Probability of success of a clinical trial: connection to other quantities
Connection to other quantities
In order to clarify terms, we list here some quantities that we have seen been used in lieu of DDCP.
Power: \(\pi(\delta, \delta_\text{suc}) \ = \ P(\widehat \delta_\text{fin} \le \delta_\text{suc})\). Computed at trial start as a function of assumed true effect \(\delta\) with no uncertainty. Conditional quantity.
Conditional power: \(P(\widehat \delta_\text{fin} \le \delta_\text{suc} \vert \hat \delta_\text{int}, \delta_\text{after int})\) with \(\delta_\text{after int}\) being the assumed effect that is present after an interim analysis. This can be considered the “updated” power after the trial has started and is a function of \(\delta_\text{after int}\), i.e. the effect assumed to hold after the interim and the interim effect estimate \(\hat \delta_\text{int}\). What to plug in for \(\delta_\text{after int}\) can be debated - it appears reasonable to use the effect the trial was powered at. Do not plug in the interim estimate \(\hat \delta_\text{int}\) for \(\delta_\text{after int}\)! See e.g. Bauer and König (2006) for a discussion. As the name says, conditional power is a conditional quantity. It predicts future trial success assuming a precise future treatment effect.
\(\text{assurance}(\delta_\text{suc})\) and \(\text{BPP}(\delta_\text{suc})\): average over power with respect to weighting distribution over \(\delta\). For assurance calculation, the average is taken over the entire range of the distribution of the effect size \(\delta\), and trial success in the power calculation is defined as \(\widehat \delta_\text{fin} \le \delta_\text{suc}\), while in the BPP calculation, trial success is defined as meeting statistical significance and clinical relevance. Both quantities need to be updated after not stopping at an interim analysis: we then average not over power but over conditional power. Marginal quantity.
Predictive probability: \(E(\)posterior of clinically meaningful effect \(\vert\) every possible future outcome\()\), defined in the context of an interim analysis as the probability of “success” at the final analysis, assuming that the data after the interim analysis follows a posterior distribution derived from a prior and the data observed at the interim analysis. So it can be interpreted as a generalization of conditional power with the aim of predicting trial success assuming a distribution of possible future treatment effects. Typically, this setup
- bears a different name,
- is typically used for a binary endpoint,
- often used to set up decision criteria in Phase 1b clinical trials, see the Unicycle tutorial or Berry et al. (2011).
However, conceptually this is identical to DDCP. The connection is nicely discussed in Trzaskoma and Sashegyi (2007), compare their Equations (3) and (4).
Summary
1.-3. above are all different quantities (some of which related to each other) that
- depend on different assumptions,
- have different properties,
- have different interpretations.
Make sure you are clear about the context in which to use these quantities, and choose them correctly for a given application. Predictive probabilities are a special case of DDCP computations for binary endpoints in small trials.