From Exposure Metrics to Classification Rules
After choosing exposure metrics and translating exposure-response evidence into candidate targets, the next task is to state exactly what counts as meeting a target. A number alone is not a complete target: the metric, units, comparison direction, time horizon, and rule for values on the boundary all matter. These details determine whether a subject is classified as an attainer or a nonattainer.
This tutorial treats a subject’s exposure summaries as given and focuses on defining the target and applying it. It does not determine which exposure metric is scientifically appropriate; that choice should be made using the exposure-response question and the earlier material on AUC, Cmax, Cmin, and time-based metrics. The classification rules below are also distinct from any later calculation of the proportion of a population that attains a target.
Write the rule in a form that can be applied to each subject’s exposure summary. For a metric \(E_i\) measured for subject \(i\), a lower-bound target with threshold \(L\) is \(E_i \geq L\); an upper-bound target with threshold \(U\) is \(E_i \leq U\). A therapeutic window with lower limit \(L\) and upper limit \(U\) is \(L \leq E_i \leq U\), where \(L \leq U\). Unless a scientific or clinical rationale explicitly specifies otherwise, the conventions in this tutorial include values exactly equal to the stated limits.
The symbol \(E_i\) is generic: in a particular application, it could represent AUC, Cmax, Cmin, or a defined time-based metric. The threshold must use the same units and definition as the metric. For example, a Cmax threshold in mg/L cannot be directly applied to an AUC value in mg·h/L. If the target depends on a particular dosing interval, treatment period, or assessment window, that time frame must also be part of the specification.
One-Sided Thresholds and Windows
A lower threshold is suitable when the target is to achieve at least a specified exposure, such as a minimum AUC or Cmin. An upper threshold is suitable when the target is to remain at or below a specified exposure, such as a maximum Cmax. These are different rules: the first treats larger values as meeting the criterion, while the second treats smaller values as meeting it. The direction is part of the target, not something to infer from the observed data.
A window is appropriate when values that are too low and values that are too high are both outside the target. Its bounds should be specified together, with units and boundary conventions. A lower bound without an upper bound is not a window; an upper bound without a lower bound is not a window. If the limits are reversed, so that \(L>U\), no value can meet the stated inclusive window, which signals a specification error rather than a meaningful target.
Worked Example: Applying a Lower Exposure Threshold
Suppose a hypothetical target requires steady-state Cmin to be at least 1.50 mg/L. Four hypothetical subjects have Cmin values of 1.42, 1.50, 1.73, and 2.10 mg/L. Apply the stated rule to each subject.
The target is \(C_{\min,i} \geq 1.50\) mg/L. For subject 1, \(1.42 < 1.50\), so the subject does not meet the target. For subject 2, \(1.50 = 1.50\), so the subject meets the target under the inclusive boundary rule. Subjects 3 and 4 also meet it because \(1.73 \geq 1.50\) and \(2.10 \geq 1.50\).
Thus, subjects 2, 3, and 4 are target attainers, and subject 1 is not. The comparison is between each subject’s Cmin and the Cmin threshold; it would be invalid to compare an AUC or a concentration measured at a different defined time with this rule.
Worked Example: Classifying Values Against a Therapeutic Window
For a hypothetical example, suppose the target is an AUC over a specified 24-hour interval from 30.0 to 50.0 mg·h/L, inclusive. Five hypothetical subjects have AUC values of 28.4, 30.0, 42.7, 50.0, and 51.2 mg·h/L.
The rule is \(30.0 \leq \mathrm{AUC}_i \leq 50.0\) mg·h/L. The value 28.4 is below the lower bound and fails. The value 30.0 equals the lower bound and passes. The value 42.7 is between the bounds and passes. The value 50.0 equals the upper bound and passes. The value 51.2 exceeds the upper bound and fails.
Therefore, the second, third, and fourth subjects meet the window target; the first and fifth do not. This classification uses the AUC for the specified interval. A value calculated over a different interval would not be interchangeable without an explicitly justified conversion or a separately defined target.
Joint Targets Require Joint Subject-Level Decisions
A single endpoint may not capture the intended exposure profile. For example, a target might require sufficient AUC while also limiting Cmax. A joint target states how those criteria combine and applies them to the same subject. In the usual conjunctive form, every listed criterion must be met. A subject meeting one criterion but failing another is not a joint attainer.
For clarity, let \(I_{ij}\) equal 1 if subject \(i\) meets criterion \(j\), and 0 otherwise. For a joint target with \(K\) required criteria, the subject’s joint classification is \(J_i = \prod_{j=1}^{K} I_{ij}\). This product is 1 only when every indicator is 1; if any criterion fails, at least one factor is 0 and the product is 0. This gives a compact mathematical expression for the logical “and” rule. If a target instead allows alternative routes to success, its rule is “or” and must be written as such; it must not be silently substituted for a conjunctive target.
