Comparing Models with Correlated and Heterogeneous Data

Why the observation model matters in a mechanism comparison

Two mechanistic models can make different predictions for the same longitudinal measurements, biomarkers or clinical endpoints. A comparison is only as meaningful as the probability model used to score those predictions. Measurements from one subject over time may be correlated; assays may have different units and precision; and distinct data sources may share samples, laboratories or other sources of variation. Treating all residuals as independent and equally precise can therefore reward the wrong predictions or make a modest difference look decisive.

The focus here is the observation likelihood used to compare mechanistic alternatives. It is not enough to compare unweighted sums of squared residuals. Each candidate model must be evaluated against the same observed data using a likelihood that represents the relevant scales and dependencies. The likelihood describes the observation process conditional on a model’s predicted means; it does not, by itself, establish that a biological mechanism is true.

This likelihood assumes that, conditional on the model predictions and any included covariates or random effects, the residual vector has mean zero and covariance \(\Sigma\), with a multivariate normal distribution. The covariance model must be scientifically defensible and applied consistently across mechanisms unless a difference in observation process is explicitly justified. If \(\Sigma\) contains estimated parameters, they must be estimated as part of the likelihood, and counted among the fitted parameters when a criterion such as AIC is used.

Longitudinal correlation changes the score

Repeated measurements on a person or experimental unit are often not independent. A shared biological state, an assay batch effect or a slowly varying measurement disturbance can make nearby observations move together. For two measurements with common standard deviation \(\sigma\) and correlation \(\rho\), a simple covariance model is $$ \Sigma=\sigma^2 \begin{pmatrix} 1 & \rho\\ \rho & 1 \end{pmatrix}, \qquad -1<\rho<1. $$ The determinant is \(\sigma^4(1-\rho^2)\), and the residual quadratic is $$ r^\mathsf{T}\Sigma^{-1}r =\frac{r_1^2+r_2^2-2\rho r_1r_2}{\sigma^2(1-\rho^2)}. $$ When \(\rho\) is positive, a pair of residuals with the same sign is less surprising than an equally large pair with opposite signs. Ignoring this structure can change both the ranking and the size of a likelihood difference.

Different assays require scale-aware scoring

A residual of 1 ng/mL and a residual of 0.01 in fractional receptor occupancy cannot be compared by their raw magnitudes. They have different units and may have very different measurement precision. The covariance matrix handles both: a residual is judged relative to its variance, and covariance terms account for dependence. With independent observations and known standard deviations, the quadratic term reduces to the sum of squared standardized residuals, \(Q=\sum_i (r_i/\sigma_i)^2\).

Standardizing residuals can be useful for interpretation, but the likelihood also contains the determinant of \(\Sigma\). That term must not be dropped when comparing observation models with different variances or covariance structures. If both candidate mechanisms use the same observations and the same fixed covariance matrix, the determinant is common and cancels from their likelihood difference. If the covariance differs, each model’s full likelihood, including its determinant, is needed.

Combining multiple data sources

For genuinely independent sources, their joint likelihood is the product of their source-specific likelihoods, so their log-likelihoods add. In covariance terms, this corresponds to a block-diagonal \(\Sigma\): each block represents the dependence within a source, and zero off-diagonal blocks assert that sources are independent conditional on the model. That assertion needs justification. Shared subjects, samples, assay batches or data-processing steps can create cross-source dependence. In that case, a joint covariance with cross-source terms, or another suitable joint observation model, is needed.

A source-specific variance can represent a measurement scale or a known difference in precision. It should not be adjusted simply to make a preferred mechanism fit better. If variance parameters are estimated, the likelihood includes their fitted values and the determinant term; AIC’s parameter count \(k\) includes them. Comparing AIC values is meaningful only when the models use the same observed outcomes and compatible likelihood definitions. AIC is \(2k-2\ell\); it penalizes fitted flexibility but cannot repair a misspecified covariance model.

Why the covariance-weighted score has this form

The quadratic term can be understood by converting correlated residuals to uncorrelated, unit-scale residuals. Because a positive-definite covariance matrix has a Cholesky factorization \(\Sigma=LL^\mathsf{T}\), define \(z=L^{-1}r\). Then $$ z^\mathsf{T}z =r^\mathsf{T}(L^{-1})^\mathsf{T}L^{-1}r =r^\mathsf{T}(LL^\mathsf{T})^{-1}r =r^\mathsf{T}\Sigma^{-1}r. $$ Thus the likelihood’s quadratic is the squared length after covariance whitening, not the unweighted squared length in the original measurement coordinates. The determinant term follows from the multivariate normal density’s normalization: the density must integrate to one, and its scale is proportional to \(|\Sigma|^{-1/2}\). Omitting it when covariance parameters differ would fail to compare the normalized probability densities.

Diagnostics, alternatives and limitations

A covariance model is an assumption, not a correction that automatically makes a comparison reliable. Check whether its form is plausible given the sampling schedule, repeated units, assay structure and data provenance. Useful checks include whether estimated variances are on plausible assay scales, whether a fitted correlation is near a boundary, and whether the covariance matrix is positive definite. Near-singular matrices can make the inverse unstable and make the likelihood highly sensitive to small changes. For multiple sources, inspect whether the assumed independence is consistent with shared subjects, samples or batches. Sensitivity analyses over scientifically plausible correlations or variance assumptions can reveal whether the ranking is robust; they should not be used to search for a preferred answer.

The main alternatives address different questions. A diagonal covariance assumes no residual correlation; it is simpler, but is not justified merely because it is easier to fit. A structured covariance, such as an autoregressive form for equally spaced measurements, can capture dependence with fewer parameters than an unrestricted covariance matrix, but its structure must fit the design and scientific context. A joint covariance with cross-source terms is needed when sources share residual variation. Predictive cross-validation can assess performance on held-out observations, but it also needs splits that respect the dependence structure: separating repeated observations from the same subject across training and test sets can create information leakage. None of these approaches makes incompatible outcomes comparable without specifying their measurement scales.

The Gaussian likelihood can be sensitive to extreme residuals and to an incorrect covariance form. It may also be difficult to estimate a large covariance matrix with sparse data. A higher likelihood under one assumed observation model is evidence only relative to the specified alternatives, data and assumptions; it is not a probability that the mechanism is correct. AIC comparisons likewise depend on an adequate likelihood and do not account for uncertainty in the chosen covariance structure unless that uncertainty is handled explicitly.