Ordered Hypotheses for Multiple Treatments: A Significance Test for Linear Ranks

Abstract In many experiments the background evidence, theories, or conditions suggest an expected ordering among the treatment effects, yet in the analysis of variance such implicit hypotheses are typically neglected. A ranking statistic L is presented as test of a monotonic relationship among the treatment groups in the two-way analysis of variance. Used with accompanying table of L, it combines considerable power with computational ease, and assumes data of only ordinal strength. L is related to the test of the linear component of the treatment sum of squares in the parametric randomized-block design, to the product-moment correlation and regression, to the normal deviate test of Lyerly's average rho, and to Friedman's chi-square of ranks. Where either L or the Friedman test may be used, L is often more accurate and appropriate, and it has some advantages over other tests of trend and monotonicity.