State Aggregation and Hidden Heterogeneity

Why a Broad State Can Hide Important Differences

A multi-state model begins with a scientific decision: what situations count as distinct states? The definition must be useful, but it can also be too broad. For example, “hospitalized” might include both ward-only care and intensive care, while “in remission” might include a first remission and a remission after relapse. Each label describes a current situation, yet the people grouped under it may have different prior histories and different prospects.

The consequence is not merely loss of detail in a diagram. If the combined situations have different transition patterns, the apparent transition rate out of the broad state can depend on the mix of situations currently represented there. That mix can change over time or differ between groups. A single rate attached to the broad label may therefore conceal clinically relevant variation.

Aggregation does not automatically make a model unsuitable. It may be reasonable when the distinctions are irrelevant to the scientific question, cannot be measured reliably, or would make the model impractically complex. The key is to understand what the broad label combines and what that combination means for interpretation.

How Mixtures Produce an Aggregate Transition Rate

Suppose a broad state \(G\) combines two subgroups, \(A\) and \(B\). For simplicity, consider one possible transition out of \(G\), with subgroup-specific instantaneous rates \(\lambda_A\) and \(\lambda_B\). An instantaneous rate describes the event rate at a moment among individuals who are still in the relevant subgroup and state. Let \(w_A(t)\) be the proportion of those currently in \(G\) who belong to \(A\); then \(w_B(t)=1-w_A(t)\).

At time \(t\), the aggregate rate is the weighted average of the subgroup rates:

This expression is a composition-weighted rate, not necessarily a fixed property of the broad state. If \(A\) and \(B\) have different rates, their proportions among those who remain in \(G\) can change as events occur. In that case, \(\lambda_G(t)\) changes even if each subgroup’s own rate stays constant.

The changing weights can be derived directly. Suppose entry to \(G\) occurs at time 0, everyone is initially in \(G\), the initial proportion in \(A\) is \(w_A(0)\), and each subgroup has a constant rate \(\lambda_A\) or \(\lambda_B\) of leaving \(G\). The proportions still in each subgroup at time \(t\), before normalizing, are proportional to \(w_A(0)e^{-\lambda_A t}\) and \(w_B(0)e^{-\lambda_B t}\). Thus:

This follows because \(e^{-\lambda_A t}\) and \(e^{-\lambda_B t}\) are the subgroup-specific probabilities of remaining in \(G\) to time \(t\) under the stated constant-rate model. Normalizing the two remaining subgroup masses gives the displayed conditional proportion. If \(A\) has the higher rate, members of \(A\) tend to leave sooner, so the remaining broad state becomes increasingly weighted toward \(B\).

With competing types of exit, the same principle applies to the rate of a particular transition: weight each subgroup’s transition-specific rate by its current share of the broad state. The weights then reflect survival in the state against all exits, not just the transition of interest. These calculations rely on having well-defined subgroup membership and appropriate subgroup-specific rates; they do not make a broad state homogeneous by themselves.

Broad Clinical Labels and Different Risks

A broad health state can combine situations with different probabilities of particular outcomes, even over a fixed follow-up interval. A useful analysis therefore asks not only “What is the risk in this state?” but also “Which clinical situations contribute to that risk, and in what proportions?” The overall risk is meaningful for a specified population and time, but it may not describe either subgroup well.

A similar issue arises when a label records current status but omits how that status was reached. “In remission” may combine first remission with remission after relapse and salvage therapy. “Discharged” may combine a first discharge with discharge after readmission. In each case, a reported risk for the combined label depends on the mixture of histories in the population being summarized.

Recognizing Hidden Heterogeneity

A broad state deserves scrutiny when its occupants have different clinical pathways, prognosis, treatment, or time since an important event. The following checks can help identify whether aggregation is masking differences:

• List what the label combines. Ask whether it includes distinct disease stages, care settings, treatment histories, or routes into the state.
• Compare transition patterns across known histories. If sufficiently observed subgroups show different risks or rates, a single broad-state summary will depend on their proportions.
• Check whether the mixture changes over follow-up. Selective exit from the state can alter subgroup shares even when each subgroup’s rate is constant.
• Compare populations with different compositions. A broad-state risk may differ between clinics or treatment groups simply because their subgroup mixes differ. That is not, by itself, evidence that the underlying subgroup-specific rates differ.
• Look for clinical meaning in the omitted distinction. A numerical difference matters especially when it changes interpretation, prediction, or a decision relevant to the research question.

These checks require care. A subgroup difference in observed rates can reflect sparse data, measurement error, or other variables that differ between groups, rather than the recorded history alone. Conversely, failure to detect a difference is not proof that histories are interchangeable, especially when subgroup samples are small. Clinical knowledge and data quality should inform the decision, not a single significance test.

Alternatives, Limitations, and a Common Mistake

The simplest approach is to keep the broad state and report its transition behavior for a clearly described target population and follow-up period. This can be appropriate when the broad average answers the scientific question. When the differences matter, alternatives include reporting subgroup-specific transition summaries, retaining a relevant history variable in the analysis, or defining more specific states. Each option requires adequate information and a clear scientific rationale; splitting states indiscriminately can create sparse data and an unnecessarily complicated model.

A broad-state model can also be useful as a deliberate simplification, provided its limitations are stated. Its average transition behavior may be population-dependent, and applying it to a population with a different mix can mislead. A single average can conceal both high- and low-risk groups, and it may be unsuitable for questions that require individual prediction or comparison of clinical pathways. The introductory material on state occupation versus event history distinguishes current-state summaries from the sequence of events that led to them; aggregation is one reason those histories can remain important.

The central practical question is not whether every possible distinction must become a separate state. It is whether the distinctions being combined are scientifically relevant to the transition questions being asked. State definitions are simplifications of clinical reality; making that simplification explicit helps prevent a broad label from being mistaken for a homogeneous clinical condition.