One Unobserved State, Several Observed Processes
A person’s underlying health may affect more than one feature of follow-up. For example, worsening health could be associated with a higher biomarker value, an increased chance of leaving scheduled follow-up, and a greater risk of a clinical event. The underlying state need not be measured perfectly—or measured at all—for these observed processes to be related.
This tutorial focuses on that common-source explanation for dependence. It is different from saying that a biomarker directly causes dropout or an event. The central idea is that an unobserved health state can help explain why these outcomes move together. The earlier material on informative dropout establishes why follow-up loss can depend on health; here the emphasis is on how the same underlying health state can also connect repeated measurements with event risk.
Let \(U_i\) represent participant \(i\)’s latent health state. For a simple illustration, higher values of \(U_i\) will mean poorer underlying health. Let \(Y_i(t)\) be a longitudinal measurement at time \(t\), let \(D_i\) indicate dropout by a specified follow-up time, and let \(E_i\) indicate a clinical event by that time. These are distinct outcomes: dropout is loss of scheduled observation, whereas an event is a clinical outcome. In the examples, assume event status can still be ascertained even if scheduled measurements stop.
Conditional Independence Can Produce Marginal Dependence
A useful starting point is a simplifying assumption: once latent health \(U_i\) is known, the measurement, dropout, and event outcome are independent of one another. This does not mean they are independent in the population. Participants with different values of \(U_i\) have different expected measurements and different probabilities of dropout and events, so combining those health groups can create associations between the observed outcomes.
The key calculation uses the law of total covariance. For two variables \(A\) and \(B\), their covariance can be decomposed into an average within-state covariance and a covariance between their conditional means:
If \(A\) and \(B\) are conditionally independent given \(U\), the first term is zero. The second term can still be nonzero: it is positive when the conditional means tend to rise together across levels of \(U\), and negative when one tends to rise as the other falls. Thus, conditional independence does not imply marginal independence.
For a binary latent state, this result has a particularly clear form. Let \(U=1\) denote poorer health, with \(P(U=1)=p\). Write \(m_1=E(A\mid U=1)\), \(m_0=E(A\mid U=0)\), and define \(q_1,q_0\) similarly for \(B\). If \(A\) and \(B\) are conditionally independent given \(U\), then:
The factor \(p(1-p)\) is positive unless one health group is absent. If both outcomes have higher conditional means in the poorer-health group, their marginal covariance is positive—even though their covariance within each health group is zero.
Worked Example: A Shared State Links Biomarker, Dropout, and Event
Consider a hypothetical population in which 40% of participants have a poorer latent health state, \(U=1\), and 60% have \(U=0\). At a planned visit, suppose the mean biomarker value is 8 units for \(U=1\) and 4 units for \(U=0\). By one year, suppose the dropout probabilities are 0.50 and 0.10, respectively, and the clinical-event probabilities are 0.30 and 0.05. Assume the biomarker, dropout indicator, and event indicator are conditionally independent given \(U\).
Let \(Y\) be the biomarker value and \(D\) the indicator of dropout by one year. First, their marginal means are:
By conditional independence, \(E(YD\mid U)\) is the product of the conditional means. Therefore:
The covariance is positive: participants with higher biomarker values tend, in this population, to be more likely to drop out. This association arises from the shared health state in the example; it does not require a direct effect of the biomarker on dropout.
For the event indicator \(E\), its marginal probability is \(0.4(0.30)+0.6(0.05)=0.15\). The same calculation gives:
Likewise, the dropout-event covariance is:
All three pairwise covariances are positive in this hypothetical example. Each number is a population quantity implied by the specified probabilities and means, not an estimate from observed study data.
A Shared State Can Affect Repeated Measurements and Risks
The binary illustration is easy to calculate, but health can vary on a continuum. One simple way to represent repeated measurements is to let a participant’s latent state shift the expected measurement at each visit:
Here, \(\mu(t)\) describes the mean pattern over time when \(U_i=0\), \(a(t)\) describes how the latent state changes the expected measurement, and \(\varepsilon_i(t)\) is measurement-level variation. For example, if higher biomarker values indicate poorer health, \(a(t)\) may be positive. This equation is a conceptual model, not a claim that every biomarker follows a straight-line pattern.
Dropout and event risk can also depend on \(U_i\). One possible representation uses hazard functions, which describe the instantaneous event rate at time \(t\) among people still at risk immediately before \(t\):
In this illustration, \(\lambda_{D0}(t)\) and \(\lambda_{E0}(t)\) are baseline hazards, and \(\gamma\) and \(\eta\) describe how the latent state is associated with dropout and event hazards. If poorer health means \(U_i\) is larger, positive values of \(\gamma\) and \(\eta\) represent higher hazards for dropout and events. A hazard ratio compares instantaneous rates; it is not itself a probability of an event.
Worked Example: Interpreting Shared-State Hazard Effects
Suppose \(U_i\) is measured in standard-deviation units, with larger values indicating poorer latent health. A hypothetical model has \(\gamma=0.70\) for dropout and \(\eta=0.90\) for the event. The dropout hazard ratio for a one-unit increase in \(U_i\) is:
At the same time \(t\), the event hazard ratio for a one-unit increase in latent health burden is:
Thus, under this hypothetical model, a one-standard-deviation increase in latent health burden is associated with about 2.014 times the dropout hazard and 2.460 times the event hazard, conditional on the model. These are rate comparisons, not statements that the dropout or event probabilities increase by those factors. Nor do they establish that changing a participant’s health state would cause those changes.
