Identifiability Of Cross-Endpoint Association

Why Cross-Endpoint Association Can Be Hard to Identify

Two endpoints may be associated because individuals differ persistently in both outcomes, because their measurement errors move together, or because both respond to a changing underlying disease state. A joint model can represent these mechanisms separately, but observing correlated endpoints does not by itself reveal which mechanism produced the association. If different parameter values imply nearly the same distribution of observed data, the components are difficult or impossible to identify separately.

This matters in pharmacometrics because the interpretation of a fitted association can affect predictions, uncertainty, and conclusions about how evidence is shared across endpoints. A high estimated association is not, on its own, evidence that any particular covariance mechanism has been identified.

Definition: A parameter is structurally identifiable if distinct values of that parameter cannot produce the same distribution of observable data under the model and design. It is practically identifiable if the available data contain enough information to estimate it with useful precision. A structurally identifiable parameter can still be practically unidentifiable when the likelihood is very flat over a substantial range.

A Covariance Decomposition for Paired Endpoints

Consider two continuous endpoints measured in the same subject \(i\) at occasions \(t=1,\ldots,T\). Write their centered values as the vector \(Y_{it}=(Y_{i1t},Y_{i2t})^\mathsf{T}\). For clarity, suppose the model is

Y_{it}=b_i\mathbf{1}+\gamma S_{it}+\varepsilon_{it}, \qquad \mathbf{1}=(1,1)^\mathsf{T}, \qquad \gamma=(\gamma_1,\gamma_2)^\mathsf{T}.

Here \(b_i\) is a subject-level shared random intercept with variance \(\tau_b^2\). The latent state \(S_{it}\) has mean zero, variance \(\omega^2\), and correlation \(\phi^{|t-s|}\) between occasions \(t\) and \(s\), where \(0\leq\phi<1\). The residual vector \(\varepsilon_{it}\) has covariance matrix \(R\), whose off-diagonal element is \(\sigma_{12}\). Assume \(b_i\), the state process, and the residuals are mutually independent; assume residuals are independent across occasions; and assume the model is jointly Gaussian when using its normal likelihood. These assumptions specify which covariance patterns each component is allowed to produce.

For \(t\ne s\), the cross-occasion covariance matrix is

\operatorname{Cov}(Y_{it},Y_{is}) = \tau_b^2\mathbf{1}\mathbf{1}^\mathsf{T} + \omega^2\phi^{|t-s|}\gamma\gamma^\mathsf{T}.

At the same occasion, residual covariance also contributes:

\operatorname{Cov}(Y_{it}) = \tau_b^2\mathbf{1}\mathbf{1}^\mathsf{T} + \omega^2\gamma\gamma^\mathsf{T} + R.

The key is that the three sources have different covariance signatures only when the design and model make those signatures distinguishable. The shared intercept persists across all occasions. The latent-state contribution changes with the time lag. Residual covariance contributes only within an occasion. If there is only one paired measurement per subject, none of the cross-occasion information is available.

Key takeaway: Cross-endpoint covariance is an observed feature of the data; shared random-effect variance, residual covariance, and latent-state variance are model components proposed to explain it. Estimating those components separately requires observable covariance patterns that distinguish them.

When Components Are Confounded

With one paired measurement per subject and \(\gamma=\mathbf{1}\), the covariance is \((\tau_b^2+\omega^2)\mathbf{1}\mathbf{1}^\mathsf{T}+R\). The two variance components multiply exactly the same matrix. Increasing one while decreasing the other by the same amount leaves the observable covariance unchanged, provided both remain valid variances. This is structural nonidentifiability: more subjects measured once each can estimate the combined contribution more precisely, but cannot separate its two parts.

Even with repeated measurements, the distinction can fail. If the latent state is effectively constant over the observed occasions, its covariance pattern resembles a subject-level random intercept. In the limiting case \(\phi=1\) and \(\gamma=\mathbf{1}\), both components contribute a constant matrix at every lag. If the residual covariance is also weakly constrained and there are few occasions, multiple allocations can remain nearly equally plausible. More parameters do not guarantee more identifiable explanations.

Worked Example: Multiple Decompositions of One Paired Covariance

Suppose a hypothetical data set with one paired measurement per subject has estimated covariance matrix

\widehat{\Sigma} = \begin{pmatrix} 4.0 & 1.8\\ 1.8 & 9.0 \end{pmatrix}.

Assume both the shared random intercept and the latent state load equally on the two endpoints, so each contributes its variance times \(\mathbf{1}\mathbf{1}^\mathsf{T}\). One possible decomposition is \(\tau_b^2=0.8\), \(\omega^2=0.5\), and

R= \begin{pmatrix} 2.7 & 0.5\\ 0.5 & 7.7 \end{pmatrix}.

