Continuous Endpoint Utility with Diminishing Returns

Why Diminishing Marginal Value Matters

A dose-response model can predict how a continuous efficacy outcome changes with dose. Turning that prediction into a decision requires more than identifying which dose produces the largest outcome: a one-unit improvement may matter greatly at low efficacy and less near a level already considered highly beneficial. A utility function makes that valuation explicit.

This tutorial considers a continuous endpoint oriented so that larger values represent better efficacy. If the recorded endpoint has the opposite direction—for example, symptom severity where lower is better—first define a benefit-oriented quantity or otherwise make the direction explicit. The utility must be defined on the endpoint’s relevant range, and its curvature should represent a stated preference rather than a feature assumed to follow automatically from the dose-response model.

Definition: A continuous efficacy utility is a function \(u(y)\) that assigns a numerical value to each possible value \(y\) of a continuous efficacy outcome. It is nondecreasing on a range if \(y_1\leq y_2\) implies \(u(y_1)\leq u(y_2)\). It is concave on that range if, for any \(y_1,y_2\) in the range and any \(0\leq\lambda\leq1\), \(u(\lambda y_1+(1-\lambda)y_2)\geq\lambda u(y_1)+(1-\lambda)u(y_2)\). Concavity expresses diminishing marginal value: as efficacy increases, an additional equal-sized improvement does not receive a greater utility increment.

Utility values are not efficacy measurements; their scale and curvature express how endpoint values are valued for the decision. The numerical examples below use hypothetical endpoint predictions and illustrative utilities. In practice, the endpoint range and utility shape should be justified for the intended decision, not chosen simply because a particular formula is convenient.

Derivative Conditions and Marginal Value

For a differentiable utility, \(u'(y)\) is the marginal utility per unit of endpoint improvement. For example, if \(y\) is measured in score points, the derivative has units of utility per point. The standard second-derivative criteria are sufficient conditions: if \(u'(y)\geq0\) throughout an interval, then \(u\) is nondecreasing there; if \(u''(y)\leq0\) throughout an interval, then \(u\) is concave there. Strict inequalities throughout an interval are sufficient for strict increase and strict concavity, respectively, but are not necessary. A zero derivative at an isolated point does not rule out strict increase, and a zero second derivative at an isolated point does not rule out strict concavity. A linear utility is concave even though its second derivative is zero everywhere.

For a twice-differentiable function, the connection between curvature and diminishing marginal value follows from the fundamental theorem of calculus. If \(a<b\) and \(u''(y)\leq0\) between them, then:

u'(b)-u'(a)=\int_a^b u''(t)\,dt\leq0.

Thus the marginal utility is nonincreasing as the endpoint increases. Equivalently, for a fixed improvement size \(h>0\), the utility gain over a later interval cannot exceed the gain over an earlier interval of the same size, when both intervals lie in the domain. To see this, write each gain as an integral of \(u'\):

u(y+h)-u(y)=\int_y^{y+h}u'(t)\,dt,\qquad u(y+2h)-u(y+h)=\int_{y+h}^{y+2h}u'(t)\,dt.

Because \(u'\) is nonincreasing, the integrand in the second interval is no greater at corresponding points than in the first. Therefore the second gain is no greater than the first. This result gives a direct check of the intended interpretation: equal endpoint improvements have diminishing utility increments.

Worked Example: Checking a Saturating Utility

Suppose a hypothetical efficacy score \(y\), measured in points, lies between 0 and 100, with larger values better. Consider \(u(y)=y/(y+30)\). Check that this utility is increasing and concave on the stated range, and calculate the utilities at 10, 30, 50, and 70 points.

For \(y\geq0\), the denominator \(y+30\) is positive, so the function is defined throughout the range. Differentiating gives:

u'(y)=\frac{30}{(y+30)^2},\qquad u''(y)=-\frac{60}{(y+30)^3}.

The first derivative is positive on this range, so the utility is strictly increasing. The second derivative is negative, so it is strictly concave. The computed values are \(u(10)=10/40=0.2500\), \(u(30)=30/60=0.5000\), \(u(50)=50/80=0.6250\), and \(u(70)=70/100=0.7000\). For successive 20-point improvements, the utility gains are 0.2500, 0.1250, and 0.07500. The endpoint improvements are equal in size, but their utility increments decline as efficacy increases.

