Continuous Endpoint Example With Baseline Adjustment

Why Baseline Adjustment Matters in a Seamless Trial

In a seamless phase 2/3 trial, data from an earlier stage may help select a treatment, while participants recruited after the transition provide additional evidence about the selected treatment. For a continuous outcome, baseline adjustment can improve precision and account for chance differences in baseline measurements. The analysis must nevertheless respect the adaptive design: a p-value for a treatment chosen using stage 1 data cannot automatically be treated as if that treatment had been specified in advance.

This tutorial works through a two-stage example with a continuous endpoint for which higher values are better. It uses a stage-specific ANCOVA contrast to adjust for baseline, a selection-adjusted stage 1 p-value, and an inverse-normal combination test. The stages use different participants, and the confirmatory analysis is prespecified. The numerical data below are hypothetical.

Definition: In a stage-specific ANCOVA, the outcome is modeled using treatment assignment and a baseline covariate. The treatment contrast is the estimated difference between treatment groups at the same baseline value. A stagewise p-value is the p-value calculated from the data in one stage; it must be valid for the relevant null hypothesis, allowing for any treatment selection that used those data.

The Stage-Specific ANCOVA Contrast

For a stage with one active treatment and one control, let \(Y_i\) be the continuous endpoint for participant \(i\), let \(T_i=1\) for active treatment and \(T_i=0\) for control, and let \(X_i\) be the baseline measurement. A basic ANCOVA model is

Y_i=\beta_0+\tau T_i+\beta_X(X_i-x_0)+\varepsilon_i, \qquad \varepsilon_i\sim N(0,\sigma^2).

Here \(x_0\) is a fixed centering value, such as the overall baseline mean. The coefficient \(\tau\) is the treatment contrast at a common baseline value; under the model’s common-slope assumption, the contrast is the same at every baseline value. If higher endpoint values are better, positive \(\tau\) favors treatment.

The usual model-based t test requires independent participants, a correctly specified linear relationship between baseline and outcome, a common baseline slope across treatment groups, and errors with mean zero, common variance, and an approximately normal distribution for exact small-sample t inference. Randomization supports causal interpretation of the treatment contrast, but does not by itself guarantee that the common-slope model is appropriate. The analysis should use the prespecified baseline variable and treatment contrast. If the treatment effect varies with baseline, a treatment-by-baseline interaction may be needed; that changes the interpretation and must be planned and handled in the testing procedure.

With multiple active treatments and a shared control in stage 1, the model includes an indicator for each active treatment. Each coefficient estimates that treatment’s baseline-adjusted contrast against control. The contrasts may be correlated because they share the control group, but a Bonferroni adjustment does not require independence.

Worked Example: Adjusting the Stage 1 Contrast

Suppose stage 1 has 60 participants, allocated equally to two active treatments, A and B, and a shared control. Higher endpoint scores are better. The stage 1 ANCOVA estimates a common baseline slope of 0.60 endpoint points per baseline point. For A, the mean endpoint is 76.8 points and the mean baseline value is 52 points; for control, the corresponding means are 74.0 and 54 points. Calculate the baseline-adjusted A-versus-control contrast using the fitted slope.

The difference in endpoint means is \(76.8-74.0=2.8\) points, and the baseline difference is \(52-54=-2\) points. Adjusting the endpoint difference to a common baseline gives

\widehat{\tau}_{A,1} =(76.8-74.0)-0.60(52-54) =2.8-(-1.2)=4.0\text{ points}.

The estimated adjusted contrast is therefore 4.0 points in favor of A. This calculation illustrates the ANCOVA contrast; the inferential analysis uses the treatment coefficient and its standard error from the fitted regression, not a standard error calculated from this mean-difference formula alone.

Selection and Valid Stagewise Evidence

Suppose A or B is selected at the phase transition using stage 1 results. A raw p-value for the selected treatment ignores the fact that two candidates were considered. One simple valid adjustment for selecting between these two treatments is Bonferroni: multiply the selected treatment’s one-sided stage 1 p-value by two, truncating at one. This adjustment is valid for a selected-null claim under the family of stage 1 comparisons, whether or not the two test statistics are correlated. More complex selection rules may require a different prespecified calibration.

