The function sets up a 2 sample one-sided decision function with an arbitrary number of conditions on the difference distribution.
decision2S( pc =0.975, qc =0, lower.tail =TRUE, link = c("identity","logit","log"))oc2Sdecision( pc =0.975, qc =0, lower.tail =TRUE, link = c("identity","logit","log"))
Arguments
pc: Vector of critical cumulative probabilities of the difference distribution.
qc: Vector of respective critical values of the difference distribution. Must match the length of pc.
lower.tail: Logical; if TRUE (default), probabilities are P(X≤x), otherwise, P(X>x).
link: Enables application of a link function prior to evaluating the difference distribution. Can take one of the values identity (default), logit or log.
Returns
The function returns a decision function which takes three arguments. The first and second argument are expected to be mixture (posterior) distributions from which the difference distribution is formed and all conditions are tested. The third 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. That is, the distance is 0 at the decision boundary, negative for a 0 decision and positive for a 1 decision.
Details
This function creates a one-sided decision function on the basis of the difference distribution in a 2 sample situation. To support double criterion designs, see Neuenschwander et al., 2010, an arbitrary number of criterions can be given. The decision function demands that the probability mass below the critical value qc of the difference θ1−θ2 is at least pc. Hence, for lower.tail=TRUE condition i is equivalent to
P(θ1−θ2≤qc,i)>pc,i
and the decision function is implemented as indicator function using the heavy-side step function H(x) which is 0 for x≤0 and 1 for x>0. As all conditions must be met, the final indicator function returns
ΠiHi(P(θ1−θ2≤qc,i)−pc,i),
which is 1 if all conditions are met and 0
otherwise. For lower.tail=FALSE differences must be greater than the given quantiles qc.
Note that whenever a link other than identity is requested, then the underlying densities are first transformed using the link function and then the probabilties for the differences are calculated in the transformed space. Hence, for a binary endpoint the default identity link will calculate risk differences, the logit link will lead to decisions based on the differences in logits corresponding to a criterion based on the log-odds. The log link will evaluate ratios instead of absolute differences which could be useful for a binary endpoint or counting rates. The respective critical quantiles qc must be given on the transformed scale.
Functions
oc2Sdecision(): Deprecated old function name. Please use decision2S instead.
Examples
# see Gsponer et al., 2010priorT <- 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=FALSEsuccessCrit <- decision2S(c(0.95,0.5), c(0,50),FALSE)# the futility criterion acts in the opposite directionfutilityCrit <- decision2S(c(0.90), c(40),TRUE)print(successCrit)print(futilityCrit)# consider decision for specific outcomespostP_interim <- postmix(priorP, n=10, m=-50)postT_interim <- postmix(priorT, n=20, m=-80)futilityCrit( postP_interim, postT_interim )successCrit( postP_interim, postT_interim )# Binary endpoint with double criterion decision on log-odds scale# 95% certain positive difference and an odds ratio of 2 at leastdecL2 <- decision2S(c(0.95,0.5), c(0, log(2)), lower.tail=FALSE, link="logit")# 95% certain positive difference and an odds ratio of 3 at leastdecL3 <- decision2S(c(0.95,0.5), c(0, log(3)), lower.tail=FALSE, link="logit")# data scenariopost1 <- postmix(mixbeta(c(1,1,1)), n=40, r=10)post2 <- postmix(mixbeta(c(1,1,1)), n=40, r=18)# positive outcome and a median odds ratio of at least 2 ...decL2(post2, post1)# ... but not more than 3decL3(post2, post1)
References
Gsponer T, Gerber F, Bornkamp B, Ohlssen D, Vandemeulebroecke M, Schmidli H.A practical guide to Bayesian group sequential designs. Pharm. Stat.. 2014; 13: 71-80
See Also
Other design2S: decision2S_boundary(), oc2S(), pos2S()