The score is measured in points, while the displayed utility values are dimensionless. The function approaches 1 as \(y\) grows, but does not reach 1 at any finite \(y\). That limiting behavior is a property of this illustrative functional form, not a universal clinical meaning of utility 1.

Choosing a Functional Form

A useful form must have the intended direction and curvature on the entire range where it will be used. The range matters: a function may be increasing and concave on one domain but fail those properties elsewhere. For nonnegative outcomes, a common saturating form is \(u(y)=y/(y+K)\), with \(K>0\). Another is \(u(y)=1-\exp(-y/K)\), also for \(K>0\). Both are increasing and concave for \(y\geq0\), and both approach an upper limit. The parameter \(K\), in endpoint units, controls how quickly the utility rises and how early diminishing returns become substantial.

A linear utility, such as \(u(y)=ay+b\) for \(a>0\), assigns the same utility gain to every equal-sized endpoint improvement. It is nondecreasing and concave, but it does not express diminishing marginal value. A saturating function instead places relatively more value on improvements at the lower end of its range. The choice among these forms should follow from the intended valuation and domain; curvature should not be inferred just from seeing a curved dose-response relationship.

Worked Example: How the Scale Parameter Changes Marginal Value

For a hypothetical nonnegative efficacy score in points, compare \(u(y)=y/(y+20)\) and \(u(y)=y/(y+60)\). Both are dimensionless, increasing, and concave. Calculate the utility gain from improving the score from 20 to 40 points under each form.

For \(K=20\), the two endpoint utilities are \(u(20)=20/40=0.5000\) and \(u(40)=40/60=0.6667\). The gain is \(0.6667-0.5000=0.1667\).

For \(K=60\), they are \(u(20)=20/80=0.2500\) and \(u(40)=40/100=0.4000\). The gain is \(0.4000-0.2500=0.1500\). At these endpoint values the smaller \(K\) form assigns a slightly larger gain to this 20-point improvement. The parameter \(K\) changes the curve’s location and marginal values; it is not a free label with no consequences. A decision analysis should therefore justify its value in the endpoint’s units and examine whether plausible alternatives change the decision.

From Efficacy Utility to a Dose Preference

Suppose each candidate dose has a deterministic predicted efficacy value. If efficacy is the only quantity used to rank doses and \(u\) is increasing, a concave transformation cannot reverse their ordering: the dose with the larger predicted efficacy still has the larger efficacy utility. Concavity changes the size of utility differences, not the ranking of these fixed predictions. To show a changed dose preference, another valued consequence must enter the decision, such as a dose-related burden expressed on a compatible utility scale.

The following example uses an additive score: efficacy utility minus a stated burden cost. This builds on the earlier material on additive utility and the marginal tradeoff condition. The numerical costs are hypothetical and are assumed to have already been placed on the same utility scale. They are not estimated by the efficacy utility formula.

Worked Example: Diminishing Efficacy Value Changes the Preferred Dose

Consider three hypothetical doses with model-predicted efficacy scores of 20, 40, and 60 points. Compare a linear efficacy utility \(u_L(y)=y/100\) with a concave utility \(u_C(y)=y/(y+20)\). Suppose the corresponding dose-related burden costs are 0.10, 0.25, and 0.40 utility units. For this calculation, assume the costs and efficacy utilities are comparable and the decision score is their difference.

With the linear utility, the efficacy utilities are 0.20, 0.40, and 0.60. Subtracting the stated costs gives total scores \(0.20-0.10=0.1000\), \(0.40-0.25=0.1500\), and \(0.60-0.40=0.2000\). The highest dose has the largest score.

For the concave utility, the efficacy utilities are \(20/40=0.5000\), \(40/60=0.6667\), and \(60/80=0.7500\). Subtracting the same costs gives \(0.5000-0.10=0.4000\), \(0.6667-0.25=0.4167\), and \(0.7500-0.40=0.3500\). The middle dose now has the largest score.

The change does not mean that concavity made 60 points less efficacious than 40 points: the efficacy utility remains higher at 60 points. Rather, under the stated concave valuation, the utility increment from 40 to 60 points is only \(0.7500-0.6667=0.08333\), while the burden cost increases by \(0.15\). That additional efficacy utility does not compensate for the added cost on the assumed scale. The conclusion depends on the cost values and their comparability with the efficacy utility; different justified values could yield a different choice.

