Absolute and Relative Efficacy Objectives

Why the Efficacy Scale Matters

A dose-response prediction does not, by itself, specify what counts as a better dose. The decision also depends on the quantity used to summarize efficacy. An increase of 5 units may be important on one endpoint and modest on another; a 20% change may represent very different numbers of units for people with different starting values. The scale is therefore part of the dose-finding objective, not merely a way to restate the same result.

The key distinction is between asking how many endpoint units change and asking how large that change is relative to a reference value. When every candidate dose is compared using one common population mean, the two summaries often rank doses identically. When individual relative changes are averaged, however, people with different baseline values contribute differently to the absolute and relative summaries. If dose effects vary across people, the rankings can diverge.

These definitions assume that \(Y(d)\) and \(Y(0)\) refer to the same endpoint, follow-up time, and target population. The reference condition must be stated: “relative to baseline,” “relative to placebo,” and “relative to an active comparator” are not interchangeable. For an endpoint where lower values are better, it is often clearer to define a positive improvement as \(Y(0)-Y(d)\), so that larger effects consistently mean greater benefit. The sign convention must be declared before comparing doses.

Three Quantities That Are Easy to Confuse

A relative effect can be summarized in more than one way. One common quantity is the ratio of population means, \(E[Y(d)]/E[Y(0)]\). Another is the mean of individual ratios, \(E[Y(d)/Y(0)]\). These are generally not equal. The first compares two marginal means; the second gives each individual’s percentage change equal weight before averaging. This tutorial uses the mean individual relative effect when discussing how relative weighting can change dose rankings.

The difference matters for a dose objective. If every candidate dose is assessed in the same population and the objective is the ratio of population means, the denominator \(E[Y(0)]\) is common to all doses. For a positive common denominator, ranking doses by \(E[Y(d)]/E[Y(0)]\) is equivalent to ranking them by \(E[Y(d)]\), and hence by \(E[Y(d)]-E[Y(0)]\). A reversal cannot occur under that comparison. A reversal can occur when the relative objective averages person-level changes, because it weights a unit of change more heavily for people with smaller positive reference values.

Why Individual Relative Effects Can Change a Ranking

Write the reference outcome for an individual as \(B=Y(0)\), and that individual’s relative change at dose \(d\) as \(r_d=[Y(d)-Y(0)]/Y(0)\). Then the individual absolute change is \(B r_d\). Averaging across the target population gives:

The absolute objective averages changes in endpoint units; the relative objective averages proportional changes. Because \(B\) varies across individuals, the absolute objective effectively gives more weight to relative changes among people with larger reference values. The relative objective does not apply that same baseline-size weighting. This difference can reverse the ordering when the dose-specific relative effects vary across individuals.

A useful way to see the condition is to compare doses A and B. Their difference in mean absolute effect is \(E[B(r_A-r_B)]\); their difference in mean relative effect is \(E[r_A-r_B]\). If \(r_A-r_B\) has the same sign for everyone, both expectations have that sign because \(B>0\). Thus, a dose that produces at least as large a relative improvement for every individual cannot be overtaken under either objective. A reversal requires a crossing pattern: one dose performs relatively better for some people, while the other performs relatively better for others.

The covariance term explains the possibility of a reversal. Even when the average relative difference favors one dose, its association with baseline size can make the average absolute difference favor the other. This is a property of the estimands and the response pattern, not a mathematical error. It is also a reason to inspect subgroup or individual-level predictions rather than relying only on two marginal summaries.

Worked Comparison for a Decreasing Endpoint

Relative effects are often used for biomarkers or other positive endpoints where a decrease is beneficial. In that setting, define improvement as the reference value minus the dose value. Using a positive improvement avoids interpreting a more negative change as “better” simply because of the sign convention.

Assumptions, Diagnostics, and Failure Modes

Before interpreting a dose ranking, verify what the predictions represent. Both dose summaries must refer to the same endpoint definition, follow-up time, reference condition, and target population. The predictions should also come from a model adequate for the decision: if the dose-response relationship or relevant dose-by-covariate variation is poorly estimated, a precise-looking ranking can be misleading. Parameter uncertainty should be carried through the chosen estimand rather than omitted after computing point predictions.

Relative summaries need additional checks. A ratio or percentage change is unstable when the reference value is near zero, undefined at zero, and can be difficult to interpret for outcomes that may be negative. A few small denominators can dominate a mean of individual ratios. If the endpoint is a positive biomarker, inspect the distribution of reference values and individual relative changes, not only their averages. If the endpoint has a meaningful zero but can take negative values, a percentage change may not be a coherent objective.

The following checks help identify problems before a scale is used to choose a dose:

• Check the direction. State whether higher or lower endpoint values indicate benefit, and define the effect sign accordingly.
• Check the denominator. Confirm that the reference value is positive and not concentrated near zero when using individual relative changes.
• Check the aggregation. Label the result as a ratio of population means, a mean of individual ratios, or another specified quantity.
• Check heterogeneity. Examine whether dose rankings cross across baseline or covariate strata. If one dose is better everywhere, a reversal between these positive-baseline scales should not occur.
• Check uncertainty. Compare uncertainty in the dose-specific estimands, especially if a small predicted difference determines the selected dose.

A practical comparison is to report both scales when each has a defensible interpretation. The absolute scale preserves clinical units and is often straightforward to relate to assay precision or a meaningful unit change. The relative scale expresses proportional change and can facilitate comparisons across settings with different baseline magnitudes, provided the denominator is meaningful. Neither scale is automatically more correct: the scientific question and decision context determine which estimand is relevant.

Common Mistake: Treating Every Percentage Summary as Equivalent

A common mistake is to calculate a percentage by dividing the mean absolute change by the mean baseline, then describe that result as the average individual percentage change. These are different calculations. In the worked reversal example, the population mean reference outcome is \(0.50(10)+0.50(100)=55\) units. Strategy A’s mean absolute gain divided by that mean is \(4.5/55\approx8.18\%\), while its mean individual relative gain is 22.5%. For Strategy B, the corresponding values are \(5.5/55=10.0\%\) and 19.0%. The ratio-of-means summary ranks B higher, while the mean of individual percentages ranks A higher.

The correction is to name the estimand before calculating it. If the objective is a ratio of population means, use that ratio consistently and recognize that a common positive denominator preserves absolute-effect rankings. If the objective is the mean individual percentage change, calculate individual or stratum-specific relative changes and average them using the target-population weights. Do not switch between the two interpretations when reporting or selecting a dose.