Choosing a Dose by Expected Utility at Fixed Model Parameters

From Predicted Outcomes to a Dose Recommendation

A response model can predict different outcomes at different doses, but predicted outcomes alone do not determine which dose to choose. A dose decision also requires a utility function that assigns value to possible outcomes. The relevant comparison is the expected utility at each candidate dose: the average utility over outcomes according to the specified response model.

In this tutorial, model parameters are held fixed at specified values. The averaging is over the variability of patient outcomes conditional on dose and those fixed parameters. It is not an average over uncertain parameter values. This distinction makes the calculation a clear, conditional decision comparison, while also limiting what its recommendation says.

Definition: Let $d$ be a candidate dose, $Y(d)$ the vector of outcomes for a patient at that dose, $U(Y(d))$ the utility assigned to those outcomes, and $\theta_0$ a specified fixed value of the response-model parameters. The conditional expected utility at dose $d$ is $$ EU(d\mid\theta_0)=E_{\theta_0}\!\left[U(Y(d))\mid d\right] =\int U(y)\,p(y\mid d,\theta_0)\,dy, $$ with a sum replacing the integral for discrete outcomes. Among a prespecified candidate set $\mathcal D$, a dose maximizing this quantity is $d^*=\arg\max_{d\in\mathcal D} EU(d\mid\theta_0)$.

The calculation requires a specified outcome distribution $p(y\mid d,\theta_0)$, a utility function on the outcomes, and a candidate-dose set. It also requires the outcome and utility definitions to match: for example, if utility depends on both clinical benefit and harm, the model must provide their joint distribution unless the utility structure makes their marginal distributions sufficient. A maximum identifies the best dose under these inputs; it does not establish that the inputs are correct or that the dose is optimal for every patient.

What Is Averaged, and What Is Held Fixed?

At a fixed dose and fixed $\theta_0$, outcomes may still vary from patient to patient. Expected utility accounts for that variability by weighting each possible outcome by its model probability. For a finite outcome set $y_1,\ldots,y_K$, with probabilities $\pi_k(d;\theta_0)$, the calculation is

EU(d\mid\theta_0)=\sum_{k=1}^{K} U(y_k)\,\pi_k(d;\theta_0), \qquad \sum_{k=1}^{K}\pi_k(d;\theta_0)=1.

The parameter value $\theta_0$ is not random in this calculation. If $\theta$ instead has a distribution representing parameter uncertainty, averaging over that distribution would produce a different quantity, such as $E_{\theta}[EU(d\mid\theta)]$. That extra averaging is deliberately outside the fixed-parameter objective here. Likewise, a dose with high expected utility can still have substantial probability of an undesirable outcome: an expectation summarizes utilities across outcomes; it does not guarantee an individual result.

A short derivation explains why additive utilities can simplify the calculation. Suppose utility has the form $U(B,H)=aB-bH$, where $B$ and $H$ are binary benefit and harm indicators and $a,b$ are fixed utility weights. By linearity of expectation,

E[U(B,H)\mid d,\theta_0] =E[aB-bH\mid d,\theta_0] =aP(B=1\mid d,\theta_0)-bP(H=1\mid d,\theta_0).

No independence assumption is needed for this identity. In contrast, for a non-additive utility, such as one assigning a special value to experiencing benefit and harm together, the joint outcome probabilities can affect expected utility and generally cannot be replaced by the two marginal probabilities alone.

Worked Calculations Across Candidate Doses

Worked Example: Additive Utility for Binary Benefit and Harm

Suppose a hypothetical response model at fixed parameter values gives the following probabilities for a clinical benefit $B=1$ and a clinically important harm $H=1$. The utility is $U(B,H)=100B-40H$ points. Assume benefit and harm are conditionally independent at each dose, so their joint probabilities are the products of their marginal probabilities.

Dose (mg)	$P(B=1\mid d,\theta_0)$	$P(H=1\mid d,\theta_0)$
20	0.30	0.05
40	0.50	0.12
60	0.62	0.25

Because utility is additive, expected utility is $100P(B=1)-40P(H=1)$. At 20 mg it is $100(0.30)-40(0.05)=30-2=28.0$ points. At 40 mg it is $100(0.50)-40(0.12)=50-4.8=45.2$ points. At 60 mg it is $100(0.62)-40(0.25)=62-10=52.0$ points. Among these three candidates, 60 mg has the largest expected utility.