Worked Example: Classifying a Joint Target
Consider a hypothetical joint target requiring all three conditions: AUC over a specified interval is at least 25.0 mg·h/L, Cmax is no more than 8.00 mg/L, and the specified time-above-threshold metric is at least 40% of the interval. Five hypothetical subjects have the following exposure summaries.
| Subject | AUC (mg·h/L) | Cmax (mg/L) | Time above threshold (%) |
|---|---|---|---|
| A | 27.0 | 7.80 | 42 |
| B | 24.0 | 7.50 | 45 |
| C | 30.0 | 8.20 | 50 |
| D | 28.0 | 8.00 | 40 |
| E | 26.0 | 7.90 | 39 |
For subject A, \(27.0 \geq 25.0\), \(7.80 \leq 8.00\), and \(42 \geq 40\), so all three criteria pass. Subject A is a joint attainer. Subject B fails the AUC criterion because \(24.0 < 25.0\), despite passing the other two; B is not a joint attainer. Subject C fails the Cmax criterion because \(8.20 > 8.00\), so C is not a joint attainer. Subject D is exactly at the upper Cmax limit and at the time-above-threshold limit, and also passes the AUC criterion; with inclusive boundaries, D is a joint attainer. Subject E fails the time criterion because \(39 < 40\), so E is not a joint attainer.
The final joint classifications are: A attains, B does not, C does not, D attains, and E does not. Each conclusion is based on all three measurements from that subject, not on separate averages or on whether the cohort as a whole appears to meet each condition.
A Practical Specification and Verification Sequence
A precise target can be checked systematically before applying it to a set of subject-level exposures. This short sequence is useful when a target has several endpoints or is being transferred into analysis code.
State what quantity is evaluated, its units, and any required definition, such as the dosing interval or the threshold used to define a time-based metric.
Use a lower bound, an upper bound, or both. Check that the values have compatible units and, for a window, that the lower limit does not exceed the upper limit.
Specify whether equality passes. If no alternative is justified, use the inclusive rules \(E_i \geq L\), \(E_i \leq U\), and \(L \leq E_i \leq U\).
For a conjunctive joint target, require every criterion to pass for the same subject. If the intended logic is different, write that logic explicitly.
Apply each comparison to the subject’s corresponding exposure metric, then verify that the joint decision follows the stated logical rule.
Common Errors, Diagnostics, and Limitations
A useful diagnostic is to inspect the target and the classification together rather than treating the threshold as a label detached from its implementation. Review a few values below, at, and above each boundary, and verify that the resulting pass/fail decisions match the written rule. For joint targets, also check subjects who pass all criteria and subjects who fail exactly one criterion. These test cases can reveal reversed inequalities, omitted endpoints, and inconsistent treatment of equality.
Several mistakes are especially consequential:
- Reversing the inequality. A lower-bound target \(E_i \geq L\) is not met by values below \(L\). Confirm that the direction reflects the intended exposure requirement.
- Ignoring units or the time definition. A threshold expressed for one metric or interval cannot be applied to a different metric or time window merely because the numerical values look similar.
- Changing the boundary rule mid-analysis. If equality is included in the written target, a value exactly equal to the boundary passes. Use an alternative convention only when specified in advance.
- Using an “or” rule for an “and” target. Passing one of several required criteria is not enough for a conjunctive joint target.
- Classifying from group summaries. A group mean inside a target range does not imply that every subject meets the target. The rule must be applied to each subject’s values.
- Treating an unavailable value as a pass. If a required metric is missing or not defined, the classification cannot be established from the stated rule. Record and handle that case explicitly rather than assigning success by default.
One-sided thresholds, windows, and joint targets answer different questions. A one-sided threshold is simpler when only a minimum or maximum is relevant. A window expresses a bounded acceptable range. A joint target makes multiple requirements explicit, but is more demanding to specify and audit because every component and the combination rule must be unambiguous. Choosing among them is a scientific decision; classification arithmetic cannot resolve whether the underlying target is clinically justified.
Finally, a target-attainment rule is an operational definition, not proof of efficacy, safety, or causal benefit. Its interpretation depends on the evidence supporting the chosen metrics and limits, the validity of the exposure summaries, and the intended population and treatment context. A precise rule makes classification reproducible; it does not remove uncertainty in the evidence behind the rule.
Check Your Understanding
Answer each question using the inclusive boundary convention unless the question states otherwise. Solutions follow the questions.
- A hypothetical upper-bound target is Cmax \(\leq 6.0\) mg/L. Do values of 5.8, 6.0, and 6.2 mg/L pass?
- A hypothetical window is \(18 \leq E_i \leq 32\) mg·h/L. Classify values 17.9, 18, 27, and 32.1 mg·h/L.
- A joint target requires AUC \(\geq 20\) mg·h/L and Cmax \(\leq 7\) mg/L. A subject has AUC 23 mg·h/L and Cmax 7.1 mg/L. Is the subject a joint attainer? Explain.
- Why does a cohort mean that lies inside a therapeutic window not establish that every subject lies inside it?
- A target uses a threshold of 10 mg·h/L, but an analyst applies it to Cmax recorded in mg/L. What is wrong with this classification?
Solutions. (1) The values 5.8 and 6.0 mg/L pass because both are at or below 6.0; 6.2 mg/L fails. (2) The value 17.9 mg·h/L is below the window and fails; 18 and 27 mg·h/L pass; 32.1 mg·h/L is above the window and fails. (3) No. The subject passes the AUC criterion but fails the Cmax criterion, so the conjunction is false. (4) A mean summarizes the cohort and does not determine each individual value; some subjects can fall below or above the bounds even when the mean is inside. (5) The analyst is comparing different metrics with incompatible units. The specified threshold cannot be applied to Cmax unless a scientifically justified, explicit target for Cmax is defined.