Parametric Proportional-Hazards Models for Transitions

Why Specify a Parametric Transition Hazard?

A proportional-hazards regression for a multi-state process describes how covariates relate to the instantaneous rate of a particular transition. A parametric baseline hazard adds a specified shape over time to that regression. This can produce smooth hazard estimates and predictions beyond observed event times, but it also makes the result depend on the chosen baseline family. The Weibull and Gompertz families are two common choices.

The model is transition-specific: a covariate can have a different coefficient, and a transition can have a different baseline hazard, for each allowed move between states. The formulas below concern one transition at a time. They do not imply that transitions from the same person are independent.

Definition: For transition $r\to s$, let $u$ be elapsed time since entry into the source state $r$, and let $X$ be the covariate vector for the individual at risk. The transition-specific proportional-hazards model is $$ h_{rs}(u\mid X)=h_{0,rs}(u)\exp(\beta_{rs}^{T}X). $$ Here $h_{rs}$ is the instantaneous transition rate per unit time, conditional on being in the transition risk set just before $u$; $h_{0,rs}(u)$ is the baseline hazard at $X=0$; and $\beta_{rs}$ is the transition-specific coefficient vector. The model assumes that covariates multiply the baseline hazard and do not change its time shape.

The clock in this definition is reset on entry to the source state. This is a sojourn-time, or clock-reset, specification. A model using time since a common study origin instead must define its time scale accordingly; the two choices need not produce the same model. If a person can enter a transition risk set more than once, each relevant spell must be represented with its start and end times and its event status.

Weibull and Gompertz Baselines

The following parameterizations fix exactly what the baseline parameters mean. In both cases, the baseline is the hazard for the reference covariate vector $X=0$, so covariates should be centered or coded deliberately if that reference is not meaningful.

Weibull proportional-hazards model: With $\lambda_{rs}>0$ and $\rho_{rs}>0$, $$ h_{0,rs}(u)=\lambda_{rs}\rho_{rs}u^{\rho_{rs}-1}, \qquad h_{rs}(u\mid X)=\lambda_{rs}\rho_{rs}u^{\rho_{rs}-1}\exp(\beta_{rs}^{T}X). $$ The baseline cumulative hazard is $H_{0,rs}(u)=\lambda_{rs}u^{\rho_{rs}}$. The parameter $\rho_{rs}$ is the shape: the baseline hazard increases if $\rho_{rs}>1$, is constant if $\rho_{rs}=1$, and decreases if $0<\rho_{rs}<1$. The parameter $\lambda_{rs}$ has units of time$^{-\rho_{rs}}$, making the hazard have units of inverse time.

Gompertz proportional-hazards model: With $a_{rs}>0$ and real-valued $\gamma_{rs}$, $$ h_{0,rs}(u)=a_{rs}\exp(\gamma_{rs}u), \qquad h_{rs}(u\mid X)=a_{rs}\exp(\gamma_{rs}u)\exp(\beta_{rs}^{T}X). $$ The baseline cumulative hazard is $H_{0,rs}(u)=a_{rs}\{\exp(\gamma_{rs}u)-1\}/\gamma_{rs}$ when $\gamma_{rs}\ne0$, and $H_{0,rs}(u)=a_{rs}u$ when $\gamma_{rs}=0$. Here $a_{rs}$ has units of inverse time and $\gamma_{rs}$ has units of inverse time. A positive $\gamma_{rs}$ gives an increasing baseline hazard, a negative one a decreasing hazard, and zero a constant hazard.

For either baseline family, the hazard ratio comparing covariate vectors $X_A$ and $X_B$ at the same transition time is $\exp\{\beta_{rs}^{T}(X_A-X_B)\}$. It is constant over $u$, even when the baseline hazard changes with time. A coefficient is a log hazard ratio per one-unit covariate difference, conditional on the model and the other covariates. It describes an instantaneous rate contrast, not a ratio of transition probabilities.

Worked Calculations

Worked Example: Weibull Model for a Transition

Suppose a hypothetical model for transition $1\to2$ uses elapsed time in months and one binary covariate $X$, coded 1 for exposed and 0 for unexposed. Let $\lambda_{12}=0.020$ month$^{-1.5}$, $\rho_{12}=1.5$, and $\beta_{12}=0.60$. Calculate the baseline hazard at 4 months, the exposed hazard, and the covariate hazard ratio.

