Why a Model Needs More Than an Exposure and an Outcome
An exposure-effect analysis can be described informally as asking whether “more exposure produces more effect.” That phrase is not yet a model specification. “Exposure” might mean a concentration at one time, an area under a concentration-time curve, or a time above a threshold. “Effect” might mean a continuous biomarker, a binary event, or a change from baseline. Even after those are chosen, the observation time and the proposed relationship between exposure and outcome still need to be stated.
A minimum specification makes those choices explicit before data are analyzed. It gives another analyst enough information to understand what each model input and output represents, what the model is meant to describe, and what assumptions are required for its interpretation. The earlier material on choosing exposure metrics established that a metric should match the pharmacologic question and effect measurement. Here, the task is to assemble that metric into a complete, interpretable model description.
A General Notation for the Specification
Let \(X_i\) denote the selected exposure metric for individual \(i\), with its units and exposure window stated in words. Let \(Y_i(t^*)\) denote that individual’s pharmacodynamic outcome measured at the specified time \(t^*\). A general mean-model statement is
Here, \(f\) names the proposed functional relationship and \(\theta\) denotes its parameters. This notation is deliberately general: the function must be chosen to suit the scientific question and outcome, rather than assumed to have a particular form. A mean relationship alone may not fully describe the data. The specification should also state the outcome’s scale and, when needed, its probability distribution or other observation model.
For example, a continuous outcome measured in concentration units may be described on its original scale or after a stated transformation. A binary outcome is not a continuous measurement: its conditional mean is a probability, so the specification must make clear how that probability relates to exposure. Naming only “effect” or “response” does not settle these choices.
Five Components to State
- Exposure variable and units. Name the metric, its units, and the time interval or sampling time to which it refers. “Exposure” by itself is insufficient. AUC over a defined interval, a concentration at a particular time, and time above a threshold have different meanings and units.
- Outcome and scale. Define what is measured and how it is represented: for example, a biomarker change in specified units, an event indicator coded 0 or 1, or a score on a defined scale. State whether the model concerns the raw outcome, change from baseline, or another clearly defined quantity.
- Observation time or window. Say when the outcome is measured and how that time relates to the exposure window. If outcomes are measured repeatedly, specify whether each time is modeled separately or the time course is part of the relationship.
- Functional relationship. State how the model allows the outcome to vary with exposure. This may involve a named function or a clearly defined comparison, but it cannot remain an unstated assumption that “higher is better” or “more exposure has more effect.” The chosen relationship should be appropriate to the outcome scale.
- Connecting assumptions. State what must hold for the exposure-effect interpretation to be meaningful. Examples include accurate exposure measurement, relevant exposure preceding the outcome, an adequate account of baseline or other important predictors, and a defensible assumption about unmeasured causes of both exposure and response if a causal interpretation is intended.
These components answer related but distinct questions. A unit identifies how the exposure is measured; a window identifies which portion of the concentration-time history contributes to it. A time point identifies when the outcome is observed; it does not by itself establish that the selected exposure window is biologically relevant.
Why the Mean-Model Notation Is Useful
A short derivation shows what the general equation says and what it does not say. Suppose a continuous outcome is represented as
where \(\varepsilon_i\) is the difference between the observed outcome and the model’s systematic component. If the conditional mean of that difference is zero at every exposure value, then
Thus, the function describes the conditional mean under that assumption. This does not say that every individual with exposure \(x\) has outcome exactly \(f(x;\theta)\), nor does it specify how variable individual outcomes are around that mean. A fuller model may need to describe that variability and its distribution. For a binary outcome, the conditional mean is the event probability; the observation model and the function for that probability must be compatible with that scale.
Worked Example: Specifying AUC and a Continuous Biomarker
Hypothetical setting. A team proposes relating total exposure during the first 24 hours after dosing to change in a biomarker at hour 24. AUC is recorded as \(720\ \text{µg·h/L}\), and the biomarker change is \(3.2\ \text{units}\). Write a minimum specification and check that the units are coherent.
Exposure. Define \(X_i\) as AUC from 0 to 24 hours, in \(\text{mg·h/L}\). Convert the recorded value using \(1\ \text{mg}=1000\ \text{µg}\):
The exposure input for this observation is therefore \(X_i=0.720\ \text{mg·h/L}\), for the stated 0-to-24-hour window.
Outcome and time. Define \(Y_i(24)\) as the change from the pretreatment biomarker value to the value at hour 24, measured in biomarker units. The observed value is \(3.2\) biomarker units. This definition specifies a change, not the absolute biomarker concentration.
Relationship and assumptions. State the proposed conditional mean as \(E\{Y_i(24)\mid X_i=x\}=f(x;\theta)\), with \(f\) to be selected and justified for the scientific question. The minimum specification is not complete if \(f\) is left unspecified before fitting. It also assumes that the 0-to-24-hour AUC is a relevant summary for the hour-24 change and that the baseline and outcome measurements are defined consistently. The example supplies an input and outcome, not enough observations to estimate \(\theta\).
Worked Example: A Binary Outcome at a Defined Time
Hypothetical setting. In a small illustrative data set, investigators record whether a prespecified response is present at 6 hours and group individuals by whether their concentration at 2 hours is below or at least \(10\ \text{mg/L}\). Among 20 individuals in the lower group, 6 respond; among 20 in the higher group, 11 respond. Specify a simple relationship and calculate the observed proportions.
