Testing linearity with applications to clines

Spatial and geographic measures of biological diversity are often represented by clines. A cline has been described as a gradient in a measurable character. Means of morphometric characters studied at different locations are plotted to provide a graphical representation of a cline often displayed as an S-shaped curve. Of special interest is testing whether two clines based on different morphometric measures are identical. By graphing one cline against another in a cline-cline plot, testing equality can be reduced to testing linearity. We consider several tests of the equality of clines by testing linearity of the cline-cline plot against S-shaped alternatives. Many of these tests are based on third differences of the cline-cline graph. The criterion of Asymptotic Relative Efficiency is used to compare the tests. Theoretical results show that a least-squares t-test performs better than the others considered. Simulations show that the one-sample t-test performs quite well when the errors used are normally distributed while the two nonparametric tests considered are likely more appropriate alternatives especially when the errors are not normally distributed. Simulations also show that in terms of power, the performance of the third-difference-based tests is very satisfactory once specific conditions and assumptions are satisfied. These conditions are equal sample sizes from each location, constant error variability, equally distanced means for these groups and independence of variables. However when these third-difference-based tests are applied to data in which some or all of these conditions are violated, the result is a considerable loss of power.