Individual Dose Decisions With Asymmetric Clinical Costs

From Forecasts to Individual Decisions

A forecast describes possible outcomes under a proposed dose; a decision specifies what to do with that forecast. Those are not the same task. A dose with a higher probability of meeting an exposure target may also carry a higher probability of toxicity, and the clinical consequences of those two outcomes need not be equally serious. Choosing a dose therefore requires both a predictive distribution and a statement of how outcomes matter.

The earlier material on posterior predictive probabilities provides the relevant starting point: for each candidate regimen, quantify the probability of clinically meaningful future outcomes conditional on the individual’s available data and the proposed regimen. Here, those probabilities are combined with outcome costs to select a dose. The costs are not statistical quantities inferred automatically from the pharmacokinetic model. They express the clinical consequences assigned to outcomes, and should be specified with clinical input.

Definition: A Bayes dose decision is a choice of candidate dose $d$ that minimizes posterior expected loss. If $Y$ denotes the future clinical or exposure outcome and $L(d,Y)$ is the loss from using dose $d$ when outcome $Y$ occurs, then $$ d^*=\arg\min_{d\in\mathcal{D}} E\{L(d,Y)\mid \text{individual data},d\}, $$ where $\mathcal{D}$ is the set of clinically feasible candidate doses. The expectation averages over uncertainty in the individual’s future outcome under each candidate dose.

A useful basic formulation assigns a cost $C_T$ to toxicity and a cost $C_U$ to inadequate exposure. Let $T$ denote the toxicity event and $U$ the inadequate-exposure event. If the losses are additive, the loss for a future outcome can be written $C_T I(T)+C_U I(U)$, where $I(\cdot)$ is one when the event occurs and zero otherwise. The posterior expected loss for dose $d$ is therefore

EL(d)=C_TP(T\mid \text{data},d)+C_UP(U\mid \text{data},d).

This expression remains valid if toxicity and inadequate exposure can occur together: in that case, both costs are incurred for that outcome. If the clinical outcome categories are defined as mutually exclusive—for example, toxicity, inadequate exposure without toxicity, and acceptable exposure—the same calculation applies using the probabilities of those categories and their corresponding losses. The model of consequences must match the clinical definition; overlapping events should not accidentally be treated as mutually exclusive.

Why Expected Loss Is the Decision Rule

Suppose a decision-maker must choose one dose before the future outcome is observed. For a particular dose, the loss is a random quantity because the outcome is uncertain. The expected loss is the probability-weighted average of the possible losses. Minimizing that average is the Bayes decision rule when the stated loss function represents the decision-maker’s objective.

Specify candidate regimens and outcomes.
Define the feasible doses, prediction horizon, and clinically relevant events under each regimen.

Obtain posterior predictive probabilities.
For each candidate dose, calculate or estimate the probabilities of toxicity, inadequate exposure, and any other outcomes assigned a loss.

Assign consequences and calculate expected loss.
Multiply each outcome probability by its loss and sum across outcomes for each dose.

Choose the minimum-loss dose and assess sensitivity.
Select the dose with the smallest expected loss, then examine whether the choice changes under plausible alternative costs or predictive assumptions.

The justification follows directly from averaging over outcomes. If dose $d$ can produce outcomes $y_1,\ldots,y_K$, with conditional probabilities $p_k(d)$ and losses $L(d,y_k)$, then $p_k(d)\geq 0$ and $\sum_{k=1}^Kp_k(d)=1$. Its expected loss is

E\{L(d,Y)\mid \text{data},d\} =\sum_{k=1}^K L(d,y_k)P(Y=y_k\mid \text{data},d) =\sum_{k=1}^K L(d,y_k)p_k(d).

This is a weighted average of the losses the dose could produce. Comparing those averages across feasible doses gives the decision rule. The rule does not assert that the selected dose will avoid harm; it says that, under the specified predictive model and loss function, no other candidate dose has a lower average loss.

Worked Example: Selecting a Dose When Toxicity Is Costlier

Suppose three candidate doses have the following hypothetical posterior predictive probabilities over a defined treatment period. For this example, toxicity, inadequate exposure, and acceptable exposure are mutually exclusive outcomes. The assigned losses are 100 units for toxicity, 20 units for inadequate exposure, and 0 units for acceptable exposure.

