We propose two test statistics based on the covariance process of the martingale residuals for testing independence of bivariate survival data. The first test statistic takes the supremum over time of the absolute value of the covariance process, and the second test statistic is a time-weighted summary of the process. We derive asymptotic properties of the two test statistics under the null hypothesis of independence. In addition, we derive the asymptotic distribution of the weighted test and construct optimal weights for contiguous alternatives to independence. Through simulations, we compare the performance of the proposed tests and the inner product of the Savage scores statistics of Clayton and Cuzick (1985, Journal of the Royal Statistical Society, Series A 148, 82-108). These demonstrate that the supremum test is generally more powerful with comparatively little power loss relative to their test when Clayton's family alternative holds, and the weighted test is more powerful when the weight is appropriately chosen.