Subject-Level Influence in Biomarker Models

Why Assess Influence at the Subject Level?

A longitudinal biomarker model uses multiple observations from each person, and those observations are generally correlated. A subject may therefore affect an estimated treatment trend or variance component through the pattern of their entire trajectory, even if no single measurement appears extreme. Subject-level influence asks a practical question: how much would the fitted model change if one subject’s complete record were omitted?

This is a sensitivity analysis, not a rule for removing data. A subject can be influential without being erroneous, and an unusual residual does not necessarily imply substantial influence on the fitted parameters. Residual diagnostics for linear mixed models examine discrepancies between observed and model-implied data; deletion diagnostics instead measure how sensitive the fit is to particular subjects.

Write the Gaussian mixed model as \(y_i\sim N(X_i\beta,V_i(\theta))\), independently across subjects \(i=1,\ldots,m\). Here \(y_i\) is the vector of biomarker observations for subject \(i\), \(X_i\) is that subject’s fixed-effect design matrix, \(\beta\) is the vector of fixed effects, and \(V_i(\theta)\) is the within-subject covariance matrix governed by covariance parameters \(\theta\). The full data have \(N=\sum_i n_i\) observations. For a deletion fit, remove \(y_i\) and \(X_i\), and refit the parameters under the same estimation method and model structure.

The comparison should focus on quantities relevant to the scientific question: a treatment-by-time contrast, a baseline effect, a random-intercept variance, or a derived quantity such as an intraclass correlation. Report both the original and deleted estimates, and interpret their difference in the original units. A raw change cannot be called large or small without considering the estimate’s scale, standard error, or scientific relevance.

What to Compare After Each Deletion

Let \(\hat\beta\) and \(\hat\theta\) denote full-data estimates, and let \(\hat\beta_{(-i)}\) and \(\hat\theta_{(-i)}\) denote estimates after deleting subject \(i\). A direct fixed-effect change is

For a coefficient with full-data standard error \(SE(\hat\beta_j)\), a useful scale-free comparison is \(\Delta_{\beta_j,i}/SE(\hat\beta_j)\). This is a descriptive diagnostic, not a test statistic with a universal cutoff or a p-value. Also compare standard errors: a deletion can leave a coefficient nearly unchanged while substantially changing its estimated precision.

Variance components are often easier to compare on their original scale and, when positive, on a relative or logarithmic scale. If a random-intercept variance changes from \(\hat\tau^2\) to \(\hat\tau^2_{(-i)}\), report the absolute change and possibly the percentage change. When a component is near zero, a percentage can be unstable or misleading; report the estimates directly and interpret the boundary carefully.

Deletion Reasoning Under ML and REML

The route to an approximate deletion calculation depends on the estimation method. Under maximum likelihood (ML), independent subject contributions yield an additive log likelihood. Under restricted maximum likelihood (REML), the restricted criterion accounts for estimation of the fixed effects using the full design and covariance structure. The simple ML deletion-score identity must not be transferred to REML.

For ML, write the log likelihood as \(\ell(\psi)=\sum_{i=1}^m\ell_i(\psi)\), where \(\psi\) collects the ML parameters and each \(\ell_i\) is a subject’s marginal log-likelihood contribution. At an interior full-data ML estimate \(\hat\psi\), the total score is zero: \(\sum_i U_i(\hat\psi)=0\), where \(U_i=\partial\ell_i/\partial\psi\). The score in the data with subject \(i\) deleted, evaluated at the full-data estimate, is then \(-U_i(\hat\psi)\). This identity relies on additive subject contributions and an interior ML solution.

Let \(J_{(-i)}\) be the negative derivative of the deleted-data score with respect to \(\psi\), evaluated at \(\hat\psi\). A first-order Taylor expansion of that score around \(\hat\psi\) gives the approximation

This is a screening approximation, not a substitute for refitting. It can be poor when deletion is substantial, the likelihood is strongly nonlinear, or the estimate is near a boundary. It also does not apply as written to REML: restricted likelihood does not estimate \(\beta\) by maximizing the restricted criterion, and its criterion does not generally split into independent subject-specific terms.

For clarity, define the full-data REML projection matrix

For covariance parameter \(\theta_a\), let \(V_a=\partial V/\partial\theta_a\). The REML score component is

The reduced-data score \(s_{R,(-i)}\) has the same form but uses \(y_{(-i)}\), \(V_{(-i)}\), \(X_{(-i)}\), and the resulting \(P_{(-i)}\). Deleting a subject changes all of these quantities, including the fixed-effect adjustment inside \(P_{(-i)}\). In particular, \(s_{R,(-i)}(\hat\theta)\) is not generally \(-s_{R,i}(\hat\theta)\).

If the reduced REML solution is interior and the negative score derivative \(J_{R,(-i)}=-\partial s_{R,(-i)}/\partial\theta'\), evaluated at \(\hat\theta\), is nonsingular, the local approximation follows by expanding the reduced score:

The corresponding fixed-effect approximation must use the reduced GLS equation, evaluated at the reduced covariance estimate, rather than treating \(\beta\) as a parameter maximized by REML. In practice, exact reduced-data REML refits are usually preferable when feasible. These equations explain how a local REML approximation can be constructed while accounting for the changed design and covariance structure.

Worked Deletion Calculations

Diagnostics, Alternatives, and Limitations

A useful influence review begins with the model fit, not a mechanical deletion threshold. Refit after removing each subject, or at least each subject flagged by a sensible preliminary screen. Check that the same fixed-effect coding, time scale, covariance structure, and estimation method are used in every comparison. Record convergence warnings and the estimates and standard errors of the quantities of interest. If a deletion fit fails or changes the estimable parameter set, report that limitation rather than treating a missing estimate as zero.

Interpret influence alongside the subject’s observed trajectory and residual diagnostics. A coherent but extreme trajectory may be scientifically important; a data-entry error calls for correction based on the source record, not deletion merely because it changes a coefficient. Conversely, a subject with modest residuals can still have high influence if their observation pattern or covariate values provide unusual leverage. Influence is about the fitted model’s sensitivity, not a judgment of the subject.

Compared with deleting individual observations, deleting a complete subject respects the independent-subject sampling structure and avoids retaining only selected pieces of a person’s trajectory. A one-case-at-a-time refit is also more interpretable than a single global fit statistic: it shows which estimates move and in what direction. Approximate score or Hessian diagnostics can reduce computation for ML screening, but exact refits remain the clearest check. For REML, use reduced-data REML criteria and GLS equations; do not use the ML additive-score identity.

Deletion diagnostics have limitations. A single-subject deletion may miss joint influence from a group of subjects, and conclusions can depend on the chosen model and target parameter. Large changes can reflect weak information or covariance misspecification as well as an unusual subject. If several deletions substantially change a conclusion, describe the sensitivity and consider scientifically justified model checks. Do not choose a preferred model or exclude subjects solely to obtain a desired result.