Target Attainment as a Dose-Finding Objective

Why Count Patients Who Meet the Goal?

A dose can increase average response without ensuring that many patients reach a clinically meaningful goal. For dose finding, it can therefore be useful to ask a direct question: what fraction of the target population would meet a prespecified clinical target at each candidate dose? This is a target-attainment objective. It focuses on the proportion of patients whose outcomes satisfy a defined criterion, rather than on the average outcome alone.

The target must be stated before calculating the rate. For a continuous endpoint, it might be a minimum improvement or a symptom score no greater than a specified value. For a binary endpoint, it might be the event of achieving clinical success. A target is not adequately defined by saying “a good response”: the outcome, threshold, direction, and assessment time must be clear. As with other dose-finding objectives, the population and outcome definition should also match the estimand.

Definition: Let $Y(d)$ denote the outcome a patient in the target population would have at dose $d$, and let $A$ be the prespecified set of outcomes that meet the clinical target. The target-attainment probability at dose $d$ is $$ p_T(d)=P\{Y(d)\in A\}. $$ For $n_d$ observed patients treated at dose $d$, the empirical target-attainment fraction is $$ \hat p_T(d)=\frac{1}{n_d}\sum_{i=1}^{n_d} I\{Y_i\in A\}, $$ where $I\{\cdot\}$ equals 1 when its condition is true and 0 otherwise.

For a higher-is-better continuous endpoint with threshold $c$, the target set is $A=[c,\infty)$, so attainment means $Y(d)\ge c$. For a lower-is-better endpoint, it may be $A=(-\infty,c]$, so attainment means $Y(d)\le c$. A target can also be an interval, such as a laboratory value between two prespecified limits. For a binary outcome coded as 1 for success and 0 otherwise, the target is typically $A=\{1\}$, and the attainment probability is the probability of success.

The probability refers to the defined target population, not merely to the patients who happened to receive a dose in a particular data set. An observed fraction estimates that probability when the dose group represents the target population and the outcome is observed as specified. If allocation was not randomized, or groups differ in prognostic characteristics, unadjusted fractions need not estimate comparable dose-specific outcomes. A model-based estimate may instead predict each dose for a common population, but then its assumptions matter.

Observed Fractions at Candidate Doses

The empirical calculation is deliberately simple: classify each patient's outcome as meeting or not meeting the target, count those meeting it, and divide by the number with an evaluable outcome at that dose. Report the numerator and denominator along with the fraction. A percentage without its denominator hides how much information supports it.

Worked Example: A Continuous Improvement Threshold

Suppose a hypothetical study compares three candidate doses for an endpoint measured as improvement in symptom score, in points, with larger values better. The prespecified target is an improvement of at least 8 points. At 2 mg, the ten observed improvements are 5.8, 6.4, 7.1, 7.5, 8.0, 8.3, 9.0, 9.2, 10.1, and 11.0 points. Across all three doses, the target counts are as follows.

Dose	Patients meeting target	Patients assessed	Empirical fraction
2 mg	6	10	0.600
5 mg	8	10	0.800
10 mg	9	10	0.900

At 2 mg, the six outcomes at or above 8 points are 8.0, 8.3, 9.0, 9.2, 10.1, and 11.0. Therefore, $$ \hat p_T(2\text{ mg})=\frac{6}{10}=0.600. $$ The observed target-attainment fraction is 60.0% at 2 mg. The corresponding observed fractions are $8/10=0.800$, or 80.0%, at 5 mg and $9/10=0.900$, or 90.0%, at 10 mg. These are estimates based on small hypothetical groups; they do not establish that the population attainment probabilities increase in exactly these increments.

The indicator formulation also gives a short justification for why the fraction estimates the target probability. If patients at a fixed dose are independent draws from the target population and outcomes are measured consistently, each indicator $I\{Y_i\in A\}$ is a Bernoulli random variable with success probability $p_T(d)$. Its expectation is $p_T(d)$. By linearity of expectation, the expected average of $n_d$ such indicators is also $p_T(d)$:

E\{\hat p_T(d)\} =\frac{1}{n_d}\sum_{i=1}^{n_d}E[I\{Y_i\in A\}] =\frac{1}{n_d}\sum_{i=1}^{n_d}p_T(d) =p_T(d).