The baseline hazard is $h_{0,12}(4)=0.020(1.5)(4^{1.5-1})=0.030(2)=0.0600$ per month.

For the exposed group, $h_{12}(4\mid X=1)=0.020(1.5)(4^{0.5})\exp(0.60)=0.0600(1.822)=0.1093$ per month. The unexposed hazard ratio reference is $X=0$, so the exposed-to-unexposed hazard ratio is $\exp(0.60)=1.822$. At the same elapsed time and conditional on the model, the exposed group has an estimated instantaneous transition rate about 1.82 times the unexposed group’s rate.

The baseline hazard increases with elapsed time because $\rho_{12}=1.5>1$. The covariate multiplies this changing baseline by the same factor at every time; it does not change the Weibull shape.

Worked Example: Gompertz Model and Rounded Calculations

Consider a hypothetical transition with elapsed time measured in months. Let $a_{rs}=0.030$ per month, $\gamma_{rs}=0.080$ per month, and let a binary covariate have coefficient $\beta_{rs}=\log(1.50)=0.4055$. Find the baseline and covariate-positive hazards at 6 months.

The baseline hazard is $h_{0,rs}(6)=0.030\exp(0.080\times6)=0.030\exp(0.48)=0.0485$ per month, rounding the model calculation to four significant figures.

The covariate-positive hazard is calculated from the unrounded model inputs: $h_{rs}(6\mid X=1)=0.030\exp(0.48)\exp(0.4055)\approx0.0727$ per month. Equivalently, the covariate multiplier is $\exp(\beta_{rs})=1.50$, so the positive group’s hazard is approximately 1.50 times the reference group’s hazard at 6 months.

Because $\gamma_{rs}>0$, the baseline hazard grows exponentially with elapsed time. The coefficient does not say that the probability of transitioning by 6 months is 50% higher; it specifies a ratio of instantaneous transition hazards at a given time among individuals in the relevant risk set.

Worked Example: Interpreting Units and a Reference Covariate

For a hypothetical transition $2\to3$, let $X$ be age in decades above age 50, so $X=0$ represents age 50 and $X=1$ represents age 60. Suppose the Gompertz parameters are $a_{23}=0.012$ per year and $\gamma_{23}=0.10$ per year, with $\beta_{23}=-0.25$. Compare the hazards at 5 years for these two ages.

At age 50, $X=0$, and the hazard is $0.012\exp(0.10\times5)=0.012\exp(0.5)=0.01978$ per year. At age 60, the hazard is $0.01978\exp(-0.25)=0.01541$ per year. The age-60-to-age-50 hazard ratio is $\exp(-0.25)=0.7788$.

Thus, at the same transition time and conditional on the model, a ten-year greater age in this coding is associated with an estimated 22.1% lower instantaneous transition rate: $1-0.7788=0.2212$. The baseline parameter $a_{23}$ refers to age 50, not age zero, because of the chosen covariate coding.

Why the Hazard-Ratio Interpretation Holds

The proportional-hazards interpretation follows directly by comparing two covariate values at the same time. Since the baseline hazard is shared for both individuals, it cancels:

\frac{h_{rs}(u\mid X_A)}{h_{rs}(u\mid X_B)} = \frac{h_{0,rs}(u)\exp(\beta_{rs}^{T}X_A)} {h_{0,rs}(u)\exp(\beta_{rs}^{T}X_B)} = \exp\{\beta_{rs}^{T}(X_A-X_B)\}.

This derivation also shows what the assumption requires: the covariate contrast is constant over transition time. The hazard may rise or fall according to a Weibull or Gompertz baseline, but the relative contrast cannot vary with $u$ unless the model includes a time-varying coefficient or another suitable extension. A transition-specific coefficient should not be carried over to another transition without a substantive and statistical basis.

