Inference for outcome probabilities in multi-state models

Abstract

In bone marrow transplantation studies, patients are followed over time and a number of events may be observed. These include both ultimate events like death and relapse and transient events like graft versus host disease and graft recovery. Such studies, therefore, lend themselves for using an analytic approach based on multi-state models. We will give a review of such methods with emphasis on regression models for both transition intensities and transition- and state occupation probabilities. Both semi-parametric models, like the Cox regression model, and parametric models based on piecewise constant intensities will be discussed.

Fig. 1

The two-state model for survival data

Fig. 2

Nelson-Aalen estimates and estimates based on a piecewise constant hazard for the two-state model for survival data

Fig. 3

The illness-death model without recovery

Fig. 4

Nelson-Aalen estimates and estimates based on piecewise constant intensities for transitions out of state 0

Fig. 5

Nelson-Aalen estimates and estimates based on piecewise constant intensities for transitions out of state 1. Both figures use 10 intervals for the piecewise constant model, with case (b) having more cut-points at the very beginning of the follow-up time

Fig. 6

Nelson-Aalen estimate for cumulative 1 → 2 intensity as a function of duration in state 1

Fig. 7

The competing risks model

Fig. 8

An extended model for BMT

Fig. 9

Kaplan-Meier estimate and estimate based on a piecewise constant hazard in the two-state model

Fig. 10

Cumulative incidence estimates (state occupation probabilities): Aalen-Johansen and piecewise constant hazards

Fig. 11

Aalen-Johansen (Kaplan-Meier) estimate for leukemia-free survival function

Fig. 12

Cumulative incidence (state occupation probability) estimates for AML patient, 32 years, BM only

Fig. 13

Estimated transition probabilities in a three-state Markov illness-death model: (a) Aalen-Johansen estimator of P₀₁(0, t), P₀₂(0, t), and P₁₂(6.22, t); (b) The predicted probabilities P₀₁(0, t) and P₀₂(0, t) under the Cox model comparing two patients with a 20-year age difference (both AML and BM only)

Fig. 14

The plug-in model for the transition probability P₁₂(6.22, t|T) for T = 6.22 and T = 1 compared to the=Aalen-Johansen estimator (assuming a Markov illness-death model)

Fig. 15

The Aalen-Johansen estimator and the Pepe Kaplan-Meier difference estimator for the state occupation probability π₁(t) in a three-state illness-death model

Fig. 16

Estimated transition probabilities in a three-state illness-death model using the Meira-Machado estimators and the Aalen-Johansen estimators: (a) P₀₁(0, t), P₀₂(0, t); (b) P₁₂(6.22, t)