State-Dependent Covariate Effects

Why the Source State Matters

A covariate can have a different association with transition intensity depending on the state an individual currently occupies. For example, a treatment could be associated with a lower rate of leaving an initial state, but a different rate of leaving a later, post-event state. That is source-state modification: the covariate effect depends on the state from which a transition begins.

This is not the same as saying the covariate has a different effect on each possible destination. If two transitions leave the same source state, a source-state model gives them the same covariate coefficient, even though their baseline intensities can be quite different. The distinction is essential when interpreting a fitted model: source-state effects allow variation across origins, not arbitrary variation across every transition.

Definition: Let $g$ denote a source state, $h$ a destination state, $X$ a covariate, and $\lambda_{gh}(t\mid X)$ the transition intensity from $g$ to $h$. A source-state-specific proportional-effects model is $$ \lambda_{gh}(t\mid X)=\lambda^0_{gh}(t)\exp(\beta_g X), \qquad g\to h\in\mathcal{E}, $$ where $\mathcal{E}$ is the set of allowed transitions. The baseline intensity $\lambda^0_{gh}(t)$ may differ for every transition, but the covariate coefficient $\beta_g$ is shared by all transitions leaving source state $g$.

For a one-unit increase in $X$, the hazard ratio for any allowed transition out of $g$ is $\exp(\beta_g)$, conditional on the model’s other covariates. The coefficient can vary with $g$. If $X$ is binary, the same ratio compares $X=1$ with $X=0$; if it is continuous, the ratio is per one-unit increase, so the unit and any scaling must be stated.

What the Model Allows—and Restricts

Suppose the states are 0 (initial), 1 (an intermediate event), and 2 (absorbing), with allowed transitions $0\to1$, $0\to2$, and $1\to2$. The model assigns coefficients $\beta_0$ to both transitions out of state 0 and $\beta_1$ to the transition out of state 1:

\begin{aligned} \lambda_{01}(t\mid X)&=\lambda^0_{01}(t)\exp(\beta_0X),\\ \lambda_{02}(t\mid X)&=\lambda^0_{02}(t)\exp(\beta_0X),\\ \lambda_{12}(t\mid X)&=\lambda^0_{12}(t)\exp(\beta_1X). \end{aligned}

The first two baseline intensities need not be equal. Thus, the model does not say that the two destinations are equally likely or have the same transition rate. It says only that a one-unit covariate increase multiplies both intensities by the same factor. By contrast, the effect can differ between transitions out of states 0 and 1 because $\beta_0$ need not equal $\beta_1$.

A transition-specific model assigns a separate coefficient $\beta_{gh}$ to every transition. The source-state model is a restriction of that model: for each source $g$, it imposes $\beta_{gh}=\beta_g$ for all allowed destinations $h$. Consequently, estimates that differ by destination within the same source state may indicate that this restriction is inadequate. A state-dependent coefficient should not be described as destination-specific unless the model actually includes destination-specific effects.

Model interpretation: A source-state effect is indexed by where the individual is at risk of moving from. A destination-specific effect is indexed by which transition is being considered. Baseline intensities may vary by transition in either model; their differences do not by themselves imply different covariate effects.

Assumptions and a Useful Derivation

The proportional-intensity interpretation requires the covariate effect to be constant over time within each source state and shared across all destinations from that state. The model also requires a correctly specified set of allowed transitions, suitable measurement and coding of covariates, and an appropriate treatment of censoring and observation. In particular, conditional on the modeled covariate and observed history used in the analysis, censoring must not selectively remove individuals in a way that distorts the transition risk sets. The covariate values must be available at the relevant times; time-varying covariates require an appropriate time-varying formulation rather than treating future values as known.

A source-state effect does not, by itself, require the entire process to be Markov. It is a model for transition intensities conditional on the covariates and history specified. If the analysis further interprets the intensity as depending only on current state and covariates, that conditional structure must also be justified. The earlier material on transition-specific Cox regression provides the risk-set framework; the key additional restriction here is sharing the covariate coefficient among transitions with a common origin.

The restriction can be seen directly from the general transition-specific model. For a fixed source $g$, the intensity ratio comparing $X+1$ with $X$ on transition $g\to h$ is

\frac{\lambda_{gh}(t\mid X+1)}{\lambda_{gh}(t\mid X)} =\frac{\lambda^0_{gh}(t)\exp\{\beta_{gh}(X+1)\}} {\lambda^0_{gh}(t)\exp(\beta_{gh}X)} =\exp(\beta_{gh}).

