Trajectory-Level Perturbations Under Repeated Measures

Why perturb a whole trajectory?

In repeated-measures data, an individual contributes an outcome record rather than a collection of unrelated values. A change at one visit can affect estimation at another because the outcome model connects measurements within a subject. In a selection model, dependence can also affect how the observed outcomes inform the probability of the subject’s missingness pattern. A perturbation that represents a plausible change to a trajectory should therefore specify which observed components move and in what relative amounts.

The aim is local: assess how a small, stated change in observed outcomes would alter the fitted parameters. This is a sensitivity diagnostic, not a claim that the perturbed values are more correct, nor a replacement for modelling missing outcomes. As in the earlier material on perturbing observed outcomes without altering missing values, the missing components remain unperturbed. The focus here is to define the direction jointly across visits and account for within-subject dependence in the resulting calculation.

Model and trajectory perturbation

For subject \(i\), let \(Y_i=(Y_{i1},\ldots,Y_{iT})^\mathsf{T}\) denote the full outcome trajectory and \(R_i\) its response indicators, with \(R_{it}=1\) if the outcome at visit \(t\) is observed. Write \(Y_i^{\mathrm{obs}}\) and \(Y_i^{\mathrm{mis}}\) for the observed and missing components. A longitudinal selection model specifies an outcome density \(f_\theta(Y_i)\) and a response mechanism \(g_\psi(R_i\mid Y_i)\). The combined parameter is \(\eta=(\theta^\mathsf{T},\psi^\mathsf{T})^\mathsf{T}\).

The subject’s observed-data likelihood contribution integrates the full-data likelihood over the missing outcomes. The integral is over the components missing under that subject’s response pattern.

Let \(d_i\) be a fixed vector with one entry for each observed visit for subject \(i\). It specifies a trajectory direction: for example, shifting every observed visit equally, shifting only a particular visit, or increasing later visits more than earlier ones. The perturbed observed outcomes are \(Y_i^{\mathrm{obs}}(\omega_i)=Y_i^{\mathrm{obs}}+\omega_i d_i\). Here \(\omega_i\) is a scalar perturbation size; the missing outcomes and the response indicators are held fixed. A perturbation involving several subjects can use a vector \(\omega\), with a specified direction across subjects and visits.

The perturbed observed log likelihood is \(\ell(\eta;\omega)=\sum_i \log L_i(\eta;Y_i^{\mathrm{obs}}(\omega_i),R_i)\), with \(\ell(\eta;0)\) the unperturbed log likelihood. This construction requires a well-defined observed-data likelihood and enough smoothness to differentiate it with respect to \(\eta\) and \(\omega\). In particular, differentiating under the integral must be justified by suitable regularity conditions. The model and response pattern are fixed during the perturbation; only the stated observed outcomes move.

Local curvature and its justification

Let \(\hat\eta\) maximize \(\ell(\eta;0)\). Assume it is an interior, locally unique maximum, the observed information \(I=-\ell_{\eta\eta}(\hat\eta;0)\) is positive definite, and the perturbed maximum \(\hat\eta(\omega)\) varies smoothly near zero. For a vector of perturbation coordinates, define the cross-derivative matrix \(\Delta=\ell_{\eta\omega}(\hat\eta;0)\). Its columns describe how the parameter score changes, to first order, for each specified trajectory perturbation.

To isolate parameter movement, use the likelihood displacement between the perturbed fit and the original estimate evaluated under the same perturbed likelihood:

This quantity is nonnegative because \(\hat\eta(\omega)\) maximizes the perturbed log likelihood. It is zero at \(\omega=0\), and it measures the loss in perturbed log likelihood from holding the parameters at their original estimate rather than refitting. The definition is especially useful for response perturbations, because the log likelihood at fixed parameters can itself change when observed values change.

For completeness, differentiate the score equation \(\ell_\eta(\hat\eta(\omega);\omega)=0\) at \(\omega=0\). Since \(\ell_{\eta\eta}=-I\), the chain rule gives \(-I\,d\hat\eta/d\omega+\Delta=0\), and hence \(d\hat\eta/d\omega=I^{-1}\Delta\). This is the first-order parameter response to the perturbation.

To obtain the curvature, expand the log likelihood near \((\hat\eta,0)\). Let \(h=\hat\eta(\omega)-\hat\eta\). The terms involving \(h\) that affect the displacement through second order are \(-\tfrac12 h^\mathsf{T}Ih+h^\mathsf{T}\Delta\omega\). The direct terms depending only on \(\omega\) occur in both log likelihoods defining \(D\), so they cancel. At the perturbed maximum, \(h=I^{-1}\Delta\omega+o(\|\omega\|)\). Substitution gives

Thus \(C\) is positive semidefinite. It is a local approximation: it describes sufficiently small perturbations around the fitted model, not the effect of a large change to the data.

How within-subject dependence changes the calculation

The information matrix and the cross-derivative are calculated from the joint observed-data likelihood, not by treating each visit as an independent record. Consequently, both within-subject covariance and the selection mechanism can change the curvature. The following Gaussian calculation isolates the covariance effect in a simple special case: two outcomes are observed, missingness is independent of the outcomes, and the mean at each visit is the common parameter \(\beta\). This is a selection-model special case with a response mechanism that contributes no outcome-dependent selection term.

Diagnostics, alternatives, and limitations

A calculated curvature is interpretable only alongside its perturbation definition and scale. Check that each component of \(d_i\) refers to an observed visit, that the response pattern is unchanged, and that the full joint likelihood—not a product that incorrectly assumes independence—was used. For incomplete records, verify that the observed-data integral and its derivatives include the response mechanism and the integration over missing outcomes. If the likelihood is not differentiable or the fitted information matrix is nearly singular, \(I^{-1}\) may be unstable; large curvature can then reflect weak identification or numerical error as well as genuine sensitivity to the chosen direction.

Useful checks include computing derivatives by more than one method, comparing the first-order parameter movement \(I^{-1}\Delta a\) with small refits in both positive and negative directions, and checking that the quadratic approximation improves as the perturbation size decreases. Material disagreement for very small perturbations can indicate an implementation error, an inaccurate Hessian, or failure of the local smoothness assumptions. Agreement at small sizes does not guarantee that the approximation is reliable for large shifts.

Trajectory perturbations answer a different question from several familiar alternatives. Exact case deletion removes a subject’s entire likelihood contribution; a trajectory perturbation preserves the subject and changes specified observed outcomes. Case-weight perturbations rescale a subject’s contribution rather than shifting visits. Covariate perturbations change predictors in the selection equation, not the outcome vector. These diagnostics can complement one another, but their curvature values are not directly comparable unless the perturbation definitions and scales are made commensurate.

A main limitation is scale dependence: multiplying \(d_i\) by a constant multiplies the curvature by the square of that constant. Curvature therefore ranks directions meaningfully only under a stated, comparable scaling. It is also a local, model-dependent diagnostic. It does not establish that a subject is erroneous, identify a causal effect, validate the selection model, or resolve sensitivity to assumptions about unobserved outcomes. A large value signals that the fitted parameters respond strongly to this specified local change under the fitted model.