The same latent state can also make measurements at different visits correlated. Consider two measurements \(Y_1=5+2U+\varepsilon_1\) and \(Y_2=6+3U+\varepsilon_2\), where \(U\) has variance 1 and the independent measurement errors have variances 1 and 4. Assume \(U\), \(\varepsilon_1\), and \(\varepsilon_2\) are mutually independent. Then:
The variances are \(\operatorname{Var}(Y_1)=2^2(1)+1=5\) and \(\operatorname{Var}(Y_2)=3^2(1)+4=13\), so the correlation is \(6/\sqrt{5(13)}\approx0.744\). Given \(U\), the two measurements are independent under the assumed independent errors; across participants, they are positively correlated because they share \(U\).
Worked Example: Checking the Direction of Dependence
Suppose the latent state still has \(P(U=1)=0.40\), but poorer health is associated with a lower mean biomarker value: \(E(Y\mid U=1)=3\), versus \(E(Y\mid U=0)=5\). Dropout probabilities remain 0.50 and 0.10. Assume conditional independence given \(U\). The binary-state covariance formula gives:
This covariance is negative because poorer health predicts a lower biomarker value but a higher dropout probability. A shared latent state does not automatically imply positive dependence; the direction depends on how each process relates to that state. The result also illustrates why the biomarker’s clinical interpretation—whether larger or smaller values represent poorer health—must be stated.
Assumptions, Alternatives, and Warning Signs
The shared-state explanation is most useful when its assumptions are made explicit. At minimum, the analyst needs a meaningful definition of the latent health construct, a defensible account of how it relates to each observed process, and a clear time scale for measurements, dropout, and events. The simple conditional-independence version additionally assumes no remaining association among those processes after conditioning on \(U_i\). Real data may violate that assumption: for example, a measured biomarker may itself help determine clinical decisions that affect follow-up.
The latent state is not directly observed, so its role cannot be confirmed by a single observed association. Diagnostics are therefore about whether the proposed model is compatible with observed data and study knowledge, not whether an unobserved state has been “proven.” Useful checks include:
- Compare observed patterns over time and across follow-up histories with patterns predicted by the model.
- Check whether the assumed direction of association is clinically plausible for each measurement and process.
- Assess whether conclusions change under reasonable alternative distributions or strengths of latent-state effects.
- Look for remaining association among measurements, dropout, or events that the proposed shared state does not account for.
A model can fit observed patterns yet still describe the latent state poorly. Several different unobserved-state specifications may imply similar observed data, particularly when measurements are sparse or dropout is common. Consequently, parameters describing \(U_i\) can depend strongly on modeling assumptions. A shared latent state is a structured explanation, not a directly measured fact.
| Approach | What it represents | Important limitation |
|---|---|---|
| Analyze each process separately | Models a longitudinal outcome, dropout, or event risk on its own. | May not represent dependence between processes that share unobserved health. |
| Adjust for observed health measures | Uses measured covariates to account for recorded differences in health. | Cannot guarantee that unmeasured health differences have been captured. |
| Use a shared latent-state representation | Allows an unobserved state to influence multiple processes. | Requires assumptions about a quantity that is not directly observed; alternative specifications may fit similarly. |
These approaches answer different modeling needs; none is automatically best in every setting. A shared-state account is especially relevant when there is a substantive reason to expect a common underlying health dimension and the analysis must represent more than one process. It does not, by itself, establish causal effects or resolve every issue created when observation depends on health.
Common pitfall: “The biomarker and dropout are associated, so the biomarker must cause dropout.” That conclusion does not follow. In the examples, both outcomes were associated with the same latent health state, and the biomarker had no direct effect on dropout in the assumed model. The observed association alone cannot distinguish that explanation from other possible structures.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- If two outcomes are independent conditional on latent health, must they be independent in the population? Explain.
- In the binary-state covariance formula, what happens to the sign of the covariance if one outcome’s conditional mean increases with poorer health and the other’s decreases?
- A one-unit increase in a latent state has a modeled event hazard ratio of \(\exp(0.5)\). What is the approximate hazard ratio, and what does it compare?
- Why does an association between biomarker values and dropout not by itself show that biomarker values cause dropout?
- Name one assumption of the simple shared-state illustration and one limitation of using it in practice.
Solutions. (1) No. Conditional independence removes the within-state covariance, but the conditional means can vary together across health states, making the covariance between those means nonzero. (2) The covariance is negative, because the two conditional-mean differences have opposite signs; the factor \(p(1-p)\) is positive when both states occur. (3) \(\exp(0.5)\approx1.649\). It compares instantaneous event hazards at the same time for two people whose latent states differ by one unit, under the model; it is not a probability ratio. (4) A common unobserved health state could affect both the biomarker and dropout, producing association without a direct causal effect of the biomarker on dropout. (5) One assumption is conditional independence of the processes given latent health. One limitation is that latent health is not directly observed, so its specification and estimated effects can depend on modeling assumptions.