Interpreting Transition-Specific Hazard Ratios

What a Transition-Specific Hazard Ratio Compares

In a multi-state analysis, a transition-specific hazard ratio answers a local question: among individuals currently eligible to make a particular transition, how does the instantaneous rate of that transition differ between two specified covariate profiles? It is not, by itself, a comparison of the probability of occupying a state or of experiencing an event by a chosen follow-up time.

The comparison is tied to a transition and its source state. For example, the rate of moving from a well state to an illness state is defined among individuals currently in the well state and at risk of that transition. It is not a rate among everyone enrolled, including people who have already left that state.

Definition: For transition $k$, let $r_k$ be its source state and $s_k$ its destination state. The transition intensity $\lambda_k(t\mid\mathcal{H}(t),X(t))$ is the instantaneous rate of moving from $r_k$ to $s_k$ at time $t$, conditional on the observed history $\mathcal{H}(t)$ and covariates $X(t)$, for a subject currently in $r_k$ and still under observation. In a transition-specific proportional-hazards model, $$ \lambda_k(t\mid\mathcal{H}(t),X(t)) = \lambda_{0k}(t\mid\mathcal{H}(t))\exp\{\beta_k^\mathsf{T}X(t)\}. $$ Here $\lambda_{0k}$ is the baseline intensity for transition $k$, and $\beta_k$ is its transition-specific coefficient vector. The form of the baseline and the conditioning history must match the chosen time scale and model.

For a Cox model, the baseline intensity is left unspecified. The proportional-hazards assumption means that, for a given transition and a fixed comparison of covariates, the hazard ratio does not vary with time on the model's time scale. When comparing two covariate vectors $x$ and $x^\ast$, the ratio is

\frac{\lambda_k(t\mid\mathcal{H}(t),X=x)} {\lambda_k(t\mid\mathcal{H}(t),X=x^\ast)} = \exp\{\beta_k^\mathsf{T}(x-x^\ast)\}.

This ratio holds when both hazards refer to the same transition, the same time and relevant history, and subjects currently in the transition's source state. Other covariates are held equal unless their differences are included in $x-x^\ast$. For a one-unit increase in a single covariate $X_j$, with all other covariates unchanged, the hazard ratio is $\exp(\beta_{kj})$. Thus $\beta_{kj}$ is a log hazard ratio, not a hazard ratio itself.

State the Reference Comparison

A reported hazard ratio is incomplete unless its covariate comparison is clear. For a binary covariate coded 0 and 1, $\exp(\beta_{kj})$ compares category 1 with category 0, provided this is the model's coding. For a continuous covariate, it compares values one unit apart; the practical meaning depends on the unit of measurement. In a model with several covariates, the comparison usually changes one covariate while holding the others fixed.

The reference profile does not mean “the baseline hazard.” The baseline intensity is a function of the transition and time (and, where relevant, history); the reference profile is the covariate value against which another profile is compared. If $x^\ast$ is the reference profile, its linear predictor is $\beta_k^\mathsf{T}x^\ast$. The corresponding ratio to $x$ is still obtained from the difference in linear predictors.

Worked Example: A One-Unit Increase in Exposure

Suppose a hypothetical transition-specific Cox model estimates the coefficient for a one-unit exposure increase on transition $A\to B$ as $\hat\beta_{AB}=0.530$, with standard error $SE=0.220$. The exposure unit is 10 mg/L. Interpret the estimated effect and its Wald confidence interval.

Comparison. The coefficient compares two people currently in state $A$, with the same modeled history and other covariates, whose exposure values differ by 10 mg/L. The hazard-ratio estimate is

\widehat{HR}_{AB}=\exp(0.530)=1.699\approx1.70.

Under the fitted proportional-hazards model, the estimated instantaneous rate of transition $A\to B$ is about 1.70 times as high for the profile with exposure 10 mg/L higher. This is a conditional rate comparison, not a 70-percentage-point increase in risk.

