Population Mean Utility Versus Mean Outcome

Why the Average Outcome May Not Be the Objective

A dose can improve the average clinical outcome while producing a less desirable distribution of outcomes for patients. The reason is that patients’ gains and losses need not have equal value: an extra unit may matter greatly near a clinically important level and little beyond it, or very high outcomes may carry disproportionate value. When preferences are represented by a nonlinear patient-level utility function, the relevant population quantity is the average of those utilities, not necessarily the utility assigned to the average outcome.

The distinction is between averaging after applying utility and applying utility after averaging. In general, these operations do not commute. As a result, the dose that maximizes mean clinical outcome can differ from the dose that maximizes mean patient utility. The calculations below treat outcome distributions as hypothetical and illustrate this distinction; they do not represent clinical-study results.

Definition: Let $Y(d)$ denote a patient’s clinical outcome under dose $d$, with outcome distribution $F_d$, and let $u(y)$ be the patient-level utility assigned to outcome $y$. The population mean utility at dose $d$ is $$ EU(d)=E\{u[Y(d)]\}=\int u(y)\,dF_d(y), $$ when this expectation exists. The mean clinical outcome is $\mu_Y(d)=E\{Y(d)\}$. These are different summaries: mean utility averages utility over patients’ outcomes, whereas $u\{\mu_Y(d)\}$ applies utility to the mean outcome.

The notation $EU(d)$ here names an estimand, not a statistical estimate. A dose selected by mean utility maximizes $EU(d)$ among the candidate doses, subject to any separate eligibility or safety constraints. The utility function must express the relevant preferences, and the outcome must be defined on a common scale and time horizon across doses. The earlier material on Patient-Level Utility for Multiple Outcomes provides the patient-level mapping; this tutorial focuses on what happens when those values are averaged over a population.

For the comparison to be meaningful, assume that each dose has a well-defined outcome distribution in the target population, that the same utility function applies at each dose, and that $E\{u[Y(d)]\}$ is finite. The mean-outcome objective likewise requires $E\{Y(d)\}$ to exist. These assumptions define the quantities being compared; they do not guarantee that either quantity is known precisely from data.

Nonlinearity and Dose Rankings

If utility is linear in the outcome, say $u(y)=a+by$ with $b>0$, then $EU(d)=a+b\mu_Y(d)$. The two objectives therefore rank doses identically. With nonlinear utility, dose rankings can diverge because the shape of $u$ determines how gains at different outcome levels are valued. The distribution matters as well as its mean: two doses with the same mean can have different mean utilities.

A short Taylor expansion gives intuition for the role of outcome variability. Suppose $u$ is twice differentiable near $\mu_Y(d)$, and outcomes are sufficiently concentrated there for a second-order approximation to be informative. Expanding around the mean gives

u(Y(d))\approx u\{\mu_Y(d)\} +u'\{\mu_Y(d)\}\{Y(d)-\mu_Y(d)\} +\frac{1}{2}u''\{\mu_Y(d)\}\{Y(d)-\mu_Y(d)\}^2.

Taking expectations, the first-order deviation term has expectation zero because $E\{Y(d)-\mu_Y(d)\}=0$, while the expected squared deviation is $\operatorname{Var}\{Y(d)\}$. Thus

EU(d)\approx u\{\mu_Y(d)\} +\frac{1}{2}u''\{\mu_Y(d)\}\operatorname{Var}\{Y(d)\}.

This is an approximation, not an exact formula in general. It shows why a concave utility function, for which $u''(y)<0$, tends to reduce mean utility when outcome variability increases around a fixed mean. A convex utility function tends to increase it. If utility is concave over the entire outcome range, Jensen’s inequality gives the exact bound $E\{u[Y(d)]\}\leq u\{E[Y(d)]\}$. Equality holds when outcomes are constant or utility is linear over the relevant range. The inequality concerns one dose’s distribution; it does not, by itself, identify which of two doses has higher mean utility.

Worked Example: A Concave Utility Reverses the Ranking

Suppose the clinical outcome $Y$ is a nonnegative score, with larger values preferred. Consider two hypothetical doses. Dose A yields an outcome of 4 for every patient. Dose B yields 0 for half of patients and 10 for the other half. Let the specified utility be $u(y)=\sqrt{y}$.

The mean outcomes are

\mu_Y(A)=4,\qquad \mu_Y(B)=\frac{1}{2}(0)+\frac{1}{2}(10)=5.

Mean-outcome maximization selects dose B because $5>4$. For the utility objective, calculate the expected utility at each dose:

EU(A)=\sqrt{4}=2,\qquad EU(B)=\frac{1}{2}\sqrt{0}+\frac{1}{2}\sqrt{10} =0+\frac{1}{2}(3.162)=1.581.

Mean utility selects dose A, despite its lower mean outcome. The square-root utility values additional outcome less at higher levels; B’s high outcomes do not compensate for its zero outcomes on this utility scale. This conclusion follows from the stated utility function and distributions, not from a general claim that dose A is clinically preferable.

Worked Distributions with Different Utility Shapes

The next examples use deliberately simple distributions so that the impact of the utility function is visible in the arithmetic. A utility scale is meaningful only in relation to the preferences it encodes; the numerical functions below are hypothetical specifications.

Worked Example: Diminishing Returns from a Capped Utility

Suppose outcome $Y$ is a clinical benefit score, larger is better, and utility is $u(y)=\min(y,6)$. Dose C produces outcomes 4 and 8 with equal probability. Dose D produces outcome 5.5 for every patient.

The mean outcomes are

\mu_Y(C)=\frac{4+8}{2}=6,\qquad \mu_Y(D)=5.5.

