Deriving Posterior Predictive Probability

Why Average Conditional Power Over a Posterior?

At an interim analysis, the future trial result depends on two sources of uncertainty: the outcomes not yet observed and the treatment effect that will govern those outcomes. Conditional power answers the first question after fixing a value for the treatment effect. Posterior predictive success probability accounts for both by averaging that conditional probability over the posterior distribution of the effect given the interim data.

This distinction matters when the interim estimate is imprecise. A single assumed effect can give a sharply different prediction from one that acknowledges uncertainty about the effect. The derivation below uses a one-sided final success boundary and a normal canonical model. The formula is exact under those assumptions, not a universal replacement for the appropriate model in every trial.

Definition: Let \(Z_1\) be the interim statistic at information \(I_1\), and let \(Z_f\) be the statistic at the planned final information \(I_f\), with \(0 \leq I_1 < I_f\). Define the information fraction \(t=I_1/I_f\), and let \(\delta\) denote the treatment effect scaled to final information. The posterior predictive success probability is the posterior average, given the interim data, of the probability that the future final statistic meets the prespecified success criterion.

Model and Assumptions

Assume the canonical normal model with known information and a fixed treatment effect. In this parameterization, the interim statistic has distribution \(Z_1\mid\delta\sim N(\sqrt{t}\delta,1)\). The final statistic has unit variance, and its mean under the effect \(\delta\) is \(\delta\). The interim and future information increments are independent conditional on \(\delta\). These assumptions are the normal independent-increments representation used for group sequential calculations.

For this tutorial, success at the final look means \(Z_f>c\), where \(c\) is the prespecified final efficacy boundary. It may be the appropriate boundary from the trial design; it need not equal a conventional unadjusted critical value. The calculation treats the observed interim result as given and asks about success if the trial continues to its planned final analysis. It does not incorporate uncertainty in variance or information, changes to the boundary, or additional future looks.

Use a normal prior \(\delta\sim N(m_0,v_0)\), with \(v_0>0\). The prior is stated on the same final-information scale as \(\delta\). Combining it with the interim likelihood gives a normal posterior. The details of prior specification are not the focus here; consistency of the prior's scale with the model is essential.

\delta\mid Z_1=z_1 \sim N(m_1,v_1), \qquad v_1=\left(\frac{1}{v_0}+t\right)^{-1}, \qquad m_1=v_1\left(\frac{m_0}{v_0}+\sqrt{t}\,z_1\right).

The likelihood contribution to posterior precision is \(t\), because \(Z_1\mid\delta\) has mean \(\sqrt{t}\delta\) and variance 1. Equivalently, the interim result contributes \(t\) units of information about \(\delta\). The posterior mean is a precision-weighted combination of the prior mean and the observed interim statistic.

Conditional Future Distribution and Its Integral

The future increment supplies proportion \(1-t\) of the final information. By independent increments, conditional on the interim statistic and \(\delta\), the final statistic is normally distributed as follows:

Z_f\mid Z_1=z_1,\delta \sim N\left(\sqrt{t}\,z_1+(1-t)\delta,\ 1-t\right).

The first part of the conditional mean, \(\sqrt{t}z_1\), is the contribution already represented by the interim statistic. The remaining mean, \((1-t)\delta\), is the expected contribution from future information. Therefore, for a fixed value of \(\delta\), the conditional probability of success is:

P(Z_f>c\mid Z_1=z_1,\delta) = \Phi\left( \frac{\sqrt{t}\,z_1+(1-t)\delta-c}{\sqrt{1-t}} \right),

where \(\Phi\) is the standard normal cumulative distribution function. Posterior predictive success probability averages this quantity over the posterior distribution of \(\delta\):

P(Z_f>c\mid Z_1=z_1) = \int_{-\infty}^{\infty} \Phi\left( \frac{\sqrt{t}\,z_1+(1-t)\delta-c}{\sqrt{1-t}} \right) \frac{1}{\sqrt{v_1}}\, \phi\left(\frac{\delta-m_1}{\sqrt{v_1}}\right) \,d\delta,

where \(\phi\) is the standard normal density. This integral is the defining posterior average: each possible effect contributes its conditional probability, weighted by how plausible that effect is after observing the interim data.

