The Product-Moment Correlation Coefficient and Linear Regression for Truncated Data

A general class of quasi-independence tests for left-truncated right-censored data

“…In this Section, we review existing methods for quasi-independence testing: conditional Kendall's tau (Tsai, 1990), conditional product-moment correlation coefficient (Chen et al, 1996), and a weighted score test for the Cox proportional hazard model (Jones and Crowley, 1992).…”

“…where V n = n−1 i=1 n j=i+1 I(Λ ij ). Chen et al (1996) proposed a test based on conditional product-moment correlation coefficient for left-truncated data. Similarly to Tsai (1990), Chen et al (1996) first defined the population version of the correlation coefficient conditioning on Ω 12 ,…”

“…Jones and Crowley (1992) presented a score-type test from the Cox proportional hazard model. Chen et al (1996) suggested a test statistic for quasi-independence based on the conditional product-moment correlation among comparable data points if failure and truncation times follow truncated bivariate normal distribution. Martin and Betensky (2005) extended the conditional Kendall's tau test to more complex truncation schemes.…”

“…Martin and Betensky (2005) extended the conditional Kendall's tau test to more complex truncation schemes. We note that both Tsai (1990) and Chen et al (1996)'s works adapted usual association measures for complete bivariate data. It may become of interest whether other measures can be adjusted for left truncation and right censoring.…”

“…The class reveals that any association measure with a form U-statistic comprising skew-symmetric transformations can induce a version for left truncated data, restricting the measure onto the mean among "comparable pairs". The class is broad enough to embed not only existing tests by Tsai (1990), Jones and Crowley (1992) and Chen et al (1996), but also Spearman's rank correlation, another well-known association measure. We derive asymptotic properties of the proposed class of statistics.…”