Why Adjust Within a Transition Risk Set?
A transition hazard compares instantaneous event rates among people who are currently eligible to make that transition. Those people may differ in prognostic characteristics as well as in the exposure of interest. For example, among people currently in a particular disease state, treatment groups might differ in baseline severity. Covariate adjustment compares their transition hazards conditional on the included characteristics; it does not change which transition is being modeled or who is at risk for it.
The distinction matters because a transition-specific hazard ratio is not a comparison of overall event probabilities. It is a conditional, instantaneous-rate comparison among people in the relevant transition risk set. Its interpretation depends on the risk set, the covariates included, and the model assumptions.
Let \(N_{i,hk}(t)\) count the \(h\)-to-\(k\) transitions made by individual \(i\) by time \(t\), and let \(Y_{i,h}(t)\) indicate whether that individual is in state \(h\) and at risk just before \(t\). For a transition-specific Cox model with fixed covariates \(Z_i\), write
Here \(\lambda_{0,hk}(t)\) is the unspecified baseline hazard for this particular transition, and \(\beta_{hk}\) is its covariate coefficient vector. The transition intensity for the counting process is \(Y_{i,h}(t)\lambda_{i,hk}(t\mid Z_i)\): it is zero for this transition when the individual is not at risk. The subscript \(hk\) emphasizes that both the baseline hazard and covariate coefficients pertain to the specified transition.
What the Adjusted Coefficient Compares
Suppose \(A_i\) is a binary exposure and \(W_i\) is a set of adjustment covariates. If the model is
then, for two individuals in the same transition risk set at the same time and with the same \(W\), the modeled hazard ratio comparing \(A=1\) with \(A=0\) is
This is a conditional hazard ratio: it compares individuals with the same modeled adjustment covariates, conditional on being at risk for the transition. It does not by itself describe a causal effect. A causal interpretation requires additional design and identification assumptions, including adequate control of confounding and appropriate handling of selection into the state. Nor does this coefficient directly give a difference in transition probabilities or cumulative incidence.
For this model to be appropriate, the transition and its risk set must be specified correctly; the covariates must be available and defined for individuals while they are at risk; and the log-hazard must have the stated covariate form. The proportional-hazards assumption says that the coefficient effects do not change with time for this transition. Censoring must not be informative about the transition, conditional on the modeled information, in a way that distorts the risk sets. The model also needs enough events and relevant covariate variation to estimate its coefficients. These conditions are assumptions to assess, not consequences of fitting the model.
Why the Partial Likelihood Uses the Transition Risk Set
At a time when one individual makes the \(h\)-to-\(k\) transition, the Cox partial likelihood compares that individual with all individuals in the risk set for this same transition. If the observed event is individual \(i\), its contribution, in the absence of tied event times, is
To see why, condition on the occurrence of exactly one target transition at that event time and on its being made by someone in the risk set. Under the model, each at-risk individual’s instantaneous rate for this transition is proportional to \(\exp(\beta_{hk}^{\mathsf T}Z_j)\). Dividing individual \(i\)'s rate by the sum of those rates gives the conditional probability that \(i\) is the one who makes the transition. The common baseline hazard cancels:
Multiplying these contributions over target-transition event times yields the partial likelihood when event times are distinct. Tied events require a specified tie-handling method. The risk set in the denominator is crucial: individuals who have already left state \(h\) are no longer at risk for this transition. A departure from \(h\) through a different outgoing transition ends that individual’s time at risk for the target-transition model; it is not counted as an \(h\)-to-\(k\) event.
Worked Example: One Adjusted Partial-Likelihood Contribution
Consider a hypothetical risk set for transition \(h\)-to-\(k\) at an event time. Four individuals are in state \(h\): the target-transition event is made by an individual with exposure \(A=1\) and severity \(W=1\). The other three have covariates \((A,W)=(1,0),(0,1),(0,0)\). Evaluate this event’s partial-likelihood contribution at \(\beta_A=\log(2)\) and \(\gamma=\log(3)\).
The event individual’s relative hazard is \(\exp\{\log(2)+\log(3)\}=6\). The other three relative hazards are \(2\), \(3\), and \(1\), respectively. Therefore the adjusted contribution is
At these trial coefficient values, the model assigns probability \(0.500\) to this individual being the one who makes the target transition, conditional on one such event occurring in this risk set. If severity were omitted, the corresponding exposure-only contribution would be \(2/(2+2+1+1)=0.333\). This difference illustrates how adjustment changes the risk-set comparison: the event individual has both covariates associated with higher modeled hazard in the adjusted model. One event contribution is not an estimate of either coefficient; estimation uses the contributions across event times.
Worked Examples of Conditional Interpretation
Worked Example: Interpreting a Fitted Adjusted Hazard Ratio
In a hypothetical study, researchers model a specified transition from a disease state to recovery. Treatment is coded \(A=1\) for the treatment group and \(A=0\) for the comparison group. The model also adjusts for baseline severity, coded \(W=1\) for high severity and \(W=0\) for low severity. Suppose the fitted treatment coefficient is \(0.58\), with standard error \(0.22\).
The adjusted hazard ratio is \(\exp(0.58)=1.786\), or about \(1.79\). A Wald 95% confidence interval for the coefficient is \(0.58\pm1.96(0.22)=(0.149,1.011)\). Exponentiating its endpoints gives the interval \((1.16,2.75)\). The Wald statistic is \(z=0.58/0.22=2.64\), giving a two-sided p-value of about \(0.0083\).