The resulting diagonal variances are \(0.8+0.5+2.7=4.0\) and \(0.8+0.5+7.7=9.0\); the cross-endpoint covariance is \(0.8+0.5+0.5=1.8\). This residual matrix is valid because its diagonal entries are positive and its determinant is \(2.7(7.7)-0.5^2=20.54>0\).

A different decomposition is \(\tau_b^2=1.0\), \(\omega^2=0.2\), and \(R=\begin{pmatrix}2.8&0.6\\0.6&7.8\end{pmatrix}\). Its diagonal totals are \(1.0+0.2+2.8=4.0\) and \(1.0+0.2+7.8=9.0\), and its cross-covariance total is \(1.0+0.2+0.6=1.8\). Both residual matrices are positive definite. Thus, these observations identify the total covariance matrix, but not a unique allocation among the three mechanisms.

How Repeated Measurements Can Help

Repeated occasions add information because the shared intercept, evolving state, and residual covariance generally behave differently over time. For the model above, the cross-covariance from endpoint 1 at occasion \(t\) to endpoint 2 at occasion \(t+k\), for \(k\geq1\), is

C_k=\tau_b^2+\gamma_1\gamma_2\omega^2\phi^k.

The residual covariance does not appear for \(k\geq1\), under the assumption that residuals are independent across occasions. If there are enough occasions and the state changes measurably over time, several lag covariances can help distinguish a persistent component from a decaying one. This requires the assumptions to be credible: for example, unmodeled serial correlation in the residuals could imitate a slowly changing latent state.

Worked Example: Using Lag Covariances in a Paired-Endpoint Simulation

Consider a hypothetical simulation of 200 subjects, each measured on two endpoints at six occasions. Data are generated with \(\tau_b^2=0.50\), \(\omega^2=0.40\), \(\phi=0.60\), \(\gamma=(1,0.8)^\mathsf{T}\), and residual covariance \(\sigma_{12}=0.18\). Residuals are independent across occasions, and the shared intercept and latent state satisfy the independence assumptions above.

For positive lags, the model-implied endpoint cross-covariance is \(C_k=0.50+0.8(0.40)(0.60)^k=0.50+0.32(0.60)^k\). Therefore, \(C_1=0.692\), \(C_2=0.6152\), and \(C_3=0.5691\). At lag zero, residual covariance is added, giving \(C_0=0.50+0.32+0.18=1.00\).

Suppose the simulated sample produces rounded cross-covariance estimates \( \widehat C_1=0.70\), \( \widehat C_2=0.62\), \( \widehat C_3=0.57\), and \( \widehat C_0=1.04\). Since \(C_1-C_2=A\phi(1-\phi)\) and \(C_2-C_3=A\phi^2(1-\phi)\), where \(A=\gamma_1\gamma_2\omega^2\), their ratio estimates \(\phi\):

\widehat{\phi} = \frac{\widehat C_2-\widehat C_3}{\widehat C_1-\widehat C_2} = \frac{0.62-0.57}{0.70-0.62} = 0.625.

Then \(\widehat A=(\widehat C_1-\widehat C_2)/[\widehat\phi(1-\widehat\phi)]=0.08/[0.625(0.375)]=0.3413\), and \(\widehat{\tau_b^2}=\widehat C_1-\widehat A\widehat\phi=0.70-(0.3413)(0.625)=0.4867\). Finally, the lag-zero estimate gives \(\widehat{\sigma}_{12}=\widehat C_0-\widehat{\tau_b^2}-\widehat A=1.04-0.4867-0.3413=0.2120\). These estimates are close to the generating values, but they do not identify \(\gamma_1\), \(\gamma_2\), and \(\omega^2\) separately: these covariance moments identify their product \(A\) unless additional information or constraints are available. The calculation illustrates what the lag pattern can distinguish, not a guarantee of precise estimates from every six-occasion design.

Likelihood Profiles and Parameter Correlations

A fitted model should be examined for evidence that the data distinguish the covariance components. A likelihood profile for one parameter is formed by fixing it at a sequence of values and, at each value, maximizing the likelihood over the remaining parameters. A flat or gently sloping profile indicates that quite different fixed values can be accommodated with little loss of fit. A long ridge in a joint profile indicates trade-offs between parameters.

Parameter correlations provide a related local diagnostic. A correlation near \(+1\) or \(-1\) between estimated variance components suggests that increasing one can be offset by changing another. Correlations are calculated from the estimated parameter covariance matrix and can depend on parameterization; they are warning signs, not proof of nonidentifiability. They can also be unreliable when estimates lie near a variance boundary or the likelihood is far from quadratic.

