Binary Responder Endpoints and Threshold Selection

From a Clinical Outcome to a Responder Event

A responder endpoint turns an outcome into a binary event: each participant either meets a prespecified response criterion or does not. This can make a result easier to communicate, but the event is not defined by the data alone. Its threshold determines which outcomes count as responses, which in turn determines the event probabilities, treatment contrast, power, and required sample size.

The threshold should represent a clinically meaningful outcome, not merely a value that happens to produce a favorable analysis. For planning, the responder definition, assessment time, and handling of the underlying outcome must be specified in terms consistent with the estimand. The earlier material on estimands and model-based endpoint contrasts is relevant here: a carefully computed contrast is not interpretable unless it represents the intended treatment effect.

For an outcome where smaller values are better, the event would instead be defined with the corresponding inequality, such as \(Y=I(X\le c)\). The direction must be explicit. Response probabilities depend on the threshold: raising a threshold for a benefit-oriented outcome cannot increase the response probability in either group. The treatment contrast may change as well, and need not move in the same direction as either group’s response probability.

Threshold Choice and Its Consequences

A threshold is clinically justified when it distinguishes a meaningful outcome from one that does not meet the intended response criterion. Evidence for that choice may come from clinical interpretation of the outcome scale, external evidence, or a prespecified clinical rationale. Statistical convenience alone—such as choosing the threshold that maximizes an observed treatment contrast—does not establish clinical meaning and risks an optimistic estimate of benefit.

The following hypothetical example illustrates the arithmetic. The normal distributions are a planning assumption for the continuous outcome, not a claim that all clinical outcomes follow a normal distribution.

Power and Sample Size on the Risk-Difference Scale

Let \(p_1\) and \(p_0\) be the planned response probabilities in the treatment and control groups. The risk difference is \(\Delta=p_1-p_0\), measured in probability units; multiplying by 100 expresses it in percentage points. For two independent groups with equal sample size \(n\), a common approximate calculation uses a two-sided test of \(H_0:p_1-p_0=0\), significance level \(\alpha\), and target power \(1-\beta\).

The approximation below uses a normal critical value, the pooled response probability for the null standard error, and the two separate probabilities for the standard error under the planned alternative. It assumes independent participant outcomes, fixed group allocation, and sufficiently large expected responder and nonresponder counts for the normal approximation to be reasonable.

This formula is a planning approximation, not a guarantee of the power of every particular analysis procedure. It should be paired with an analysis plan that tests the same contrast and uses the same endpoint definition. For small samples or probabilities near 0 or 1, use a method suited to the planned test or confirm performance by simulation rather than relying uncritically on a normal approximation.

Planning on Relative Scales

A risk ratio compares response probabilities multiplicatively: \(RR=p_1/p_0\). An odds ratio compares odds, where the odds in a group are \(p/(1-p)\). These measures are not interchangeable with a risk difference. For example, the same relative effect can correspond to different absolute differences at different baseline probabilities. In addition, odds ratios can differ from risk ratios even when the outcome is not rare; choosing an odds-ratio scale is not simply a relabeling of the risk ratio.

For planning a risk ratio with a large-sample Wald test of \(H_0:RR=1\), the approximate variance of the estimated log risk ratio in two independent groups is

The expression follows from the approximate variance of a sample proportion, \(p(1-p)/n\), and the delta method for the logarithm: the derivative of \(\log(p)\) is \(1/p\), so the approximate variance of \(\log(\hat p)\) is \(\{p(1-p)/n\}/p^2=(1-p)/(np)\). Independent groups contribute additive variances. Under the planned alternative, the log risk ratio is \(\log(RR)\); dividing its squared value by the variance gives the approximate squared signal-to-noise ratio. Setting this ratio to \((z_{1-\alpha/2}+z_{1-\beta})^2\) yields the planning expression below.

For an odds ratio \(\theta\), a planned control probability \(p_0\) and target odds ratio imply the treatment probability

For a large-sample Wald calculation of \(\log(\widehat{\theta})\), its approximate variance with equal group sizes is \(\{1/[p_1(1-p_1)]+1/[p_0(1-p_0)]\}/n\). The resulting approximate sample size is the sum of these variance coefficients multiplied by \((z_{1-\alpha/2}+z_{1-\beta})^2\), then divided by \(\{\log(\theta)\}^2\). This relies on adequate counts in all four cells of the two-by-two table.

Diagnostics, Alternatives, and Limitations

Before accepting a calculation, check that the endpoint, analysis contrast, and planning assumptions align. The threshold and assessment time should be clinically defensible and fixed in advance. The planned probabilities should be plausible for the intended population and treatment context. Check expected responders and nonresponders in each group; for instance, the risk-ratio and odds-ratio approximations can become unstable when a probability is near zero or one. Also verify that the sample-size formula matches the planned test, sidedness, allocation ratio, and significance level.

The normal approximations are less dependable with sparse data, severe imbalance, or very high or low probabilities. Alternatives include score-based procedures for comparing two proportions, exact procedures for sparse tables, and regression models when covariate adjustment is planned. They may have different finite-sample operating characteristics, so sample size should be based on the intended analysis rather than chosen from a formula solely because it is convenient. For complex designs or uncertain response probabilities, simulation can estimate power directly by generating binary outcomes under the planned design and applying the planned analysis to each simulated trial.

A binary endpoint also discards distinctions among participants on the same side of the threshold. Someone just below the response cutoff is classified with someone whose outcome is far worse, while two outcomes just on opposite sides of the cutoff are treated as different response categories. If the full outcome scale is clinically informative, a continuous-endpoint analysis may retain more information; the appropriate choice depends on the estimand, interpretation, and design. Threshold-based results can be sensitive to the cutoff, so sensitivity analyses using clinically plausible thresholds can help describe that dependence, but they do not justify selecting the most favorable threshold after observing results.

• Pitfall: “The risk ratio is 1.5, so the treatment adds 50 percentage points to the response rate.” This confuses a relative increase with an absolute difference. If the control probability is 0.20, an RR of 1.5 implies a treatment probability of 0.30 and a 10-percentage-point difference.
• Correction: State the scale and its units: a risk difference is in probability units or percentage points, while a risk ratio and odds ratio are dimensionless relative measures. Translate to probabilities where possible for clinical interpretation.
• Pitfall: Choosing the threshold that yields the smallest planned sample size or the largest observed treatment contrast.
• Correction: Justify the cutoff clinically before using trial outcomes to compare groups; assess how plausible alternative cutoffs affect planning without treating them as outcome-driven replacements.