Thus the empirical fraction is unbiased under these sampling assumptions. Independence also gives $SE\{\hat p_T(d)\}=\sqrt{p_T(d)\{1-p_T(d)\}/n_d}$; in practice, an estimated standard error substitutes $\hat p_T(d)$ for the unknown probability. With small samples, uncertainty can be substantial even when the observed rate looks high.

Model-Based Attainment for Continuous Outcomes

A model can estimate attainment probabilities at candidate doses, including doses or populations for which direct observations are limited. The probability is obtained from the outcome distribution by integrating its density over the target set. For a higher-is-better outcome with threshold $c$, this is the upper-tail probability $P\{Y(d)\ge c\}$. The model must represent both the dose-specific location and the variability around it: a predicted mean alone does not determine the target rate.

Worked Example: Normal Predictions Across Three Doses

Consider a hypothetical model for improvement in points. Assume that, at each dose $d$, an outcome for a patient from the specified population follows $Y(d)\sim N(\mu(d),\sigma^2)$, with a common residual standard deviation $\sigma=2.0$ points. Suppose the model predicts means of 7.0, 8.5, and 9.0 points at 2, 5, and 10 mg, respectively. The clinical target is $Y(d)\ge 8$ points.

Under the stated normal model, standardizing $Y(d)$ gives $Z=\{Y(d)-\mu(d)\}/\sigma\sim N(0,1)$. Therefore, $$ p_T(d)=P\{Y(d)\ge 8\} =1-\Phi\left(\frac{8-\mu(d)}{2.0}\right), $$ where $\Phi$ is the standard normal cumulative distribution function.

At 2 mg, the standardized threshold is $(8-7.0)/2.0=0.500$, so $p_T(2\text{ mg})=1-\Phi(0.500)=1-0.6915=0.3085$. At 5 mg, it is $(8-8.5)/2.0=-0.250$, giving $p_T(5\text{ mg})=1-\Phi(-0.250)=0.5987$. At 10 mg, it is $(8-9.0)/2.0=-0.500$, giving $p_T(10\text{ mg})=1-\Phi(-0.500)=0.6915$.

The model therefore predicts attainment probabilities of 30.9%, 59.9%, and 69.2% at 2, 5, and 10 mg. These are population probabilities conditional on the assumed model and population specification—not observed percentages and not certainties about individual patients. The calculation assumes a normal outcome distribution and a common standard deviation; if either assumption is unsuitable, these tail probabilities may be inaccurate.

In a pharmacometric model with patient-level covariates or random effects, predictions may vary across patients. A population target rate should average the target probability across the target population, not report only a probability for a “typical” patient. In general, if $X$ denotes patient characteristics and $b$ denotes individual random effects, a model-based population rate can be expressed as

p_T(d)=\int P\{Y(d)\in A\mid X=x,b\}\,f(x,b)\,dx\,db,

where $f(x,b)$ represents the specified population distribution. This averaging matters when covariates or individual variability affect the chance of reaching the target. A conditional rate for one covariate profile answers a different question from a marginal rate for the whole target population.

Binary Outcomes and the Meaning of a Rate

For a binary endpoint, the target rate is the probability of the event coded as success. If $B_i(d)$ is 1 when patient $i$ meets the target and 0 otherwise, then the empirical rate is the number of successes divided by the number assessed. A model might estimate this probability directly, for example through a dose-response model for binary outcomes. Regardless of the estimation method, the event definition and assessment time must stay fixed across doses.

Worked Example: Binary Clinical Success

In a hypothetical study, clinical success at a prespecified assessment visit is coded as 1 and lack of success as 0. At each candidate dose, 60 patients have an assessed outcome. The observed success counts are 24 at 5 mg, 39 at 10 mg, and 42 at 20 mg.

The target-attainment fractions are: $$ \hat p_T(5\text{ mg})=\frac{24}{60}=0.400,\qquad \hat p_T(10\text{ mg})=\frac{39}{60}=0.650,\qquad \hat p_T(20\text{ mg})=\frac{42}{60}=0.700. $$

Thus, the observed fractions are 40.0%, 65.0%, and 70.0%. For the 10 mg group, under independent Bernoulli sampling, the estimated standard error is $$ SE(\hat p_T)=\sqrt{\frac{0.650(1-0.650)}{60}} =\sqrt{0.003792}=0.0616. $$ The approximate standard error is 6.16 percentage points. It reflects sampling variability under the stated sampling model; it does not include uncertainty from model choice, missing outcomes, or a mismatch between the trial sample and the target population.

