Why Compare Nested Transition Models?
A multi-state regression model can allow a covariate to have one shared effect across transitions or a different effect for each transition. The more flexible model can fit the observed data at least as well, but a better fit alone does not establish that the additional transition-specific terms are useful. A likelihood-ratio test asks whether the observed improvement is large relative to the improvement expected from fitting extra parameters when the restrictions in the simpler model are true.
The comparison must be between nested models fitted to the same data under the same likelihood framework. In a transition-specific proportional-hazards model, the transition intensity for transition \(r\) can be written \( \lambda_r(t\mid X)=\lambda_{0r}(t)\exp(X^\mathsf{T}\beta_r) \), where \(X\) is the covariate vector, \(\lambda_{0r}(t)\) is the baseline intensity for transition \(r\), and \(\beta_r\) is its regression coefficient vector. A reduced model might constrain some entries of the vectors \(\beta_r\), while retaining the same baseline-hazard specification.
Constructing the Test
Let \(\ell_F\) and \(\ell_R\) denote the maximized log likelihoods under the full and reduced models. The full model must contain all the reduced model's possibilities, so \(\ell_F\) cannot be smaller than \(\ell_R\), apart from numerical optimization error. The test statistic and its reference distribution are
Here \(df\) is the rank of the restrictions imposed by the reduced model on the full model. If the restrictions are linear and independent, this is the number of independent coefficient constraints. More generally, it is the rank of the restriction Jacobian at the null parameter value. Parameters estimated in both models without restriction do not add degrees of freedom just because they are numerous.
For example, suppose a covariate has coefficient \(\beta_r\) on each of \(K\) transitions. Testing the shared-effect restriction \(\beta_1=\cdots=\beta_K\) imposes \(K-1\) independent constraints, not \(K\): the common coefficient remains freely estimated. Testing that all \(K\) coefficients equal zero imposes \(K\) constraints. With \(q\) covariates, these counts become \((K-1)q\) and \(Kq\), respectively, provided all constraints are identifiable and independent.
Assumptions and a Short Justification
A usual chi-square likelihood-ratio test requires that the models be nested, fitted to the same analyzed observations and event histories, and based on the same likelihood definition. The null parameter value should be identifiable and lie in the interior of the parameter space; the likelihood should be sufficiently smooth, and the information for the tested parameters should be nonsingular. Independent subjects are ordinarily assumed, with censoring handled under the model's censoring assumptions. For transition models, covariate values, transition risk sets, event definitions, and any treatment of tied event times must be handled consistently in both fits.
The justification is Wilks' likelihood-ratio result. Under the null, the full model's extra parameter estimates fluctuate around the restrictions by amounts of order \(n^{-1/2}\). A second-order expansion of the log likelihood around the null estimate makes the gain from those fluctuations a quadratic form. With regular, identifiable parameters, that form converges to a chi-square distribution whose dimension is the number of independent directions released by the restrictions.
The sum on the right has a \(\chi^2_{df}\) distribution. This is an asymptotic result, not an exact finite-sample guarantee. For a Cox transition model, \(\ell\) in the comparison is the maximized partial log likelihood, and the familiar partial-likelihood ratio test also relies on large-sample regularity. For a fully parametric transition model, use the maximized likelihood specified by that model.
Worked Example: Testing Whether Two Transition Effects Are Shared
In a hypothetical study of 360 patients, investigators model two possible transitions out of an initial state. A binary treatment has coefficient \(\beta_1\) for transition 1 and \(\beta_2\) for transition 2. There are 92 and 74 observed transitions of the respective types. The full Cox model estimates separate coefficients; the reduced model imposes \(\beta_1=\beta_2\). Both fits use the same patients, risk-set definitions, covariates, and tie handling. Assume independent patient trajectories, adequate event information for both coefficients, and the usual non-informative censoring conditions for the fitted models.
The hypothetical maximized partial log likelihoods are \(\ell_F=-424.19\) and \(\ell_R=-426.70\). Thus
The null restriction \(\beta_1=\beta_2\) is one independent constraint, so \(df=1\). The upper-tail probability is \(P(\chi^2_1\ge 5.02)\approx 0.025\). At the 0.05 significance level, this is evidence against a shared treatment coefficient: the data show a statistically significant improvement in fit when the two transition effects are allowed to differ. It does not, by itself, identify which transition has the larger effect or establish that either effect is nonzero.
Choosing the Restrictions and Degrees of Freedom
State the scientific question as a set of coefficient restrictions before fitting the models. This prevents the common error of choosing degrees of freedom from a model label, or from the number of transitions alone. The comparison below uses a single covariate; with several covariates, count the independent restrictions across all tested coefficients.
| Question | Reduced-model restriction | Degrees of freedom |
|---|---|---|
| Does the covariate have any effect on either of two transitions? | \(\beta_1=0,\ \beta_2=0\) | 2 |
| Is one shared effect adequate for two transitions? | \(\beta_1=\beta_2\) | 1 |
| Is one shared effect adequate for three transitions? | \(\beta_1=\beta_2=\beta_3\) | 2 |
| Are all three transition-specific effects zero? | \(\beta_1=\beta_2=\beta_3=0\) | 3 |
The shared-effect test and the joint zero-effect test answer different questions. A shared effect can be large and nonzero, while a joint zero-effect test asks whether all transition coefficients vanish. Likewise, testing equality does not test whether the common effect is zero.
