The Conditional Error Principle and Adaptive Designs

The Conditional Error Principle (CEP) is the foundation of many frequentist adaptive designs. It asserts that any new statistical test chosen after an adaptation must have a conditional error rate no greater than that of the original, pre-specified test. This allows for flexible mid-course modifications to a trial, such as sample size increase, while still controlling the overall experimental-wise type I error rate.

Formally, the CEP framework can be described as follows.

• Start with a preplanned level-α design 𝐷₀ and an interim (Stage-1) test statistic 𝑇₁.

• Define the conditional error function (computed under the original design 𝐷₀)

• CEP rule: After observing 𝑇₁ = t, one can implement any new test statistic on future (Stage-2) data such that its size ≤ α(t). Then by the law of total expectations, the overall test has size ≤ α. That is,

An example of a two-stage 𝑧-test with adaptation:

Practical applications of the CEP include many well-known adaptive design approaches, such as:

• Group Sequential Designs: naturally embedded within CEP framework (e.g., O’Brien-Fleming boundaries derived from fixed conditional error spending).

• Sample Size Re-estimation: adjusts the second-stage sample size based on conditional power or precision.

• Other Adaptive Designs: including seamless phase II/III trials, multi-arm, two-stage trials with arm selection, etc.