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The association between two event times is of scientific importance in various fields. Due to population heterogeneity, it is desirable to examine the degree to which local association depends on different characteristics of the population. Here we adopt a novel quantile-based local association measure and propose a conditional quantile association regression model to allow covariate effects on local association of two survival times. Estimating equations for the quantile association coefficients are constructed based on the relationship between this quantile association measure and the conditional copula. Asymptotic properties for the resulting estimators are rigorously derived, and induced smoothing is used to obtain the covariance matrix. Through simulations we demonstrate the good practical performance of the proposed inference procedures. An application to age-related macular degeneration (AMD) data reals interesting varying effects of the baseline AMD severity score on the local association between two AMD progression times.