Beta Regression for Bounded Biomarkers

Why Use Beta Regression for a Biomarker?

Some longitudinal biomarkers are proportions or normalized measurements that can take any value strictly between zero and one. Examples include a normalized laboratory measure or the fraction of a specified component in a sample. Their range is bounded, and their variability often changes with their mean. A Gaussian model can assign positive probability to impossible values below zero or above one, and a constant residual variance may be a poor description of the data. Beta regression addresses these features by modeling a response on the unit interval with a beta distribution.

In a longitudinal model, the response is recorded repeatedly for subject \(i\) at occasion \(j\), and its conditional distribution may depend on time, covariates, and subject-specific random effects. The beta distribution handles the response’s bounded continuous scale; it does not, by itself, account for within-subject dependence. That dependence must be represented in the longitudinal specification.

Mean and Precision Parameterization

Write \(\alpha_{ij}=\mu_{ij}\phi_{ij}\) and \(\beta_{ij}=(1-\mu_{ij})\phi_{ij}\). The conditional beta density is

The parameter \(\mu_{ij}\) is the conditional mean, while \(\phi_{ij}\) is the precision. In this parameterization, the conditional variance is

Thus, at a fixed mean, increasing precision reduces variance. Precision does not mean that observations are more likely to be near one particular endpoint; it controls concentration around the mean. The variance also depends on the mean: even at the same precision, a mean near 0.5 has a larger variance than a mean near either endpoint.

A standard regression specification uses a logit link for the mean and a log link for precision. Let \(x_{ij}\) be a vector of predictors for the mean and \(w_{ij}\) a vector for precision. In a repeated-measures setting, a subject-specific mean model can include random effects \(b_i\):

Here \(\beta\) and \(\gamma\) are regression coefficient vectors, and \(z_{ij}\) selects the random effects entering the mean. The logit ensures that the fitted mean remains strictly between zero and one; the log link ensures positive precision. A constant-precision model is the special case in which \(\phi_{ij}=\phi\) for all observations. A precision model with covariates is useful when conditional dispersion changes systematically with time or other predictors.

For example, if \(x_{ij}\) includes time, its coefficient describes a change in the log odds of the conditional mean per time unit, holding other modeled terms fixed. It is not a change of that size in the biomarker itself. Likewise, a coefficient in the precision equation describes a change in log precision, not a direct change in the mean.

Worked Examples

Assumptions and a Short Justification

Beta regression requires a continuous response strictly inside the unit interval. Conditional on the modeled predictors and random effects, the beta distribution must be an adequate description of the response distribution. In a repeated-measures model, observations are typically treated as conditionally independent given the subject-specific effects and modeled predictors; dependence left over after conditioning can invalidate standard uncertainty calculations. The chosen mean and precision equations must also be adequate: a suitable mean model cannot compensate for a systematic misspecification of dispersion, or vice versa.

The mean and variance expressions follow directly from the beta distribution’s shape parameters. For a beta random variable with shapes \(\alpha\) and \(\beta\), its mean is \(\alpha/(\alpha+\beta)\) and its variance is \(\alpha\beta/[(\alpha+\beta)^2(\alpha+\beta+1)]\). Here \(\alpha+\beta=\phi\), so the mean is \(\mu\phi/\phi=\mu\). Substituting the shapes into the variance gives

This derivation explains why precision controls concentration and why the variance changes with the mean even when precision is constant.

Diagnostics, Alternatives, and Limitations

A fitted beta model should be checked for more than convergence. Compare observed and fitted biomarker distributions across time and important covariate groups, and examine residual patterns for systematic changes in mean or spread. Residuals with trends over time, group-specific spread not captured by the precision equation, or pronounced departures from the fitted distribution suggest that the specification may be inadequate. In repeated measurements, remaining subject-level patterns can indicate that the random-effects structure or other representation of dependence is insufficient.

Exact zeros and ones are not values in the support of the ordinary beta distribution. A recorded endpoint may reflect a genuine boundary value, a detection limit, rounding, or a data-processing convention; these situations are not interchangeable. Check how the measurements were generated and recorded before choosing a model. If zeros or ones are genuine and occur with positive probability, a model designed for boundary masses, such as a zero-inflated or hurdle biomarker model, may be more appropriate. Do not simply insert an endpoint into the beta density, or silently alter the observations without justifying the change.

A Gaussian longitudinal model is an alternative when the response is effectively unbounded or when its conditional distribution is adequately approximated by a Gaussian distribution. It can be convenient, but fitted values or predictive distributions may extend outside zero and one, and a constant residual variance may fail. Beta regression respects the interior support and can represent mean-dependent variance, but it cannot accommodate genuine endpoint mass without an extension. A transformation followed by a Gaussian model is another possibility, but it changes the scale of interpretation and does not automatically solve endpoint problems.

Beta regression also has limits. The standard model is for continuous interior responses, and its adequacy depends on the chosen mean and precision predictors and on the dependence structure used for repeated data. A good fit for the marginal shape at one occasion does not establish that within-subject dependence is correctly represented. In a joint longitudinal and time-to-event analysis, the beta model specifies the longitudinal response distribution; it does not alone specify the event-time model or the relationship between the longitudinal process and event risk.