Conditional Versus Marginal Marker Associations

Why Conditional and Marginal Associations Differ

When biomarkers are correlated, a hazard coefficient for one marker depends on what else is included in the hazard model. A coefficient from a model containing several markers describes the association of one marker with the event hazard holding the other modeled markers fixed. A model containing only that marker describes a different quantity: its association when the other markers have been left out and their variation is not explicitly controlled.

Neither quantity is automatically the right one for every scientific question. The conditional association can help describe the contribution of a marker beyond the other markers in the model. The single-marker association can describe how that marker relates to risk in the population without adjustment for the rest of the panel. They need not have the same size or even the same direction.

The Conditional Hazard Model

Let \(q_{ik}(t)\) denote the value of the modeled association summary for marker \(k\) in subject \(i\) at time \(t\). It may be a current marker value or another specified summary of its longitudinal trajectory. Let \(x_i\) denote covariates included in the hazard model. A simultaneous-marker proportional hazards model is

Here \(h_0(t)\) is the baseline hazard, and \(\beta_k\) is the conditional log-hazard coefficient for marker \(k\). The model says that, at a given time, the hazard is proportional to the exponential of the modeled marker summaries and covariates. In a joint model, the marker summaries may be latent quantities informed by the longitudinal measurements; the hazard model is interpreted conditional on those quantities, with their uncertainty handled by the joint analysis.

The coefficient is conditional on the specified marker panel, not on every biomarker that could have been measured. Its interpretation also depends on the chosen association summary and scale. For example, a coefficient for a current-value summary is not a coefficient for a slope or cumulative burden.

The model assumes the stated hazard form is appropriate over the time range being analyzed. It also relies on the joint longitudinal-survival model adequately representing marker trajectories and their relationship with event times, and on the event and censoring processes being handled under suitable assumptions. In particular, censoring should be independent of the event process conditional on the modeled information, or the censoring mechanism must be modeled appropriately. Estimating a distinct conditional association also requires enough information to distinguish the markers’ contributions; severe lack of marker variation or overlap can make that difficult.

What “Marginal” Means Here

A single-marker analysis may include marker 1 but omit marker 2. Its hazard is not generally obtained simply by deleting \(\beta_2q_{i2}(t)\) from the simultaneous model. If marker 2 is associated with marker 1, or if marker 2 predicts the event, then omitting it leaves its variation mixed into the observed association for marker 1.

More formally, suppose the conditional hazard depends on marker 1 and on other marker information \(U\), which may include the trajectory of marker 2. Among subjects with marker 1 summary \(q\) who are still at risk at time \(t\), the hazard averaged over the omitted information is

The distribution \(f(u\mid q,T_i\ge t)\) is the distribution of the omitted marker information among the people still at risk, not necessarily its distribution at study entry. Subjects with marker values associated with higher hazards tend to experience events earlier, changing the composition of the risk set over time.

This equation follows from averaging the instantaneous event probability over the omitted information among those still at risk. For a short interval of length \(dt\), the conditional event probability is approximately \(h(t\mid q,u)dt\). Averaging over \(u\) gives approximately \(dt\int h(t\mid q,u)f(u\mid q,T_i\ge t)du\); dividing by \(dt\) yields the marginal hazard.

As a result, a marginal hazard need not retain the proportional hazards form, even when the conditional model does. Its relationship with marker 1 can vary over time as the risk-set distribution changes. A coefficient from a fitted single-marker proportional hazards model is therefore generally a model-dependent summary of that marginal relationship, rather than a universal transformation of the conditional coefficient.

Correlation Can Also Reverse the Marginal Direction

A marker’s conditional association can be positive while its unadjusted association is negative. This can occur when the marker is positively correlated with another marker that has a sufficiently strong protective conditional association. The next example uses the same entry-time averaging logic, so it does not assume that the marginal hazard ratio remains constant after risk-set selection.

A Continuous-Marker Calculation

For continuous markers, a familiar correlation-based calculation can describe the marginal association at entry under specific assumptions. Suppose \(M_1\) and \(M_2\) are standardized, jointly normal, and have correlation \(\rho\). Suppose the conditional hazard is \(h_0(t)\exp(\beta_1M_1+\beta_2M_2)\), and consider the instantaneous hazard at entry, before event-related selection changes their distribution.

Under joint normality, \(M_2\mid M_1=m\) is normal with mean \(\rho m\) and variance \(1-\rho^2\). For a normal random variable \(Z\sim N(\mu,\sigma^2)\), \(E[\exp(aZ)]=\exp(a\mu+a^2\sigma^2/2)\). Applying this result gives

The second term in the exponent does not depend on \(m\). Consequently, the ratio of the entry-time marginal hazards at \(M_1=m+1\) and \(M_1=m\) is \(\exp(\beta_1+\rho\beta_2)\). This calculation relies on the stated normality and entry-time conditions; it is not a general formula for a fitted marginal coefficient during follow-up.

How to Compare the Modeling Choices

The main approaches answer different questions. A simultaneous-marker model estimates conditional associations for the included markers. A single-marker joint model estimates an association without explicitly adjusting the hazard for the other panel markers. A composite marker, such as a weighted burden, instead defines a combined predictor; its coefficient is about that specified combination, not the separate contribution of each marker.

A useful analysis begins by stating the target interpretation, then fitting a model whose included markers and association summaries match that target. Comparing single-marker and simultaneous-marker estimates can be informative, but their difference is not, in general, a clean numerical measure of confounding or of a marker’s “share” of the association. In hazard models, averaging over omitted predictors and changing risk sets complicate such decompositions.

Diagnostics, Limitations, and a Common Mistake

When conditional and marginal estimates differ, first check that the comparison is meaningful: the models should use comparable populations, time scales, marker summaries, and covariate adjustments. Confirm the marker units and whether they were scaled, since a coefficient per raw unit cannot be compared directly with one per standard deviation. Examine uncertainty intervals as well as point estimates; a change in a coefficient alone does not establish that the corresponding associations differ precisely.

A large change in estimate after adding another marker may reflect real correlation and different estimands, but it can also signal that the data provide limited information to separate the markers. Look for very high marker correlation, sparse combinations of marker values, unstable estimates across reasonable model specifications, and wide uncertainty intervals. Also consider whether the conditional proportional hazards specification is plausible. If the marginal risk-set association changes over time, a constant-coefficient single-marker model may summarize that pattern imperfectly.

Common mistake: “The simultaneous model gives a smaller coefficient for marker 1, so marker 2 explained exactly the amount by which marker 1’s effect was inflated.” This reasoning treats hazard coefficients like coefficients in a simple linear decomposition. The correction is that the models generally target different conditional and marginal quantities; risk-set selection and hazard-ratio non-collapsibility can change the relationship. Report each estimate with its model, scale, and intended interpretation rather than treating their difference as a universal adjustment formula.

These approaches have limits. A conditional coefficient is conditional only on the markers and covariates actually modeled, and it can be imprecise when correlated markers carry similar information. A marginal association can depend on the marker distribution in the population and on follow-up time. Neither coefficient alone establishes causality: causal interpretation requires additional design and assumptions beyond a correctly fitted joint association model.