A note on the application of simple linear regression methods for trend detection at multiple sites and visits.

In comparing running median, tolerance, cusum, and regression methods for trend detection over a small number of visits, Yang et al. found that application of multiple Z-tests on the basis of a simple linear regression for each site separately was the most efficient for detection of trends at several sites simultaneously. Because the use of multiple Z-tests completely ignores the covariance among measurements taken from different sites, to improve the power we propose a global chi 2-test. Assuming the covariance matrix known, we have found that the proposed chi 2-test procedure is more powerful than multiple Z-tests for two-sided alternatives when both the correlation among measurements and the number of sites are small. We also have found that the former procedure can have power uniformly larger than the latter when ratios of slopes to standard deviations of measurements at different sites vary and the number of sites is large. In fact, in the latter situation, the proposed global chi 2-test procedure, usually used only for two-sided alternatives, can even have power larger than that of multiple Z-tests for one-sided alternatives. In the situation where the ratios of slopes to standard deviations of measurements are all equal, however, the proposed multivariate approach based on the chi 2-test distribution is the least efficient, especially when the number of sites and the correlation are moderate or large. Finally, to account for the effect of multiple tests over a series of visits on the overall alpha-level, on the basis of Monte Carlo simulations, we compute critical values for sequential use of the proposed multivariate test procedure.