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Generalized log-rank tests for interval-censored failure time data. (English) Zbl 1092.62107

A class of tests is considered for the $k$-samples homogeneity hypothesis by interval censored failure time data. I.e., for each subject a random interval $\left({L}_{i},{R}_{i}\right)$ is observed to which its failure time belongs. The test statistics is

${U}_{\xi }=\sum _{i=1}^{n}{x}_{i}\frac{\xi \left({\stackrel{^}{G}}_{n}\left({L}_{i}\right)\right)-\xi \left({\stackrel{^}{G}}_{n}\left({R}_{i}\right)\right)}{{\stackrel{^}{G}}_{n}\left({L}_{i}\right)-{\stackrel{^}{G}}_{n}\left({R}_{i}\right)},$

where $n$ is the number of subjects in the union of all samples, ${x}_{i}$ is the vector of indicators of the sample (its $l$-th element equals 1 iff the $i$-th subject belongs to the $l$-th sample and is 0 otherwise), ${\stackrel{^}{G}}_{n}\left(x\right)$ is a nonparametric estimator of the survival function under the null hypothesis (homogeneity), $\xi$ is a fixed function. (E.g., for $\xi \left(x\right)=xlog\left(x\right)$ this is the score statistics). The asymptotic normality of ${U}_{\xi }$ under ${H}_{0}$ is demonstrated. Simulation results and real breast cancer data application are considered.

##### MSC:
 62N03 Testing (survival analysis) 62G10 Nonparametric hypothesis testing 62G20 Nonparametric asymptotic efficiency