The independence assumption can also be checked against a joint calculation. At 60 mg, the probability of benefit without harm is $0.62(0.75)=0.465$; benefit with harm is $0.62(0.25)=0.155$; harm without benefit is $0.38(0.25)=0.095$; and neither is $0.38(0.75)=0.285$. Their probabilities sum to 1. Utilities for those four outcomes are 100, 60, $-40$, and 0 points, respectively. Thus the joint calculation is $100(0.465)+60(0.155)-40(0.095)+0(0.285)=46.5+9.3-3.8=52.0$ points, as before. The choice is conditional on the specified utility and fixed-parameter probabilities; it is not a claim that 60 mg gives the best outcome to every patient.

Worked Example: Expected Utility for a Threshold Response

Consider a second hypothetical model. A continuous efficacy measurement $X$, in units, follows $X\mid d,\theta_0\sim N(\mu_d,4^2)$, with fixed residual standard deviation 4 units. Define success as $X\geq 15$. Utility is 50 points for success and 0 otherwise, minus 30 points if a binary harm occurs. The model gives the following fixed-parameter predictions.

Dose (mg)	$\mu_d$ (units)	$P(H=1\mid d,\theta_0)$
10	13	0.02
20	20	0.10
30	16	0.25

For a normally distributed $X$, the success probability is $P(X\geq15)=1-\Phi((15-\mu_d)/4)$, where $\Phi$ is the standard normal cumulative distribution function. The expected utility is 50 times that probability minus 30 times the harm probability. Independence between efficacy and harm is not required because the utility is additive in the two indicators.

At 10 mg, $P(X\geq15)=1-\Phi(0.50)\approx0.3085$, so expected utility is $50(0.3085)-30(0.02)=15.425-0.600=14.83$ points. At 20 mg, $P(X\geq15)=1-\Phi(-1.25)=\Phi(1.25)\approx0.8944$, giving $50(0.8944)-30(0.10)=44.72-3.00=41.72$ points. At 30 mg, $P(X\geq15)=1-\Phi(-0.25)=\Phi(0.25)\approx0.5987$, giving $50(0.5987)-30(0.25)=29.935-7.500=22.44$ points. The fixed-parameter expected-utility maximum among these candidates is 20 mg.

The model’s residual standard deviation matters even though its parameters are fixed: it determines how much probability lies above the threshold. Replacing the outcome distribution with the mean alone would classify the 20 mg mean as above the threshold, but would not calculate the probability of success or the expected utility.

Worked Example: Why Joint Outcome Probabilities Can Matter

Suppose a hypothetical utility table assigns 100 points to benefit without harm, $-50$ points to harm without benefit, 20 points when both occur, and 0 points when neither occurs. At two candidate doses, the fixed-parameter model has the same marginal probabilities: $P(B=1)=0.60$ and $P(H=1)=0.20$. The joint probabilities differ.

Dose (mg)	$P(B=1,H=1)$	$P(B=1,H=0)$	$P(B=0,H=1)$	$P(B=0,H=0)$
25	0.18	0.42	0.02	0.38
50	0.02	0.58	0.18	0.22

Each row sums to 1 and has the stated marginals. At 25 mg, expected utility is $100(0.42)-50(0.02)+20(0.18)+0(0.38)=42-1+3.6=44.6$ points. At 50 mg it is $100(0.58)-50(0.18)+20(0.02)+0(0.22)=58-9+0.4=49.4$ points. This utility and model therefore favor 50 mg. The identical marginal benefit and harm probabilities would not distinguish the doses; their different joint distributions do.

For comparison, if utility were instead additive, $100B-50H$, both doses would have expected utility $100(0.60)-50(0.20)=50$ points, regardless of the joint probabilities. The utility structure, not merely the outcome labels, determines what information the calculation needs.

Interpreting and Checking the Result

A fixed-parameter dose recommendation is a conditional decision result: it identifies the candidate with the largest expected utility under the supplied model, utility scale, and parameter values. It does not quantify how certain the parameter values are, and it does not by itself show how stable the recommendation is to alternative assumptions. The expected utility is measured in the units of the utility scale. If the utility is defined in points, the result is points; those points are not automatically a clinical measurement such as milligrams or probability.

