Modeling Repeated Biomarker Measures Within Participants

Why Repeated Biomarker Measures Need a Participant-Level Model

When a biomarker is measured several times in each participant, observations from the same person are generally related. A participant may have a persistently higher or lower biomarker level than others, even at comparable exposures. In addition, each measurement may contain occasion-specific biological variation and assay error. Treating every observation as independent can therefore misrepresent both the uncertainty in exposure-response estimates and the sources of variation.

The aim here is to model those sources separately. The exposure-response function describes the systematic relationship of interest; a participant-level random effect represents persistent differences between participants; and a residual term represents deviations that vary from measurement to measurement. This framework builds on the earlier material on direct-effect, turnover, and effect-compartment models: the exposure-response function can be selected from that material, while the repeated-measure structure is specified separately.

A Random-Intercept Model for Repeated Biomarkers

Let \(Y_{ij}\) be the biomarker measurement for participant \(i\) at occasion \(j\), and let \(C_{ij}\) be the corresponding exposure. A useful starting point is a random-intercept model:

Here \(f(C_{ij};\boldsymbol{\theta})\) is the population exposure-response function, with parameters \(\boldsymbol{\theta}\); \(b_i\) is participant \(i\)'s random intercept; and \(\varepsilon_{ij}\) is the residual for that measurement. For example, \(f\) might be a linear function or a previously specified pharmacodynamic function. The repeated-measure structure does not itself determine the shape of \(f\).

A standard Gaussian version assumes \(b_i\sim N(0,\omega^2)\) and \(\varepsilon_{ij}\sim N(0,\sigma^2)\), with random effects independent between participants and residuals independent across occasions and participants. It also assumes that random effects and residuals are independent of one another and, in this basic model, that the random intercept is independent of the exposures. The mean function and variance structure should be appropriate for the biomarker scale. If residual variability changes systematically with the predicted response or exposure, a constant \(\sigma^2\) may be inadequate.

The zero mean for \(b_i\) defines the population-level mean function: across participants, random intercept deviations average to zero. The variance \(\omega^2\) describes between-participant variability in persistent level, while \(\sigma^2\) describes within-participant residual variability under the assumed model. Neither variance should automatically be labelled “biological” or “assay” error alone; the residual may combine several unmodelled sources.

What the Random Intercept Implies

The covariance structure follows directly from the model. Suppose two observations are from the same participant but different occasions. Their shared random intercept contributes covariance, while their independent residuals do not. For a single observation, both terms contribute variance.

Thus, under this model, repeated observations have positive within-participant correlation even after accounting for exposure. The intraclass correlation coefficient (ICC), the correlation between two distinct observations from the same participant under the stated assumptions, is:

For participant \(i\), the vector of observations has mean vector \(\mathbf{f}_i\) and covariance matrix \(\omega^2\mathbf{1}\mathbf{1}^{\mathsf T}+\sigma^2\mathbf{I}\), where \(\mathbf{1}\) is a vector of ones and \(\mathbf{I}\) is an identity matrix of the appropriate size. More generally, if a model has random effects with design matrix \(\mathbf{Z}_i\) and covariance matrix \(\mathbf{G}\), and residuals with covariance matrix \(\mathbf{R}_i\), then its marginal covariance matrix is \(\mathbf{Z}_i\mathbf{G}\mathbf{Z}_i^{\mathsf T}+\mathbf{R}_i\). This makes explicit how both participant effects and residual assumptions determine dependence.

Within-Person and Between-Person Exposure Effects

A key interpretive issue is that exposure varies both within participants over time and between participants. These patterns need not have the same relationship with the biomarker. A participant with a higher average exposure may differ in other ways from a participant with a lower average exposure; that between-person association does not, by itself, show how the biomarker changes when an individual’s exposure changes.

For a linear exposure-response analysis, one way to distinguish the associations is to decompose exposure into its participant mean and the occasion-specific deviation from that mean:

Here \(\bar{C}_i\) is participant \(i\)'s mean exposure over the included occasions, and \(\bar{C}\) is the average of the participant means in the analysis. The coefficient \(\beta_W\) describes the modelled within-person association: the expected biomarker difference associated with a one-unit exposure difference from that participant’s own mean, holding the other model terms fixed. The coefficient \(\beta_B\) describes the between-person association with participant mean exposure. These are distinct estimands, not two interchangeable estimates of a single universal effect.

