Why Transition Hazard Ratios Are Not Overall Risk Ratios

Why a Transition Effect Is Not an Overall Risk Effect

A transition hazard ratio describes a relative rate for a specified move between states, among individuals currently eligible to make that move. A cumulative incidence or state-occupation probability instead describes how much probability has accumulated along all relevant paths by a chosen time. Those are different quantities: other outgoing transitions, later transitions, and time spent in each state all affect the overall probability.

This distinction matters in interpreting transition regression. A coefficient can be a clear, transition-specific association without being a risk ratio for a clinical endpoint. To compare overall probabilities, one must specify the endpoint and time horizon, and account for the full transition structure. The calculations below use simple continuous-time models so that the difference is visible directly.

For a binary covariate \(z\), a proportional transition-hazards model for one transition can be written as \(\lambda_{rs}(t\mid z)=\lambda_{rs,0}(t)\exp(\beta_{rs}z)\). Its hazard ratio comparing \(z=1\) with \(z=0\) is \(\exp(\beta_{rs})\). This comparison is conditional on occupying the origin state; it does not compare the marginal probabilities of reaching or occupying the destination state by time \(t\).

A Competing-Destinations Calculation

Start with a person in state 0, from which two absorbing destinations, state 1 and state 2, are possible. Assume constant transition hazards \(\lambda_{01}\) and \(\lambda_{02}\), no return transitions, and no censoring in the probability calculation. The time to leave state 0 then has total rate \(\lambda_{01}+\lambda_{02}\). The cumulative incidence of transition \(0\to1\) by time \(t\) is the probability of still being in state 0 at time \(u\), multiplied by the transition rate to state 1, integrated over possible transition times \(u\).

The rate \(\lambda_{02}\) enters this probability even though it is not the hazard for transition \(0\to1\). Increasing the hazard of the competing transition can reduce the chance of reaching state 1 first. As a result, even a proportional change in \(\lambda_{01}\) does not generally produce the same proportional change in \(F_1(t)\).

State Occupation in an Illness-Death Model

The distinction is even more pronounced when the destination state is not absorbing. Consider three states: 0 is healthy, 1 is ill, and 2 is dead. The allowed transitions are \(0\to1\), \(0\to2\), and \(1\to2\), with constant rates \(a\), \(c\), and \(d\), respectively. Assume everyone starts in state 0, the rates are constant, and the transition intensities depend only on current state and the fixed covariate. This is a time-homogeneous Markov illness-death model. Death is absorbing.

To occupy state 1 at time \(t\), a person must enter state 1 at some time \(u\leq t\), then avoid the \(1\to2\) transition until \(t\). The probability of remaining in state 0 until \(u\) is \(e^{-(a+c)u}\), the rate of entering state 1 at \(u\) is \(a\), and the probability of remaining in state 1 from \(u\) to \(t\) is \(e^{-d(t-u)}\). Integrating these contributions gives the state-occupation probability.

The expression includes both entry into illness and subsequent exit from illness. It therefore is not simply the cumulative incidence of \(0\to1\). If \(a+c=d\), the integral instead gives \(P_1(t)=at e^{-dt}\), the continuous limiting case. In either form, changing \(a\) changes the time of entry and the amount of time available for the \(1\to2\) transition.

The stated values in the preceding example require care: applying the formula gives \(0.80/0.90\) multiplied by \(0.3599\), which is approximately \(0.3199\). For the baseline group, the same formula gives \(0.1909\). These figures illustrate that the illness-occupation ratio is about 1.676 under the specified rates. They do not imply a general ordering between a transition hazard ratio and an occupation ratio.

What the Calculations Do, and Do Not, Establish

The equations demonstrate a general point rather than a universal numerical relationship. The overall probability for a chosen endpoint depends on all transition rates along paths that can lead to, or away from, that endpoint. It also depends on the time horizon. A hazard ratio is an instantaneous, conditional comparison; a risk ratio is a ratio of probabilities accumulated by a specified time. A state-occupation ratio concerns presence in a state at that time, not necessarily whether that state was ever visited.

The assumptions used above are deliberately restrictive: constant rates, a fixed baseline covariate, the stated allowed transitions, and a Markov model in the illness-death example. Under these assumptions, the formulas follow directly from the transition process. In applications, transition hazards may vary over time, covariates may change, and history beyond the current state may matter. The transition-specific hazard interpretation still needs a well-defined risk set and model, while overall probabilities then require calculations appropriate to the fitted transition structure.

A useful way to avoid conflation is to identify the estimand before interpreting a coefficient:

• Transition effect: the hazard ratio for a named transition, conditional on occupying its origin state.
• Cumulative incidence effect: a comparison of probabilities of having experienced a specified event by a named time, accounting for competing transitions.
• State-occupation effect: a comparison of probabilities of being in a specified state at a named time, accounting for entry and exit paths.

These are not interchangeable summaries. If the scientific question concerns one-year incidence or occupation, the fitted transition model must be translated into that probability and the resulting contrast calculated directly. The transition hazard ratio alone does not supply it.

Diagnostics, Alternatives, and Common Pitfalls

Before interpreting transition regression, check that the transition definitions and risk sets match the scientific question. For each modeled transition, confirm who is eligible to make it and how censoring and state history are represented. A proportional-hazards specification also requires the covariate effect on that transition’s hazard to be adequately represented as proportional over time; a single hazard ratio can be misleading if the effect varies materially over time. In a Markov model, check whether conditioning on current state and included covariates is a defensible description of transition rates. Dependence on time since entry or earlier history may require a richer model.

A common mistake is to read \(\exp(\beta_{01})=2\) as “twice the probability of illness” or “twice the one-year risk.” The reasoning is wrong because the coefficient compares instantaneous \(0\to1\) rates among people currently in state 0, not probabilities at a horizon. The correction is to specify the probability of interest, include the competing and subsequent transitions, and calculate that probability under each covariate value before forming a risk ratio or difference.

Transition-specific regression is useful when the scientific question concerns how a covariate relates to a particular move, and it allows different effects for different transitions. An alternative is to report model-based cumulative incidence or state-occupation probabilities at clinically meaningful times, with absolute differences or ratios as appropriate. These probability summaries answer different questions and depend on the whole transition model. Neither approach makes the other unnecessary; the appropriate summary is determined by the estimand, not by the form of a regression coefficient.

Finally, a mismatch between a transition hazard ratio and an overall risk ratio is not evidence that the regression is faulty. It is often the expected consequence of competing paths and transitions out of the destination state. Model diagnostics address whether the transition model is credible; they do not turn a conditional rate contrast into a marginal probability contrast.