After selection, stage 2 enrolls new participants and compares the selected treatment with control. Provided stage 2 data are independent of stage 1 and the stage 2 model and test are prespecified, its p-value is valid conditional on the stage 1 selection under the selected treatment’s null hypothesis. Reusing stage 1 participants in stage 2, or changing the endpoint or analysis after seeing the data without valid calibration, would undermine this justification.

Worked Example: Stage 1 Selection-Adjusted P-Value

Continue the hypothetical stage 1 trial. The adjusted contrasts are 4.0 points for A and 2.5 points for B, and both have standard error 2.0 points. The selection rule chooses the larger adjusted contrast, so A is selected. For this illustration, assume the stage 1 ANCOVA residuals meet the model assumptions and use a one-sided test of improvement. The model estimates four parameters: an intercept, two treatment coefficients, and a baseline slope. Thus \(df=60-4=56\).

For A, the treatment statistic is

t_{A,1}=\frac{4.0\text{ points}}{2.0\text{ points}}=2.00.

The raw one-sided p-value is \(P(t_{56}\ge 2.00)\approx 0.02518\). Applying the Bonferroni adjustment for two candidates gives

p_1=\min(1,2\times 0.02518)=0.05036.

Thus 0.05036, rather than the unadjusted 0.02518, is the stage 1 p-value used for A in this example. The adjustment reflects the selection opportunity; it does not change A’s estimated contrast of 4.0 points.

Combining the Two Stages

A common way to combine stagewise evidence is the inverse-normal method. Let \(p_1\) and \(p_2\) be the valid one-sided p-values from stages 1 and 2. Let \(\Phi\) denote the standard normal cumulative distribution function, and choose prespecified weights \(w_1,w_2\) satisfying \(w_1\ge 0\), \(w_2\ge 0\), and \(w_1^2+w_2^2=1\). Define

Z_{\mathrm{comb}} =w_1\Phi^{-1}(1-p_1)+w_2\Phi^{-1}(1-p_2).

For a one-sided test at level \(\alpha\), reject the null hypothesis when \(Z_{\mathrm{comb}}\ge \Phi^{-1}(1-\alpha)\). The combined p-value is \(1-\Phi(Z_{\mathrm{comb}})\). The weights should be set in advance, often to reflect planned information from each stage; choosing weights after observing results can invalidate the stated calibration.

The justification is straightforward when stagewise p-values are independent and exactly uniform under the null. Each transformed value \(\Phi^{-1}(1-p_j)\) then has a standard normal distribution. The weighted sum is standard normal because the weights are fixed, their squared values sum to one, and the transformed values are independent. If a stagewise p-value is conservative rather than exactly uniform, its transformed value is stochastically no larger than a standard normal variable. With nonnegative weights and independence, replacing either standard normal contribution by such a stochastically smaller contribution cannot increase the upper-tail rejection probability. This explains both why conservative stagewise p-values can be used and why nonnegative weights matter for that claim.

In an adaptive trial, the relevant independence is supported by using distinct participants in the two stages and by having a stage 2 p-value valid conditional on the stage 1 selection. Selection adjustment is needed for the stage 1 evidence; a valid stage 2 test does not retrospectively remove the selection effect from stage 1.

Worked Example: Stage 2 ANCOVA and Final Combined Test

Suppose stage 2 enrolls 100 new participants: 50 receive selected treatment A and 50 receive control. In the stage 2 ANCOVA, the mean endpoint scores are 80.0 points for A and 77.1 points for control; the baseline means are 51.0 and 51.5 points, respectively. The fitted common baseline slope is again 0.60 points per baseline point. The ANCOVA treatment coefficient has standard error 1.60 points. Assume the model conditions described above are reasonable.

The stage 2 adjusted contrast is

\widehat{\tau}_{A,2} =(80.0-77.1)-0.60(51.0-51.5) =2.9-(-0.30)=3.20\text{ points}.

The stage 2 model estimates an intercept, a treatment coefficient, and a baseline slope, so \(df=100-3=97\). Its statistic and one-sided p-value are

t_{A,2}=\frac{3.20\text{ points}}{1.60\text{ points}}=2.00, \qquad p_2=P(t_{97}\ge 2.00)\approx 0.02415.

