Why Utility Units Matter
A utility scale assigns numerical values to clinical consequences so that uncertain outcomes can be compared through their expected utility. The numerical origin and unit may be conventional: one stakeholder may express the same utility differences in a scale ten times larger, or add a constant to every consequence. If the change is applied consistently, those choices of origin and unit should not alter which dose has the greater expected utility. But not every change that preserves the ordering of individual outcomes preserves the ordering of uncertain dose options.
This distinction matters when interpreting elicited utilities and when converting them for calculation or presentation. A change in units is not necessarily a change in preference, whereas a nonlinear remapping can change how variation in outcomes affects an expected score. The key question is whether the same transformation is applied to every possible utility outcome at every dose, and whether that transformation is positive affine.
The expectation is taken over the outcome distribution at each dose. The transformation in the definition acts on each possible utility value before averaging. For a nonlinear \(g\), this generally differs from applying \(g\) to the original expected utility: \(E[g(U_d)]\) need not equal \(g(E[U_d])\). This distinction is central to understanding why the positive affine result holds and why a general increasing transformation need not.
The Positive Affine Invariance Result
Assume the candidate doses are evaluated using the same utility scale, each dose has a well-defined finite expected utility, and one common transformation \(g(u)=au+b\) is applied to all possible outcomes. Linearity of expectation gives the transformed score directly.
To compare two doses \(d_1\) and \(d_2\), subtract their transformed expected utilities. The common additive constant cancels, and the common positive multiplier scales the original difference.
The theorem also preserves all ties and therefore preserves the set of doses that maximize expected utility. The constant \(b\) changes the level of every dose score but not any pairwise difference. The multiplier \(a\) changes the numerical size of each difference while retaining its sign. Thus the ranking is invariant, but the numerical values and the magnitudes of differences are not generally unchanged.
Worked Examples
Worked Example: Changing the Utility Unit and Origin
Suppose two hypothetical doses have the following outcome utilities. At Dose A, utility is 2 with probability 0.25 and 8 with probability 0.75. At Dose B, utility is 6.5 with probability 1. The probabilities sum to 1 for each dose, so these distributions define the expected utilities.
Before transforming the scale:
The doses tie. Now use the common positive affine transformation \(g(u)=4u+3\). For Dose A, its possible transformed utilities are \(4(2)+3=11\) and \(4(8)+3=35\). For Dose B, the transformed utility is \(4(6.5)+3=29\). Therefore:
The tie is preserved. In general, the pairwise difference after transformation is \(4\) times the original difference; here both are zero. This illustrates why it would be incorrect to say that every difference changes when \(a\ne1\): a zero difference remains zero.
Worked Example: A Nonlinear Transformation Reverses the Ranking
Consider hypothetical Dose A, which yields utility 0 or 10 with probability 0.5 each, and hypothetical Dose B, which yields utility 4 with certainty. All utilities are nonnegative, so the strictly increasing transformation \(g(u)=\sqrt{u}\) is defined for every possible outcome.
On the original scale:
Dose A has the greater expected utility. After transforming each possible outcome and then taking its expectation:
The transformed expected utility is greater for Dose B, so the ranking reverses. The transformation preserves the order of individual utility values—for example, \(10>4>0\)—but it does not preserve the expected-utility ranking of these uncertain outcomes. This is not a contradiction: expected utility averages the transformed outcome utilities, and the averaging operation is affected by the nonlinear remapping.
Worked Example: Negative Scaling and Dose-Specific Offsets
Suppose two hypothetical doses have expected utilities \(EU(A)=7\) and \(EU(B)=5\). First consider the common transformation \(g(u)=-2u+1\). It is affine, but its slope is negative, so it is not a positive affine transformation.
Dose B now has the greater transformed score: the ranking has reversed. The difference changes from \(7-5=2\) to \(-13-(-9)=-4\), which is \(-2\) times the original difference. A zero slope would instead map every utility to the same constant and tie all doses, regardless of their original ranking.