Under the fitted proportional-hazards model, at the same time and within the same severity level, the treatment group has an estimated instantaneous recovery-transition hazard about \(1.79\) times the comparison group’s hazard among individuals currently at risk for that transition. The confidence interval is a procedure that, over repeated samples under the model and its assumptions, would cover the conditional hazard ratio 95% of the time. It is not a 95% probability statement about this fixed parameter. The p-value describes how incompatible a test statistic at least this extreme is with a zero treatment coefficient under \(H_0\); it is not the probability that \(H_0\) is true. This interpretation is conditional on the specified risk set and model and is not, on its own, a causal conclusion.
Worked Example: Different Covariate Mixes Can Reverse a Crude Comparison
Suppose a hypothetical study records person-time at risk for one transition in two severity groups. The rates below are event counts divided by person-time; all person-time is measured while individuals are eligible for that transition.
| Exposure group | Severity | Events | Person-time | Rate per 100 person-time units |
|---|---|---|---|---|
| Exposed | Low | 18 | 900 | 2.00 |
| Exposed | High | 8 | 100 | 8.00 |
| Unexposed | Low | 1 | 100 | 1.00 |
| Unexposed | High | 36 | 900 | 4.00 |
Within the low-severity group, the rate ratio is \(2.00/1.00=2.00\). Within the high-severity group, it is \(8.00/4.00=2.00\). But the exposed group contributes \(900+100=1{,}000\) person-time units and \(18+8=26\) events, for a crude rate of \(26/1{,}000=0.026\) per unit, or \(2.60\) per 100 units. The unexposed group contributes \(1{,}000\) person-time units and \(37\) events, for \(3.70\) per 100 units. The crude rate ratio is therefore \(2.60/3.70=0.703\), even though the within-severity rate ratios are both 2.00.
The groups have very different severity distributions, and severity is associated with the transition rate. A model that adjusts for severity targets a within-severity comparison rather than this crude aggregate comparison. These are person-time rates, not necessarily the marginal hazard ratio from a Cox model: marginal hazards can change as the composition of surviving risk sets changes over time. The example shows why crude and adjusted comparisons can differ, not that one number is automatically causal or universally preferable.
Model Checks, Alternatives, and Limitations
Before interpreting an adjusted coefficient, verify that the analysis still corresponds to the intended transition. Check that each event is assigned to the correct transition, that people enter and leave its risk set at the right times, and that covariates are represented consistently. Examine whether the covariate form and proportional-hazards assumption are plausible for the transition. Sparse events, near-constant covariates within risk sets, or little overlap between exposure groups can make estimates unstable or dependent on extrapolation. If people contribute multiple transitions, dependence among their observations can also affect uncertainty estimates; the variance method must be suitable for the data structure.
A crude transition-specific analysis omits adjustment and can be useful as a descriptive comparison, but it does not control for measured covariate differences. A transition-specific adjusted Cox model allows the covariates to explain differences in hazards while retaining the same target transition. A pooled model that imposes common effects across transitions answers a different modeling question unless transition-specific effects are allowed. Alternatively, stratifying on a covariate permits separate baseline hazards across its strata but does not estimate a single coefficient for that covariate in the same way as adjustment. The choice depends on the scientific target and the assumptions one is prepared to make.
Adjustment has limits. It cannot correct confounding by unmeasured or poorly measured variables, and it does not guarantee comparable exposure groups within every risk set. Conditioning on having entered state \(h\) can itself create selection, so the adjusted hazard ratio is specific to those who are currently in that state and at risk. Adjusting for a variable affected by the exposure can also change the question being answered and may introduce bias. A coefficient’s interpretation should name the transition, the risk-set condition, and the covariates held fixed.
Practical Sequence for an Adjusted Transition Model
Specify the origin state, destination state, and event that counts as the target transition.
Include individuals in the origin state and eligible for the transition immediately before each event time.
Identify the covariates, their forms, and the transition-specific coefficients; state the proportional-hazards and censoring assumptions.
Check event coding, covariate support, model form, and whether the assumptions are plausible for this transition.
Report the conditional hazard ratio with uncertainty, naming the transition and the covariates held fixed; avoid turning it into a probability or causal claim without further justification.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- For a model of transition \(h\)-to-\(k\), who belongs in its risk set immediately before time \(t\)?
- A fitted binary-exposure coefficient is \(\log(1.5)\). State the adjusted hazard-ratio interpretation when the model also includes severity.
- Why does the common baseline hazard disappear from a transition-specific Cox partial-likelihood contribution?
- In the rate example, why can the crude rate ratio be below 1 when both within-severity rate ratios equal 2?
- Does adjusting for measured covariates alone make the exposure coefficient causal? Give one reason for your answer.
Solutions. (1) Individuals in state \(h\) who are eligible to make the \(h\)-to-\(k\) transition immediately before \(t\). (2) At the same time, among individuals currently at risk for that transition and with the same modeled severity, the exposure group’s hazard is estimated to be 1.5 times the unexposed group’s hazard. (3) It is a common multiplicative factor in every at-risk individual’s transition rate, so it cancels when the event individual’s rate is divided by the sum of risk-set rates. (4) The exposed group contributes mostly low-severity person-time, while the unexposed group contributes mostly high-severity person-time; aggregation therefore compares differently composed groups. (5) No. Unmeasured confounding, selection into the origin state, or other violations of causal identification assumptions can remain after adjustment.