Translating Intermediate Contrasts Into Final-Endpoint Predictions

Why Translate an Intermediate Contrast?

In a multi-arm, multi-stage trial, a treatment arm may have an informative intermediate outcome before its final time-to-event endpoint is mature. A prespecified joint model can use the observed intermediate contrast to predict that arm’s final-endpoint effect. This prediction is not the final effect itself: it is a model-based summary of what the intermediate result implies about the final effect, together with the uncertainty that remains.

The translation depends on how intermediate and final treatment effects are related across arms, and on how precisely the intermediate contrast has been measured. A strong association in the model does not make an individual arm’s final effect known. Nor does a favorable predicted effect establish final-endpoint efficacy. This tutorial develops the conditional prediction and its uncertainty under a specific joint model. It does not determine an arm-retention rule or calculate conditional power.

A Joint Model for Treatment Effects and an Observed Contrast

Let $\theta_{Ij}$ and $\theta_{Fj}$ denote the latent treatment effects for arm $j$, relative to the common control, on the intermediate and final endpoints. For example, both effects could be log hazard ratios, with negative values favoring the experimental arm. The model below describes variation in treatment effects across arms; it is not a model for individual patients’ outcomes.

Definition: For each arm $j$, suppose the pair of latent effects follows a bivariate normal distribution: $$ \begin{pmatrix} \theta_{Ij}\\ \theta_{Fj} \end{pmatrix} \sim N\left[ \begin{pmatrix} \mu_I\\ \mu_F \end{pmatrix}, \begin{pmatrix} \tau_I^2 & \rho\tau_I\tau_F\\ \rho\tau_I\tau_F & \tau_F^2 \end{pmatrix} \right]. $$ The observed intermediate contrast is $X_j=\theta_{Ij}+\varepsilon_{Ij}$, where $\varepsilon_{Ij}\sim N(0,v_I)$ is sampling error independent of the latent effects. Here $\mu_I,\mu_F$ are the across-arm mean effects, $\tau_I,\tau_F$ are across-arm standard deviations, $\rho$ is the correlation between the latent effects, and $v_I$ is the sampling variance of the intermediate contrast.

This specification assumes a common distribution across the arms being modeled, a valid normal approximation for the intermediate contrast, and known model parameters for the calculations that follow. In practice, the joint model and its parameters must be prespecified or estimated with appropriate care. If parameters are estimated, treating them as known produces plug-in prediction uncertainty and may understate total uncertainty. The independent-error assumption also matters: if the intermediate contrast’s estimation error is associated with the latent effects in a way the model omits, the conditional calculation need not be valid.

The target here is the latent final effect $\theta_{Fj}$, not a future noisy estimate of it. Observing $X_j=x$ gives information about $\theta_{Ij}$, but because $X_j$ contains sampling error, it gives less information than observing the latent intermediate effect exactly.

Conditional Prediction and Its Derivation

The pair $(\theta_{Fj},X_j)$ is jointly normal. Its means are $\mu_F$ and $\mu_I$; the variance of $X_j$ is $\tau_I^2+v_I$; and their covariance is $\rho\tau_I\tau_F$, by the independence of the measurement error from the latent effects. A standard conditional-normal result therefore gives the prediction distribution directly.

Result: Under the stated model, conditional on observing $X_j=x$, the latent final effect has distribution $$ \theta_{Fj}\mid X_j=x \sim N\left( \mu_F+ \frac{\rho\tau_I\tau_F}{\tau_I^2+v_I}(x-\mu_I), \; \tau_F^2- \frac{(\rho\tau_I\tau_F)^2}{\tau_I^2+v_I} \right). $$ The first argument is the conditional mean; the second is the conditional variance.

The mean starts at the across-arm mean $\mu_F$ and shifts according to how far $x$ lies from $\mu_I$. The multiplier \[ \frac{\rho\tau_I\tau_F}{\tau_I^2+v_I} \] is the amount of change in predicted final effect per unit change in the observed intermediate contrast. Its sign follows the sign of $\rho$. Its magnitude is reduced by measurement error: relative to the error-free slope $\rho\tau_F/\tau_I$, it is multiplied by $\tau_I^2/(\tau_I^2+v_I)$, a factor between zero and one. Thus measurement error attenuates the slope toward zero, whether the underlying association is positive or negative.

The conditional variance is smaller than or equal to the across-arm variance $\tau_F^2$, because the subtracted term is nonnegative. The intermediate result can reduce uncertainty when the model links the endpoints, but it does not generally eliminate it. If $\rho=0$, observing the intermediate contrast does not update the final-effect prediction under this model.

