Worked Predictive Success Analysis for a Continuous Endpoint

From Interim Evidence to a Final Go Decision

For a continuous endpoint, an interim estimate can look encouraging without making final success likely. The remaining participants still contribute uncertain data, and the final decision is based on the posterior after those data are observed. Predictive success quantifies the probability, given current evidence and a specified model for the remaining observations, that the final posterior will satisfy a prespecified go criterion.

This calculation has three linked parts: update the treatment-effect posterior with interim data, predict the future data using that posterior, and identify which future outcomes would make the final posterior cross the decision threshold. The calculation below uses a normal model with known sampling variances so each step can be shown directly.

Here \(v_1\) and \(v_2\) are sampling variances of estimates, not variances of individual outcomes. In an actual trial, they may be determined from participant-level variance assumptions and the numbers allocated to each treatment arm. The simple model also assumes that the endpoint is analyzed on the stated scale, that the planned final estimator is appropriate, and that no unmodeled changes to enrollment or analysis occur after the interim look.

Interim Posterior and Final Posterior

With a normal prior and a normal likelihood of known variance, the interim posterior is normal. Its variance \(V_1\) and mean \(m_1\) are obtained by adding prior and data precisions, where precision is the reciprocal of variance:

After the remaining estimate \(Y_2\) is observed, its likelihood contributes precision \(1/v_2\). Thus the final posterior has variance \(V_F=(1/V_1+1/v_2)^{-1}\) and mean \(m_F=V_F(m_1/V_1+Y_2/v_2)\). These expressions can also be written as a precision-weighted combination:

The weights sum to one. More precise information receives more weight: if the interim posterior variance \(V_1\) is small relative to \(v_2\), then \(m_1\) receives the larger weight. The final posterior go rule considered here is \(P(\theta>\Delta\mid Y_1,Y_2)\geq\gamma\), where \(\Delta\) is a clinically meaningful treatment-effect threshold and \(\gamma\) is a prespecified posterior probability requirement.

Predicting the Remaining Data

The predictive distribution averages over uncertainty in the treatment effect that remains after the interim data. One can express the future estimate as \(Y_2=\theta+\varepsilon_2\), where, conditional on \(\theta\), \(\varepsilon_2\sim N(0,v_2)\) and is independent of interim data. Since the current posterior for \(\theta\) is \(N(m_1,V_1)\), the future estimate has distribution

This is a predictive distribution: its variance includes both uncertainty about the treatment effect, \(V_1\), and sampling uncertainty in the remaining participants, \(v_2\). Using only \(v_2\) would treat the unknown treatment effect as if it were already known. Conversely, using only \(V_1\) would omit noise in the future estimate.

For the go rule, the final posterior probability is a function of \(m_F\) and \(V_F\). In the normal model, \(P(\theta>\Delta\mid Y_1,Y_2)=\Phi((m_F-\Delta)/\sqrt{V_F})\). Because \(\Phi\) is increasing, meeting the rule is equivalent to a threshold on the final posterior mean:

where \(z_\gamma\) is the \(\gamma\) quantile of the standard normal distribution. Substituting the expression for \(m_F\) converts this into a threshold on \(Y_2\). The predictive success probability is then the probability, under \(Y_2\mid Y_1\), that \(Y_2\) lies on the success side of that threshold.

Checking the Final Decision for Possible Future Results

The threshold calculation can be checked by substituting possible future estimates back into the final posterior. This is useful because a sign error in the weights or an incorrectly translated posterior criterion can reverse the decision. In this example, higher \(Y_2\) produces higher \(m_F\), so the successful future values are above the cutoff.

Derivation and Interpretation

The predictive distribution follows directly from adding independent normal uncertainties. Conditional on the interim data, write \(\theta=m_1+U\), where \(U\sim N(0,V_1)\). Write the future sampling error as \(\varepsilon_2\sim N(0,v_2)\), independent of \(U\). Then \(Y_2=m_1+U+\varepsilon_2\). The sum of independent normal variables is normal, with means added and variances added, so \(Y_2\mid Y_1\sim N(m_1,V_1+v_2)\). This establishes the predictive distribution used above.

The final-posterior weighting also follows from normal precision addition. Combining a normal distribution \(N(m_1,V_1)\) with a likelihood proportional to \(\exp(-(Y_2-\theta)^2/(2v_2))\) gives a posterior whose quadratic term in \(\theta\) has coefficient \(1/V_1+1/v_2\), and whose linear term is \(m_1/V_1+Y_2/v_2\). Completing the square yields variance \(V_F=(1/V_1+1/v_2)^{-1}\) and mean \(V_F(m_1/V_1+Y_2/v_2)\). This is why the final mean uses precision weights rather than an unweighted average.

In general, if \(m_F=a+bY_2\) with \(b>0\), the go condition is \(Y_2\geq(\Delta+z_\gamma\sqrt{V_F}-a)/b\). If \(Y_2\mid Y_1\sim N(m_1,V_1+v_2)\), then predictive success equals one minus the standard normal cumulative probability evaluated at that cutoff, standardized by \(\sqrt{V_1+v_2}\). If the future data instead move the final mean in the opposite direction, the inequality direction must be changed accordingly.

Diagnostics, Alternatives, and Limitations

Several checks help identify a calculation that is going wrong. Confirm that each variance is on the scale of the estimator and has squared endpoint units; the standard deviations and decision threshold then have endpoint units. Confirm that \(V_F\) is smaller than \(V_1\), because adding independent information should reduce posterior variance. Check that the final-mean weights are positive and sum to one, and verify the go cutoff by substituting a value just above and just below it. Finally, confirm that the predictive variance includes both \(V_1\) and \(v_2\).

The result depends on the model and decision rule, not just on the observed interim estimate. The normal known-variance model may be unsuitable when endpoint distributions are strongly skewed, heavy-tailed, or have outliers; when treatment-arm variances differ materially; or when the endpoint requires a transformation or a different likelihood. An uncertain variance, missing outcomes, nonadherence, or changes to the remaining allocation can also require a richer model. In those settings, posterior prediction can still be used, but the future data must be generated from a model that represents those features.

Predictive success differs from a plug-in calculation that treats \(\theta=m_1\) as known and calculates a future success probability at that single value. It also differs from current posterior evidence, such as \(P(\theta>\Delta\mid Y_1)\), which concerns the treatment effect given data already observed. A frequentist conditional-power calculation typically evaluates future success under a specified fixed effect or a specified alternative; Bayesian predictive success instead averages over the current posterior distribution of \(\theta\). These methods can answer related planning questions, but they are not interchangeable.

The calculation is conditional on the stated continuation plan, remaining information, model, prior, and final criterion. It is not by itself a guarantee about the trial's eventual conclusion, nor does it establish that the decision rule is well calibrated across possible true effects. Those broader properties require evaluation of the trial procedure under alternative data-generating scenarios.