From Clinical Priorities to a Decision Scale
A dose can produce several kinds of outcomes: efficacy without important toxicity, efficacy with toxicity, toxicity without efficacy, or neither. Choosing among doses therefore requires more than listing response and toxicity rates. A dose utility makes explicit how the trial’s decision-makers value those outcomes relative to one another. The resulting score is not an objective property of a dose; it represents stated clinical priorities on a defined scale.
The earlier material on joint efficacy and toxicity outcomes supplies a useful starting point: describe each patient by the outcomes relevant to the decision. Here, the task is to assign values to those outcome states and combine them with their probabilities. Probabilities may come from an analysis or be provided as hypothetical planning quantities; constructing or fitting a probability model is not the purpose here.
For a simple binary-outcome illustration, let \(E=1\) denote efficacy and \(T=1\) denote a specified toxicity during the assessment period; zero denotes the absence of the corresponding event. The four possible states are \((E,T)=(0,0),(1,0),(0,1),(1,1)\). The toxicity definition and follow-up period must be clinically explicit: a utility for any-grade toxicity, for example, need not represent preferences about severe or irreversible toxicity.
Defining the Utility Scale
A practical elicitation begins by describing the outcome states in language clinicians and patients can interpret, including the relevant time horizon and severity. Decision-makers then compare states or assign scores. One approach is to anchor the least-preferred state at zero and a reference state at a convenient higher value. Other states are assigned values that reflect their relative desirability. The numerical spacing matters: a difference of 20 utility units is intended to represent twice the preference difference of 10 units on the chosen scale.
The scale may use arbitrary points, such as 0 to 100, or another explicitly defined unit. Utility differences, not the absolute zero or unit label, drive comparisons. Multiplying every utility by a positive constant leaves rankings unchanged; adding the same constant to every state also leaves rankings unchanged. These transformations do not make the choices objective: the relative values assigned to clinically distinct states still encode preferences.
Worked Example: Assigning Values to Four Outcome States
Suppose a hypothetical committee considers a specified assessment period and agrees on these preference statements: neither efficacy nor toxicity has value 0; efficacy without toxicity is worth 70 points; toxicity without efficacy is worth −50 points; and experiencing both is worth 20 points. These scores are hypothetical elicited preferences, not observed patient data.
The assigned utility table is \(U(0,0)=0\), \(U(1,0)=70\), \(U(0,1)=-50\), and \(U(1,1)=20\). The positive score for both outcomes means the committee judges efficacy to outweigh the specified toxicity in that state. A different committee might reasonably assign a lower value, especially if the toxicity were more severe.
The points also happen to have an additive form: \(U(e,t)=70e-50t\). This reproduces all four assignments, since \(70-50=20\) when both events occur. That equality is a substantive preference assumption, not a mathematical necessity. If the committee instead assigned \(U(1,1)=5\), the value of experiencing both would not equal the sum of the separate efficacy and toxicity contributions.
A score table can preserve important non-additive preferences. For example, toxicity might be especially undesirable when efficacy is absent, or a particular combination might be judged worse than the sum of its separate components. In those cases, retain the four state-specific values rather than forcing an additive formula. Elicitation should also check that the resulting ordering and spacing actually match the stated preferences.
Expected Utility for a Dose
Let \(p_{et}(d)=P(E=e,T=t\mid d)\) be the probability of outcome state \((e,t)\) at dose \(d\). These four probabilities are nonnegative and sum to one. If the utility scale is fixed for the decision, the dose’s expected utility is
The expected utility summarizes the average preference score across repeated patients treated at that dose, under the specified outcome probabilities and utility values. Selecting the dose with the largest expected utility is a decision rule only when maximizing this quantity is the stated objective and any applicable eligibility or safety conditions have also been addressed.
Why the Probability-Weighted Formula Holds
The formula follows directly from the definition of an average over mutually exclusive outcome states. Imagine a large group of patients treated at dose \(d\), with the four state proportions close to their probabilities. The average utility in that group is approximately the sum of each state’s proportion multiplied by its utility. In expectation, those proportions are replaced by the corresponding probabilities.
The states \((e,t)\) are mutually exclusive and exhaustive, so exactly one occurs for each patient and \(\sum_{e,t}p_{et}(d)=1\).
The utility \(U(E,T)\) equals \(U(e,t)\) whenever that patient has state \((e,t)\).
By the expectation rule for a discrete random variable, \(E\{U(E,T)\mid d\}=\sum_{e,t}P(E=e,T=t\mid d)U(e,t)\), which is the stated expected utility.
This calculation needs probabilities for the states and utilities on a common scale. It does not require the efficacy and toxicity events to be independent: the joint state probabilities can reflect any dependence. Nor does the formula imply that an individual patient receives an “average” outcome; it is a population-level decision summary.
Worked Example: Comparing Two Doses With State Utilities
Use the four scores from the first example. Suppose the hypothetical joint outcome probabilities are as follows. Each row gives probabilities in the order no efficacy/no toxicity, efficacy only, toxicity only, and both.
| Dose | \(p_{00}\) | \(p_{10}\) | \(p_{01}\) | \(p_{11}\) |
|---|---|---|---|---|
| Low | 0.35 | 0.45 | 0.10 | 0.10 |
| High | 0.20 | 0.45 | 0.10 | 0.25 |
Both rows sum to 1.00. For the low dose,
For the high dose,
On this hypothetical scale, the high dose has expected utility 3.0 points greater than the low dose. The difference is driven by the larger probability of the “both” state, which has positive utility under these preferences. This is a preference-based comparison, not evidence that the high dose is clinically acceptable under every safety requirement.