The source-state model sets $\beta_{gh}=\beta_g$ for each destination $h$ reachable from $g$. Therefore, for any two such destinations $h_1$ and $h_2$, their covariate hazard ratios are equal:

\exp(\beta_{gh_1})=\exp(\beta_g)=\exp(\beta_{gh_2}).

This equality concerns relative effects, not absolute intensities. Because $\lambda^0_{gh_1}(t)$ and $\lambda^0_{gh_2}(t)$ can differ, the corresponding transition intensities can remain very different at every covariate value.

Worked Example: Identifying the Effect Being Modeled

Consider allowed transitions $0\to1$, $0\to2$, and $1\to2$, and a binary treatment indicator $X$. A fitted source-state model estimates $\beta_0=-0.945$ and $\beta_1=0.350$. Identify the treatment hazard ratio for each transition and explain which comparisons the model does and does not permit.

For transitions $0\to1$ and $0\to2$, the source is state 0, so the estimated hazard ratio for treated versus untreated individuals is $\exp(\beta_0)$:

\exp(-0.945)\approx 0.3887.

Under the model, treatment is associated with an estimated transition intensity about 0.3887 times as large for each of those two transitions, conditional on the model. Equivalently, the intensity is estimated to be about 61.13% lower, since $1-0.3887=0.6113$. For $1\to2$, the source is state 1, so the estimated hazard ratio is

\exp(0.350)\approx 1.419.

Thus, treatment is associated with an estimated intensity about 1.419 times as large for $1\to2$. The model allows this difference between source states. It does not allow the treatment hazard ratio for $0\to1$ to differ from the ratio for $0\to2$. It does allow the baseline intensities for those transitions to differ.

Worked Example: A Confidence Interval for a Source-State Effect

In a hypothetical analysis, the estimated treatment coefficient for transitions out of state 0 is $\hat{\beta}_0=-0.945$, with standard error $SE=0.2059$. Assume the large-sample normal approximation for the coefficient is adequate. Calculate a 95% confidence interval for the coefficient and for the hazard ratio.

The coefficient-scale Wald interval is the estimate plus or minus 1.96 standard errors:

-0.945\pm1.96(0.2059) =-0.945\pm0.403564 =(-1.348564,-0.541436).

Exponentiating the estimate gives $\exp(-0.945)\approx0.3887$. Exponentiating both endpoints gives the 95% confidence interval for the hazard ratio:

\left(\exp(-1.348564),\exp(-0.541436)\right) \approx(0.2596,0.5819).

Under the model, this interval estimates the treatment-to-no-treatment intensity ratio for each transition out of state 0. Its endpoints are below 1, indicating that the interval is compatible with lower transition intensities under treatment for these transitions. In repeated sampling under the model and regularity conditions, 95% of intervals constructed this way would cover the true coefficient; this is not a 95% probability statement about this particular fixed parameter.

For a two-sided test of $H_0:\beta_0=0$, the Wald statistic is $z=-0.945/0.2059\approx-4.590$, giving a two-sided $p$-value below 0.00001. At conventional significance levels, the data provide evidence against a zero treatment coefficient for transitions out of state 0. This test does not establish causation and does not show that the two destination-specific effects are different; the model constrains them to be equal.

Comparing Effects Across Source States

To ask whether the covariate effect differs by source state, compare the relevant coefficients. For two sources $g$ and $k$, the null hypothesis of equal log-intensity effects is $H_0:\beta_g-\beta_k=0$. The corresponding ratio of hazard ratios is $\exp(\beta_g-\beta_k)$. A valid standard error for the difference must use the covariance between the estimates:

SE(\hat{\beta}_g-\hat{\beta}_k) =\sqrt{\operatorname{Var}(\hat{\beta}_g)+\operatorname{Var}(\hat{\beta}_k) -2\operatorname{Cov}(\hat{\beta}_g,\hat{\beta}_k)}.

Ignoring the covariance is not generally justified when the estimates come from the same fitted model. The contrast and its uncertainty should be computed from the model’s joint covariance estimate.

Worked Example: Testing Whether Two Source-State Effects Differ

Suppose a hypothetical model gives $\hat{\beta}_0=-0.945$ with standard error 0.2059 and $\hat{\beta}_1=0.350$ with standard error 0.1800. Suppose the estimated covariance is 0.00600. Calculate the estimated contrast $\hat{\beta}_1-\hat{\beta}_0$, its standard error, and a Wald interval. Assume the joint large-sample normal approximation is appropriate.

The estimated difference in log-intensity effects is $0.350-(-0.945)=1.295$. Its standard error is

\sqrt{0.2059^2+0.1800^2-2(0.00600)} =\sqrt{0.06279481} \approx0.2506.

