Quantifying Uncertainty in Early-Phase Biomarker Effects

Why quantify uncertainty in an early-phase effect?

An estimated exposure-response curve is not known exactly merely because a model has converged or its fitted line appears plausible. Uncertainty in its parameters can affect conclusions about the size and location of the biomarker response, and can be especially important when the observed concentrations cover only part of the curve. A useful summary therefore reports not only a point estimate, but also how uncertain the estimate or a response prediction is.

The earlier material on direct-effect Emax models defines the response curve. Here, let \(C\) be concentration, let \(E_0\) be the baseline or reference response, and let \(\Delta R(C)\) denote the modeled change from that reference. For a direct-effect model with no Hill coefficient,

Here \(E_{\max}\) is the maximum modeled change as concentration becomes large, and \(EC_{50}\) is the concentration at which the modeled change is half of \(E_{\max}\). This tutorial concerns uncertainty in these quantities and in predictions from the curve; it does not establish whether the model form is suitable for a particular data set.

Intervals for Emax and EC50

A common approximation is to treat an estimated parameter as normally distributed around its true value, with standard error \(SE\). For a two-sided 95% confidence interval, the estimate plus or minus 1.96 standard errors gives an approximate interval. This is most defensible when the likelihood is reasonably well behaved, the estimate is not near a boundary, and the sampling distribution is close to normal. For a positive parameter such as \(EC_{50}\), an interval constructed on the log scale is often more sensible: it cannot have a negative lower endpoint and allows uncertainty to be asymmetric on the original scale.

For \(E_{\max}\), a symmetric interval on the original scale can be reasonable if its sampling distribution is approximately normal and the estimate is well away from zero. If the parameter is constrained to be positive or its uncertainty is strongly skewed, an interval from a transformed scale, a likelihood profile, or a bootstrap may better reflect the fitted model. Merely reporting a narrow standard error does not resolve weak identification: for example, if concentrations scarcely approach the plateau, \(E_{\max}\) may be poorly determined even when the curve fits observed values.

Propagating parameter uncertainty to the response curve

A response at a specified concentration is a function of multiple fitted parameters. Its uncertainty depends both on uncertainty in each parameter and on their covariance. The delta method provides a local approximation: if \(\hat{\boldsymbol{\theta}}\) estimates a parameter vector with covariance matrix \(\widehat{\Sigma}\), and \(g(\boldsymbol{\theta})\) is a smooth derived quantity, then a first-order Taylor expansion gives

The justification is that, near the estimate, \(g(\hat{\boldsymbol{\theta}})\) is approximately \(g(\boldsymbol{\theta})+\nabla g(\boldsymbol{\theta})^T(\hat{\boldsymbol{\theta}}-\boldsymbol{\theta})\). The variance of this linear term is the gradient multiplied on both sides of the parameter covariance matrix, as displayed. The approximation can be poor when the function is highly curved over the plausible parameter range or the parameter distribution is markedly non-normal.

The delta-method interval relies on local linearity and an approximately normal estimate of the derived response. It should be checked against the shape of the response function and the uncertainty in the fitted parameters. When the interval is asymmetric, reaches an implausible region, or depends strongly on a boundary or sparse concentration coverage, parameter draws or a likelihood-based method can reveal behavior that a symmetric approximation hides.

Uncertainty in an individual response prediction

An interval for the fitted mean curve and an interval for an individual outcome answer different questions. To predict one future measurement, uncertainty can include parameter uncertainty, between-participant variability, and residual measurement or within-participant variability. Which components belong in the prediction depends on the target. A prediction for a new participant generally includes between-participant variability; a prediction for a repeat measurement on a participant with an informative individual history may instead condition on that participant's estimated effect. Residual variability is included when predicting a measured outcome, but not when describing the latent modeled mean.

A useful distinction is that the model's estimated population mean at a concentration is not the same as the response expected for every individual. Individual predictions may also be conditional on observed data for that person. A conditional prediction should state what information was used; it should not be presented as an unconditional prediction for a new participant.

Probability that an effect exceeds a threshold

A probability statement about an unknown effect requires care about the inferential framework. In a Bayesian analysis, the posterior distribution describes uncertainty about parameters conditional on the data, model, and prior. A posterior probability that a biomarker effect exceeds a threshold is the posterior mass above that threshold. In a frequentist analysis, a confidence interval does not itself assign a probability to a fixed parameter. A point estimate and confidence interval can show the estimate's precision relative to a threshold, but they are not a posterior probability.

The posterior probability depends on the posterior distribution, which in turn depends on the likelihood, model, and prior. It should be accompanied by enough information to understand that conditioning. If a normal approximation is used, inspect whether posterior draws are skewed, multimodal, or constrained near a boundary. In those cases, calculate the fraction of posterior draws exceeding the threshold rather than relying on a normal tail approximation.

Choosing and checking an uncertainty method

The Wald or delta-method interval is convenient and transparent when estimates are well identified and approximately normal. It is not automatically reliable simply because software reports a standard error. The following alternatives are useful when the local approximation is doubtful:

• Profile likelihood: evaluates the likelihood over fixed values of a parameter or derived quantity while re-estimating other parameters. It can represent asymmetry and is useful for checking whether a nominally precise estimate is weakly identified.
• Bootstrap: repeatedly resamples data or simulates and refits under a specified sampling scheme, then summarizes the resulting estimates or predictions. Its validity depends on a resampling scheme that reflects the study design and on successful, representative refits.
• Posterior simulation: draws from the fitted posterior and calculates parameter intervals, response intervals, or the fraction of draws above a threshold. This directly represents posterior uncertainty, but results depend on the model and prior and require adequate sampling and convergence.

These approaches answer related but not identical questions. A confidence interval has a repeated-sampling coverage interpretation under its assumptions; a credible interval summarizes posterior probability; and a prediction interval concerns future outcomes. Comparing methods can be informative, but numerical agreement does not prove that the exposure-response model is scientifically adequate.

Uncertainty summaries deserve extra scrutiny when the estimated \(EC_{50}\) is outside or near the edge of the observed concentration range, when the data do not approach a plateau needed to identify \(E_{\max}\), or when parameters are strongly correlated. Warning signs include very wide or highly asymmetric intervals, estimates that change substantially under small modeling choices, bootstrap refits that fail frequently, or posterior distributions that remain sensitive to prior choices. These signs do not dictate one alternative method; they indicate that the available data may not identify the requested quantity precisely.

Common mistake and practical interpretation

In an early-phase setting, uncertainty should be reported on the scale that answers the scientific question, with units and a clear target: a model parameter, a population mean response at a specified concentration, or an individual measurement. A threshold probability is meaningful only when the threshold refers to that same target. Precision is also not the same as relevance: a narrow interval can describe an effect that is small, while a potentially important estimated effect can remain highly uncertain.