Interpreting Dose Uncertainty Intervals and Decision Curves

Two Summaries, Two Questions

A dose uncertainty interval and a dose-specific probability curve can be displayed together, but they answer different questions. An interval summarizes variation in a dose quantity across an uncertainty distribution. A curve asks, at each fixed dose, how often the target is attained under that distribution. Neither is automatically a recommended dose, and neither by itself describes the clinical consequences of missing the target.

This distinction matters when comparing candidate doses. An interval may show that a dose threshold is uncertain, while the probability curve shows how attainment changes as the chosen dose changes. The curve can help assess a stated probability requirement or compare options, but selecting an action also requires a decision rule, such as a minimum acceptable attainment probability or a utility criterion.

Definition: Let $d$ denote dose, $\theta$ the uncertain model parameters, and $T(d,\theta)$ an indicator equal to 1 when a specified target is attained at dose $d$, and 0 otherwise. Given an uncertainty distribution $Q(\theta\mid \text{data})$, the dose-specific target-attainment probability is $$ p(d)=P\{T(d,\theta)=1\mid \text{data}\} =\int T(d,\theta)\,dQ(\theta\mid \text{data}). $$ A dose uncertainty interval is an interval formed from a specified dose-valued quantity that varies under the same uncertainty distribution, such as a minimum dose meeting a target. The quantity, uncertainty distribution, and interval construction must be stated to interpret it.

The distribution $Q$ might be a posterior distribution or another explicitly defined distribution used to represent parameter uncertainty. The probability $p(d)$ is conditional on the model, target definition, data, and chosen $Q$. If the target concerns both efficacy and safety, $T(d,\theta)$ must indicate that both required conditions hold. If patient variability is also incorporated, the probability must say whether it is averaged over individuals, covariates, or a target population. These choices affect the interpretation; they are not supplied by the words “attainment probability” alone.

For a simple monotone target, define $D^*(\theta)$ as the minimum dose that attains the target for parameter value $\theta$. If attainment holds at all doses at or above this threshold, then $T(d,\theta)=1$ exactly when $D^*(\theta)\le d$. In that special case, the dose-specific curve is the cumulative distribution function of $D^*$. But many dose problems are not this simple: a safety limit can make attainment fail again at high doses, or the target-attaining doses can form several separate regions. Then a single threshold distribution does not fully describe the curve.

How to Read the Interval and the Curve

An interval for $D^*$, such as the 10th to 90th percentiles under $Q$, describes the spread of the uncertain threshold. It does not say that every dose inside the interval is equally likely to attain the target. Nor does it say that doses outside the interval cannot attain it. The meaning depends on the quantity being summarized: an interval for a minimum effective dose differs from an interval for a utility-maximizing dose or for the upper boundary of a therapeutic window.

A curve $p(d)$ instead gives the probability at each dose. With monotone threshold attainment, a higher dose can only increase or preserve $p(d)$. With a therapeutic window or joint efficacy-safety target, the curve can rise, plateau, and fall. If probabilities are estimated from parameter draws $\theta^{(1)},\ldots,\theta^{(M)}$, a direct estimate is

\widehat p(d)=\frac{1}{M}\sum_{m=1}^{M}T(d,\theta^{(m)}).

This is the fraction of simulated parameter draws for which the target is attained at the specified dose. It is an estimate under the simulated uncertainty distribution, not a frequency guaranteed in future patients and not a probability that the model is correct. Monte Carlo error, uncertainty about the model, the target definition, and any unrepresented patient variability remain relevant.

Worked Example: A Threshold Interval and Monotone Attainment Curve

Suppose ten hypothetical parameter draws imply minimum target-attaining doses, in mg, of 20, 22, 24, 25, 27, 28, 30, 32, 35, and 40. Assume the target is attained at dose $d$ for a draw precisely when $d\ge D^*$. Use the empirical distribution of these draws, and define the 20th and 80th percentiles by the nearest-rank order statistics, ranks $\lceil 10(0.20)\rceil=2$ and $\lceil 10(0.80)\rceil=8$.

