Predictive probability of success

• Classification based on type of end point: Normal, binary, time to event.
• Classification based on the relationship between the trial providing data and the trial to be predicted

1. Cross trial PPOS: using data from one trial to predict the other trial
2. Within trial PPOS: using data at interim analysis to predict the same trial at final analysis

• Classification based on the relationship between the end point(s) with data and the end point to be predicted

1. 1 to 1 PPOS: using one end point to predict the same end point
2. 1 to 1* PPOS: using one end point to predict another different but correlated end point

Conditional power is the probability of observing a statistically significance assuming the parameter equals to a specific value.[2] More specifically, these parameters could be treatment and placebo event rates that could be fixed in future observations.[3] This is a frequentist statistical power. Conditional power is often criticized for assuming the parameter equals to a specific value which is not known to be true. If the true value of the parameter is known, there is no need to do an experiment.

Predictive power addresses this issue assuming the parameter has a specific distribution. Predictive power is a Bayesian power. A parameter in Bayesian setting is a random variable. Predictive power is a function of a parameter(s), therefore predictive power is also a variable.

Both conditional power and predictive power use statistical significance as success criteria. However statistical significance is often not enough to define success. For example, health authorities often require the magnitude of treatment effect to be bigger than statistical significance to support a registration decision.

To address this issue, predictive power can be extended to the concept of PPOS. The success criteria for PPOS is not restricted to statistical significance. It can be something else such as clinical meaningful results. PPOS is conditional probability conditioned on a random variable, therefore it is also a random variable. The observed value is just a realization of the random variable.[4]

Posterior probability of success is calculated from posterior distribution. PPOS is calculated from predictive distribution. Posterior distribution is the summary of uncertainties about the parameter. Predictive distribution has not only the uncertainty about parameter but also the uncertainty about estimating parameter using data. Posterior distribution and predictive distribution have same mean, but former has smaller variance.

PPOS is a conditional probability conditioned on randomly observed data and hence is a random variable itself. Currently common practice of PPOS uses only its point estimate in applications. This can be misleading. For a variable, the amount of uncertainty is an important part of the story. To address this issue, Tang[5] introduced PPOS credible interval to quantify the amount of its uncertainty. Tang advocates to use both PPOS point estimate and credible interval in applications such as decision making and clinical trial designs. Another common issue is the mixed use of posterior probability of success and PPOS. As described in the previous section, the 2 statistics are measured in 2 different metrics, comparing them is like comparing apples and oranges.

PPOS can be used to design futility interim for a big confirmatory trials or pilot trials.

In interim analysis, Predictive Probability of Success can also be calculated through the use of simulations through the following method:[1]

1. Sample the parameter of interest from the posterior distribution attained from the currently available set of data
2. Complete the dataset by sampling from the predictive distribution which holds values not yet observed in the data under interim analysis
3. Use the newly completed dataset to calculate criteria used to calculate success which could be things like p-values, posterior probabilities, etc. This can then be used to categorized if a trial was a success or not.
4. These three steps then get repeated a total of n number of times. The PPOS is determined by getting the proportion of trials that were successes in the dataset.

Using simulation to calculate PPOS makes it possible to test statistics with complex distributions since it alleviates the computing complexity that would otherwise be required.[3]