Candidate dose	Probability of toxicity	Probability of inadequate exposure	Probability of acceptable exposure
50 mg	0.02	0.35	0.63
75 mg	0.08	0.15	0.77
100 mg	0.22	0.05	0.73

Each row’s probabilities sum to one. The expected loss is $100P(T)+20P(U)$. For 50 mg,

EL(50)=100(0.02)+20(0.35)=2+7=9\text{ loss units}.

For 75 mg,

EL(75)=100(0.08)+20(0.15)=8+3=11\text{ loss units}.

For 100 mg,

EL(100)=100(0.22)+20(0.05)=22+1=23\text{ loss units}.

The minimum is 9 loss units, so 50 mg is preferred under these predictive probabilities and costs. Although inadequate exposure is more common at 50 mg, its assigned cost is relatively modest compared with toxicity. This is not a universal recommendation for 50 mg: changing the costs or the predictive probabilities can change the preferred dose.

How Unequal Costs Change the Preferred Dose

A cost ratio often makes the trade-off easier to interpret than costs considered separately. For two candidate doses, suppose dose A has toxicity probability $t_A$ and inadequate-exposure probability $u_A$, while dose B has corresponding probabilities $t_B$ and $u_B$. Dose A is preferred when

$$ C_Tt_A+C_Uu_A<C_Tt_B+C_Uu_B. $$

When $C_T>0$, divide both sides by $C_T$ and define $r=C_U/C_T$, the cost of inadequate exposure relative to toxicity. The comparison becomes $t_A+ru_A<t_B+ru_B$. Thus, a dose choice can depend on the relative cost $r$, not on either probability in isolation. A threshold value of $r$ marks where the ranking changes, provided the two doses trade off toxicity and inadequate-exposure risk.

Worked Example: Finding the Cost-Ratio Threshold

Consider two hypothetical doses with mutually exclusive outcome categories. Dose A has toxicity probability 0.04 and inadequate-exposure probability 0.30. Dose B has toxicity probability 0.12 and inadequate-exposure probability 0.10. Assume zero loss for acceptable exposure, and let $r=C_U/C_T$.

After dividing expected loss by the positive quantity $C_T$, the scaled losses are

EL_A/C_T=0.04+0.30r,\qquad EL_B/C_T=0.12+0.10r.

Dose A is preferred when $0.04+0.30r<0.12+0.10r$. Subtract $0.04+0.10r$ from both sides to obtain $0.20r<0.08$, so A is preferred when $r<0.40$. Dose B is preferred when $r>0.40$, and the doses tie when $r=0.40$.

For a numerical check, set $C_T=100$ and $C_U=25$, giving $r=0.25$. Then $EL_A=100(0.04)+25(0.30)=4+7.5=11.5$ units, while $EL_B=100(0.12)+25(0.10)=12+2.5=14.5$ units; A is preferred. If instead $C_U=50$, then $r=0.50$: $EL_A=4+15=19$ units and $EL_B=12+5=17$ units, so B is preferred. The switch reflects a change in the relative valuation of inadequate exposure, not a change in either dose’s predicted probabilities.

Using Posterior Predictive Simulation

Dose-specific event probabilities may be available directly, or they may be estimated from posterior predictive simulation. For the latter, simulate future outcomes under each candidate regimen using the individual’s posterior uncertainty and the assumed future dosing and observation conditions. Count the simulated outcomes in each event category, divide by the number of simulations, and apply the loss function. In simulation, the average loss across simulated outcomes estimates the posterior expected loss directly.

Worked Example: Comparing Simulated Outcomes Across Doses

Suppose 1,000 hypothetical posterior predictive outcomes are simulated for each of three candidate doses. The event categories are mutually exclusive. The assigned losses are 100 units for toxicity and 50 units for inadequate exposure, with zero loss for acceptable exposure.

Dose	Toxicity outcomes	Inadequate-exposure outcomes	Acceptable outcomes
60 mg	20	300	680
80 mg	80	120	800
100 mg	220	40	740

For 60 mg, the estimated probabilities are $20/1000=0.020$ for toxicity and $300/1000=0.300$ for inadequate exposure. Its estimated expected loss is

\widehat{EL}(60)=100(0.020)+50(0.300)=2+15=17\text{ loss units}.

For 80 mg, the estimates are $0.080$ and $0.120$, so

\widehat{EL}(80)=100(0.080)+50(0.120)=8+6=14\text{ loss units}.

