Conditional Independence Assumptions In Marker Panels

Why Conditional Independence Matters

A panel of markers contains several kinds of dependence. Measurements from different markers may move together persistently within a person, may share short-term errors at the same visit, or may each be associated with that person’s event time. A joint model must say which sources of dependence it represents. Conditional independence assumptions make this specification explicit: they state which parts of the data are independent after conditioning on the subject-specific latent quantities and observed covariates.

The central distinction is between the marker measurements themselves and the latent marker trajectories they inform. In a shared-random-effects model, an event process can depend on those latent trajectories while remaining conditionally independent of the marker measurement errors. This does not say that marker measurements and events are independent in the population. Their marginal association is generally induced by their dependence on the same random effects.

Definition: Let $Y_i$ denote all marker measurements for subject $i$, $T_i$ the event time, $b_i$ the vector of that subject’s marker random effects, and $X_i$ the observed covariates. The shared-random-effects conditional independence assumption is $$ Y_i \perp T_i \mid b_i,X_i. $$ It means that, conditional on $b_i$ and $X_i$, the marker measurement process and event-time process contribute separate probability factors. It does not require the markers to be independent of one another, nor does it imply marginal independence of $Y_i$ and $T_i$.

A Model That Makes the Assumptions Visible

Suppose there are $K$ markers. Write $Y_{ikj}$ for the measurement of marker $k$ on subject $i$ at visit $j$, and let $b_i=(b_{i1},\ldots,b_{iK})^\mathsf{T}$ collect marker-specific random effects. A representative longitudinal model is

Y_{ikj}=m_{ik}(t_{ikj};b_{ik})+\epsilon_{ikj}, \qquad m_{ik}(t;b_{ik})=x_{ik}(t)^\mathsf{T}\beta_k+z_{ik}(t)^\mathsf{T}b_{ik}.

Here $m_{ik}$ is the latent mean trajectory, $\beta_k$ contains fixed effects, and $\epsilon_{ikj}$ is measurement-level error. A common specification is $b_i\sim N(0,G)$, where $G$ can include covariances between markers’ random effects. Thus, conditioning on the full vector $b_i$ removes dependence due to those random effects; it does not make the random effects independent in the population.

Residual assumptions are a separate part of the model. For example, residuals can be independent across visits conditional on $b_i,X_i$, while residuals for different markers observed at the same visit have a non-diagonal covariance matrix $R$. In that case, markers at a visit are not conditionally independent given $b_i,X_i$. If $R$ is diagonal as well, then the marker errors at that visit are conditionally independent under a Gaussian residual model. The model’s longitudinal likelihood must respect whichever structure is specified.

A typical event model specifies a hazard conditional on the latent marker trajectories or their random effects, such as

\lambda_i(t\mid b_i,X_i) = \lambda_0(t)\exp\left\{\gamma^\mathsf{T}w_i+ \sum_{k=1}^{K}\alpha_k m_{ik}(t;b_{ik})\right\}.

In this expression, $\lambda_0(t)$ is a baseline hazard, $w_i$ are event-model covariates, and $\alpha_k$ link the latent value of marker $k$ to the hazard. The key assumption is that the event process depends on the marker information through the specified latent quantities and covariates, not through additional unmodelled dependence on the measurement errors. Other event links may use slopes, contrasts, or cumulative marker values; the conditional-independence question is the same.

Given $b_i,X_i$, the joint contribution therefore factors as

p(Y_i,T_i\mid b_i,X_i) = p(Y_i\mid b_i,X_i)\,p(T_i\mid b_i,X_i).

This factorization does not necessarily split the first factor into one independent density per marker. If same-visit residuals are correlated, the longitudinal contribution must retain that covariance. If event times are right-censored, the event factor uses the observed event or censoring contribution; interpreting that factor also requires an appropriate conditional censoring assumption.

Worked Example: Factoring a Conditional Contribution

Consider two markers whose measurement errors are conditionally independent given $b_i,X_i$, and an event model satisfying the shared-random-effects assumption. For one hypothetical subject, suppose the two conditional marker likelihood contributions are $0.30$ and $0.25$. Let $b_{i1}=1$, $b_{i2}=-0.5$, $\alpha_1=0.4$, $\alpha_2=0.2$, and let the constant baseline hazard be $0.05$ per month. Calculate the conditional contribution for four months without an event.

