Trend tests for monotone trend or umbrella trend (monotone upward changing to monotone downward or vise versa) in count data are proposed when the data exhibit extra-Poisson variability. The proposed tests, which are called the GS1 test and the GS2 test, are constructed by applying an orthonormal score vector to a generalized score test under an rth-order log-linear model. These tests are compared by simulation with the Cochran-Armitage test and the quasi-likelihood test of Piegorsch and Bailer (1997, Statistics for Environmental Biology and Toxicology). It is shown that the Cochran-Armitage test should not be used under the existence of extra-Poisson variability; that, for detecting monotone trend, the GS1 test is superior to the others; and that the GS2 test has high power to detect an umbrella response.