Now return to the original scores and apply different offsets by dose: add 0 to Dose A's score and 3 to Dose B's score. The resulting scores are \(7+0=7\) and \(5+3=8\), so Dose B ranks higher. This is not a common transformation \(g(u)=au+b\): the offset depends on the dose and therefore does not cancel in pairwise comparisons. A scale conversion must be common across the alternatives for the invariance result to apply.
Why General Increasing Transformations Are Different
If each dose produced a single certain utility value, any strictly increasing transformation would preserve the ranking of those values. With uncertain outcomes, however, expected utility compares averages across possible consequences. A nonlinear transformation changes the distances between utility values, and so changes their contributions to those averages. The square-root example shows this directly: the transformed utilities remain ordered outcome by outcome, but their expected values rank the doses differently.
There is another possible calculation that should not be confused with transforming the utility scale. One might first calculate \(EU(d)\) and then apply an increasing function \(g\) to that single number. In that calculation, \(EU(d_1)>EU(d_2)\) implies \(g(EU(d_1))>g(EU(d_2))\). But redefining the utility of each possible clinical consequence means calculating \(E[g(U_d)]\), not \(g(E[U_d])\). For nonlinear \(g\), these expressions need not be equal, and only the first represents expected utility under the transformed outcome scale.
The distinction also clarifies what preference information is sufficient. A strictly increasing transformation preserves the ordering of individual consequences, so it preserves an ordinal ranking of those consequences. Expected-utility comparisons across uncertain outcomes require more than that ordering: the numerical spacing of utility values affects the expectation. Under the expected-utility framework, positive affine transformations preserve those comparisons, but arbitrary increasing transformations need not.
Assumptions, Diagnostics, and Limits
The invariance result is exact, not approximate, when its conditions hold. It does not require normally distributed outcomes or equal outcome variability across doses; it requires finite expectations and one common positive affine transformation. If dose scores are calculated by averaging over additional uncertainty, the same linearity argument applies provided the same transformation is applied to the same utility outcomes throughout that calculation.
When checking a claimed scale change, use the following sequence:
Check whether the transformation is applied to each possible outcome utility before taking expectations, rather than only to the already-calculated expected utility.
The same \(a\) and \(b\) must apply to every utility value at every dose. Dose-specific weights, offsets, or nonlinear rules are not covered by the theorem.
The multiplier must satisfy \(a>0\). A negative multiplier reverses strict rankings, while a zero multiplier collapses all scores to a tie.
For each pair, verify that the transformed difference equals \(a\) times the original difference. This checks the ranking and catches arithmetic or implementation errors.
A common error is to argue that a nonlinear transformation must preserve dose choice because it is increasing. That reasoning considers only the order of possible utility outcomes, not the expectations of their transformed values. Another error is to treat a dose-specific adjustment as a harmless change of origin. A shared additive constant cancels; different constants do not. Finally, numerical differences should not be described as unchanged under a rescaling: they are multiplied by \(a\), although ties remain ties.
Positive affine invariance is useful when utility values are reported in different positive units or with a different common zero. It does not establish that the underlying utility assignments are clinically appropriate, nor does it make a nonlinear re-elicitation equivalent to a unit conversion. The rankings are invariant only relative to the specified expected-utility model and the common transformation conditions above.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- Dose A and Dose B have expected utilities 3 and 5. Under the common transformation \(g(u)=2u-1\), what are the transformed expected utilities and their ranking?
- If two doses are tied before applying \(g(u)=6u+4\), must they remain tied? Explain using the pairwise-difference formula.
- Why can a strictly increasing nonlinear transformation reverse an expected-utility ranking even though it preserves the order of every pair of individual utility values?
- Two dose scores are 8 and 6. An analyst adds 0 to the first and 3 to the second. Is this a common additive shift, and what ranking results?
Solutions. (1) Dose A becomes \(2(3)-1=5\), and Dose B becomes \(2(5)-1=9\); Dose B remains higher. (2) Yes. Their original difference is zero, and the transformed difference is \(6(0)=0\). (3) The expectation averages transformed outcome utilities; a nonlinear transformation changes the spacing between possible utilities, so the averages can change order. (4) No. The offsets differ by dose. The new scores are \(8+0=8\) and \(6+3=9\), so the second dose ranks higher.