Closed-Form Result

The integral has a closed form under the normal prior and canonical normal model. Conditional on the interim data, represent the posterior effect as \(\delta=m_1+\sqrt{v_1}U\), where \(U\sim N(0,1)\). The future noise can be represented by an independent \(V\sim N(0,1)\). Then the final statistic has the posterior predictive representation:

Z_f=\sqrt{t}\,z_1+(1-t)m_1 +(1-t)\sqrt{v_1}\,U+\sqrt{1-t}\,V.

The two random terms are independent and normal. Their variances add, so the predictive distribution is normal with mean \(\sqrt{t}z_1+(1-t)m_1\) and variance \((1-t)^2v_1+(1-t)\). Consequently:

Result: Under the stated model and prior, the posterior predictive probability of crossing the one-sided final boundary \(c\) is

P(Z_f>c\mid Z_1=z_1) = \Phi\left( \frac{\sqrt{t}\,z_1+(1-t)m_1-c} {\sqrt{(1-t)+(1-t)^2v_1}} \right).

The term \((1-t)\) in the predictive variance is future sampling variation. The term \((1-t)^2v_1\) is the additional variation induced by posterior uncertainty about the treatment effect.

This result can also be justified by conditioning on \(\delta\) first, as in the integral, or by deriving the predictive normal distribution directly. Both arguments give the same answer because they describe the same conditional probability. If the final criterion is two-sided, success means \(Z_f>c\) or \(Z_f<-c\); with predictive mean \(\mu_f\) and predictive standard deviation \(\sigma_f\), the probability is \(\Phi((\mu_f-c)/\sigma_f)+\Phi((-c-\mu_f)/\sigma_f)\).

Worked Calculations

Worked Example: Integrating an Uncertain Effect

Suppose a hypothetical trial has information fraction \(t=0.50\), interim statistic \(z_1=1.50\), and one-sided final boundary \(c=1.96\). Use the prior \(\delta\sim N(0,1)\). Calculate the posterior predictive probability of final success.

The posterior variance and mean are:

v_1=\left(1+\frac{1}{2}\right)^{-1}=0.6667,\qquad m_1=0.6667\left(0+\sqrt{0.50}(1.50)\right)=0.7071.

The predictive mean of \(Z_f\) is \(\sqrt{0.50}(1.50)+(0.50)(0.7071)=1.4142\). Its variance is \((1-0.50)+(1-0.50)^2(0.6667)=0.6667\), giving a standard deviation of \(0.8165\). Thus:

P(Z_f>1.96\mid Z_1=1.50) = \Phi\left(\frac{1.4142-1.96}{0.8165}\right) = \Phi(-0.6687) \approx 0.252.

Under this model and prior, the probability of crossing the final boundary, conditional on the interim result, is about 25.2%. This is not a probability that the treatment is effective; it is a predictive probability for a future trial statistic.

Worked Example: An Informative Prior and Later Interim Information

For a second hypothetical calculation, let \(t=0.60\), \(z_1=1.20\), \(c=1.96\), and \(\delta\sim N(1,0.25)\). The prior is centered at a positive treatment effect and has variance \(0.25\), on the final-information scale.

The posterior parameters are:

v_1=\left(4+0.60\right)^{-1}=0.2174,\qquad m_1=0.2174\left(4+\sqrt{0.60}(1.20)\right)=1.0716.

The predictive mean is \(\sqrt{0.60}(1.20)+0.40(1.0716)=1.3582\). The predictive variance is \(0.40+(0.40)^2(0.2174)=0.4348\), so the standard deviation is \(0.6594\). Therefore:

P(Z_f>1.96\mid Z_1=1.20) = \Phi\left(\frac{1.3582-1.96}{0.6594}\right) = \Phi(-0.9127) \approx 0.181.

The interim statistic is positive, but the modeled probability of crossing the boundary is about 18.1%. The numerical result depends on both the interim evidence and the specified prior; it should not be reported without identifying the prior and its scale.

