Testing hypothesis of parameters of generalized proportional hazards models under unknown lifetime distribution

The paper deals with the construction of the Cox proportional hazards model and its generalizations. The considered generalizations of the proportional hazards model are the Hsieh model and the simple cross-effect model (SCE-model), which allow decreasing, increasing or non-monotonic behavior of the ratio of hazard rate functions. The algorithm of the estimation of regression parameters and unknown baseline distribution for the generalized models is developed by using the partial likelihood function. The research on statistical properties of estimates carried out with computer simulations, has shown that the bias and the variance of obtained estimates decrease with the sample size growth. However, the bias and the variance of obtained estimates increase with the censoring degree growth. The likelihood ratio test and the Wald test are used for testing hypothesis about parameters of considered models. In this paper, the expressions of elements of the matrix of the second partial derivatives by regression parameters of the Hsieh model and SCE-model have been obtained. In the case of testing hypothesis about parameters of proportional hazards model and the Hsieh model, the distributions G (S | H ₀) of likelihood ratio test statistic and the Wald statistic are independent of covariates values or regression parameters values. The difference between the simulated test statistic distributions and the corresponding % -distributions decreases with the sample size growth. The dimension of the covariate vector and the number of estimated parameters affect not only the number of degrees of freedom of the limiting distribution, but also the closeness of simulated test statistic distributions to the theoretical distributions: the smaller the dimension of the covariate vector the smaller the difference between empirical and limiting distributions for the same sample size. In the case of testing hypothesis of insignificance of parameters P and у of SCE-model, the statistic distributions G (S |H ₀) of ₂ the Wald test are not close to the corresponding limiting % -distributions even for the large sample sizes. Basing on the obtained results of research on statistic distributions and the power of considered tests, it is advisable to use the test proposed by M.S. Nikulin and likelihood ratio test for checking proportional hazard assumption against the competing hypothesis corresponding to the SCE-model. The application of the Wald test can result in inaccurate computation of p-value because empirical statistic distributions significantly differ from corresponding limiting distributions.

Keywords

Authors

References