From Clinical Outcomes to Preference-Valued Scores
Clinical outcomes are usually first described in clinical terms: response category, symptom severity, or whether a toxicity occurred. A utility function maps such outcomes to numerical scores representing their value to a decision maker. The mapping makes a preference structure explicit; it does not turn a clinical outcome into a directly measured physical quantity.
For example, a patient may prefer a complete response without toxicity to a partial response with moderate toxicity. A utility function can assign scores to these states, but the numbers are meaningful only to the extent justified by the preference information and scale interpretation behind them. The introductory material on aligning outcome scales explains why outcomes must be defined consistently before they are combined or compared.
An outcome state may be a single category or a combination of attributes. For benefit-risk assessment, a joint state can specify both response and toxicity. For example, “partial response with moderate toxicity” is distinct from either “partial response” or “moderate toxicity” considered alone. A function defined on joint states can represent the preference ordering directly without assuming that each attribute has an independent score.
Ordinal Ranking and Cardinal Utility
A utility function can be used at different levels of numerical interpretation. An ordinal utility assignment records only which states are preferred to which. If \(u(x)>u(y)\), state \(x\) ranks above state \(y\); the size of the difference \(u(x)-u(y)\) has no established interpretation. Any strictly increasing transformation of the scores preserves the ranking.
A cardinal utility representation carries more information: differences between scores have a defined meaning within the specified model. In expected-utility theory, when preferences over lotteries satisfy the relevant representation assumptions, a utility representation is unique up to a positive affine transformation. Thus, the origin and unit are conventional, but the relative size of utility differences is preserved by valid changes of scale. Cardinal utility is not the same as a ratio scale: a score of 0.8 is not thereby “twice as good” as a score of 0.4.
The invariance result can be checked directly. Let \(v(x)=a+b\,u(x)\), where \(a\) is any constant and \(b>0\). Then
The sign of every difference, and the ratio of any two nonzero differences, is unchanged. By contrast, a nonlinear increasing transformation can preserve the order while changing the differences. For instance, scores \(0,\ 0.5,\ 0.6,\ 1\) become \(0,\ 0.25,\ 0.36,\ 1\) when squared: the ordering remains, but the gaps do not. Therefore, an ordinal scale cannot be treated as cardinal merely because its categories have been assigned numbers.
Worked Example: Ranking Response Categories
Suppose a hypothetical trial uses three ordered response states: no response, partial response, and complete response. A clinical panel agrees that complete response is preferred to partial response, which is preferred to no response. Consider the assignments \(0, 1, 2\), respectively.
These scores correctly encode the agreed order. But, without additional evidence, they do not show that the preference difference between no response and partial response equals the difference between partial and complete response. The scores are ordinal.
The increasing transformation \(v=u^2\) gives scores \(0, 1, 4\). It preserves the order, so it is equally valid as an ordinal coding. Yet the transformed gaps are 1 and 3 rather than 1 and 1. The original and transformed scores therefore cannot both be interpreted as preserving the same cardinal differences.
Conclusion: The data support a ranking, not a claim about the relative magnitudes of preference differences. Calling the values “utility points” does not change that conclusion.
Defining Utilities for Response and Toxicity States
Before assigning scores, specify the outcome states precisely. State definitions should make clear what counts as response, how toxicity severity is classified, and the assessment period to which each state refers. If the definition leaves out a clinically important distinction, the utility function cannot restore it later.
A direct assignment to joint states is one way to represent response and toxicity together. The following hypothetical scores illustrate a cardinal scale for a defined decision-maker perspective. They are teaching values, not estimates from patient data or recommendations for clinical use.
| Response state | Toxicity state | Utility score |
|---|---|---|
| Complete response | None | 1.00 |
| Complete response | Moderate | 0.72 |
| Partial response | None | 0.68 |
| Partial response | Moderate | 0.42 |
| No response | None | 0.30 |
| No response | Moderate | 0.08 |
The table places more utility on better response when toxicity is held fixed, and less utility on moderate toxicity when response is held fixed. It also encodes trade-offs between attributes: complete response with moderate toxicity scores above partial response without toxicity. That ordering is part of the hypothetical specification, not a mathematical consequence of the clinical labels.
Worked Example: Interpreting a Joint-State Utility Difference
Using the hypothetical table, compare complete response with moderate toxicity to partial response without toxicity. The respective scores are 0.72 and 0.68 utility points.
On this specified cardinal scale, the first state has a score 0.04 utility points higher than the second. This is a small difference on the table’s scale, but it is not automatically clinically negligible: that judgment depends on how the scale was established and what decision is being made.
The score difference does not mean a 4% increase in response, a 4% increase in survival, or a 4% probability of preferring one state. It is a difference on the stated utility scale. If only the ordering had been justified, the defensible conclusion would be merely that the first state ranks above the second.
