome of the trial
# here with HR of 0.8 for 40 events
decComb(postmix(flat_prior, m=log(0.8), n=40))
#> Using default prior reference scale 2
#> [1] 1