Worked Example: Jointly Removing Two Covariates Across Three Transitions
In a hypothetical cohort of 510 patients, a model includes age and baseline severity on each of three transitions. The observed transition counts are 140, 88, and 61. The full model estimates six coefficients: two covariates for each transition. The reduced model removes all six terms but retains the same transition-specific baselines and other adjustment terms. Assume both fits use the same patient histories and likelihood, the six terms are identifiable, and the model and censoring assumptions are adequate for this analysis.
The maximized log likelihoods are \(\ell_F=-607.00\) and \(\ell_R=-612.40\). The joint likelihood-ratio statistic is
There are two independent restrictions per transition and three transitions, giving \(df=3\times2=6\). Thus \(P(\chi^2_6\ge10.80)\approx0.095\). At \(\alpha=0.05\), the test does not reject the joint null that these six coefficients are zero. This is not proof that all effects are absent; it means the observed improvement does not meet the chosen threshold for rejecting that joint null. The test also does not isolate which covariate or transition might contribute an effect.
Worked Example: A Two-Restriction Test in a Parametric Model
Suppose a hypothetical analysis models one continuous covariate across three transitions using a piecewise-exponential model. The full model has separate coefficients \(\beta_1,\beta_2,\beta_3\), with the same time intervals and baseline structure in both fits. Investigators ask whether the first two effects can be treated as equal while the third effect is zero. The reduced model imposes \(\beta_1=\beta_2\) and \(\beta_3=0\). Assume independent patient trajectories, adequate event information in the relevant intervals, and a correctly specified intensity and censoring model.
The hypothetical maximized log likelihoods are \(\ell_F=-301.820\) and \(\ell_R=-305.495\). The statistic is
The equality restriction contributes one independent constraint, and setting the third coefficient to zero contributes another, so \(df=2\). The p-value is \(P(\chi^2_2\ge7.350)\approx0.0253\). At the 0.05 level, the data are inconsistent with this combined restriction under the model. This test does not separately establish that the first two coefficients differ or that the third is nonzero; those are distinct hypotheses requiring their own comparisons.
Diagnostics, Alternatives, and Limitations
Before interpreting a likelihood-ratio result, verify that the full model actually has at least as high a maximized log likelihood as the reduced model. A violation often indicates failed convergence, inconsistent data processing, or an incorrect comparison. Check that both fits used the same subjects, event histories, covariate definitions, time scale, baseline-hazard structure, and tie method. Also inspect whether the tested coefficients are estimable: empty transition risk sets, very few events, or covariate patterns that perfectly predict a transition can produce unstable estimates, singular information, or unreliable chi-square approximations.
A likelihood-ratio comparison is not appropriate for models that are not nested. Nor is the ordinary chi-square reference automatically valid when a null value lies on a parameter-space boundary, when parameters are unidentified under the null, or when data are too sparse for the asymptotic approximation to be credible. If estimates diverge or the likelihood is nearly flat, do not treat a computed p-value as a remedy for weak identification. The model structure and data support need investigation.
The score test for shared transition effects evaluates the likelihood's local slope at the restricted fit, whereas the likelihood-ratio test compares maximized likelihoods. A Wald test instead uses the estimated coefficients and their estimated covariance. These tests can differ in finite samples, especially with sparse data or strong nonlinearity; none removes the need to specify the correct restrictions. Information criteria such as AIC compare fit with a complexity penalty but do not provide the same null-hypothesis test. If robust, subject-clustered variance estimates are needed for dependence across transition contributions, an ordinary likelihood-ratio reference may not provide robust inference; use an inferential procedure appropriate to the dependence structure rather than assuming a sandwich variance changes the likelihood-ratio test.
Common Mistake
The practical value of the test is disciplined model comparison: it quantifies whether relaxing a specified set of transition-specific restrictions materially improves fit. Its conclusion is conditional on the chosen transition model, covariate specification, event-history data, and regularity assumptions. It does not replace checking whether the transition intensities and covariate effects are scientifically plausible.
Check Your Understanding
Answer each question, then compare with the solutions below.
- For four transitions, a single covariate has coefficients \(\beta_1,\ldots,\beta_4\). How many degrees of freedom test whether a shared coefficient is adequate?
- For three transitions and two covariates, what are the degrees of freedom for jointly testing that all six transition-specific coefficients are zero?
- A full and reduced model have log likelihoods \(-210.6\) and \(-214.1\). Calculate the likelihood-ratio statistic.
- Why is a comparison of two models with different analyzed patient histories not a standard nested likelihood-ratio test?
Solutions. (1) The shared-effect restriction imposes \(4-1=3\) independent equality constraints, so \(df=3\). (2) Setting six identifiable coefficients to zero imposes six independent restrictions, so \(df=6\). (3) \(G^2=2\{(-210.6)-(-214.1)\}=2(3.5)=7.00\). The degrees of freedom cannot be determined without knowing the restrictions. (4) The maximized likelihoods then summarize different analyzed data, so their difference does not isolate the fit improvement from relaxing the model restrictions on a common data set.