Assumptions, Diagnostics, and Limitations

Before using a continuous endpoint utility in dose comparison, check the conditions that give the calculation meaning:

Endpoint direction and domain: larger \(y\) must represent better efficacy for an increasing benefit utility, and the function must be defined over every value that the dose-response predictions may take.
Curvature: the utility must be nondecreasing and concave over the decision-relevant range if diminishing marginal value is intended. Check derivatives where available, or compare equal-sized increments across the range.
Units and scale: the outcome units and any scale parameters must be explicit. Costs or other utility components must use a scale that permits the intended comparison.
Decision inputs: dose-specific predictions and any additional valued consequences must be stated. A utility function alone does not specify the full decision when other outcomes matter.

A diagnostic plot of \(u(y)\) over the full outcome range can reveal unexpected decreases, an implausible sharp bend, or a plateau that begins too early. A plot of \(u'(y)\), when derivatives are available, should remain nonnegative for a nondecreasing benefit utility and should not rise if concavity is intended. Numerical checks at clinically meaningful endpoint values are useful as well: compute the utility gain for equal-sized changes at low, middle, and high values, and verify that the pattern matches the stated preference.

Several limitations remain even when the mathematics checks out. Concavity does not by itself establish that the selected degree of diminishing returns is clinically appropriate. A saturating utility may understate the value of high outcomes if the upper range has important consequences. Likewise, utility differences used to balance efficacy against a burden depend on how both components have been specified. Sensitivity analysis over defensible utility forms and parameters can show whether the preferred dose is stable or driven by a particular choice.

Common pitfall: “The utility is concave, so the lower dose must be preferred.” This is not generally correct. Concavity says that marginal efficacy utility is nonincreasing; it does not, by itself, rank a dose with a larger deterministic efficacy prediction below a dose with a smaller one. With efficacy alone and an increasing utility, the ordering is preserved. A preference change requires another decision component, such as the dose-related cost in the example, and depends on its stated value.

Practical Comparison and Use

A linear utility is the appropriate contrast when equal endpoint improvements are intended to have equal value. A concave utility encodes diminishing marginal value, while threshold or hinge utilities encode a different feature: a change in value at or around a specified target. These are not interchangeable shapes. The choice should match the decision question, and the resulting dose ranking should be checked over plausible endpoint ranges and utility specifications. If the dose predictions are not deterministic, evaluating utility of outcomes requires additional care; the calculations here compare stated predicted endpoint values and do not establish that utility at a predicted mean equals average utility across patients.

Key takeaway: Concavity formalizes diminishing marginal efficacy value: equal endpoint improvements contribute progressively smaller utility gains. It can alter a dose choice when efficacy is weighed against another valued consequence, but a monotone transformation alone preserves the ordering of deterministic efficacy predictions.

Check Your Understanding

Try the questions first, then check the brief solutions below.

For a twice-differentiable utility on an interval, what do \(u'(y)\geq0\) and \(u''(y)\leq0\) each imply? Are strict inequalities necessary for nondecreasing utility and concavity?
For \(u(y)=y/(y+10)\) with \(y\geq0\), calculate \(u(10)\), \(u(20)\), and the utility gains for the two successive 10-point improvements.
Two doses predict deterministic efficacy scores of 25 and 50 points. If the only ranking criterion is an increasing efficacy utility, can a concave transformation make the 25-point dose rank higher?
In the dose comparison example, why did the concave utility favor the middle dose even though the highest dose had the largest predicted efficacy?

Solutions. (1) The first condition implies nondecreasing utility; the second implies concavity. Strict inequalities throughout are sufficient for strict increase and strict concavity, but they are not necessary for the non-strict properties. (2) \(u(10)=10/20=0.5000\), \(u(20)=20/30=0.6667\), and \(u(30)=30/40=0.7500\). The gains are \(0.6667-0.5000=0.1667\) and \(0.7500-0.6667=0.08333\), so the later equal-sized improvement has the smaller gain. (3) No. An increasing function preserves the ordering, so the 50-point prediction has greater efficacy utility. (4) The highest dose’s additional efficacy utility was smaller than its increase in the stated burden cost; after subtraction, the middle dose had the largest total score.