Worked Example: Predicting a Final Log Hazard Ratio

Suppose, hypothetically, that the intermediate and final treatment effects are both measured as log hazard ratios. Across arms, let $\mu_I=-0.20$, $\mu_F=-0.15$, $\tau_I=0.30$, $\tau_F=0.25$, and $\rho=0.70$. The sampling variance of the observed intermediate contrast is $v_I=0.0144$, and the observed contrast is $x=-0.35$. Assume the model parameters are known for this calculation and the specified independent-error model is appropriate.

The total variance of the observed intermediate contrast and its covariance with the latent final effect are $$ \tau_I^2+v_I=0.30^2+0.0144=0.1044, \qquad \rho\tau_I\tau_F=0.70(0.30)(0.25)=0.0525. $$

The conditional mean is $$ \mu_F+\frac{\rho\tau_I\tau_F}{\tau_I^2+v_I}(x-\mu_I) =-0.15+\frac{0.0525}{0.1044}[-0.35-(-0.20)] =-0.15+(0.5029)(-0.15) =-0.2254. $$ Thus the predicted final log hazard ratio is $-0.2254$, more favorable than the across-arm mean of $-0.15$, because the observed intermediate contrast is below its across-arm mean and the modeled association is positive.

The conditional variance and standard deviation are $$ 0.25^2-\frac{0.0525^2}{0.1044} =0.0625-0.02640 =0.03610, \qquad \sqrt{0.03610}=0.1900. $$ A model-based 95% prediction interval for the latent final log hazard ratio is therefore $$ -0.2254\pm1.96(0.1900)\approx(-0.5978,\;0.1470). $$ This interval includes zero. It describes uncertainty about the arm’s latent final effect under the specified model; it is not a 95% probability statement about a fixed effect and does not establish final-endpoint benefit.

Measurement Error and the Direction of Translation

The model’s slope should not be interpreted as a deterministic conversion factor. In particular, a negative intermediate contrast need not imply a negative final contrast with certainty. The conditional distribution retains residual variation, and the predicted mean depends on the across-arm means as well as on the observed contrast. The following hypothetical calculation isolates the effect of measurement error when the modeled association is negative.

Worked Example: Negative Association With Noisy Intermediate Data

Suppose effects are standardized, with $\mu_I=\mu_F=0$, $\tau_I=1$, $\tau_F=0.80$, and $\rho=-0.50$. Let $v_I=1$, and suppose an arm has observed intermediate contrast $x=1$. Assume the bivariate normal model and independent sampling error hold.

The covariance is $(-0.50)(1)(0.80)=-0.40$, and the variance of $X$ is $1^2+1=2$. The conditional mean is $$ 0+\frac{-0.40}{2}(1-0)=-0.20. $$ The error-free slope would be $\rho\tau_F/\tau_I=(-0.50)(0.80)/1=-0.40$. With measurement error, the multiplier is $-0.20$, which is less negative and closer to zero. This is attenuation toward zero; it would be incorrect to claim that the signed slope must always be less than the error-free slope.

The conditional variance is $$ 0.80^2-\frac{(-0.40)^2}{2} =0.64-0.08=0.56, $$ so the conditional standard deviation is $\sqrt{0.56}=0.7483$. A 95% interval for the latent final effect is $$ -0.20\pm1.96(0.7483)\approx(-1.667,\;1.267). $$ Although the predicted mean is negative, the interval is wide and includes effects of either sign. The observed intermediate contrast has shifted the prediction, but substantial uncertainty remains.

Predicting an Effect Is Not Predicting a Future Estimate

A prediction interval for $\theta_{Fj}$ quantifies uncertainty about the latent final treatment effect. If the question instead concerns a future final-endpoint estimate, its sampling variability must also be included. Suppose that estimate is $Y_j=\theta_{Fj}+\varepsilon_{Fj}$, with independent future sampling error $\varepsilon_{Fj}\sim N(0,v_F)$, independent of the quantities in the intermediate model. Then, conditional on $X_j=x$, its mean is the same as the latent-effect conditional mean, while its variance is the conditional variance of $\theta_{Fj}$ plus $v_F$.

Worked Example: Adding Final-Readout Sampling Variance

Continue the first hypothetical example, and suppose a future final estimate has sampling variance $v_F=0.0225$. Assume this future sampling error is independent of the intermediate data and latent effects. The conditional mean remains $-0.2254$, but the variance for the future estimate is $$ 0.03610+0.0225=0.05860, \qquad \sqrt{0.05860}=0.2421. $$

A 95% prediction interval for the future final estimate is $$ -0.2254\pm1.96(0.2421)\approx(-0.6999,\;0.2490). $$ This interval is wider than the interval for the latent final effect because it includes final-readout sampling variability as well as uncertainty about the effect. The two intervals answer different questions and should not be substituted for one another.

