Nonparametric estimator for the survival function of quality adjusted lifetime (QAL) in a three-state illness–death model

In this work, we consider nonparametric estimation of QAL distribution in a three-state illness–death model. In our approach, we first write down the expression for the distribution of QAL in terms of the joint distribution of the sojourn times in the three states. The estimate of the QAL distribution is obtained by substituting the estimates of sojourn time distributions in the expression of the QAL distribution. The proposed nonparametric estimate, assuming independence between time to illness and sojourn time in the state of illness, is uniformly consistent. Asymptotic normality has also been established. An estimate of asymptotic variance has been obtained. The performance of the proposed estimator is investigated by simulation. A data set of the Stanford Heart Transplant program has been analyzed for illustration.