For 100 mg, they are $0.220$ and $0.040$, giving

\widehat{EL}(100)=100(0.220)+50(0.040)=22+2=24\text{ loss units}.

Among these candidates, 80 mg has the lowest estimated expected loss. This is a simulation-based estimate, not an exact value: with a finite number of draws, the estimated probabilities and losses have Monte Carlo error. If competing doses have very similar estimated losses, increase simulation precision and check whether the ranking is stable before treating a small difference as decisive.

Assumptions, Diagnostics, and Alternatives

The decision rule is mathematically straightforward, but its usefulness depends on several substantive conditions:

Relevant predictions: Each candidate regimen’s predictive probabilities should correspond to the individual’s current information, the proposed dose, and the clinical time horizon used to define the outcomes.
Meaningful outcomes: Toxicity and inadequate exposure need operational definitions. If they can co-occur, decide whether the consequences are additive or whether a joint outcome has its own loss.
Defensible costs: The loss values should represent the clinical consequences relevant to the decision. Arbitrary numerical weights can create a precise-looking answer without a defensible clinical interpretation.
Feasible options: The minimization is over doses that can actually be prescribed and administered, including any dose limits or formulation constraints relevant to the decision.
Comparable horizon: Probabilities and costs should refer to compatible outcome periods. A short-term toxicity probability should not be combined casually with a long-term inadequate-exposure cost without specifying how the time scales are reconciled.

Useful diagnostics focus on the inputs and on the stability of the result. Check that mutually exclusive outcome probabilities sum to one, or that the joint-event formulation correctly allows probabilities to overlap. Confirm that predicted probabilities are computed for every candidate dose rather than transferred from a different regimen. For simulation-based comparisons, assess Monte Carlo precision; a dose ranking that changes across repeated runs is not yet numerically stable. Finally, calculate the preferred dose over a plausible range of $C_U/C_T$. If small, reasonable changes in the cost ratio reverse the choice, the decision is sensitive and should be reported as such.

This approach differs from choosing the dose with the largest probability of meeting an exposure target. That alternative focuses on one target event and may not account for toxicity or the unequal consequences of different failures. A rule based on a fixed acceptable toxicity probability can be useful as a clinical constraint, but it does not by itself rank doses that all satisfy the constraint. Choosing from a point estimate of an individual parameter also differs: it can omit uncertainty in the future outcome. Expected-loss selection uses the full posterior predictive probabilities, but it does not make the predictions or loss values automatically correct.

A common mistake is to select the dose with the lowest toxicity probability, or to add the toxicity and inadequate-exposure probabilities and treat them as equally important. The first ignores inadequate exposure; the second implicitly assigns equal costs and can double-count outcomes if the events overlap. The correction is to define the outcome structure and use the stated loss function explicitly. Expected loss also has limitations: a single average can conceal rare but severe outcomes, and a single cost ratio may not capture patient preferences or clinical constraints. In practice, present the selected dose together with its expected loss, predicted event probabilities, assumptions, and sensitivity to plausible costs. A minimum-loss dose is a decision conditional on those inputs, not a guarantee of safety or efficacy.

Key takeaway: Individual dose selection under asymmetric clinical costs is a probability-weighted comparison of consequences. The preferred dose minimizes posterior expected loss, and its interpretation is inseparable from the predictive model, the outcome definitions, and the costs used.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions below.

A candidate dose has posterior predictive toxicity probability 0.05 and inadequate-exposure probability 0.20. Toxicity costs 80 units and inadequate exposure costs 10 units. Assuming additive losses, calculate expected loss.
Two doses have $(P(T),P(U))=(0.03,0.25)$ and $(0.10,0.08)$, respectively. With $C_T=100$ and $C_U=20$, which dose has lower expected loss?
Why can a dose with the lowest predicted toxicity probability fail to minimize expected loss?
If toxicity and inadequate exposure can occur together, what must be checked before applying the weighted-probability formula?

Solutions. (1) $80(0.05)+10(0.20)=4+2=6$ loss units. (2) The first dose has loss $100(0.03)+20(0.25)=3+5=8$ units; the second has loss $100(0.10)+20(0.08)=10+1.6=11.6$ units, so the first is preferred. (3) It may have a higher probability of inadequate exposure, and the cost assigned to that outcome may make its total expected loss larger. (4) Establish whether losses are additive when both events occur, or define mutually exclusive outcome categories with appropriate joint-event losses; do not assume the events are exclusive if they can overlap.