# 9.7 - Futility Assessment with Conditional Power; Adaptive Designs

9.7 - Futility Assessment with Conditional Power; Adaptive Designs

As an alternative to the above methods, we might want to terminate a trial when the results of the interim analysis are unlikely to change after accruing more patients (futility assessment/curtailed sampling). It just doesn't look like there could ever be a significant difference!

Unconditional power, as we have used in earlier sample size calculations is the probability of acheiving a significant result at a pre-specified alpha under a pre-specified alternative treatment effect as calculated at the beginning of a trial. Conditional power is an approach that quantifies the probability of rejecting the null hypothesis of no effect once some data are available. If this quantity is very small, a conclusion can be reached that it would be futile to continue the investigation.

As a simple example, consider the situation in which we want to determine if a coin is fair, so the hypothesis testing problem is:

$$H_0: p = Pr[\text{Heads}] = 0.5 \text{ versus } H_1: p = Pr[\text{Heads}] > 0.5$$.

The fixed sample size plan is to toss the coin 500 times, count the number of heads, X. But do we actually need to flip the coin 500 times? Using this futility assessment procedure we could reject $$H_0$$ at the 0.025 significance level if:

$$Z=\dfrac{X-250}{\sqrt{(500)(0.5)(0.5)}} \ge 1.96$$

This is equivalent to rejecting $$H_0$$ if X ≥ 272. Suppose that after 400 tosses of the coin there are 272 heads. It is futile to proceed further because even if the remaining 100 tosses yielded tails, the null hypothesis still would be rejected at the 0.025 significance level. The calculation of the conditional power in this example is trivial (it equals 1) because no matter what is assumed about the true value of p, the null hypothesis would be rejected if the trial were taken to completion.

You can also look at this in the other direction. Suppose that after 400 tosses of the coin there are 200 heads. The null hypothesis will be rejected if there are at least 72 heads during the remaining 100 tosses.

Even if p = 0.6 (arbitrary assignment), the conditional power is:

$$Pr[X \ge 72 | n=100, p=0.6]$$

$$= Pr\left[\dfrac{X-60}{\sqrt{(100)(0.6)(0.4)}} \ge \dfrac{72-60}{\sqrt{(100)(0.6)(0.4)}} \right]$$

$$= Pr[X \ge 2.45] = 0.007$$

The probability based on a standard normal table is calculated to be .007, a very small probability. Thus, it is futile to continue because there is such a small chance of rejecting $$H_0$$.

Similarly, two clinical trial scenarios can be envisioned:

1. A trend in favor of rejecting $$H_0$$ is observed at $$t < T$$, with intervention $$>$$ control. Compute conditional probability of rejecting $$H_0$$ at T given current data. If probability is sufficiently large, one might argue trend not going to disappear.
2. A negative trend consistent with $$H_0$$ at t . Compute conditional probability of rejecting $$H_0$$ at end of trial at T given some alternative $$H_1$$ is true. How large does the true effect need to be before the negative trend is reversed? If the probability of trend reversal is highly unlikely, termination might be considered.