A 95% Wald interval is first calculated for the log coefficient:

0.530\pm1.96(0.220) = 0.530\pm0.4312 = (0.0988,\ 0.9612).

Exponentiating both endpoints gives the interval on the hazard-ratio scale:

\left(\exp(0.0988),\exp(0.9612)\right) \approx(1.10,\ 2.61).

The data and model yield a 95% Wald confidence interval of approximately 1.10 to 2.61 for the transition-specific hazard ratio per 10 mg/L higher exposure. In repeated use of this interval procedure under its assumptions, 95% of such intervals would cover the true model parameter. The interval does not assign a 95% probability to this particular fixed parameter.

From Coefficients to Comparisons of Profiles

With multiple covariates, use the entire difference between the two profiles. Suppose the model includes exposure and age, with transition-specific coefficients $\beta_{k1}$ and $\beta_{k2}$. Comparing profiles $(x_1,x_2)$ and $(x_1^\ast,x_2^\ast)$ gives

HR_k = \exp\{\beta_{k1}(x_1-x_1^\ast)+\beta_{k2}(x_2-x_2^\ast)\}.

If age is held fixed, its term is zero. If both covariates differ, the ratio reflects both changes jointly under the model; it should not be presented as the effect of exposure alone.

Worked Example: Comparing Two Covariate Profiles

In a hypothetical model for transition $C\to D$, suppose $\hat\beta_{k1}=0.12$ per year of age and $\hat\beta_{k2}=-0.40$ for a treatment indicator coded 1 versus 0. Compare a 50-year-old untreated person with a 45-year-old treated person, assuming each is currently in state $C$ and the modeled histories otherwise match.

Take the 45-year-old treated profile as $x^\ast=(45,1)$ and the 50-year-old untreated profile as $x=(50,0)$. The linear-predictor difference is

$$ 0.12(50-45)+(-0.40)(0-1) = 0.60+0.40 = 1.00. $$

Therefore, the estimated ratio of transition intensities is

\widehat{HR}_k=\exp(1.00)=2.718.

Under this model, the estimated instantaneous rate of $C\to D$ for the specified 50-year-old untreated profile is 2.72 times that for the specified 45-year-old treated profile, conditional on being in state $C$ and the same modeled history. This joint comparison does not isolate an age effect: age and treatment both differ. Reversing the comparison would give $1/2.718=0.3679$, apart from rounding.

Why the Source-State Condition Matters

The transition intensity concerns individuals who are in the relevant source state and eligible to make the transition. For transition $A\to B$, someone already in $B$ or in another state is not in that transition's risk set at that moment. The hazard ratio therefore compares rates within the appropriate risk-set context, not unconditional rates in the original cohort.

This distinction matters because people can enter and leave states over time. The characteristics of those currently in a source state may differ from the characteristics of the original cohort. A conditional transition hazard ratio does not directly give a marginal effect on state-occupation probabilities, cumulative incidence, or expected time in a state. Obtaining those quantities generally requires combining transition behavior over paths, not simply relabeling one transition coefficient.

The transition-specific model also matters. A coefficient for one transition applies to that transition's hazard, with its own baseline intensity and risk set. The same covariate may have a different coefficient for another transition. This tutorial interprets each coefficient in place; comparing coefficients across transitions is a separate inferential question.

Assumptions, Uncertainty, and Diagnostics

The hazard-ratio interpretation depends on the fitted model and its data conditions. In particular, the transition-specific proportional-hazards model must adequately represent the intensity over the analysis time scale; the covariate comparison must be within the model's stated form; and event times, transitions, covariates, and risk sets must be recorded and constructed appropriately. For a Cox analysis, censoring must be non-informative conditional on the modeled information, or handled by a justified method. Covariates that change over time must be defined using information available at the time of the transition, rather than future information.

