Worked Efficacy Exposure-Response Analysis

From Efficacy Objective to an Exposure-Response Analysis

An efficacy exposure-response analysis asks how an efficacy outcome varies with patients’ measured or model-derived drug exposure. The analysis can describe a pattern within the data and help quantify predicted response at observed exposure values. It does not, by itself, establish that changing an individual patient’s exposure would cause the predicted change in response: exposure is generally not randomized, even when treatment assignment is.

The analysis begins by making the clinical question operational. Consider a hypothetical trial of an oral drug for a chronic inflammatory condition. The endpoint is whether a patient achieves at least a 30% improvement from baseline in a disease-activity score at week 12. For this worked analysis, assume week-12 outcomes are observed for all 120 patients included in the efficacy analysis. Let \(Y_i=1\) indicate that patient \(i\) meets the response definition and \(Y_i=0\) otherwise.

This definition specifies the population, endpoint, time point, and exposure summary. It also makes assumptions about events that could affect interpretation. Here, assume that the week-12 response is assessed regardless of treatment discontinuation or dose modification, and that the specified exposure summary is available for each included patient. In a real analysis, the handling of discontinuation, rescue medication, and unavailable outcomes must be prespecified. Different choices can target different estimands; the earlier material on missing outcomes and informative dropout explains why missingness cannot simply be ignored without justification.

For patient \(i\), define \(C_i\) as the time-averaged steady-state concentration over weeks 4–12, measured in mg/L. This exposure measure must be constructed using a consistent rule, with the concentration model, sampling information, and relevant dosing records specified. It should represent exposure before or during the outcome window in a clinically defensible way. Using exposure measured after the response assessment, or choosing an exposure summary only because it produced a favorable association, can distort the analysis.

Model, Assumptions, and Interpretation

For a binary response, a logistic regression model relates the probability of response to exposure while keeping predictions between 0 and 1. Center exposure at 20 mg/L and scale it in 10 mg/L units:

Here \(p_i=P(Y_i=1\mid C_i)\). The intercept \(\beta_0\) is the log odds of response at 20 mg/L, and \(\beta_1\) is the change in log odds associated with a 10 mg/L increase in exposure under this model. Exponentiating \(\beta_1\) gives the odds ratio per 10 mg/L increase. An odds ratio is not a risk ratio and does not give a constant change in response probability.

The linearity assumption concerns the log odds, not the probability itself. The resulting probability curve is generally curved even when the log odds change linearly. The model also does not account for baseline predictors unless they are added. An adjusted model might include prespecified baseline severity and other prognostic variables, but its exposure coefficient would then describe the exposure association conditional on those included variables. Adjustment does not guarantee elimination of residual confounding.

Worked Logistic Analysis Across Observed Exposures

Assume the hypothetical trial has 120 treated patients, with measured or modeled average exposures from 10 to 40 mg/L. A patient-level logistic regression gives the illustrative fitted coefficients \(\hat{\beta}_0=0.600\) and \(\hat{\beta}_1=0.450\). These are hypothetical rounded model estimates, not results from a real study. Predictions should be emphasized within the observed exposure range; predictions outside it require extrapolation.

The fitted odds ratio per 10 mg/L increase is \(\exp(0.450)=1.568\). In this model, the odds of response are estimated to be 1.568 times as large for a 10 mg/L higher exposure, holding the model structure fixed. This statement is about odds, not a 56.8 percentage-point increase in response probability. The probability change depends on the starting probability.

Justification for the Probability Interval

The uncertainty calculation starts with the estimated linear predictor at the exposure value of interest. For a logistic regression with coefficient vector \(\hat{\boldsymbol{\beta}}\), let \(\mathbf{z}\) be the corresponding design vector. The estimated predictor is \(\hat{\eta}=\mathbf{z}^{T}\hat{\boldsymbol{\beta}}\). If the estimated coefficient covariance matrix is \(\widehat{\operatorname{Var}}(\hat{\boldsymbol{\beta}})\), then the estimated variance of that linear combination is \(\mathbf{z}^{T}\widehat{\operatorname{Var}}(\hat{\boldsymbol{\beta}})\mathbf{z}\). Taking its square root gives the standard error.

At 20 mg/L, \(x=0\) and the design vector for this one-predictor model is \(\mathbf{z}=(1,0)^T\). The variance expression therefore selects the estimated variance of the intercept; its square root is the stated 0.381. Under a large-sample normal approximation, \(\hat{\eta}\pm1.96SE(\hat{\eta})\) gives an approximate 95% interval on the log-odds scale. Since the inverse-logit is monotone increasing, applying it to the two endpoints preserves their order and yields the probability interval. This transformation does not make the interval symmetric on the probability scale.

Diagnostics, Alternatives, and Limitations

Before interpreting the curve, check whether the data support the model. Plot observed response proportions across exposure ranges alongside fitted probabilities, while recognizing that a coarse grouping can conceal within-range patterns. Examine whether the log-odds trend is plausibly linear, whether a few patients have unusually influential exposure values, and whether the response or nonresponse counts are too sparse at the range boundaries. Check that the exposure distribution has meaningful support around each value for which a prediction is reported. The earlier material on predictive checks and validation provides broader approaches to assessing predictive performance; here, the key question is whether the chosen exposure-response structure is credible for this endpoint and dataset.

Warning signs include systematic differences between observed and fitted response rates, a sharp pattern in residual diagnostics, unstable estimates when a small number of observations are removed, and very wide intervals at sparsely observed exposure values. Complete or near-complete separation—where exposure values nearly divide responders from nonresponders—can produce unstable or extremely large coefficient estimates. A straight line in log-odds may also be inappropriate if the response plateaus or changes more sharply over part of the exposure range. These issues call for investigation, not automatic selection of a more favorable model.

The main alternatives answer related but different questions. A model using exposure categories avoids imposing a straight-line log-odds trend across the whole range, but discards information within categories and can give imprecise estimates when some categories contain few outcomes. A flexible curve, such as a prespecified spline, can represent curvature but uses more parameters and may be unstable where data are sparse. A comparison by randomized dose preserves the randomized assignment contrast, but it describes response by assigned dose rather than by achieved exposure. None of these choices removes the need to define the endpoint, exposure window, analysis population, and missing-outcome strategy.

The most important limitation is the distinction between exposure and randomized assignment. Patients with higher exposure may differ in adherence, physiology, disease severity, concomitant treatment, or other characteristics related to response. A fitted exposure coefficient can therefore combine a pharmacologic exposure association with differences between patients. The earlier material on sensitivity analysis for residual confounding is relevant when considering how robust an interpretation is to unmeasured factors. Even a well-fitting, validated model remains descriptive unless the assumptions that identify a causal exposure effect are justified.