Score Tests for Shared Transition Effects

Why Test for a Shared Transition Effect?

A covariate may have a different association with the hazard of each transition. For example, a treatment indicator might have separate regression coefficients for progression and death. A shared-effect model assumes these coefficients are equal. A score test asks whether the data provide evidence against that restriction, using the fit under the shared-effect assumption rather than fitting the fully transition-specific model.

The test is especially useful when the shared-effect model is relatively simple to fit, while the unrestricted model has many transition-specific parameters. Its central calculation is the score and information evaluated at the restricted fit. The resulting statistic measures how strongly the likelihood points away from the shared-effect restriction, after accounting for nuisance parameters.

Model, Hypothesis, and Assumptions

For clarity, parameterize the unrestricted model with a shared coefficient \(\beta\) and deviations \(\delta_2,\ldots,\delta_K\): set \(\beta_1=\beta\) and \(\beta_j=\beta+\delta_j\) for \(j=2,\ldots,K\). Equality is then the null hypothesis \(\delta=0\), where \(\delta\) stacks all deviation coefficients. Let \(\gamma\) collect nuisance parameters, including the shared coefficient and any transition-specific baseline or other nuisance terms. Write the log likelihood as \(\ell(\gamma,\delta)\).

The score \(U\) is the vector of first derivatives of \(\ell\), and the expected information \(I\) is the negative expected second-derivative matrix. The observed information, the negative Hessian evaluated at the fitted parameters, is also used in standard implementations. The model and information calculation must be specified consistently.

The usual large-sample score test assumes a correctly specified likelihood, a well-defined restricted maximum-likelihood estimate, differentiability near that estimate, and nonsingular information for the tested directions after nuisance adjustment. Independent subjects, or a correctly specified likelihood for their contributions, are needed for the conventional model-based reference distribution. The transition risk sets and covariate histories must also be represented correctly. In a piecewise-exponential model, for example, the person-time exposure and event counts must correspond to the stated intervals and transition risks.

The restrictions must be identifiable: the data must contain information that distinguishes the transition-specific coefficients. A coefficient for a transition with no events, or with no covariate variation in its risk set, may not be estimable. In that situation, the nominal number of restrictions is not automatically the test’s degrees of freedom.

Deriving the Score Statistic

Fit the model under \(H_0\), obtaining the restricted estimate \(\tilde{\gamma}\) with \(\delta=0\). Because \(\tilde{\gamma}\) maximizes the restricted likelihood, its nuisance score satisfies \(U_\gamma(\tilde{\gamma},0)=0\). The tested score \(U_\delta(\tilde{\gamma},0)\), however, need not be zero: it records the local direction in which freeing the transition deviations could improve fit.

The nuisance parameters can change as the deviations move away from zero. To account for that, use a local quadratic expansion of the log likelihood around the restricted estimate. For small changes \(a\) in \(\gamma\) and \(d\) in \(\delta\), its change is approximately \(U_\delta^\mathsf{T}d-\frac12(a^\mathsf{T}I_{\gamma\gamma}a+2a^\mathsf{T}I_{\gamma\delta}d+d^\mathsf{T}I_{\delta\delta}d)\). Maximizing this quadratic over the nuisance change gives \(a=-I_{\gamma\gamma}^{-1}I_{\gamma\delta}d\). Substitution shows that the information for the tested direction, after allowing nuisance parameters to adjust, is the Schur complement below.

This adjustment removes information about \(\delta\) that is attributable to its association with nuisance parameters. At the restricted estimate, the nuisance score is zero, so the efficient score for \(\delta\) is \(U_\delta\) itself. The score statistic and its usual large-sample reference distribution are:

With a scalar tested parameter \(\delta\) and a scalar nuisance parameter \(\theta\), the efficient information is \(I_{\delta\delta}-I_{\delta\theta}^2/I_{\theta\theta}\). This simple form makes clear why using \(I_{\delta\delta}\) alone can give an incorrect statistic: it ignores the nuisance adjustment.

Worked Tests With Multiple Restrictions

For several restrictions, the efficient information is a matrix and the score statistic combines their directions. The degrees of freedom are the rank of the tested information, not necessarily the number of transitions by itself. They equal \((K-1)p\) only when all the intended transition contrasts are estimable.

Practical Interpretation, Diagnostics, and Alternatives

The score test is attractive because it requires the restricted fit but not the unrestricted maximum-likelihood estimate. It is a local test: it assesses the likelihood’s slope in the directions that would separate the transition effects. Like other large-sample tests, it can be unreliable when event counts are sparse, when a tested coefficient is near a boundary or poorly identified, or when information is nearly singular. A finite score statistic does not fix these problems.

Check that the restricted model converged, that event and exposure records were constructed correctly for every transition, and that the tested information matrix is positive definite in the directions being tested. Very large standard errors, extreme fitted coefficients, or a rank-deficient information matrix indicate that the nominal chi-square calculation may not be trustworthy. Also check substantive model assumptions such as the chosen baseline hazard form and covariate specification; a score test of equality does not validate those assumptions.

The likelihood-ratio test compares maximized likelihoods under restricted and unrestricted models, while the Wald test evaluates the estimated unrestricted coefficient contrasts relative to their estimated uncertainty. In regular large samples the three tests are often similar, but they can differ in finite samples. The score test avoids relying on a potentially unstable unrestricted estimate, whereas a Wald test requires it. None of these tests makes a poorly specified transition model correct. For clustered or otherwise dependent transition contributions, the model-based information and chi-square calibration may fail unless the likelihood properly represents that dependence; an appropriate variance and testing procedure must be chosen for the sampling structure.

The p-value is the probability, under the shared-effect null and the stated model assumptions, of a score statistic at least as large as the observed \(Q\). It is not the probability that the null hypothesis is true. A nonsignificant result indicates insufficient evidence against shared effects, not proof that the effects are identical. The test also does not establish that a covariate causes a transition hazard to change.