Assumptions, Diagnostics, and Alternatives

The calculation is useful only to the extent that the joint model represents the setting. Check, at minimum, whether the endpoint effects and their directions are defined consistently across arms; whether a common bivariate distribution is credible; whether the intermediate estimates have plausible standard errors; and whether the correlation and across-arm variances were specified or estimated using information appropriate to the prediction. A narrow interval can be misleading if it relies on an overconfident correlation or ignores parameter uncertainty.

The normal model also deserves scrutiny when there are few arms, highly skewed treatment effects, influential arms, or an intermediate effect near a boundary. With sparse arms, estimating $\rho$, $\tau_I$, and $\tau_F$ precisely may be difficult. If intermediate and final estimators share data, their estimation errors may be correlated. That feature does not automatically invalidate prediction of the latent final effect, but it must be represented if the target or joint likelihood involves the final estimator. Model checks should examine fit and sensitivity to plausible parameter values rather than treating one fitted correlation as certain.

The main alternatives answer related but different questions. Using the across-arm mean $\mu_F$ alone avoids updating from the intermediate outcome, but discards the information the joint model says it contains. A direct empirical regression of final effects on intermediate effects may be attractive when sufficient arm-level data support it, but its fitted slope and prediction uncertainty must also be justified; it is not automatically safer than a prespecified joint model. A fully Bayesian or other joint parameter-uncertainty analysis can propagate uncertainty in model parameters, whereas the calculations above condition on fixed parameters. The introductory material on Cox models for intermediate-to-final prediction and multistate intermediate outcomes addresses outcome-specific modeling structures; the present calculation concerns translating a contrast once an appropriate joint distribution has been specified.

A predicted final effect should therefore be reported with its scale, direction, model assumptions, and uncertainty. It is a model-based projection, not a causal guarantee that the intermediate outcome captures the final treatment effect. The introductory material on selection accuracy from a joint model motivates why such projections can inform arm comparisons, but prediction alone does not establish a selection rule or a final success probability.

Common Pitfalls

Using the latent intermediate effect as if it were observed without error. Replacing $\tau_I^2+v_I$ with $\tau_I^2$ overstates how strongly a noisy contrast updates the final prediction.
Ignoring the sign of the association. When $\rho<0$, an intermediate contrast above its mean shifts the predicted final effect downward, not upward.
Making a signed attenuation claim for every correlation. Measurement error reduces the absolute slope toward zero. For a negative slope, the attenuated value is greater, not less, than the error-free slope.
Calling a prediction interval a confidence interval for final success. The interval describes uncertainty about an effect under the model; it is not a test result or a probability that the null hypothesis is true.
Reporting only a point prediction. The conditional variance is essential: two arms with similar predicted means can still have substantial uncertainty about their final effects.

Key takeaway: Under a prespecified joint normal model, an observed intermediate contrast updates the final-effect prediction in proportion to its modeled covariance with the final effect, relative to the total variance of the observed contrast. Sampling error attenuates that update, and the conditional variance quantifies uncertainty that remains.

Check Your Understanding

Answer each question, then compare with the solutions below.

In the first worked example, what is the predicted latent final effect if the observed intermediate contrast is $x=-0.10$, with all other model inputs unchanged?
Under the stated model, what happens to the conditional mean and variance when $\rho=0$?
As $v_I$ becomes very large, how does the prediction based on $x$ behave?
Why is a future-estimate prediction interval wider than the latent-effect prediction interval when $v_F>0$ and future sampling error is independent?

Solutions. 1. The slope is $0.5028736$, so the predicted mean is $-0.15+0.5028736[-0.10-(-0.20)]=-0.09971$. The conditional variance remains $0.03610$, since the model inputs are unchanged. 2. The covariance term is zero, so the conditional mean is $\mu_F$ and the conditional variance is $\tau_F^2$: the intermediate contrast provides no update. 3. The multiplier tends to zero as $v_I$ increases, so the predicted mean tends to $\mu_F$; the conditional variance tends to $\tau_F^2$. A very noisy intermediate contrast contributes almost no information. 4. The future estimate includes independent sampling error in addition to uncertainty about the latent effect, so its conditional variance is the latent conditional variance plus $v_F$, making its interval wider.