Mean-outcome maximization selects C. But utility stops increasing above 6. Therefore

EU(C)=\frac{\min(4,6)+\min(8,6)}{2} =\frac{4+6}{2}=5, \qquad EU(D)=\min(5.5,6)=5.5.

Mean utility selects D. The outcome of 8 under C does not add utility beyond the cap, while D gives every patient utility 5.5. If the cap is not defensible as a representation of preferences, this ranking is not defensible either; utility specification is a substantive part of the decision.

Worked Example: Convex Utility Rewards a Spread

Suppose a higher score is preferred and stakeholders specify $u(y)=y+0.1y^2$ over the outcome range in question, so that utility rises more steeply at higher scores. Dose E produces a score of 4 for every patient. Dose F produces scores of 2 and 6 with equal probability.

Both doses have the same mean outcome:

\mu_Y(E)=4,\qquad \mu_Y(F)=\frac{2+6}{2}=4.

For dose E, mean utility is

$$ EU(E)=u(4)=4+0.1(4^2)=4+1.6=5.6. $$

For dose F, calculate utility at each possible outcome before averaging:

EU(F)=\frac{u(2)+u(6)}{2} =\frac{\{2+0.1(2^2)\}+\{6+0.1(6^2)\}}{2} =\frac{2.4+9.6}{2}=6.0.

Mean outcome ties the doses, while mean utility selects F. The specified convex utility places extra value on the high score. This is a preference consequence, not evidence that variability is generally desirable; a different, concave utility could penalize the same spread.

Calculation, Diagnostics, and Alternatives

For a discrete outcome distribution, calculate mean utility by multiplying each possible utility by its probability and summing. For a continuous distribution, integrate utility against the outcome distribution. In a model-based dose analysis, the same principle applies: obtain the predicted outcome distribution at each candidate dose, transform outcomes using the specified patient-level utility, and average on the utility scale. Computing utility only at the predicted mean outcome generally substitutes $u\{\mu_Y(d)\}$ for the target $EU(d)$.

A practical calculation can be organized as follows:

Specify the outcome and preferences.
Define the clinical outcome, its scale and time horizon, and the utility function for individual outcomes.

Obtain each dose’s outcome distribution.
Use the same target population and outcome definition for every candidate dose.

Transform and average.
Evaluate utility across the distribution and compute $EU(d)$; do not replace this with utility at the mean unless utility is linear or the distribution is degenerate.

Compare dose objectives.
Report mean outcome and mean utility separately, with their units or scales, and explain any difference in the rankings.

Diagnostics should check that probabilities are valid and sum to one, that simulated or integrated outcomes cover the clinically relevant range, and that utility is evaluated at individual outcomes before averaging. Inspect the utility curve and the outcome distributions together: a ranking driven by a sharp threshold, cap, or extreme tail deserves particular scrutiny. Recalculate under plausible alternative utility specifications when preference uncertainty could change the selected dose.

Mean clinical outcome is an appropriate alternative when the mean itself is the decision objective or when utility is a positive affine transformation of outcome. It is easier to communicate in clinical units, but it cannot represent nonlinear patient preferences by itself. Mean utility directly represents a stated patient-level preference mapping, but depends on that mapping and may be less immediately interpretable in clinical units. A rate-based objective answers a different question about the frequency of a specified event; it should not be treated as interchangeable with either mean outcome or mean utility.

These calculations also have limits. A selected dose is only as credible as the outcome distributions and utility specification used to evaluate it. Estimation error in either can affect rankings, especially when doses have similar mean utilities. A population mean can also conceal differences among patient groups; this tutorial’s calculation does not resolve how such differences should influence a decision. Finally, maximizing mean utility does not automatically impose a separate safety constraint: if a constraint is required, it must be stated and assessed separately.

Common Mistake

Wrong reasoning: “Dose B has the higher mean outcome, so it must have the higher mean utility.” That conclusion assumes, without justification, that outcome and utility are interchangeable or that utility is linear. With nonlinear utility, the expected utility depends on the full outcome distribution, as well as on the utility assigned to each outcome.

Correction: Calculate $E\{u[Y(d)]\}$ for each dose by applying utility to outcomes first and then averaging. Use $u\{E[Y(d)]\}$ only when it is the actual objective or when the conditions make it equivalent, such as positive affine utility or a constant outcome.

Key takeaway: Mean outcome and population mean utility are distinct dose-finding objectives. Nonlinear utility makes the distribution of outcomes matter, so a dose with a lower or equal mean outcome can still have higher mean patient utility.

Check Your Understanding

Answer each question using the definitions and calculations in this tutorial.

Dose G yields outcomes 3 and 7 with equal probability; dose H yields outcome 5 for everyone. If $u(y)=y$, compare their mean outcomes and mean utilities.
For a concave utility function, what does Jensen’s inequality say about $E\{u(Y)\}$ and $u\{E(Y)\}$? Does that inequality alone rank two doses?
Two doses have the same mean outcome but different outcome distributions. Must they have the same mean utility? Explain.
A proposed analysis calculates utility at the model-predicted mean outcome for each dose. State a condition under which this gives the mean utility, and explain what to do otherwise.

Solutions. (1) Both doses have mean outcome 5. With $u(y)=y$, $EU(G)=\{u(3)+u(7)\}/2=(3+7)/2=5$, and $EU(H)=u(5)=5$; both objectives tie. (2) For concave $u$, $E\{u(Y)\}\leq u\{E(Y)\}$, under the conditions for Jensen’s inequality. This is a within-dose comparison and does not by itself rank two dose distributions. (3) No. Nonlinear utility can assign different average utility to distributions with the same mean, as shown by the convex-utility example. (4) It gives mean utility if utility is affine over the relevant outcome range or the outcome is constant. Otherwise, apply utility to outcomes over the full dose-specific distribution and average the resulting values.