Regression Prediction for Arm Selection

Why Regress Intermediate Effects on Final Effects?

An intermediate outcome can arrive before the final endpoint is sufficiently mature for a reliable treatment-effect estimate. The earlier material on conditional prediction and predictive probability explains how interim information can inform final-outcome prospects. Regression offers a way to learn a relationship between intermediate and final treatment effects from comparable completed trials, then apply that relationship to an arm under consideration. Trial-level covariates can adjust predictions when, for example, baseline risk or follow-up differs across trials.

This is a prediction problem, not a claim that an intermediate effect causes a final effect. The model estimates how final effects have varied with intermediate effects and covariates in a specified collection of trials. Its usefulness for arm selection depends on whether that relationship transports to the current trial and whether prediction uncertainty is sufficiently small for the intended decision.

Model and Assumptions

A basic linear specification is

The coefficients describe the fitted conditional mean: \(\beta_1\) is the expected change in final effect for a one-unit increase in intermediate effect, holding the included covariates fixed. It is not automatically a causal effect. For ordinary least-squares prediction intervals, the usual working assumptions are a correctly specified linear conditional mean, independent residuals with common variance, and—when using exact small-sample \(t\) intervals—normally distributed residuals. The predictor values are treated as known. Historical trials must also be sufficiently comparable to the current trial for their fitted relationship to be relevant.

In practice, \(Y_j\) and \(X_j\) are often estimates rather than error-free true effects. A simple regression that treats estimated intermediate effects as exact can understate uncertainty and may distort the slope when those estimates are noisy. Also, multiple arms from one trial can have correlated contrasts; they should not be treated as independent observations without appropriate modelling. These are substantive design and analysis issues, not details that can be repaired by reporting a narrower interval.

Write the predictor vector, including the intercept, as \(\mathbf{x}_0=(1,X_0,Z_0^\mathsf{T})^\mathsf{T}\), and the fitted coefficient vector as \(\hat{\boldsymbol{\beta}}\). The predicted mean final effect is \(\hat{Y}_0=\mathbf{x}_0^\mathsf{T}\hat{\boldsymbol{\beta}}\). For ordinary least squares, the estimated variance of this mean prediction is \(s^2h_0\), where \(s\) is the residual standard deviation and \(h_0=\mathbf{x}_0^\mathsf{T}(X^\mathsf{T}X)^{-1}\mathbf{x}_0\) is the prediction’s leverage. A future arm effect has additional residual variation.

Mean Prediction Versus Predicting a Future Effect

A confidence interval for the conditional mean answers how uncertain the estimated regression mean is at the specified predictor values. A prediction interval answers how uncertain the final effect for a new arm is, including both uncertainty in the estimated mean and the arm-to-arm residual variation. Selection concerns a future arm effect, so a confidence interval for the mean alone generally understates the relevant uncertainty.

If there are \(m\) fitted regression coefficients and the residual assumptions hold, a \(100(1-\alpha)\%\) prediction interval is

Here \(n\) is the number of independent historical observations used in the regression. The \(1\) under the square root represents residual variation for a new arm; \(h_0\) represents uncertainty in the estimated regression mean at that arm’s predictor values. The interval is wider at predictor combinations with higher leverage, especially those far from the historical data.

Carrying Interim-Effect Uncertainty Forward

The usual prediction formula treats the current arm’s predictor values as fixed. In an interim analysis, however, \(X_0\) is itself estimated. Ignoring its uncertainty can make the final-effect prediction appear more precise than it is. A first-order approximation follows by applying the delta method to the regression prediction. In a model with one uncertain intermediate predictor, its contribution to prediction variance is approximately \(\hat{\beta}_1^2\operatorname{Var}(\hat{X}_0)\), provided the interim estimate is independent of the historical regression fit.

Thus, an approximate variance for a future final effect is the regression prediction variance plus the propagated input variance. This approximation assumes the coefficient estimates and current intermediate estimate are independent and treats the fitted slope as fixed for the input-uncertainty contribution. If the historical fit and current estimate are dependent, or the predictor is highly uncertain, a joint model or simulation that represents those dependencies is preferable.

Using a Covariate-Adjusted Prediction

A trial-level covariate should be included because it is scientifically relevant and measured comparably, not simply because it improves fit in the historical data. A covariate can shift the predicted final effect or improve precision if it explains systematic differences between trials. The current arm’s covariate value must be available at the selection look and expressed on the same scale used to fit the model.

Diagnostics, Alternatives, and Limitations

Before using a prediction to inform arm selection, check whether the model is behaving plausibly and whether its uncertainty is credible. Useful checks include residual plots for curvature and changing variance, examination of influential trials, and comparison of predicted with observed final effects in held-out trials or through cross-validation. Calibration matters: among cases assigned similar predicted effects or probabilities, observed outcomes should agree reasonably with those predictions. Validation should preserve trial-level independence; splitting correlated arms from one trial between training and validation sets can give an overly optimistic assessment.

Look especially for extrapolation. A prediction at an intermediate effect or covariate combination far beyond the historical range depends strongly on the assumed regression form. Large leverage is a warning that the estimate is sensitive to the data and model. In that situation, a prediction interval may be wide, while a narrow interval can indicate an inadequately represented source of uncertainty rather than reliable precision.

Regression is not the only approach. A survival model can use event times and censoring directly when the required data and assumptions are available; an intermediate-to-final translation model can encode a prespecified relationship; and conditional-power or predictive-probability calculations address related questions using interim trial information. These approaches have different targets and assumptions. Regression is particularly useful for estimating an empirical relationship across trials with covariates, but it does not replace careful treatment of time-to-event data, censoring, or the trial design.

A regression association at the trial level need not describe individual patients. Differences between trials, unmeasured effect modifiers, changes in standard care, or selective availability of completed trials can all undermine transportability. Uncertainty in the intermediate-effect estimates, dependence between arms, and a small number of independent trials can also make ordinary least-squares intervals unreliable. A model-based prediction should therefore be treated as conditional evidence, not as a guaranteed forecast.