The 20th–80th percentile interval for the minimum target-attaining dose is therefore $[22,32]$ mg: the second ordered value is 22 mg and the eighth is 32 mg. This interval summarizes uncertainty in the threshold; it is not a range of doses with equal attainment probability.

At 25 mg, four of the ten thresholds are at or below the dose: 20, 22, 24, and 25 mg. Thus $\widehat p(25)=4/10=0.40$. At 30 mg, seven thresholds are at or below the dose, so $\widehat p(30)=7/10=0.70$. At 35 mg, nine are at or below it, giving $\widehat p(35)=0.90$.

The interval and curve are consistent but serve different purposes. The interval locates the middle 60% of simulated thresholds under the stated convention. The curve gives the estimated probability at each particular dose. For example, 30 mg lies within the interval, yet its estimated attainment probability is 0.70, not 0.50 merely because it is near the interval’s upper end.

Why the Cumulative-Distribution Link Can Fail

The link between the curve and a threshold distribution follows directly from the event defining attainment. If every draw has one minimum threshold $D^*$, and attainment is monotone in dose, then

\{T(d,\theta)=1\}=\{D^*(\theta)\le d\}, \qquad p(d)=P(D^*\le d\mid \text{data}).

Thus $p(d)$ is the cumulative distribution function of $D^*$, conditional on the specified uncertainty distribution. A probability level can then be associated with a threshold quantile: for a continuous distribution, the dose at which $p(d)=0.80$ is the 80th percentile of $D^*$. This identity requires a single threshold and monotone attainment. It should not be carried over automatically to targets with both lower and upper dose limits.

For a target-attaining set $A(\theta)=\{d:T(d,\theta)=1\}$, the general curve is $p(d)=P\{d\in A(\theta)\mid\text{data}\}$. Each dose is assessed separately against the uncertain set. A distribution of one selected dose, such as the minimum feasible dose, discards information about the rest of $A(\theta)$, so it generally cannot reconstruct this curve.

Worked Example: A Therapeutic Window

Consider five hypothetical parameter draws. Under each draw, the target is met only between a lower efficacy boundary $L$ and an upper safety boundary $U$, inclusive. The simulated feasible windows, in mg, are $[10,25]$, $[15,35]$, $[20,40]$, $[25,45]$, and $[30,50]$. Assume equal weight for the five draws.

At 15 mg, the first two windows contain the dose, so $\widehat p(15)=2/5=0.40$. At 25 mg, the first four windows contain it, so $\widehat p(25)=4/5=0.80$. At 35 mg, the second, third, and fourth windows contain it, giving $\widehat p(35)=3/5=0.60$. At 45 mg, two windows contain it, giving $\widehat p(45)=2/5=0.40$.

The curve rises and then falls because increasing dose can improve efficacy attainment while eventually violating the safety boundary in some draws. Reporting only the distribution of the minimum feasible dose would describe the lower boundaries, 10, 15, 20, 25, and 30 mg; it would omit the upper boundaries and could not yield the probabilities at 35 or 45 mg. For this joint target, display or compute the dose-specific curve directly, and state that the event requires both efficacy and safety criteria.

Using the Curve in a Decision

A curve becomes decision-relevant only when paired with a decision criterion. For example, a chance constraint may require $p(d)\ge 0.80$; this defines eligible doses, but it does not rank eligible doses unless another rule is supplied. A rule that chooses the lowest eligible dose is different from one that maximizes expected utility. In particular, a high attainment probability is not by itself evidence that a dose has the best benefit-harm tradeoff.

Worked Example: Applying a Stated Probability Requirement

Use the ten thresholds from the first example, but suppose the candidate doses are 25, 30, and 35 mg. A hypothetical decision rule requires an estimated attainment probability of at least 0.80 and, among eligible candidates, selects the lowest dose. The estimates are $4/10=0.40$ at 25 mg, $7/10=0.70$ at 30 mg, and $9/10=0.90$ at 35 mg.