Before interpreting a maximum, check the ingredients of the calculation. Verify that outcome probabilities are valid and sum to one when a joint distribution is used; confirm the dose and outcome units; and ensure that the utility assigned to each possible outcome reflects the intended decision objective. For continuous outcomes, check that the distribution and any threshold or transformation used in utility are applied consistently. If the utility includes interactions between outcomes, verify that the response model supplies the needed joint distribution.

A useful justification for the discrete calculation is that expected utility is a probability-weighted average. If $U(y_k)$ is the utility at outcome $y_k$, and $\pi_k$ is its probability, then each outcome contributes $U(y_k)\pi_k$, and adding all contributions gives the expectation. This is why an outcome with a large utility can contribute little when it is unlikely, and why an undesirable outcome can lower the expectation even when it is not the most likely outcome.

Specify the comparison.
List the candidate doses and the outcome vector whose values determine utility.

Fix the model inputs.
State the parameter values and the conditional outcome distribution at each dose. Do not silently average over parameter uncertainty.

Calculate the expectation.
Weight each outcome utility by its probability, using joint probabilities whenever the utility requires them.

Compare and qualify.
Choose a maximum over the stated candidate set, report utility units, and state that the result is conditional on the model and utility inputs.

This approach differs from selecting a dose by its mean response alone: a mean does not generally determine expected utility, especially for thresholds, nonlinear utilities, or multiple outcomes. It also differs from marginal benefit-risk balancing at one dose: here the same utility calculation is performed across candidates to identify a maximum. A hard safety constraint is another decision rule; if such a constraint is imposed, doses violating it should be excluded or otherwise handled according to that rule rather than treated as ordinary utility trade-offs.

Important limitations remain. A fixed-parameter maximum may change if parameter values, residual variability, or utility weights change. A finite candidate set only supports a comparison within that set; it does not establish an optimum at an unlisted dose. A population-level recommendation also requires the outcome distribution and utility to represent the intended population or target patient. Finally, expected utility can conceal the distribution of individual outcomes, so it should not be interpreted as the outcome a typical patient is certain to experience.

Common Mistakes and Failure Modes

Confusing outcome variability with parameter uncertainty. Integrating over possible patient outcomes at fixed parameter values is not the same as averaging across uncertain parameters. State explicitly which expectation is being calculated.
Using utility of the mean outcome. In general, $U(E[Y])$ is not equal to $E[U(Y)]$. Calculate utility for outcomes and then average, particularly for threshold or nonlinear utilities.
Assuming independence without checking. Independence is unnecessary for additive utility in separate outcomes, but can change expected utility for non-additive utility. Use the joint distribution when required.
Calling the maximizing dose universally best. The maximum is conditional on the specified candidates, response model, fixed parameter values, and utility. Changes to these inputs may change the ranking.
Reporting a utility score without its scale or context. State the units or interpretation of utility points and avoid treating them as a clinical endpoint unless the scale was defined that way.

Key takeaway: At fixed model parameters, expected utility is the response model’s probability-weighted average of utility across possible outcomes. Compare that quantity across candidate doses, and distinguish the resulting conditional ranking from uncertainty about the parameters or certainty about an individual outcome.

Check Your Understanding

Try each question, then check the concise solutions below.

A binary benefit has probability 0.40 and binary harm has probability 0.10 at a candidate dose. Utility is $80B-30H$ points. Calculate expected utility and state whether independence is needed.
Why can the utility of the mean efficacy measurement differ from expected utility when utility awards a fixed score only if efficacy exceeds a threshold?
For a non-additive utility that assigns a special score when benefit and harm occur together, what outcome probabilities may be needed?
A dose maximizes expected utility at one specified parameter value. Does that result account for uncertainty about the parameter? Explain.
What does a maximum over three listed candidate doses establish, and what does it not establish about unlisted doses?

Solutions. 1. By linearity of expectation, $80(0.40)-30(0.10)=32-3=29$ points. Independence is not needed for this additive utility. 2. A threshold utility depends on the probability of crossing the threshold, not just the mean. In general, applying the utility to the mean does not average the utilities of possible outcomes. 3. The joint probabilities for benefit and harm together and separately may be required; the two marginal probabilities alone may not determine the expectation. 4. No. It averages outcome variability conditional on the specified parameter value, but does not average over or otherwise quantify uncertainty about that parameter. 5. It establishes the largest calculated expected utility among those three candidates under the stated inputs. It does not establish that the best dose over a broader or continuous dose range is one of those candidates.