Identifiability of Transition-Specific Effects

Why Identifiability Is a Data-Support Question

A regression model can assign a separate coefficient to every transition and covariate, but writing down a coefficient does not ensure the data can determine it. If two terms make exactly the same contribution to every observed transition likelihood, the data identify only their combined effect. If a transition has events but no covariate variation in the relevant risk sets, its transition-specific covariate effect may likewise be inseparable from the baseline hazard.

Identifiability is therefore about what distinct parameter values imply for the distribution of the observed data. It is not the same as whether an algorithm converges, whether a standard error is small, or whether an event count exceeds a rule of thumb. The relevant evidence is the pattern of covariates and transition events, considered together with the model’s baseline and other regression terms.

Model and Rank Condition

Let \(r\) index an allowed transition, and let \(x\) denote a vector of covariates. A transition-specific proportional-hazards model can be written as

where \(\lambda_{0r}(t)\) is the baseline transition hazard and \(\beta_r\) is the vector of coefficients for transition \(r\). The coefficient for a particular covariate may differ across transitions. The model presumes that the stated transition hazards and covariates are correctly defined, that subjects’ observation and censoring mechanisms are handled appropriately by the likelihood, and that the chosen baseline and coefficient parameterization is used consistently. Identifiability is assessed within that parameterization; changing constraints or sharing parameters can change which combinations are identified.

For a piecewise-exponential representation, divide follow-up into intervals in which each transition’s baseline hazard is constant. If \(Y_{rg}\) is the event count and \(E_{rg}\) the person-time exposure for transition \(r\) and covariate pattern \(g\), then the Poisson likelihood has mean

for an interval \(k\), where \(\alpha_{rk}\) is its log baseline hazard. The offset is \(\log E_{rg}\). In this formulation, the design matrix includes columns for baseline interval terms and for the regression terms. A rank deficiency means some linear combination of columns is redundant.

More precisely, write the linear predictor as \(Z\gamma+X\beta\), where \(Z\gamma\) contains nuisance baseline terms and \(X\beta\) contains the coefficients of interest. The coefficients of interest are identifiable from this design if every vector \((a,b)\) satisfying \(Za+Xb=0\) has \(b=0\). If a nonzero \(b\) can be offset by changing the nuisance parameters \(a\), at least one coefficient combination is aliased. Equivalently, after removing the part of \(X\) explained by \(Z\), the remaining columns must have full rank.

Why Rank Detects Aliasing

The rank condition has a direct likelihood justification. In a Poisson log-linear model, the likelihood depends on the parameters through the fitted means \(\mu_i=\exp(o_i+X_i^\mathsf{T}\theta)\), where \(o_i\) is a known offset and \(X_i\) is a design row. If \(Xv=0\), then for any scalar \(c\),

Thus \(\theta\) and \(\theta+cv\) give the same fitted means and the same likelihood. The data cannot select one value along that direction. With nuisance baseline terms present, apply this argument to the combined design \([Z\;X]\): a null direction that changes a coefficient of interest while compensating with baseline terms establishes nonidentifiability of that coefficient combination.

For a Cox model, event counts and covariate values among the corresponding risk sets matter. At an event time for transition \(r\), the partial likelihood compares the covariate value of the subject making the transition with values among subjects still at risk of that transition. A direction \(v\) contributes no information if \(x^\mathsf{T}v\) is constant within every relevant risk set: it cancels from each event-time comparison. The information matrix is built from weighted covariate variation within those risk sets. Consequently, counting event records or checking rank of the event subjects’ covariate values alone is not sufficient for a Cox-model identifiability assessment.

Examples of Aliasing and Missing Contrasts

Event Counts, Diagnostics, and Failure Modes

Event counts are important because a transition coefficient is informed by events for that transition. If no events occur for a transition, its Cox partial likelihood has no event contribution for that transition. In a parametric model with an unrestricted transition-specific baseline, the data also provide no ordinary event-based contrast for its regression coefficient; boundary estimates of a baseline rate do not create information about the coefficient. But a positive count is only a prerequisite for useful information in many settings, not proof of identifiability. As the example above shows, events can occur without a covariate comparison.

A practical assessment should be aligned with the fitted likelihood and parameterization. For a piecewise-exponential model, inspect the design columns for the event-exposure cells actually represented, including baseline interval terms, and check whether the coefficients of interest retain full rank after nuisance terms are accounted for. For a Cox model, inspect covariate variation among the transition-specific risk sets at event times, not just the covariate distribution among people who experienced events. In either model, look for empty transition-by-covariate cells, covariates that are constant within a transition, and terms that are identical or exact linear combinations.

A rank check is not the only diagnostic. Near-dependence rather than exact dependence can produce very large standard errors and estimates that change substantially with small data changes. Few events or little covariate variation can yield weak information even when the relevant matrix technically has full rank. Complete or quasi-complete separation can also lead to an estimate tending to infinity: that is a boundary or existence problem, distinct from exact rank deficiency. Convergence messages alone cannot distinguish these problems, so examine the design, estimates, standard errors and event patterns together.

What Alternatives Can and Cannot Do

A shared-effect model, in which a covariate has one coefficient across transitions, uses a restriction to pool information. That can make a common effect estimable when separate transition-specific effects are unsupported, but it changes the scientific assumption: the transition effects are constrained to be equal. A model that removes an aliased covariate or combines redundant terms likewise changes the parameterization. Such decisions should be justified by the scientific question, not made solely to obtain a numerical answer.

Penalized methods, including ridge or lasso penalties, can return finite numerical solutions in rank-deficient or sparse settings. The penalty selects among otherwise equivalent or weakly distinguished values; it does not make the original unpenalized, separately specified effects identifiable from the data. Likewise, a robust variance estimate changes uncertainty estimation, not the rank of the information supplied by the observed event patterns.

The main limitation of rank-based reasoning is that it addresses exact distinguishability, not adequacy of the model, precision, or causal interpretation. A full-rank design does not guarantee a reliable estimate when events are sparse, nor does it validate the baseline hazard, proportional-hazards assumption, censoring assumptions or covariate specification. Identifiability is a necessary foundation for interpreting a transition-specific coefficient, not a substitute for the rest of the analysis.

A Practical Assessment