The proportional-hazards assumption is consequential: if the coefficient changes with time, a single exponentiated coefficient may summarize an average or model-dependent effect rather than a constant transition-specific hazard ratio. Diagnostics should therefore be examined for the transition being interpreted, not only pooled across transitions. Sparse events or little covariate overlap in a transition-specific risk set can also make estimates unstable or comparisons depend heavily on extrapolation.

For a coefficient estimate $\hat\beta_{kj}$ with standard error $SE$, the usual large-sample Wald interval on the log scale is $\hat\beta_{kj}\pm1.96SE$ for an approximate 95% interval. Exponentiating its endpoints gives an interval for the hazard ratio. The approximation can be unreliable with limited information, especially when a transition has few events or estimates are extreme. A confidence interval that includes 1 indicates that the data are compatible, at this confidence level and under the model, with equal transition intensities for the specified comparison; it does not establish equality.

Name the transition and source state.
Identify which transition coefficient is being interpreted and the state-specific risk set to which its intensity applies.

Write the covariate contrast.
State the two profiles, including the reference profile, and specify which other covariates are held fixed.

Exponentiate the linear-predictor difference.
Calculate $\exp\{\hat\beta_k^\mathsf{T}(x-x^\ast)\}$, not the coefficient itself, as the estimated hazard ratio.

Qualify the interpretation.
Report the transition, units, conditional source-state comparison, uncertainty, and any relevant model limitations.

Common Mistakes and Alternative Targets

Calling the coefficient a hazard ratio. A coefficient of 0.53 is on the log-hazard scale; its hazard ratio is $\exp(0.53)$, approximately 1.70.
Leaving the reference comparison implicit. “The hazard ratio is 1.70” is incomplete without stating the transition, covariate difference, units, and held-fixed covariates.
Ignoring source-state eligibility. The ratio concerns transition rates among those currently in the source state, not everyone in the cohort.
Interpreting a hazard ratio as a risk ratio or probability difference. An instantaneous rate ratio does not directly state the probability of making the transition by a fixed time.
Giving a causal interpretation automatically. A regression hazard ratio describes a model-based association. Causal interpretation requires additional design and identification assumptions.

A common-effect model may impose a shared covariate coefficient across transitions, while a transition-specific model allows coefficients to vary by transition. These models answer differently constrained questions; a coefficient from the latter should not be described as a common effect. Likewise, a transition hazard ratio and a predicted cumulative risk are distinct targets: the latter reflects the combined process of entering and leaving states over time.

Key takeaway: Interpret $\exp\{\beta_k^\mathsf{T}(x-x^\ast)\}$ as a hazard ratio for a clearly stated covariate contrast on one specified transition, conditional on being in that transition's source state and on the model's history and assumptions.

Check Your Understanding

Answer each question by naming the transition and the covariate comparison where relevant.

For a coefficient of $0.20$ on a continuous covariate measured in years, what is the hazard ratio for a one-year increase, with other covariates fixed?
A binary covariate has coefficient $-0.30$ and is coded 1 versus 0. Interpret the estimated hazard ratio for transition $R\to S$.
Why is a transition-specific hazard ratio not, on its own, the probability of reaching the destination state by one year?
A model includes age and treatment. If only age differs between two profiles, what happens to the treatment term in the hazard-ratio formula?

Solutions. (1) $\exp(0.20)=1.221$: per additional year, the estimated instantaneous rate is about 1.22 times as high for individuals in the transition's source state, at the same other covariate values. (2) $\exp(-0.30)=0.7408$: the coded-1 group has an estimated transition intensity about 0.741 times that of the coded-0 group, conditional on being in $R$, the model history, and other covariates. (3) The hazard is an instantaneous conditional rate among those currently eligible for that transition; a one-year probability depends on how subjects move through the multi-state process over time. (4) The treatment difference is zero, so its coefficient multiplied by the treatment contrast contributes zero to the linear-predictor difference.