Worked Chronic Disease Progression and Mortality Predictions

Landmark Predictions in a Chronic Disease Model

At a landmark time, prediction starts from a person’s state at that time, not from their state at the beginning of follow-up. For a chronic disease, a useful simplified model distinguishes people who are healthy, people whose disease has progressed, and people who have died. A clinician may want to know the probabilities of occupying each state one or more years after a particular assessment, conditional on the person’s current state.

The key calculation is to follow each possible path through the states over the prediction horizon. A person currently healthy may remain healthy, progress and remain alive, or die before progression or after it. Someone already in the progressed state cannot return to the healthy state under this model. Predictions for a target group can then be formed by averaging the conditional predictions using the group’s actual distribution of landmark states.

For the calculations below, assume a time-homogeneous continuous-time Markov model with transitions \(H\to P\), \(H\to D\), and \(P\to D\), and no other transitions. Let the corresponding constant transition rates be \(\alpha\), \(\beta\), and \(\gamma\), respectively. Conditional on the current state, the model assumes that future transitions depend on the present state and not on earlier history, and that these rates remain constant throughout the prediction period. State \(D\) is absorbing: a person who has died remains dead.

The total rate of leaving \(H\) is \(\lambda_H=\alpha+\beta\). The time to leave \(H\) is therefore exponentially distributed with rate \(\lambda_H\). In particular, a healthy person can still be healthy at horizon \(u\) only if no transition out of \(H\) has occurred.

Deriving the State-Occupation Probabilities

Starting in \(H\), the probability of remaining healthy through time \(u\) is \(e^{-\lambda_Hu}\). To occupy \(P\) at time \(u\), the person must progress from \(H\) at some time \(s\) between 0 and \(u\), and then remain in \(P\) until \(u\). The density of progression at time \(s\), with no earlier exit from \(H\), is \(\alpha e^{-\lambda_Hs}\). The probability of surviving in \(P\) from \(s\) to \(u\) is \(e^{-\gamma(u-s)}\). Multiplying and integrating over possible progression times gives the probability of occupying \(P\).

The last equality follows by factoring out \(e^{-\gamma u}\) and integrating \(e^{(\gamma-\lambda_H)s}\) from 0 to \(u\). If \(\gamma=\lambda_H\), the integral instead gives \(\alpha u e^{-\lambda_Hu}\), the continuous limiting case. Finally, because the three states are mutually exclusive and exhaustive at horizon \(u\), the probability of occupying \(D\) is the remainder.

Starting in \(P\), the only possible transition is to death. Thus \(p_{PP}(u)=e^{-\gamma u}\) and \(p_{PD}(u)=1-e^{-\gamma u}\); both probabilities of occupying \(H\) are zero. Starting in \(D\), the probabilities are \(p_{DD}(u)=1\) and \(p_{DH}(u)=p_{DP}(u)=0\). Each starting state therefore has a complete row of state-occupation probabilities that sums to one.

Predictions Depend on Both Current State and Horizon

A prediction is not specified by a transition model alone: it also requires a current state and a future horizon. In this model, the same rates produce different probabilities for a person currently healthy and a person currently progressed. Changing the horizon also changes every state-occupation probability, even when the rates stay fixed.

Combining Predictions for a Target Group

A group prediction must use the distribution of current states in the target population at the landmark. If a fraction \(w_H\) is healthy, \(w_P\) is progressed, and \(w_D\) is dead, with \(w_H+w_P+w_D=1\), then the group probability of occupying state \(j\) after \(u\) is \(w_Hp_{Hj}(u)+w_Pp_{Pj}(u)+w_Dp_{Dj}(u)\). For a prediction among people alive at the landmark, \(w_D=0\) and \(w_H+w_P=1\).

Assumptions, Diagnostics, and Alternatives

The calculations are exact for the stated constant-rate model, but that does not establish that the model describes a particular chronic disease well. Before interpreting predictions, check whether the model’s structure and assumptions are credible for the intended target population and landmark:

• Allowed transitions. The model rules out recovery from \(P\) to \(H\) and any other unlisted transition. If such transitions are clinically possible and relevant, the three-state structure may be inadequate.
• Constant rates. The rates are assumed not to vary over the prediction interval. If transition rates change with time since the landmark, age, treatment, or disease duration, constant-rate predictions may be systematically inaccurate.
• Markov conditioning. The future is assumed to depend on the current state, not on how long the person has already been in it or on earlier history. The earlier material on semi-Markov prediction addresses duration-conditioned predictions when this assumption is unsuitable.
• Target population and landmark information. The current-state weights and any covariates used for prediction must represent the target population at the landmark. Information arising after that landmark must not be used as if it were already available.

Diagnostics should include checking that all calculated probabilities lie between zero and one and that each starting-state row sums to one. For fitted rates, examine whether the model’s implied transition patterns are plausible across time and relevant subgroups, and compare predicted state proportions with observed follow-up patterns when suitable data are available. A probability sum that is materially different from one usually signals an arithmetic error, omitted state, or incorrect weighting; it is not evidence of an unusual disease process.

Transition-based prediction, as used here, builds state-occupation probabilities from transition-specific rates and the possible paths through the model. A direct landmark approach instead models the probability of the future state at a specified horizon without requiring the same transition decomposition. These approaches answer closely related prediction questions but rely on different modeling structures. The direct approach is not a substitute for checking the transition model’s assumptions, and a transition model’s path interpretation does not by itself guarantee better predictive performance.

The main limitation of the worked calculations is their simplicity: constant rates and three states may be too restrictive for real chronic disease. More flexible models can allow time-varying rates or additional predictors, but predictions still need a clearly defined landmark, target population, current state, and horizon. Uncertainty in estimated rates also means that model-based probabilities are estimates, not known individual outcomes.

Common Mistakes

• Using the wrong starting state. A prediction conditional on being healthy at the landmark is not interchangeable with one conditional on progressed disease. Start with the state the person actually occupies at the landmark.
• Confusing state occupation with a transition probability. The probability of being dead at \(u\) includes every allowed path ending in death by \(u\), not only direct death from the starting state.
• Applying population weights twice. If 30% of the target group is currently progressed, the progressed-state prediction is weighted by 0.30—not by \(0.70\times0.30\).
• Ignoring probability conservation. For each starting state, and for a valid weighted group prediction, probabilities across \(H\), \(P\), and \(D\) must sum to one.
• Treating model output as a personal certainty. A predicted probability describes a model-based distribution for people with the specified landmark information; it does not determine an individual’s future state.