The log-hazard contribution from the markers is

\alpha_1b_{i1}+\alpha_2b_{i2} =(0.4)(1)+(0.2)(-0.5)=0.30.

Thus the hazard is $0.05\exp(0.30)=0.06749$ per month. Under a constant hazard, the probability of surviving four months conditional on these random effects is

S_i(4\mid b_i)=\exp\{-4(0.06749)\}=0.7634.

The conditional joint contribution is the product of the two marker contributions and the event-free contribution:

$$ (0.30)(0.25)(0.7634)=0.05726. $$

This multiplication is justified here because the example assumes independent marker errors conditional on $b_i,X_i$, as well as conditional independence of the marker process and event process. If the two marker errors had within-visit covariance, the marker factor would instead be a joint bivariate contribution; multiplying the two separate marker factors would not be justified.

Persistent Dependence and Visit-Level Dependence

Correlated random effects and correlated residuals describe different patterns. Correlation in $G$ represents persistent subject-level association: a person with a higher-than-average latent trajectory for one marker may tend to have a higher-than-average trajectory for another. Residual covariance represents additional association between measurements after accounting for those trajectories, often at the same visit. Either source may be present, both may be present, or neither may be supported by the modeling context.

A short derivation shows how these assumptions determine marginal marker association. For simplicity, take two markers measured once at a common time and write $Y_1=b_1+\epsilon_1$, $Y_2=b_2+\epsilon_2$, with mean-zero random effects and errors. Assume the errors are independent of the random effects. Expanding the covariance gives

\begin{aligned} \operatorname{Cov}(Y_1,Y_2) &=\operatorname{Cov}(b_1+\epsilon_1,b_2+\epsilon_2)\\ &=\operatorname{Cov}(b_1,b_2)+\operatorname{Cov}(\epsilon_1,\epsilon_2), \end{aligned}

because the cross-covariances between random effects and errors are zero under the stated assumption. Consequently, setting the residual covariance to zero does not eliminate marginal association if the random effects are correlated. Conversely, independent random effects do not guarantee marginally independent measurements if their residuals are correlated.

Worked Example: Separating Two Sources of Covariance

Suppose two hypothetical markers have random-effect variances $0.80$ and $1.25$, random-effect covariance $0.60$, residual variances $0.25$ and $0.36$, and same-visit residual covariance $0.20$. Assume the residuals are independent of the random effects. Find the marginal covariance and correlation of the measurements, and the conditional residual correlation.

The marginal covariance is the sum of the two covariance components:

\operatorname{Cov}(Y_1,Y_2)=0.60+0.20=0.80.

The marginal variances are $0.80+0.25=1.05$ and $1.25+0.36=1.61$. Hence the marginal correlation is

\operatorname{Corr}(Y_1,Y_2) = \frac{0.80}{\sqrt{(1.05)(1.61)}} = \frac{0.80}{1.300} \approx 0.615.

After conditioning on the random effects, the remaining correlation is the residual correlation:

\operatorname{Corr}(\epsilon_1,\epsilon_2) = \frac{0.20}{\sqrt{(0.25)(0.36)}} = \frac{0.20}{0.30} \approx 0.667.

The measurements are not conditionally independent given the random effects in this example, because the residual covariance is nonzero. A model that set that covariance to zero would omit a specified source of same-visit dependence, even though it retained the correlated random effects.

Why Conditional Independence Does Not Mean Marginal Independence

Integrating over the random effects generally reintroduces dependence. The event distribution varies across subjects with $b_i$, and the marker trajectories also vary with $b_i$. As a result, marker observations carry information about event risk, even when their errors are conditionally independent of the event process. This is the mechanism that allows a shared-random-effects model to represent marker–event association.

Worked Example: Dependence Induced by a Shared Latent Effect

Consider a hypothetical binary latent effect $B$, with $P(B=1)=P(B=0)=0.5$. Let a binary marker $Y$ equal $B$, and let $D$ indicate an event by a fixed time. Suppose $P(D=1\mid B=1)=0.8$ and $P(D=1\mid B=0)=0.2$. Conditional on $B$, the event is independent of the marker’s measurement process; here the marker is a perfect indicator of $B$.

