The Ordered-Heterogeneity Family of Tests

Abstract

SUMMARY A new testing procedure is described concerning ordered alternative hypotheses when three or more populations are categorized by class. The test is specifically designed for use in complex testing applications frequently found in biological research where the isotonic regression and JonckheereTerpstra test cannot readily be applied. The test has both parametric and nonparametric applications. Significance tables for the test are provided for both large and small sample sizes, and a power analysis relative to isotonic regression is described for the case of the one-way layout. The problem of testing ordered alternative hypotheses in the context of comparing several populations is common in biological research. The techniques of isotonic regression [reviewed in Barlow et al. (1972); Robertson, Wright, and Dykstra (1988)] and the Jonckheere-Terpstra test [reviewed in Hollander and Wolfe (1973)] have been developed for the context of the one-way layout, but these techniques frequently cannot be pragmatically applied in more complex biological applications. The reasons for this are twofold. First, it is common in biological research to test ordered alternatives in the context of complex designs where concomitant variates require the use of multi-way contingency, ANOVA, ANCOVA, or mixed-effects ANOVA models. An example of this situation from our own research concerns filter feeders in marine ecosystems and the identification of the factors that cause variation in demographic parameters both among and within populations. Many marine invertebrates capture microscopic food with appendages that act like nets. Their feeding rate is therefore a flux that may depend on the concentration of particles in the water and its velocity. These two components of particle flux can vary dramatically among sites and microhabitats. This should lead to predictable variation in growth rates and other demographic parameters. We have measured growth rates of transplanted barnacles from a common source population at numerous sites in New England and within particular microhabitats at each site (tidal height, which varies the amount of time barnacles are submerged and can feed, and exposure to waves, which alters velocities). The experimental design is a three-way repeated-measures ANCOVA, with initial body size serving as a covariate. All three factors have a priori expectations on HA. Sites are ordered by their predicted ranks in particle concentrations. Tidal heights are ordered by their ranked period of submergence (low zone > mid zone > high zone). Orientations are ordered by their expected ranked velocities (exposed > intermediate > protected). Although the isotonic regression and Jonckheere-Terpstra test can be extended to two-factor designs, they are not easily applied to more complex models such as this. Second, biologists have been, and will continue to be, confronted with a broad spectrum of new k-sample heterogeneity tests. Many of these represent specialized extensions of ANOVA- and ANCOVA-like statistical models and do not evaluate ordered alternative hypotheses, despite the fact that a priori ordering is commonly implicit in the hypotheses biologists seek to test [see for review Gaines and Rice (1990)]. The null distributions of the test statistics for these tests are frequently idiosyncratic and complex. For example, Excoffier, Smouse, and Quattro (1992) describe