A Short Derivation: Which Rescalings Preserve Cardinal Meaning?
Suppose a utility representation is transformed to \(v=a+bu\), with \(b>0\). For any two states, their difference changes by the same positive factor \(b\). Consequently, every difference keeps its sign, and for two pairs with a nonzero denominator,
Thus, adding a constant changes the zero point and multiplying by a positive constant changes the unit, but neither changes the relative sizes of differences. This is why a cardinal utility scale is interpreted as an interval scale rather than a ratio scale. Multiplication by a negative constant reverses preferences, while a nonlinear transformation generally changes difference sizes and is not an equivalent cardinal rescaling.
Worked Example: Converting Utility Units Correctly
Consider three hypothetical outcome states with cardinal utility scores 0.20, 0.50, and 0.80. Convert them to a scale defined by \(v=10+20u\).
The first-to-second difference is \(20-14=6\), and the second-to-third difference is \(26-20=6\). On the original scale both differences were \(0.30\); the new unit is 20 times the old unit. Their equality and the ordering are preserved.
Now consider replacing \(u\) with \(u^2\). The scores become 0.04, 0.25, and 0.64, with gaps 0.21 and 0.39. This nonlinear change preserves the ranking but not the relative size of differences. It is permissible if the numbers are used only as ordinal codes, but it is not an equivalent representation of the same cardinal scale.
Assumptions, Diagnostics, and Common Pitfalls
A utility function is useful only in relation to a clear purpose and a defensible preference interpretation. The relevant decision maker might be an individual patient or a defined group, but those perspectives should not be silently interchanged. The outcome states should be mutually distinguishable for the intended comparison, and their definitions should be sufficiently complete to capture the features that matter. If cardinal differences are claimed, the method used to support those differences must justify that stronger interpretation.
Practical checks include asking whether the ordering matches the stated preferences, whether important response-toxicity combinations are missing, and whether the numerical spacing has an evidential basis. Check that the utility scale is being used at the level it supports: rank comparisons for ordinal scores, and difference comparisons only for cardinal scores. Sensitivity to plausible alternative assignments can reveal whether an apparent conclusion depends on a fragile numerical choice.
- Numbers mistaken for cardinal evidence. Assigning 1, 2, and 3 to low, medium, and high toxicity creates an order; it does not establish equal preference gaps.
- Zero mistaken for “no value.” Zero is a scale anchor. A state scored zero may still be acceptable or preferred to states with negative scores, depending on the scale definition.
- Utility mistaken for a clinical measurement. A utility score is preference-valued, not a direct measure of symptom severity, response probability, or treatment effect.
- Attributes assumed to combine independently. A score for each attribute does not by itself justify adding those scores. Interactions or trade-offs may require direct consideration of joint states.
- One perspective treated as universal. A score representing one decision maker’s preferences need not represent another person or population.
A utility function is not the only way to represent outcomes. An ordinal ranking is more modest when only relative order is supported. A vector of clinical outcomes preserves the separate response and toxicity information without compressing it into one score, though it does not itself provide a single preference-valued summary. A composite clinical endpoint is another distinct construction: it classifies or combines clinical events according to endpoint rules, rather than necessarily encoding preference differences. The appropriate representation depends on the decision question and on the evidence available for the proposed interpretation.
The main limitation is that a score can look more precise than its foundations. Utility values depend on outcome definitions, perspective, and the way preferences are represented. Cardinality should therefore be stated as an assumption or supported property, not inferred from decimal places. A transparent ordinal scale may be more defensible than an unsupported cardinal one.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- Scores of 1, 2, and 3 are assigned to mild, moderate, and severe toxicity. What can be concluded from these values if no further preference evidence is available?
- A utility function assigns 0.3 to state A and 0.7 to state B. What does the score difference establish if the function is ordinal only?
- For a cardinal scale, use \(v=5+10u\) to transform scores \(u=0.2\) and \(u=0.6\). Calculate both transformed scores and their difference.
- Why does a direct score for “response with toxicity” require a precise definition of both attributes?
- What is the error in saying that a utility of 0.8 means an outcome is twice as good as one with utility 0.4?
Solutions. (1) Only the ordering mild \(<\) moderate \(<\) severe is established; equal one-point gaps have not been justified. (2) The scores show that B ranks above A. The numerical difference 0.4 has no established preference-difference meaning on an ordinal scale. (3) \(v(0.2)=5+10(0.2)=7\), and \(v(0.6)=5+10(0.6)=11\), so the transformed difference is \(11-7=4\). (4) Without definitions, different events or severities could be grouped under the same label, making the state represented by the score unclear. (5) Utility is not a ratio scale: its zero point is conventional, so the ratio \(0.8/0.4\) does not establish a “twice as good” interpretation.