Additive Weights and Ranking Sensitivity
When the utility is additive, it can be written \(U(E,T)=bE-cT\), where \(b\) is the value assigned to efficacy and \(c\) is the penalty for toxicity. For nonnegative \(b\) and \(c\), a larger \(b\) gives more weight to efficacy and a larger \(c\) makes toxicity more costly. Taking expectations gives
This equality follows by linearity of expectation: \(E(bE-cT\mid d)=bE(E\mid d)-cE(T\mid d)\), and a binary indicator has expectation equal to its event probability. It holds whether or not efficacy and toxicity are associated. It also shows why a utility weighting is a decision choice: changing the relative size of \(b\) and \(c\) can change the preferred dose.
Worked Example: Finding the Weight at Which Rankings Switch
Consider two hypothetical doses with efficacy and toxicity probabilities summarized as follows: dose A has efficacy probability 0.60 and toxicity probability 0.15; dose B has efficacy probability 0.75 and toxicity probability 0.40. Use the additive utility \(bE-cT\), with \(b>0\).
The expected utilities are \(EU(A)=0.60b-0.15c\) and \(EU(B)=0.75b-0.40c\). Subtracting gives
Thus B ranks above A exactly when \(0.15b-0.25c>0\), or \(c/b<0.60\). They tie when \(c/b=0.60\), and A ranks above B when \(c/b>0.60\). For example, with \(b=1,c=0.5\), \(EU(A)=0.525\) and \(EU(B)=0.550\), so B ranks higher. With \(b=1,c=1\), the utilities are 0.45 and 0.35, so A ranks higher. The probabilities alone do not determine a unique ranking without the preference weights.
A sensitivity analysis should examine a clinically defensible range of utility values or weights, not merely one convenient alternative. A ranking that persists throughout that range is robust to the examined preference uncertainty. A ranking that changes identifies the tradeoff requiring further discussion; it does not by itself show which weight is correct.
Diagnostics, Alternatives, and Limits
Before interpreting a ranking, check the outcome definitions, probability inputs, and score assignments. The four joint probabilities should be between zero and one and sum to one. Confirm that outcome states are described over the same follow-up period at every dose, and that the utility scale gives sensible values to important combinations. If utilities were elicited from more than one stakeholder group, compare the resulting rankings rather than silently averaging incompatible preferences.
A common mistake is to treat a toxicity penalty as a universal constant and announce that the dose with the most efficacy is best after subtracting toxicity. The correction is to state the scale and the rationale for its relative weights, then examine whether plausible alternatives change the decision. Another mistake is to infer that a high expected utility makes a dose safe. Utility-based ranking and safety restrictions answer different questions: a protocol may first exclude doses that fail a safety requirement, then compare utility among the remaining doses.
| Approach | What it does | Important limitation |
|---|---|---|
| Expected utility | Combines outcome probabilities with explicit preference values. | Results depend on the utility scale and probability estimates. |
| Safety-constrained choice | Excludes doses that do not meet a specified safety condition before comparing eligible doses. | Does not, by itself, rank eligible doses by the value of their efficacy-toxicity tradeoff. |
| Separate efficacy and toxicity summaries | Reports each outcome probability without collapsing them to one score. | Leaves the tradeoff to a later decision and may not yield a single ranking. |
Expected utility reduces a multidimensional outcome to one number, so it can hide differences between doses with similar scores. It also depends on the adequacy of the probabilities and the relevance of the chosen outcome states. When preferences are uncertain, report the utility specification and sensitivity results alongside the ranking. When no defensible common scale can be elicited, presenting outcome probabilities separately may be more transparent than asserting a precise utility-based ordering.
Check Your Understanding
Use the definitions above. The solutions are included after the questions so that each calculation can be checked.
- Suppose \(U(0,0)=0\), \(U(1,0)=60\), \(U(0,1)=-40\), and \(U(1,1)=20\). Is this table additive in efficacy and toxicity under the form \(bE-cT\)? Explain.
- A hypothetical dose has joint probabilities \(p_{00}=0.30\), \(p_{10}=0.50\), \(p_{01}=0.10\), and \(p_{11}=0.10\). Using utilities \(0,60,-40,20\) in the corresponding state order, calculate its expected utility.
- Under additive utility \(bE-cT\), dose C has efficacy probability 0.70 and toxicity probability 0.20; dose D has efficacy probability 0.80 and toxicity probability 0.35. Find the condition on \(c/b\) for D to rank above C.
- Why does computing expected utility from joint state probabilities not require efficacy and toxicity to be independent?
Solutions. (1) Yes. The additive values are \(b=60\) and \(c=40\), which give \(U(1,1)=60-40=20\), matching the table. (2) \(EU=(0.30)(0)+(0.50)(60)+(0.10)(-40)+(0.10)(20)=30-4+2=28\) utility points; the probabilities sum to 1. (3) \(EU(D)-EU(C)=(0.80-0.70)b-(0.35-0.20)c=0.10b-0.15c\). For \(b>0\), D ranks above C when \(c/b<0.10/0.15=0.667\) (approximately). (4) The calculation sums the probability of each joint state multiplied by its utility. Those joint probabilities already describe how outcomes occur together; independence is not needed.