Assumptions, Estimation, and Diagnostics

A parametric transition regression is interpretable only relative to its risk sets, time scale, covariate coding, and baseline family. Common assumptions and checks include:

Correct transition definition and time scale. Identify the source and destination states, the risk-set entry time, and whether the clock resets on entry to the source state. Incorrect spell boundaries or time origins alter the modeled hazard.
Proportional covariate effects. At fixed covariates, the hazard ratio is constant over elapsed time. Transition-specific residual or graphical diagnostics can reveal systematic time patterns inconsistent with this assumption.
Suitable baseline shape. The Weibull imposes a monotone hazard (or a constant one); the Gompertz imposes an exponentially changing hazard. Compare fitted hazards with nonparametric or flexible summaries over the observed time range and inspect whether conclusions depend strongly on the family.
Appropriate censoring and observation. Censoring and loss to follow-up should be non-informative conditional on modeled information for the likelihood used. The transition and censoring times must be observed or handled by a valid observation model.
Independent subject trajectories and adequate information. Subjects are typically assumed independent, while multiple transitions within one subject belong to that subject’s trajectory and are not independent observations. Sparse events, especially for a particular transition, can make baseline and covariate estimates unstable.

In a fully specified transition-intensity model, a path likelihood accounts for both observed jumps and time spent at risk. In brief, an observed transition contributes its hazard at the jump time, while exposure to all possible exits from the current state contributes the exponential of minus their integrated hazards. Consequently, a transition regression is not fitted by treating every transition record as an unrelated person. The likelihood or inferential method must respect each subject’s observed path.

Diagnostics should be conducted transition by transition. Look for lack of fit in the baseline time pattern, influential observations, implausible fitted hazards, and covariate effects that change over time. Check estimates and uncertainty on their natural scale as well as the log-hazard scale. Predictions far beyond the observed time range deserve particular caution: a Gompertz hazard with positive $\gamma_{rs}$ grows rapidly, while a negative $\gamma_{rs}$ tends to zero; a Weibull hazard continues its power-law pattern indefinitely.

Choosing Between the Parametric Model and Alternatives

A semi-parametric Cox proportional-hazards model leaves the baseline hazard unspecified while retaining a proportional covariate effect. It can reduce dependence on a chosen baseline shape, but it does not provide the same direct parametric baseline extrapolation. Weibull and Gompertz models are useful when their shapes are credible and smooth predictions are needed. They can be efficient under correct specification, but a wrong baseline form can bias predictions and may affect covariate estimates as well.

Earlier material on baseline-hazard choices and stratification distinguishes modeling a shared covariate effect from allowing different baseline hazards. In this setting, choose a baseline family for each transition in light of its time scale and event information; do not assume one family or one parameter value is appropriate across all transitions. A model with a simple, interpretable shape can be preferable to a more flexible one when data are limited, but fit and sensitivity should still be assessed.

Common pitfall: It is incorrect to interpret a Weibull shape parameter as a covariate effect, or to treat the Gompertz parameter $\gamma_{rs}$ as a hazard ratio. These are baseline-time-shape parameters. Covariate hazard ratios come from $\exp(\beta_{rs}^{T}(X_A-X_B))$. Also, a hazard ratio is not a probability ratio and does not by itself determine a transition probability, which depends on the full multi-state process and all relevant transition hazards.

Check Your Understanding

Try each question, then compare your reasoning with the solutions below.

A Weibull transition model has $\rho_{rs}=0.7$. Is its baseline hazard increasing, decreasing, or constant with elapsed time?
In a Gompertz model, $a_{rs}=0.020$ per year and $\gamma_{rs}=0$. What is the baseline hazard at 3 years?
A covariate coefficient is $\beta_{rs}=0.30$ for a one-unit increase in $X$. Give the hazard ratio and its time interpretation.
Why can a fitted Gompertz model with positive $\gamma_{rs}$ be risky for long-range extrapolation?

Solutions. (1) Since $0<\rho_{rs}<1$, the Weibull baseline hazard decreases with elapsed time. (2) When $\gamma_{rs}=0$, the Gompertz baseline is constant, so the hazard at 3 years is $a_{rs}=0.020$ per year. (3) The hazard ratio is $\exp(0.30)=1.350$ per one-unit increase in $X$, conditional on the model and at the same transition time; it is constant over time under proportional hazards. (4) A positive Gompertz slope makes the baseline hazard grow exponentially, so modest parameter error can lead to very large extrapolated hazards far beyond the observed follow-up period.