This decomposition does not guarantee causal interpretation. For example, time-varying factors could influence both exposure and biomarker, and participant means could be related to unmeasured participant characteristics. It does make the two sources of exposure variation explicit, so the fitted coefficients are easier to interpret than a single pooled slope when within- and between-person relationships differ.

Fitting Choices and Model Assumptions

In a Gaussian mixed-effects model, the fixed-effect parameters and variance components can be estimated by maximum likelihood or restricted maximum likelihood (REML). In either case, the repeated observations from one participant contribute jointly through their mean vector and covariance matrix; they are not treated as independent records. A fitted model should state which exposure-response function was used, which participant-level effects were included, and what residual covariance structure was assumed.

A random intercept is a parsimonious starting point when participants appear to differ in level but the data do not support more complex subject-specific patterns. If the exposure-response slope plausibly differs between participants, a random-slope model may be considered. For a linear exposure term, one possible extension is:

The random intercept \(b_{0i}\) and random slope \(b_{1i}\) are typically modelled jointly, allowing a covariance between them. This represents participant-specific deviations in both level and exposure slope. It requires enough informative repeated measurements and exposure variation to estimate the additional variance components. A more elaborate model is not automatically more reliable.

The simple model assumes that, after accounting for the mean function and random effects, residuals have the specified distribution and dependence. In particular, independent residuals may be unrealistic if measurements close in time remain more alike than measurements far apart. A time-dependent residual covariance model may then be needed. The exposure and response should also be aligned appropriately in time; a model that pairs the wrong exposure with a biomarker measurement can produce a misleading relationship even if its statistical structure is otherwise well specified.

Useful initial checks include plotting participant-level trajectories, comparing observed and model-predicted values over time and exposure, and examining residuals for systematic trends, changing spread, and serial patterns. Large participant-specific departures, curvature, or residual autocorrelation can signal that the mean function, random-effects structure, or residual model is inadequate. These checks identify concerns to investigate; they do not prove that assumptions hold.

Participant Averages and Their Precision

Repeated observations can improve the precision of a participant’s estimated level, but averaging does not remove all variation. Under the random-intercept model, for \(m\) observations with independent residuals and the same participant-level variance, the variance of a participant’s average around its modelled mean is:

The persistent participant component remains, while the independent residual contribution decreases as more measurements are averaged. If residuals are correlated, the residual contribution does not generally reduce to \(\sigma^2/m\). Thus, counting measurements as if each provided an independent participant does not correctly quantify the information in a study.

Alternatives, Limitations, and Common Pitfalls

The random-effects model is not the only way to handle repeated outcomes. A model with participant fixed effects can control for stable participant characteristics while estimating within-person associations, but it does not estimate a population distribution of participant effects and cannot estimate a between-person effect in the same way. Generalized estimating equations (GEE) instead target a population-average relationship and use a working correlation structure; their interpretation and variance estimation differ from a likelihood-based mixed model. The choice depends on the scientific estimand, outcome distribution, design, and assumptions—not simply on which method produces a preferred result.

A basic Gaussian model can be unsuitable for a bounded, skewed, or otherwise non-Gaussian biomarker, or when residual variance changes across the response range. Random-effect distributions and residual independence are also assumptions, not facts guaranteed by fitting software. Sparse measurements, little exposure change within participants, and a small number of participants can make variance components or random slopes difficult to estimate. Model complexity should be supported by the design and the data.

• Pitfall: treating all observations as independent. The reasoning “there are 100 measurements, so there are 100 independent units” ignores shared participant effects. Model participant-level dependence; the number of participants and the number of repeated measurements convey different information.
• Pitfall: interpreting a pooled exposure slope as a within-person response. A pooled slope can combine within-person changes with differences in average exposure across participants. Separate these terms when their interpretations matter.
• Pitfall: assuming the residual is only assay error. Residual variation can include measurement error, short-term biological variation, and misspecification. Interpret \(\sigma^2\) as residual variation under the specified model unless its sources are independently established.
• Pitfall: adding random slopes because they seem more realistic. A random slope requires sufficient exposure variation and repeated information to estimate. Check whether the data support it and whether the fitted model behaves sensibly.