Worked Example: Neutral Interim Evidence

Suppose \(t=0.50\), \(z_1=0\), \(c=1.96\), and \(\delta\sim N(0,1)\). The posterior has \(m_1=0\) and \(v_1=1/(1+0.50)=0.6667\). The predictive mean is zero, while its variance is \(0.50+0.25(0.6667)=0.6667\). Hence:

P(Z_f>1.96\mid Z_1=0) = \Phi\left(\frac{-1.96}{\sqrt{0.6667}}\right) = \Phi(-2.400) \approx 0.0082.

For comparison, evaluating conditional power only at the posterior mean effect \(\delta=0\) gives \(\Phi(-1.96/\sqrt{0.50})\approx0.0028\). The posterior predictive probability is larger here because it includes posterior mass on positive effects as well as negative effects. This comparison is not a general rule that predictive probability must exceed plug-in conditional power; the direction and size of the difference depend on the data, prior, and success criterion.

Checks, Comparisons, and Limitations

A useful calculation starts by checking that the model and inputs match the trial. The information fraction must refer to the planned final information, the effect parameter and prior must use the same scaling, and the success boundary must be the one intended for the final decision. The normal independent-increments model assumes known information and a fixed effect. Material departures from these assumptions can make the formula misleading.

The main alternative is conditional power under a specified effect: insert that effect into the conditional probability and do not average over an effect distribution. A plug-in calculation using an interim estimate is also conditional power, but it fixes the effect at the estimate. Posterior predictive probability instead propagates the full posterior distribution. It can be calculated by numerical integration when a convenient closed form is unavailable, but numerical calculation does not remove dependence on the model or prior.

The closed-form derivation is for one future final look. If the event of interest is crossing any one of several future boundaries, the success event involves the joint distribution of multiple correlated future statistics; applying this single-look formula to each boundary separately does not, by itself, give the probability of success across the sequence. Likewise, if information is estimated, the treatment effect varies over time, or the endpoint is not adequately represented by the normal model, the assumptions and predictive distribution need to be revised.

At \(t=1\), there is no future information: the final statistic equals the observed statistic, so the probability of crossing a strict final boundary is determined by that observation, not by posterior uncertainty about \(\delta\). The formula above is intended for \(t<1\). At very early interim looks, posterior predictions may be particularly sensitive to the prior because the likelihood contributes little information. A prior sensitivity check is therefore an important diagnostic, not a cosmetic refinement.

Common pitfall: Substituting the posterior mean \(m_1\) into the conditional-power formula and calling the result posterior predictive probability ignores uncertainty in \(\delta\). The correction is to integrate over the posterior, or equivalently to use the predictive variance \((1-t)+(1-t)^2v_1\). Also check the effect scale: using a prior for an unscaled effect where the likelihood is written for \(\delta\) changes both the posterior and the prediction.

Check Your Understanding

Try each question, then check the solution directly below it.

Why does posterior predictive variance contain both \((1-t)\) and \((1-t)^2v_1\)?
Solution: The future sampling increment has variance \(1-t\). Uncertainty in the effect contributes the term \((1-t)\delta\) to the final statistic, so posterior variance \(v_1\) for \(\delta\) contributes \((1-t)^2v_1\). The independent variances add.
Suppose \(t=0.50\), \(z_1=0\), \(m_0=0\), and \(v_0=1\). Find the posterior mean and variance of \(\delta\).
Solution: \(v_1=(1+0.50)^{-1}=0.6667\), and \(m_1=0.6667(0+\sqrt{0.50}\cdot0)=0\).
In the setting of Question 2, with \(c=1.96\), what is the posterior predictive success probability?
Solution: The predictive mean is zero and the variance is \(0.50+0.25(0.6667)=0.6667\). Thus the probability is \(\Phi(-1.96/\sqrt{0.6667})=\Phi(-2.400)\approx0.0082\).
Does a posterior predictive probability of \(0.25\) mean there is a 25% probability that the null hypothesis is true?
Solution: No. It means that, under the specified model and prior and conditional on the interim data, the predictive probability that the future final statistic meets the success criterion is 25%. It is not a posterior probability for \(H_0\).