Individualized State Probabilities From Transition-Specific Models

From Transition Models to Individualized Predictions

A transition-specific model describes how the rate of moving from one state to another depends on a person's covariates. To make a prediction, however, we usually need a different quantity: the probability that this person occupies each state at a specified future time. This tutorial connects those quantities. It treats the fitted models as inputs and focuses on translating their predictions into a coherent distribution over states.

Suppose the observed state at a landmark is \(r\), the prediction horizon is \(t\) time units later, and the person's covariate profile is \(z\). An individualized prediction is conditional on that profile and the landmark information used to define the target. It is a model-based probability, not a claim that the person's future state is known with certainty.

Here time zero denotes the prediction landmark. The covariates may be fixed at their landmark values, or a prespecified covariate path may be used if the models allow time-varying covariates. We first describe the fixed-profile case. The state process must be Markov conditional on the covariates being used: given the current state and that covariate information, the future transition law does not also depend on unmodeled earlier state history.

Transition-Specific Models and the Prediction Calculation

For distinct states \(i\) and \(j\), write \(q_{ij}(u\mid z)\) for the transition intensity from \(i\) to \(j\) at elapsed time \(u\), conditional on covariates \(z\). It is the instantaneous transition rate among individuals currently in state \(i\), in the sense that, over a short interval of length \(h\), the transition probability is \(q_{ij}(u\mid z)h+o(h)\). Set \(q_{ii}(u\mid z)=-\sum_{j\ne i}q_{ij}(u\mid z)\); entries for impossible transitions are zero. The resulting matrix \(Q(u\mid z)\) is the transition intensity matrix.

A common transition-specific proportional hazards specification is \(q_{ij}(u\mid z)=q_{ij,0}(u)\exp(\beta_{ij}^{T}z)\), with a distinct baseline intensity and covariate coefficient vector for each modeled transition. Other parametric or semiparametric transition-specific specifications may be used. For a particular person, substituting their covariates into each fitted transition model gives their predicted transition intensities.

Those intensities must then be combined across possible paths. If \(P(a,b\mid z)\) denotes the matrix whose \((i,j)\) entry is the probability of being in state \(j\) at time \(b\), given state \(i\) at time \(a\) and covariates \(z\), the individualized prediction from starting state \(r\) is row \(r\) of that matrix. For a time-homogeneous intensity matrix, \(P(a,b\mid z)=\exp\{(b-a)Q(z)\}\). With time-varying intensities, the transition matrix is obtained by solving the forward transition equations, or equivalently by the corresponding product integral.

Here \(\mathbf{e}_r^{T}\) is a row vector with a 1 in the starting-state position and 0 elsewhere. If the interval is split at an intermediate time \(u\), the Markov property gives \(P(0,t\mid z)=P(0,u\mid z)P(u,t\mid z)\). In components, the probability of ending in \(j\) is the sum over every possible intermediate state \(k\) of the probability of reaching \(k\) by \(u\), multiplied by the probability of reaching \(j\) from \(k\) by \(t\).

This identity explains why the calculation accounts for all possible routes through the state system. It also shows why a single transition's fitted hazard is generally not itself the desired state probability: a person can enter, leave, or revisit states before the horizon.

Worked Examples

Assumptions, Diagnostics, and Alternatives

The multiplication or integration is only as credible as its inputs and assumptions. In particular, check that the modeled state set includes all outcomes relevant to the prediction; the person starts in the stated landmark state; the transition models apply to the covariate profile; and the conditional Markov assumption is plausible for the covariates and state history represented. If a model uses fixed covariates, it also assumes that the prediction appropriately treats those covariates as fixed over the horizon. The fitted transition models must represent the relevant transition-specific hazards, and the data must provide adequate support for the profile and transitions being predicted.

Useful numerical and statistical checks include:

• Each transition-probability matrix should have entries between 0 and 1, with each row summing to 1 up to rounding error.
• The final state probabilities should be nonnegative and sum to 1. A meaningful discrepancy suggests a matrix-orientation error, omitted state, or faulty calculation.
• Check whether predicted transition rates are plausible and whether fitted transition-specific models show serious lack of fit, such as an untenable proportional-hazards assumption when that model was used.
• Examine whether the person's covariate profile lies within the data's support and whether the relevant transitions have enough observations. Sparse or absent transitions can make profile-specific predictions unstable or unsupported.
• Compare predicted state occupation with observed outcomes in appropriate validation data. A coherent set of probabilities can still be poorly calibrated.

A common mistake is to calculate the probability of the final state using only the transition model leaving the starting state, as though intermediate transitions could not occur. The correction is to combine all modeled paths through intermediate states, using matrix multiplication or the corresponding continuous-time transition equations. Another mistake is to read a predicted probability as a personal guarantee. A value of 0.716 means the model assigns probability 0.716 to state \(A\) for people with the specified information and covariates under its assumptions; it does not make the individual's eventual state certain.

Individualized predictions differ from pooled state-occupation estimates, which average over or otherwise summarize a target group rather than applying transition models to a specified profile. They also differ from direct horizon models, which model the state at the target time without necessarily constructing transition paths. Direct models may avoid some transition-process assumptions, but they do not automatically provide a coherent account of intermediate states or a full trajectory. Transition-specific models make path structure explicit, at the cost of relying on the adequacy of each transition model and their joint combination.

These are plug-in predictions: they use fitted model estimates as if fixed. They do not, by themselves, quantify uncertainty from estimating the transition models. They may also be unreliable when the Markov assumption fails, when covariates evolve in ways not represented by the models, or when predictions require extrapolation beyond observed follow-up or covariate support.