Exposure and outcome. Let \(X_i\) be concentration at 2 hours, in \(\text{mg/L}\). Let \(Y_i(6)=1\) if the response is present at 6 hours and \(0\) otherwise. The outcome is binary, and its conditional mean is a probability.
Observed group proportions. In the lower group, the observed response proportion is \(6/20=0.300\), or 30.0%. In the higher group, it is \(11/20=0.550\), or 55.0%. The difference in these observed proportions is \(0.550-0.300=0.250\), or 25.0 percentage points.
Relationship and assumptions. A simple specification could assign one response probability to concentrations below \(10\ \text{mg/L}\) and another to concentrations at or above that threshold, with the outcome measured at 6 hours. In this sample, the corresponding observed proportions are 0.300 and 0.550. This is a descriptive, threshold-group comparison, not evidence by itself that concentration caused the difference. A probability model must keep predicted probabilities between 0 and 1. Interpretation also requires a defensible threshold, consistent outcome ascertainment, and consideration of factors that may differ between exposure groups.
Worked Example: Repeated Outcomes and a Mismatched Window
Hypothetical setting. A study records concentration at 1 hour and biomarker change at both 1 hour and 8 hours. A proposed exposure metric is AUC from 0 to 8 hours, measured in \(\text{mg·h/L}\). For one participant, AUC is \(16\ \text{mg·h/L}\); the changes are \(0.4\) units at 1 hour and \(1.1\) units at 8 hours.
Specify the target. If the question concerns the 8-hour response, define \(Y_i(8)\) as change from baseline at hour 8, in biomarker units, and state \(E\{Y_i(8)\mid X_i=x\}=f(x;\theta)\), where \(X_i\) is the 0-to-8-hour AUC. The participant’s model input and target outcome are \(16\ \text{mg·h/L}\) and \(1.1\) units, respectively.
The 1-hour change of \(0.4\) units is a different outcome at a different time. Pairing it with the full 0-to-8-hour AUC would use exposure accrued after that outcome was measured. That may create a temporal mismatch and cannot be justified merely because both measurements belong to the same participant. If the 1-hour response is the target, the exposure metric and window must instead be chosen and justified for that question. The example illustrates why time is part of the specification, not an optional label.
Assumptions, Diagnostics, and Alternatives
The assumptions should reflect the intended interpretation. A descriptive exposure-effect model can summarize how outcomes vary with measured exposure without claiming that exposure caused the variation. A causal interpretation requires stronger conditions: for example, important common causes of exposure and outcome must be adequately addressed, and the exposure contrast must correspond to a meaningful intervention or comparison. Randomization may support some of these conditions, but it does not repair a poorly defined outcome or an exposure metric that is irrelevant to the response time.
Before fitting, check that the exposure has a clear unit and window, the outcome has a defined scale, and each outcome is matched to an exposure value that could plausibly precede it. After fitting, examine whether the proposed relationship is compatible with the observed data, whether predictions are possible on the outcome scale, and whether residual patterns or systematic differences across time or subgroups suggest a missing feature. These checks can reveal a poor specification; they do not prove that the assumptions are true.
There is no single alternative that replaces all five components. Different choices answer different questions. An instantaneous concentration can target a contemporaneous or near-term relationship; a cumulative metric summarizes exposure across a window; and repeated-outcome modeling represents response over multiple times rather than reducing it to one time point. These are alternative exposure or outcome structures, not interchangeable labels. For the same outcome, a model may use its original scale or a justified transformed scale, but the scale must be reported because it changes what the modeled quantity means.
The minimum specification also has limits. It does not establish that the selected exposure metric is biologically correct, that the functional relationship is identifiable from the available data, or that the data contain enough exposure variation to estimate it precisely. It is the necessary starting description, not a guarantee of a valid or useful fitted model.
Common Mistake
Wrong reasoning: “We measured concentration and response in every participant, so the model is fully specified; the software can determine the relationship.” This leaves the concentration time, units, response scale, response time, and intended relationship unstated. Software cannot decide which exposure window answers the scientific question, and an unspecified scale makes model predictions difficult to interpret.
Correction: State the exposure metric with units and window, define the outcome and its scale, identify when it is observed, specify the proposed functional relationship, and list assumptions needed for interpretation. Then assess whether the data can support that specification. If a choice is unresolved, label it as a modeling decision rather than implying it is determined automatically by the data.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- A model uses AUC but does not state the interval over which it was calculated. Which component is missing, and why does it matter?
- A binary response is measured at 12 hours. What must be stated about the outcome scale, and what does its conditional mean represent?
- Concentration at 8 hours is proposed to explain an outcome measured at 4 hours. What is the main concern?
- Does a well-written mean relationship alone fully describe the distribution of a continuous outcome? Explain.
- What additional conditions are needed to interpret an observed exposure-response pattern causally rather than descriptively?
Solutions. (1) The AUC interval is missing. AUC over different windows summarizes different exposure histories, so its interpretation and units cannot be fully understood without the interval. (2) State the response definition and coding, such as 1 for response and 0 for no response. Its conditional mean is the probability of response at 12 hours for the stated exposure. (3) The proposed exposure measurement occurs after the outcome, so it may include information unavailable before the response; the temporal ordering and relevance must be reconsidered. (4) No. A mean relationship describes the conditional mean but not the variability or full distribution around it; an observation model may also be needed. (5) A causal interpretation needs defensible control of common causes of exposure and outcome and a meaningful exposure contrast, among other design-specific assumptions. Without them, describe the pattern as an association rather than a causal effect.