Regression for Competing Destinations From One State

Why Separate Destination Effects Matter

A person in one state may leave it through either of two destinations—for example, a disease state may end in recovery or death. A covariate can affect the instantaneous rate of these transitions differently. A regression model that forces the effect to be the same for both destinations may hide that difference; separate models, in turn, need a formal comparison if the question is whether the effects differ.

This tutorial develops that comparison for two transitions from a shared source state. The destinations are absorbing, so a person who reaches either destination no longer contributes time at risk in the source state. The focus is on transition hazards and their regression coefficients, not on predicting destination probabilities.

A Piecewise-Exponential Regression Model

Divide follow-up time into intervals \(I_j\). Within interval \(I_j\), specify a separate baseline intensity for each destination and a possibly different coefficient vector for each transition:

Here \(\lambda_{kj}>0\) is the baseline intensity for destination \(k\) in interval \(j\), \(x\) is the covariate vector, and \(\beta_k\) is the vector of transition-specific coefficients. A one-unit increase in covariate \(x_r\), holding other covariates fixed, multiplies the intensity of transition \(0\to k\) by \(\exp(\beta_{kr})\). This is a hazard ratio for that transition, not a ratio of overall destination probabilities.

The model assumes that each transition intensity is constant within each specified interval, that the chosen covariate effects and their functional forms are correctly specified, and that censoring is independent of the unobserved transition times conditional on the modeled covariates. Covariates must be defined at the appropriate time: a baseline covariate stays fixed, whereas a time-varying covariate must be updated using information available at that time. The two transition intensities jointly determine the rate of leaving state 0; they do not describe two independent opportunities to remain at risk after departure.

Likelihood and the Coefficient-Comparison Question

For a person observed in state 0 until time \(T\), let \(\delta_k=1\) if the observed exit is to destination \(k\), and let both indicators be zero if follow-up is censored without an observed transition. Conditional on the covariates, the likelihood contribution is

The exponential term accounts for remaining in state 0 up to time \(T\), and the event factor identifies which destination occurred. Taking logarithms gives a sum of one term for each transition. In piecewise-exponential data, if \(E_{ikj}\) denotes person \(i\)'s time at risk in interval \(j\) and \(d_{ikj}\) indicates an observed transition to \(k\) there, the contribution has the form

This factorization explains how each destination contributes event and exposure information to its transition-specific regression. It does not mean each person can experience both competing events during the same spell in state 0: after the first transition, that person leaves the risk set. If the scientific question is whether covariate \(x_r\) has the same effect on both intensities, the null hypothesis is \(H_0:\beta_{1r}=\beta_{2r}\). The alternative is that the two coefficients differ.

Testing Whether the Effects Differ

A Wald test uses the estimated contrast and its estimated covariance. In general, for \(D=\hat\beta_{1r}-\hat\beta_{2r}\),

Under \(H_0\) and standard large-sample regularity conditions, \(Z\) is approximately standard normal. The covariance term must be included when available; treating estimates as independent without justification can give the wrong standard error. A confidence interval for the coefficient difference is \(D\pm z_{1-\alpha/2}SE(D)\), where \(z_{1-\alpha/2}\) is the standard normal quantile. Exponentiating its endpoints gives an interval for the ratio of the transition-specific intensity ratios.

Constrained Models and a Likelihood-Ratio Test

An alternative is to fit two nested models with the same data, covariates, time intervals, and baseline-intensity specification. The unrestricted model estimates \(\beta_{1r}\) and \(\beta_{2r}\) separately. The constrained model imposes \(\beta_{1r}=\beta_{2r}\), while retaining destination-specific baseline intensities. The likelihood-ratio statistic is \(2(\ell_{\text{unrestricted}}-\ell_{\text{constrained}})\). Under \(H_0\), regularity conditions, and adequate information, it is approximately chi-squared with one degree of freedom for this single equality constraint.

Diagnostics, Alternatives, and Limitations

The comparison is only as credible as the two transition models. Examine event counts and person-time separately by destination, covariate pattern, and time interval. Very few events can produce unstable estimates, large standard errors, or near-separation. Check whether the chosen time intervals are adequate: marked variation in estimated rates within an interval can indicate that a piecewise-constant baseline is too coarse. Inspect the covariate specification as well; a linear effect on the log-intensity scale may be inappropriate for a continuous predictor, and an effect may vary over time even when the model assumes a constant coefficient.

A standard alternative is a stacked transition-regression design with a transition indicator and interactions between that indicator and the covariates. The interaction for \(x_r\) represents the difference between its transition-specific coefficients, so testing that interaction is equivalent to testing equality in the separate-coefficient parameterization. Whichever representation is used, retain destination-specific baseline terms unless a shared baseline is scientifically justified. If subjects contribute multiple spells or dependence is otherwise expected, use an appropriate variance estimator, such as a subject-clustered sandwich estimator; account for covariance in the contrast.

A common mistake is to compare the two coefficient estimates by eye, or to conclude that effects differ because one transition’s coefficient is statistically significant and the other’s is not. Those are not tests of \(H_0:\beta_{1r}=\beta_{2r}\). The correct comparison estimates the difference and its uncertainty, or directly compares constrained and unconstrained likelihoods. Another mistake is to describe \(\exp(\beta_{kr})\) as the relative probability of reaching destination \(k\). It is a transition-specific hazard ratio conditional on being in the source state; it is not an overall risk ratio.

Finally, this model does not make competing exits independent in the sense of allowing a person to remain at risk for both after the first exit. It describes the two intensities while the person occupies the source state. The fitted coefficients are conditional, model-based associations; causal interpretation requires additional design and confounding assumptions beyond those stated here.