What the Target Rate Adds—and What It Does Not

The earlier material on absolute and relative efficacy objectives distinguishes outcomes such as mean change and differences between doses. Target attainment answers a different, threshold-based question. A mean response summarizes the center of an outcome distribution; a target rate describes the share of that distribution on the clinically specified side of a threshold. Neither generally determines the other.

For example, two hypothetical populations can have the same mean improvement of 10 points but different variability. With target improvement $Y\ge 8$, suppose one has $Y\sim N(10,1^2)$ and the other has $Y\sim N(10,4^2)$. Their target probabilities are $$ P(Y\ge 8)=1-\Phi\left(\frac{8-10}{1}\right)=\Phi(2)=0.9772 $$ and $$ P(Y\ge 8)=1-\Phi\left(\frac{8-10}{4}\right)=\Phi(0.5)=0.6915, $$ respectively. The means are identical, but the predicted attainment rates differ by about 28.6 percentage points. This is why a mean-only comparison cannot substitute for calculating a target rate when the clinical objective concerns the proportion meeting a threshold.

Conversely, a target rate does not describe how far above or below the threshold patients fall, and it ignores distinctions among outcomes that are all classified on the same side. A rate of 70% could arise from outcomes just above the threshold or substantially above it. The target definition therefore determines what information is retained and what is discarded.

Assumptions, Diagnostics, and Common Pitfalls

A defensible dose-specific rate requires a clearly defined outcome and threshold, a consistent measurement time, and a population relevant to the decision. Empirical rates additionally rely on adequate outcome ascertainment and representative dose groups. Model-based rates require a suitable model for the outcome distribution, dose relationship, and relevant patient variability. If observed dose groups differ in covariates, raw fractions can confound dose differences with population composition; model-based standardization to a common covariate distribution can address that issue only if the model and available data support it.

Useful checks include comparing model-predicted and observed target rates by dose, checking whether predicted continuous distributions reproduce the observed spread and tails, and inspecting rates across clinically relevant covariate groups. A good fit to the mean does not guarantee a good fit near the clinical threshold. Sparse data near the threshold, few patients at a dose, missing outcomes, and strong dependence on distributional assumptions can all make estimated rates unstable. Sensitivity analyses using plausible alternative outcome distributions or missing-outcome assumptions can reveal whether a dose comparison depends on fragile choices.

Changing the target after seeing the results. This can make a rate look more favorable without reflecting a prespecified clinical goal. Define the threshold and direction first.
Calling a sample fraction the population probability. The observed $6/10$ is an estimate, not proof that exactly 60% of the target population would attain the target.
Using the mean as if it were the rate. A mean above a threshold does not tell how many individual outcomes exceed it; distributional information or patient-level data are needed.
Comparing unadjusted rates from unlike groups. Differences may reflect covariate composition or selection as well as dose. State the population each rate represents and standardize when the estimand requires a common population.
Ignoring uncertainty or denominator size. Similar observed rates can have very different precision if one is based on 20 patients and another on 200.

Target attainment is most useful when the clinical decision is naturally framed as the chance of reaching a defined goal. It should not be treated as a complete benefit-risk objective: toxicity, burden, and other outcomes are not represented unless explicitly included in the target definition. The next step in a broader decision analysis may combine efficacy and safety criteria, but the attainment calculation itself remains the probability of meeting its stated target.

Check Your Understanding

Work through each question, then check the solutions below.

A lower-is-better symptom score has a target of 4 points or less. In a group of 25 patients at one dose, 18 meet the target. What is the empirical attainment fraction?
A higher-is-better outcome follows $Y(d)\sim N(12,3^2)$ at a candidate dose. The target is $Y(d)\ge 9$. Calculate the model-based attainment probability.
Two doses have the same predicted mean outcome. Name one reason their target-attainment probabilities could differ.
Why may an unadjusted target rate comparison be misleading when dose groups have different covariate distributions?

Solutions. (1) The target is $Y\le 4$, and the observed fraction is $18/25=0.720$, or 72.0%. (2) Standardize the threshold: $(9-12)/3=-1$. Thus $P(Y\ge 9)=1-\Phi(-1)=\Phi(1)=0.8413$, or 84.1%, under the stated normal model. (3) The dose-specific outcome distributions could have different variances or shapes; the amount of probability beyond the threshold can then differ despite equal means. (4) Patient characteristics may influence the outcome and differ in prevalence between groups, so the observed rates can reflect population composition as well as dose. A comparison for a common target population requires appropriate standardization or another justified adjustment.