Construction of probabilistic survival models from left truncated and right censored data

In the lifetime data analysis, the obtained samples of observations turn out to be censored, as a rule. Moreover, there is often such a situation when selection of devices (or individuals) into a sample is carried out according to some condition on the lifetime. In this case, the obtained lifetime data are truncated. In this paper, the problem of construction of parametric proportional hazards model, introduced by Cox, on the basis of left truncated and right censored data has been considered. Selection of factors, influencing significantly on the survival function, is carried out on the basis of the semiparametric model, in which the lifetime distribution is supposed to be unknown. The Wald test is used for testing hypothesis on equality of regression parameters to zero. By means of computer simulation methods, the distributions of Wald statistic for testing the parametric hypotheses for the Cox model from left truncated and right censored data have been studied. The convergence of the distributions of the Wald statistic to the corresponding chi-squire distribution has been analyzed for various censoring degrees. An approach for testing goodness-of-fit of the parametric Cox model from left truncated and right censored data on the basis of Cox–Snell residuals, which under true null hypothesis belong to the standard exponential distribution, has been proposed. For testing the hypothesis of exponential distribution of residuals, the modified Kolmogorov, Cramer–von Mises–Smirnov and Anderson–Darling goodness-of-fit tests are suggested to be used. On the basis of the obtained statistical regularities, we have carried out the statistical survival analysis of nonnative-born population in the north regions of industrial development. On the basis of semiparametric proportional hazards model, the predicting factors, significantly influencing on the lifetime of people in the North are determined. Then, the baseline hazard rate function corresponding to the generalized gamma distribution is parameterized. The goodness-of-fit of the obtained parametric Cox model is tested.