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In statistical literature, several dependence measures have been extensively established and treated, including Pearson's correlation coefficient, Spearman's  ρ  and Kendall's  τ . In the context of survival analysis with length-biased data, a measure of dependence between survival time and covariates appears to have not received much intention in the literature. The purpose of this paper is to extend Kent's [Information gain and a general measure of correlation. Biometrika. 1983;70(1):163–173.] dependence measure, based on the concept of information gain, to length-biased survival data. Specifically, we develop a new approach to measure the degree of dependence between survival time and several continuous covariates, without censoring, when the relationship is linear. In this regard, kernel density estimation with a regression procedure is proposed. The consistency for all proposed estimators is established. In particular, the performance of the dependence measure for length-biased data is investigated by means of simulations studies.

Dependence measure for length-biased survival data using kernel density estimation with a regression procedure