Use equal inverse-normal weights \(w_1=w_2=1/\sqrt{2}\). With \(p_1=0.05036\), the transformed stage 1 value is approximately \(\Phi^{-1}(0.94964)=1.641\); with \(p_2=0.02415\), the stage 2 value is approximately \(\Phi^{-1}(0.97585)=1.975\). Therefore,

Z_{\mathrm{comb}} =\frac{1.641+1.975}{\sqrt{2}} \approx 2.557, \qquad p_{\mathrm{comb}}=1-\Phi(2.557)\approx 0.0053.

At a prespecified one-sided level of 0.025, the combined test rejects the null hypothesis of no improvement for the selected treatment under the design’s selection-adjustment and independence conditions. The result is evidence of benefit; it is not a 0.53% probability that the null hypothesis is true. The stage-specific adjusted estimates, 4.0 and 3.20 points, remain useful for describing the observed effects, but the combined test is a combination of evidence, not an estimate obtained by averaging those contrasts.

Diagnostics, Alternatives, and Limitations

Before relying on the calculation, inspect the model and the adaptive design together. For each stage, examine residual plots for curvature, unequal spread, influential observations, and departures from the assumed baseline relationship. Check whether treatment groups have sufficient baseline overlap and whether the estimated slope is unduly influenced by a few observations. A strong treatment-by-baseline pattern challenges the common-slope model; a treatment-by-baseline interaction or a justified alternative model may be appropriate, but the contrast and testing plan must then be defined accordingly. Missing endpoint data can also bias the analysis if the method used to handle missingness does not support the required interpretation.

The inverse-normal method is not the only possible approach. A prespecified conditional-error method can preserve the original type I error while allowing adaptation, but it requires the conditional-error rule to be specified and followed. A stagewise combination test is transparent when valid stagewise p-values and weights are available. In contrast, treating all observations as one ordinary pooled ANCOVA can fail to account for treatment selection or changes made at the transition. A pooled model may be justified only when its assumptions and adaptive calibration are established; simply adding stage 1 and stage 2 records is not a substitute for that justification.

The approach also has limits. Bonferroni adjustment can be conservative, especially with many candidates or correlated comparisons, and may reduce power. The example assumes a one-sided direction, a fixed analysis, independent stage samples, and a continuous outcome analyzed by a suitable ANCOVA model. It does not make an incorrect model correct, repair biased handling of missing outcomes, or permit unplanned changes to endpoint definitions. Any adaptation that affects the stagewise tests must be reflected in the p-value calibration.

Common pitfall: “A was selected because its stage 1 result was best, so I can use its ordinary one-sided p-value in the final test.” This ignores the opportunity to select the most favorable of multiple candidates. Use a prespecified selection-adjusted stage 1 p-value, and ensure the stage 2 test is valid conditional on the selection. Also require nonnegative combination weights when relying on the conservative-p-value argument: weights with a negative component can reverse the ordering of evidence and invalidate that justification.

Fit the stage 1 ANCOVA.
Estimate each baseline-adjusted treatment contrast and calculate stage 1 p-values under the prespecified model.

Apply the selection rule and adjustment.
Select the treatment as planned and use a stage 1 p-value calibrated for considering the candidate treatments.

Analyze new stage 2 participants.
Fit the prespecified ANCOVA for the selected treatment and obtain a p-value valid conditional on the stage 1 decision.

Combine and interpret.
Apply the prespecified nonnegative weights, calculate the combined p-value, and interpret it alongside the stage-specific adjusted contrasts.

Check Your Understanding

Answer each question, then compare your reasoning with the brief solutions below.

In an ANCOVA, the treatment contrast is 3.0 points, the standard error is 1.5 points, and the test is one-sided for improvement. What is the test statistic?
Why is an unadjusted stage 1 p-value for the selected treatment potentially misleading when two treatments were considered?
For inverse-normal weights \(w_1=0.6\) and \(w_2=0.8\), check whether the squared-weight condition holds and whether both weights meet the sign condition.
What feature of the stage 2 data supports validity of its p-value after treatment selection?
Why is the inverse-normal combined test not automatically a pooled estimate of the treatment effect?

Solutions: (1) \(t=3.0/1.5=2.00\). (2) Selecting the most favorable result creates a multiplicity opportunity, so the raw p-value does not account for selection. (3) \(0.6^2+0.8^2=0.36+0.64=1\), and both weights are nonnegative. (4) Stage 2 uses new, independent participants and a test valid conditional on the stage 1 selection. (5) The method combines transformed p-values, which measure evidence against the null; it does not average the stage-specific contrasts or their standard errors.