Worked Example: Reading a Nearly Flat Profile

Suppose a hypothetical repeated-endpoint fit profiles the shared variance \(\tau_b^2\) while reoptimizing the latent-state variance and other parameters. Along the best-fitting ridge, the pairs \((\tau_b^2,\omega^2)\) are \((0.20,0.80)\), \((0.40,0.60)\), \((0.60,0.40)\), and \((0.80,0.20)\). Their corresponding increases in \(-2\) log likelihood above the minimum are 0.7, 0.1, 0.0, and 0.6.

All four pairs have the same sum, \(\tau_b^2+\omega^2=1.00\), which is the combination that would determine their joint contribution if the two effects had identical covariance patterns. The small profile changes show that, in this hypothetical data set, the observed departures from that exact confounding provide little information about the allocation. If regular large-sample likelihood assumptions hold and the parameter is an interior scalar parameter, a 95% profile-likelihood interval uses an increase in \(-2\) log likelihood of about 3.84. The displayed values are all below that cutoff, so this profile does not sharply exclude the shown allocations. Variance boundaries or other irregularities can invalidate that simple cutoff, so it must not be applied automatically.

Suppose the estimated correlation between \(\tau_b^2\) and \(\omega^2\) is \(-0.98\). This strong negative correlation is consistent with the ridge: larger shared variance tends to accompany smaller state variance. It does not mean the two components are negatively associated in the population; it describes uncertainty in their estimates under this model.

Assumptions, Diagnostics, and Remedies

A covariance decomposition is informative only if its structural assumptions are defensible. In this example, residual independence across occasions is what prevents residual covariance from contributing to positive-lag covariance. The assumed state correlation \(\phi^k\) creates a particular decay pattern. If either assumption is wrong, an apparently distinctive pattern may be attributed to the wrong component.

Useful checks include comparing profiles for the variance components, examining their estimated correlations, and checking whether the fitted covariance pattern follows the observed same-occasion and lagged cross-covariances. Warning signs include a profile that stays nearly flat over broad ranges, a parameter correlation close to a boundary, estimates that change substantially when a covariance component is fixed or removed, or variance estimates repeatedly landing at zero. These signs call for investigating identifiability and model assumptions, not simply accepting the optimizer's reported point estimates.

Design can help when it produces the information needed to separate covariance signatures. More repeated occasions provide more lag patterns; a wider range of intervals can help reveal the rate of state decay; and measurements that distinguish persistent subject differences from time-varying responses can add useful information. But adding subjects with the same one-time paired measurement cannot resolve exact structural confounding. Fixing a parameter, constraining loadings, or dropping a component can yield an estimable model, but the resulting separation comes from the modeling assumption, not from evidence in the data. Sensitivity analyses should make that dependence clear.

The main alternative is to model endpoint dependence with a different structure, such as a different shared-effect specification or a copula. An alternative can represent dependence more flexibly, but flexibility does not automatically identify its parameters; its observable implications still need to be distinguished by the data and design. Separate endpoint fits avoid estimating a cross-endpoint covariance decomposition, but they do not estimate a joint model of association. Choose among these approaches based on the scientific question and the information the measurements actually contain.

Common pitfall: “The model converged and returned separate estimates, so each covariance component must be identified.” Convergence only says that an algorithm found a numerical solution under its settings. A nearly flat likelihood can have a reported optimum while allowing many materially different component values. Inspect profiles and parameter correlations, and report when the data identify only a combined covariance contribution.

Check Your Understanding

Answer each question, then compare with the solutions below.

With one paired observation per subject and equal loadings, which combination of shared-intercept and latent-state variances appears in the covariance?
Under the stated model, why does residual covariance not contribute to \(C_2\), the cross-endpoint covariance at lag two?
A likelihood profile remains nearly flat as one variance increases and another decreases. What does this suggest, and what does it not establish?
Can increasing the number of subjects, while retaining only one paired measurement per subject, resolve exact confounding between two components with identical covariance signatures?

Solutions. (1) The covariance contains \((\tau_b^2+\omega^2)\mathbf{1}\mathbf{1}^\mathsf{T}\), so it identifies the sum rather than the two variances separately. (2) Residuals are assumed independent across occasions, so their covariance is zero between distinct occasions; \(C_2=\tau_b^2+\gamma_1\gamma_2\omega^2\phi^2\). (3) It suggests weak information about the separate values and a trade-off between them. It does not by itself prove structural nonidentifiability; the exact model and design determine whether the profile is truly flat or merely imprecise. (4) No. More subjects can improve precision for the observable covariance, but identical signatures leave the component allocation structurally confounded.