History Dependence in Illness-Death Data

Why the Same State May Not Mean the Same Future

A state records a person’s current status, but it may not record how that status was reached. Two people who are both ill at a particular time might differ in how long they have been ill, how many previous episodes they have had, or what treatment they have received. If these differences predict what happens next, the current state alone does not summarize the information relevant to their future.

This tutorial focuses on that pattern in illness-death data. The central question is not whether a person’s history is associated with an outcome in general, but whether people with the same current state at the same time have different future outcomes depending on their earlier histories. Fixing the observation time matters: a transition probability can change over calendar time in a time-inhomogeneous model without depending on earlier history.

In the basic illness-death structure, state 0 denotes being free of the illness, state 1 denotes being ill, and state 2 denotes death, which is absorbing. The usual transitions are \(0\to1\), \(0\to2\), and \(1\to2\). Extensions can include recovery or repeated episodes. The issue here is not which transitions are allowed; it is whether the current state and time are enough to describe their future occurrence.

For a time-inhomogeneous Markov model, the future conditional on the past depends on the current state and the current time, not on the earlier path. In intensity notation, this means that, once \(X(t)=i\) and \(t\) are fixed, earlier histories do not further change \(\lambda_{ij}\). History dependence is a failure of that state-only condition. It does not, by itself, identify the cause of the difference or show that history has a causal effect.

Three Ways Histories Can Differ

The following examples use hypothetical data to illustrate comparisons within the same current illness state. In each example, suppose all 40 people in each group are in state 1 at the stated landmark, are followed for the same horizon, and have complete follow-up. The displayed proportions are descriptive probabilities over that horizon in the hypothetical group; they are not fitted transition-intensity estimates. For simplicity, assume the illness-death structure has no recovery during the follow-up window. If recovery is possible, the outcome would need to be defined to account for that competing transition.

Why History Composition Changes a Pooled Risk

A simple calculation shows why history dependence matters even when the goal is only to describe outcomes for everyone currently ill. Suppose the two duration groups in the first example have observed 12-month death proportions of 0.15 and 0.45. If the groups are equally represented, the pooled proportion is 0.30. But if the mix of durations changes, the pooled proportion changes too, even when the group-specific proportions stay the same.

For an equal mixture of short- and long-duration histories, the calculation is

If 80% of the currently ill group has the longer-duration history and 20% the shorter-duration history, then

The pooled proportion rises from 0.30 to 0.39 because the history mix changes; the two group-specific proportions have not changed. This is a mixture identity, not evidence that one history caused another outcome. It explains why a prediction based only on current state can be inadequate when the distribution of earlier histories differs across populations or times.

The identity follows by partitioning the group currently in state 1 into history categories \(h\). The probability of the outcome for the whole group is the sum, over those categories, of the category-specific probability multiplied by the fraction of the group in that category. If the category-specific probabilities differ, changing those fractions changes the pooled probability. If the probabilities are equal across histories, this particular source of variation disappears.

How to Assess the Pattern

A useful descriptive assessment begins by defining the landmark time, current state, future outcome, follow-up horizon, and history feature before comparing groups. For example, “death within 12 months among people in state 1 on the landmark date, by duration in state 1 at that date” is more precise than “death by illness duration.” The groups must be compared at the same time scale and under a clear rule for transitions such as recovery.

A practical sequence is:

If histories with the same current state show systematically different future transition patterns, that is evidence against the adequacy of a state-only Markov description for the question and data being considered. Relevant checks include whether the apparent contrast persists at multiple landmark times and whether groups have comparable measurement and follow-up. Sparse observations, inaccurate onset dates, or different loss-to-follow-up patterns can make a contrast unstable or misleading. A single small-sample difference is a reason to investigate, not a definitive diagnosis.

Interpretation, Alternatives, and Limitations

A state-only Markov model treats people in the same state at the same time as having the same transition behavior, regardless of how they arrived there. When duration, recurrence, or treatment history is informative, that assumption may be too restrictive. One response is to describe or model the relevant history alongside current state; another is to use a time scale or model structure suited to the feature under study. These alternatives require clear definitions and adequate data. They do not make confounding, measurement error, or incomplete follow-up disappear.

The comparison is not simply between “Markov” and “wrong.” A state-only model may be a useful approximation for a particular prediction target, population, and time horizon even if some history differences exist. Conversely, a strong association between history and outcome does not alone identify the best model. The choice depends on the scientific question, the histories observed, and the predictions or summaries required. The introductory material on Markov assumptions distinguishes dependence on current state from dependence on earlier history; here the illness-death examples make that distinction concrete.

A central pitfall is to compare two groups at different calendar times and attribute their different outcomes to illness duration. Calendar-time changes in care or population composition can produce differences even without duration dependence. Another is to treat crude group proportions as causal treatment effects. A third is to assume that a transition out of illness is impossible because it was not tabulated. Define the state process and outcome explicitly, and distinguish descriptive evidence from causal claims.