Additive-Hazards Regression for Multi-State Transitions

Why Add Rates Rather Than Multiply Them?

A proportional-hazards model describes covariate effects through relative rates: a coefficient changes a transition intensity by a multiplicative factor. An additive-hazards model instead describes an absolute difference in instantaneous transition rates. This can be useful when a change in rate per unit of a covariate is the scientific quantity of interest, or when a constant relative effect is not a suitable working assumption.

In a multi-state model, the outcome is a path through a set of states, and each allowed move has its own transition intensity. Additivity is therefore specified transition by transition. A covariate can increase one transition rate, decrease another, or have no effect on a particular transition. A coefficient describes a conditional instantaneous rate contrast; it is not, by itself, a contrast in the probability of being in a state at a later time.

Model Specification and Interpretation

This equation is used for individuals at risk of the transition: just before \(t\), they are in state \(h\). If \(Y_{ih}(t)\) is the indicator that individual \(i\) is in state \(h\) immediately before \(t\), the counting-process intensity for observed \(h\to k\) jumps is \(Y_{ih}(t)\lambda_{hk,i}(t)\). Individuals not in state \(h\) do not contribute to that transition’s risk set at that time.

For two individuals with covariate vectors \(z_a\) and \(z_b\) at the same time, the model implies $$ \lambda_{hk}(t\mid z_a)-\lambda_{hk}(t\mid z_b) =\beta_{hk}(t)^{\mathsf T}(z_a-z_b). $$ Thus, if \(Z\) is a binary indicator, its coefficient is the modeled instantaneous rate difference between \(Z=1\) and \(Z=0\), holding the other modeled covariates fixed. If \(Z\) is measured in years, the coefficient’s units are rate per year of that covariate; if time is measured in years, the rate itself has units per year. A positive coefficient means a higher transition intensity at that time, not necessarily a higher probability of reaching the destination by a specified follow-up time.

The baseline intensity is the rate at the reference covariate values, \(Z=0\). Centering a continuous covariate changes which covariate value is the reference but does not change the modeled rate contrasts. Categorical covariates can be represented with indicator variables; each coefficient then compares its category with the chosen reference category, conditional on the other terms.

Conditions for a Meaningful Model

The model must be defined on a specified state graph: only transitions permitted by that graph receive transition intensities. Covariates used at time \(t\) must be known just before \(t\), so that the intensity does not use future information. For interpretation as a conditional intensity, the model also presumes a well-defined observation process and transition history. If the intensity is intended to depend only on the current state, time, and covariates, that is a Markov specification; additive hazards do not require that simplification, and history-dependent covariates or summaries can be included when appropriate.

Most importantly, every modeled intensity must be nonnegative over the covariate values and times where the model is intended to apply: $$ \lambda_{hk,0}(t)+\beta_{hk}(t)^{\mathsf T}z\geq 0. $$ Unlike an exponential link, a linear additive form does not guarantee this automatically. The condition should be checked over the covariate support, especially for combinations of covariates that are rare or near the edge of the observed range. Additivity also asserts a particular functional form: for a continuous covariate, a constant coefficient gives a constant rate difference per unit, not a constant rate ratio.

When rates are estimated from right-censored data, the observation and censoring assumptions must support the chosen estimation procedure. In particular, censoring that carries information about an unobserved transition, even after conditioning on the modeled information, can distort the risk sets. These conditions concern the data and estimator; they do not turn an associational coefficient into a causal effect. A causal interpretation needs additional design and identification assumptions.

Accumulated Rate Contrasts and a Short Derivation

The instantaneous interpretation follows directly by subtracting the two model equations: the baseline term cancels for individuals with the same transition and time. A useful extension is to integrate that difference over time. For a fixed covariate contrast \(z_a-z_b\), the difference in cumulative transition-specific intensities over \([0,\tau]\) is $$ \int_0^\tau \left\{\lambda_{hk}(t\mid z_a)-\lambda_{hk}(t\mid z_b)\right\}\,dt =\int_0^\tau \beta_{hk}(t)^{\mathsf T}(z_a-z_b)\,dt. $$ This is a contrast in integrated intensity, not generally a probability difference. The distinction matters because actual state occupancy and destination probabilities depend on all allowed transitions and on how long individuals remain at risk for each transition.

Diagnostics, Alternatives, and Limitations

A specification should be checked transition by transition. First verify that each equation corresponds to the intended edge and uses the appropriate risk set. Then inspect whether the additive form is plausible over the observed covariate range: a straight-line rate contrast may be inadequate for a continuous covariate, and a coefficient that changes over time may be needed when the rate difference is not stable. The earlier material on spline effects and time-varying coefficients provides ways to represent those departures. Sparse observations at particular covariate values can make both the estimated contrasts and the nonnegativity check unreliable there.

Negative fitted intensities are a direct warning that the additive model is not a valid intensity at those covariate-time combinations. They should not be silently interpreted as negative rates or repaired by treating them as probabilities. Restricting interpretation to supported covariate ranges, revising the covariate form, or choosing a different model may be necessary. Residual patterns or poor agreement between predicted and observed transition behavior can also signal a misspecified covariate form or time pattern, but no single diagnostic establishes that the model is correct.

The main alternative in regression for transition intensities is a multiplicative proportional-hazards specification. It ensures positive modeled rates through its link, while additive hazards give a direct absolute-rate interpretation. Neither scale is universally preferable: a constant rate difference and a constant rate ratio are different scientific assumptions. Both approaches require appropriate transition definitions and covariate timing, and neither coefficient alone gives a state-occupation probability. Additive models can also be sensitive to extrapolation because linear rate contrasts continue beyond the observed covariate range.

A frequent mistake is to call a positive additive coefficient a “hazard ratio” or to say it raises the event probability by that amount. The correction is to name the transition, state the rate units, and describe a conditional instantaneous difference. For example, “the coefficient is \(0.025\) per year” means a \(0.025\)-per-year higher instantaneous rate for a one-unit covariate contrast, at the stated time and among individuals in the source state. It does not mean a 2.5% increase in risk or a causal effect without further assumptions.