The marginal event probability is $P(D=1)=(0.5)(0.8)+(0.5)(0.2)=0.5$, and $P(Y=1)=0.5$. Their joint probability is

$$ P(Y=1,D=1)=(0.5)(0.8)=0.40. $$

If $Y$ and $D$ were marginally independent, that joint probability would be $(0.5)(0.5)=0.25$. Instead, their indicator covariance is $0.40-0.25=0.15$. Thus conditional independence given the shared latent effect is compatible with positive marginal dependence. It is precisely the shared latent effect, not direct dependence of measurement error on the event, that produces this association in the example.

Assumptions, Diagnostics, and Alternatives

The assumptions should be stated at the level of the model actually fitted. A useful checklist is:

Random-effects distribution: specify the distribution of the full vector $b_i$, including which cross-marker covariances are allowed. A diagonal $G$ assumes no random-effect covariance; it is not a consequence of conditioning on $b_i$.
Residual structure: specify whether marker errors are independent across markers and visits, or whether covariance is allowed within a visit or in other defined blocks.
Event link: state which latent marker features enter the hazard. Conditional independence is questionable if event risk also depends on an unmodeled marker feature or directly on measurement error.
Conditional separation: justify why, given the modeled latent effects and covariates, remaining marker errors do not carry additional information about event times.
Censoring: where censoring is present, state the conditional independence assumption used for its contribution. Dependence between censoring and event time that remains after conditioning can bias the event component.

Diagnostics can reveal tension with these assumptions, though they cannot prove conditional independence. Persistent cross-marker patterns left unexplained by the fitted random effects suggest that the latent covariance structure or mean trajectories may be inadequate. Residual cross-marker association at visits suggests that an independent-residual specification may be too restrictive. A remaining association between marker residual patterns and event outcomes can indicate an omitted shared latent feature, an inadequate event link, or another source of dependence. These are prompts to examine the model, not automatic proof of a particular alternative.

The main alternatives represent dependence in different places. A model with independent marker random effects and independent residuals is parsimonious but rules out both persistent and same-visit cross-marker dependence. A model with correlated random effects but independent residuals captures persistent association only. Adding within-visit residual covariance can represent short-term association as well. A model with marker-specific random effects but no shared random effects in the event hazard does not, by itself, connect latent marker trajectories to event risk. The appropriate choice depends on the scientific structure and the data; a more flexible covariance specification is not automatically better identified or more reliable.

Conditional independence is a modeling approximation, not a biological law. It can fail if an unmeasured process affects both marker measurement errors and event risk, if the event model omits relevant marker history, or if marker observations are influenced by health changes not represented in the latent trajectory. Furthermore, selection into the observed risk set can alter observed marker associations. Conclusions about marker–event relationships should therefore be understood as conditional on the chosen trajectory, covariance, and hazard specifications.

Common pitfall: “The markers are independent given the random effects, so their residual covariance can be ignored.” This is incorrect unless the model explicitly assumes independent residuals. Conditioning removes dependence explained by the random effects, but same-visit residual covariance may remain. Check the residual structure separately from the random-effects covariance.

Check Your Understanding

Answer each question using the distinction between latent effects, measurement errors, and event processes.

If $Y_i\perp T_i\mid b_i,X_i$, must $Y_i$ and $T_i$ be marginally independent?
Two markers have random-effect covariance $0.4$ and residual covariance $0$. What is their marginal covariance in the simple additive model $Y_k=b_k+\epsilon_k$, assuming random effects and errors are mutually independent?
If the same-visit residual covariance is nonzero, may the longitudinal likelihood be factored into separate marker densities conditional on $b_i,X_i$?
Name one kind of unexplained dependence that could indicate a problem with the conditional-independence assumptions.

Solutions. (1) No. Integrating over $b_i$ can induce marginal dependence because both the marker distribution and event distribution vary with $b_i$. (2) The marginal covariance is $0.4+0=0.4$, by covariance decomposition. (3) Not into independent marker densities: the same-visit joint marker contribution must retain the nonzero residual covariance. (4) Examples include residual cross-marker association at visits, or an association between marker residual patterns and event outcomes after conditioning on the modeled random effects and covariates. Either suggests that the specified residual or event-link structure may be inadequate.