The Wald statistic for equality is $1.295/0.2506\approx5.167$. The 95% confidence interval for the coefficient difference is

1.295\pm1.96(0.2506) \approx(0.8038,1.7862).

Exponentiating the contrast gives $\exp(1.295)\approx3.651$, the estimated ratio of the hazard ratio for transitions out of state 1 to the hazard ratio for transitions out of state 0. The exponentiated interval is approximately $(2.234,5.967)$. The interval excludes 1, consistent with evidence that the covariate effects differ by source state. This is a comparison of source-state effects, not evidence that destination-specific effects differ within either source.

Intensities, Destinations, and Diagnostics

The distinction between relative and absolute rates matters in practical interpretation. Suppose, hypothetically, that for untreated individuals at a particular time the baseline intensities out of state 0 are 0.0400 per year for $0\to1$ and 0.0200 per year for $0\to2$. With $\hat{\beta}_0=-0.945$, the corresponding treated intensities under the model are

0.0400\exp(-0.945)\approx0.01555\ \text{per year}, \qquad 0.0200\exp(-0.945)\approx0.007774\ \text{per year}.

The treatment multiplier is the same, but the absolute intensities remain different. Also, an intensity is an instantaneous transition rate among individuals currently at risk of that transition; it is not itself a probability of eventually reaching the destination. In competing-risks settings, the other outgoing intensities also affect which transition occurs first.

A source-state model should be questioned when the equal-effect restriction within a source is implausible or inconsistent with the data. Useful checks include examining transition-specific estimates or plots, comparing their direction and magnitude within each origin, and assessing whether estimated effects appear to change over time. Sparse events for one transition can make apparent differences unstable, so uncertainty matters. A strong discrepancy among well-estimated effects sharing a source state may motivate a more flexible transition-specific model; a time pattern may instead suggest that the proportional-effects assumption needs attention.

These checks do not turn a visual discrepancy into a formal test, and failure to see a discrepancy does not prove the restrictions are true. The source-state model can be more precise when the shared-effect assumption is reasonable, because it pools information across outgoing transitions. That gain comes at the cost of bias if the true covariate effects differ by destination. A fully transition-specific model avoids that equality restriction but estimates more parameters and may be imprecise, particularly for rare transitions. The model should be selected to match the scientific question and supported by the available event information.

Common Mistakes and Limits

Calling a source-state effect destination-specific. The coefficient is indexed by the state an individual leaves. Transitions from that source share it, even when their destinations differ.
Equating baseline intensities. Sharing $\beta_g$ does not require the baseline intensities for outgoing transitions to be equal.
Comparing coefficients without their covariance. A difference test needs the variance of the contrast, including the covariance term.
Interpreting an intensity ratio as a probability ratio. A hazard ratio compares instantaneous intensities among individuals at risk, not cumulative probabilities of transition.
Overstating an observational association. An estimated covariate effect is conditional on the model and does not, by itself, identify a causal effect.

The model also has limited scope: it assumes a proportional covariate effect over time within each source state, and a shared effect over all destinations from that source. If either restriction is scientifically untenable, the coefficient may be a misleading summary. Interpretation further depends on the chosen covariate scale, correct transition structure, and adequate information in the relevant risk sets.

Key takeaway: In a source-state-specific model, the covariate hazard ratio changes with the origin state, while remaining common across destinations from the same origin. Separate baseline intensities preserve transition-specific rate differences; comparing source-state effects requires a contrast with its joint covariance.

Check Your Understanding

Answer each question, then check the concise solutions below.

A model gives $\beta_2=0.20$ for all transitions out of state 2. What hazard ratio corresponds to a one-unit increase in the covariate?
Two transitions leave the same state but have very different baseline intensities. Does the source-state model require their covariate hazard ratios to differ?
For two source-state coefficients, what information beyond the two standard errors is needed to calculate the standard error of their difference?
Why is a transition intensity ratio not generally the same as a ratio of cumulative transition probabilities?

Solutions. (1) The hazard ratio is $\exp(0.20)\approx1.221$: a one-unit increase multiplies each modeled intensity out of state 2 by about 1.221. (2) No. The baseline intensities can differ, while the common source-state coefficient imposes the same covariate hazard ratio on both transitions. (3) The estimated covariance between the two coefficient estimates is needed, using $SE(\hat{\beta}_g-\hat{\beta}_k)=\sqrt{\operatorname{Var}(\hat{\beta}_g)+\operatorname{Var}(\hat{\beta}_k)-2\operatorname{Cov}(\hat{\beta}_g,\hat{\beta}_k)}$. (4) An intensity is an instantaneous rate among individuals still at risk; cumulative transition probabilities also depend on time and on competing outgoing transitions.