Only 35 mg meets the rule, since $0.90\ge0.80$, whereas $0.70<0.80$ and $0.40<0.80$. The rule therefore selects 35 mg from these candidates. This is a conclusion about the stated rule and the empirical estimates, not a general claim that 35 mg is clinically optimal. With only ten draws, the estimated probabilities are coarse, and the rule does not account for consequences beyond the target indicator.

The 20th–80th percentile interval $[22,32]$ mg would not, by itself, produce this selection. Its midpoint is not an attainment-based decision rule, and the interval does not state which candidate satisfies the 0.80 requirement.

Assumptions, Diagnostics, and Alternatives

Before interpreting either summary, verify that the target event is defined precisely, the dose scale and units are consistent, and the uncertainty distribution represents the intended source of uncertainty. If the target includes multiple criteria, confirm that the event is their required joint occurrence, rather than separate marginal probabilities. For simulation-based calculations, verify that parameter draws were generated and weighted as intended, that every dose was evaluated using the same target definition, and that the number of draws is adequate for the precision needed.

Several patterns can flag a problem. A supposedly monotone threshold curve that decreases as dose increases is inconsistent with the stated single-threshold model and may indicate coding, dose-grid, or simulation error. A curve based on a joint efficacy-safety event may legitimately be nonmonotone, so monotonicity should not be imposed on it automatically. A jagged curve can arise from a small number of draws or a coarse dose grid; report the simulation resolution and avoid implying precision unsupported by the calculation. If interval endpoints are unstable under modest changes in the draws or quantile convention, the interval should be interpreted cautiously.

The main alternatives answer related but distinct questions. A threshold interval is compact and useful when a clinically meaningful threshold dose exists, but it can hide shape, skewness, and multiple feasible regions. A dose-specific curve preserves how attainment changes across doses, but can be visually dense and depends on a clearly defined event. A single recommended dose is easier to communicate, but suppresses uncertainty unless accompanied by an interval or curve. Expected utility can incorporate different consequences of efficacy and toxicity, whereas a binary target curve treats all target-attaining outcomes alike and all failures alike.

A further limitation is that a narrow interval or steep curve does not validate the model or target. Both may be precise conditional summaries under assumptions that are wrong or incomplete. A curve evaluated over a limited dose range also cannot support conclusions beyond that range; extrapolation requires separate justification. When there are several disjoint target-attaining regions, report their structure or the full curve rather than compressing the result into a single interval that may suggest continuity.

Common pitfall: “The dose interval is 22–32 mg, so any dose in that range has about the same chance of success.” This is not justified: the interval summarizes quantiles of a dose-valued quantity, not the probability at each dose. Calculate or inspect $p(d)$ at the doses of interest, then apply an explicitly stated decision rule.

Check Your Understanding

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

In a monotone threshold model, what event does the dose-specific curve $p(d)$ measure in terms of the random threshold $D^*$?
Suppose 12 equally weighted draws give threshold doses, in mg, of 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, and 32. What is the empirical attainment probability at 20 mg?
Why can a distribution of minimum feasible doses fail to describe attainment probabilities when there is an upper safety boundary?
A rule requires estimated attainment probability of at least 0.75. Candidate doses have estimated probabilities 0.60, 0.75, and 0.90. If the rule chooses the lowest eligible candidate, which dose is selected, and what does that conclusion not establish?

Solutions. 1. Under monotone threshold attainment, success at $d$ is the event $D^*\le d$, so $p(d)=P(D^*\le d\mid\text{data})$. 2. Six of the 12 thresholds are at or below 20 mg (10, 12, 14, 16, 18, and 20), so the empirical probability is $6/12=0.50$. 3. The minimum feasible dose records a lower boundary but omits the upper safety boundary; a dose above that upper boundary can fail even when it exceeds the minimum. The full dose-specific event must be evaluated. 4. The second candidate is selected because 0.75 meets the requirement and is the lowest probability-qualified option. This establishes only the result under the stated rule and estimates; it does not establish that the dose is optimal under every clinical objective or that the model is correct.