Propagating Uncertainty From Score to Risk

Why Uncertainty Must Travel With the Risk Estimate

An absolute-risk estimate combines several quantities: a baseline incidence, the estimated effect of a genetic score, and often clinical predictors and fitted model coefficients. Each may be uncertain. Reporting a single risk without considering that uncertainty can make a borderline estimate appear more definite than it is. This matters especially when a clinical rule uses a threshold: a small change in the estimated score effect or baseline incidence may move an individual from one side of the threshold to the other.

The goal here is to quantify uncertainty in the estimated risk, not to predict with certainty whether a particular person will develop disease. The two questions differ. A person with an estimated 10% risk still has an uncertain future outcome; meanwhile, the 10% estimate itself may also be uncertain because its model was fitted using finite data.

Risk Models and the Delta Method

For a binary outcome over a specified horizon, consider a logistic risk model for an individual with score \(S\) and clinical predictors \(Z\):

Here \(\eta\) is the log odds, \(\alpha\) is the intercept, \(\beta\) is the score coefficient, \(\gamma\) is the vector of clinical coefficients, and \(p\) is the model-estimated absolute risk. The model must be appropriate for the target population and time horizon; in particular, its baseline incidence and calibration should be relevant to the setting in which the risk is reported. The earlier material on calibration and local baseline incidence addresses those requirements. Uncertainty calculations quantify imprecision in the fitted quantities; they do not repair a misspecified or poorly calibrated model.

Let \(\hat{\theta}\) collect the estimated model parameters, with estimated covariance matrix \(\widehat{\Sigma}\). If \(g(\theta)\) is a smooth function of those parameters, the delta method approximates its variance by multiplying the covariance matrix on both sides by the gradient of \(g\), evaluated at the estimates. For risk, the derivative of the expit function is \(p(1-p)\). Thus, if the linear predictor estimate has standard error \(SE(\hat{\eta})\),

A normal interval formed directly as \(\hat{p}\pm1.96SE(\hat{p})\) can extend below 0 or above 1. A common alternative is to form the interval on the log-odds scale, then transform its endpoints. If the estimated linear predictor is approximately normal, an approximate 95% interval for risk is

This transformed interval remains between 0 and 1 and is generally asymmetric around the point estimate. It describes uncertainty in the model-estimated risk under the stated approximation and fitted model. It is not a 95% probability that the person’s future outcome will match the estimated risk.

Uncertainty in Baseline Incidence

A genetic relative effect becomes an absolute risk only after it is anchored to baseline incidence. Consequently, uncertainty in that incidence contributes to uncertainty in an individual risk even if the genetic effect is treated as known. For example, if \(q\) is a baseline risk and an odds ratio \(r\) applies to the individual, the corresponding risk is

If \(q\) is estimated, its uncertainty passes through the logit transformation and then through the risk transformation. For a score model with \(\eta=\operatorname{logit}(q)+\beta S\), the derivative of \(\eta\) with respect to \(q\) is \(1/\{q(1-q)\}\). If the baseline estimate and score-effect estimate are independent, their contributions to the variance of \(\hat{\eta}\) add. If they are not independent, the covariance term must also be included.

Multiple Coefficients, Correlations, and Score Uncertainty

With multiple predictors, it is important to propagate the full covariance matrix rather than treating each coefficient as independent by default. Write \(\eta=\alpha+\beta S+\gamma^\mathsf{T}Z\), and let \(x=(1,S,Z^\mathsf{T})^\mathsf{T}\) be the predictor vector matching the coefficient vector. Then \(\hat{\eta}=x^\mathsf{T}\hat{\theta}\), and its estimated variance is

This expression includes all variances and covariances. For example, uncertainty in an intercept and a score effect may reinforce or partly offset one another depending on the sign of their covariance and the person’s score. Omitting covariance terms can therefore either overstate or understate uncertainty.

The score itself may also be uncertain. If a score is computed using estimated weights, uncertainty in those weights can contribute to uncertainty in the person’s score. In a first-order approximation, if \(S=w^\mathsf{T}G\), where \(G\) is the person’s fixed genotype vector and \(w\) is an estimated weight vector, then the score variance from weight estimation is approximately \(G^\mathsf{T}\widehat{\operatorname{Var}}(\hat{w})G\). Its contribution to the linear predictor variance is approximately \(\beta^2\) times that score variance, with additional covariance terms if the weight estimates and risk-model coefficients are dependent. This is different from variation in scores across people: a population score standard deviation is not, by itself, uncertainty about one person’s score.

Practical Workflow and Interpretation

A reliable analysis makes the source and meaning of uncertainty explicit. The following sequence applies whether uncertainty is propagated analytically or with simulation.

Simulation is useful when the model is nonlinear, several uncertainty sources must be combined, or the distribution of risk is noticeably asymmetric. Each simulation draw should preserve the fitted coefficients’ joint dependence, use a plausible draw for uncertain baseline incidence or score weights, and recalculate the risk. Quantiles of the resulting risk draws can then summarize an interval. Drawing every coefficient independently when the fitted coefficients are correlated does not reproduce the fitted model’s uncertainty.

Common Pitfall: Confusing Three Different Uncertainties

A frequent error is to treat a risk interval, a threshold-crossing probability, and the chance of a future event as interchangeable. They answer different questions. A risk interval describes uncertainty in an estimated conditional risk due to estimation, under its model and inferential assumptions. A threshold-crossing probability describes how often an uncertainty distribution for the estimate falls above a chosen cut point. The individual outcome remains a future random event whose probability is the person’s risk, conditional on the model and predictors.

For example, an estimated risk of 23.9% with a 38.2% threshold-crossing probability does not mean the person has a 38.2% chance of disease. Nor does it mean there is a 38.2% objective probability that the true risk exceeds the threshold unless a probability model with that interpretation has been specified. The correct interpretation is tied to how uncertainty was quantified.