A note about using ranked outcomes in t-tests and ANOVAs

Conover and Iman (1981) showed that the Mann-Whitney test is equivalent to a t-test on ranked responses. Huitema (1980) in Chapter 12 and Shirley (1981) illustrate the use and interpretation of an analysis of covariance on ranked outcomes with worked examples. Shirley (1987) extends Conover and Iman's work to investigating interactions by presenting a worked example involving a full factorial between subjects analysis of variance. Blair, Sawilowsky and Higgins (1987) and Wilcox (2003, p.604) do not advocate using ranked outcomes to test for interactions 'in more complex designs'. Akritas (1991) shows that rank transformed responses are not applicable to repeated measures ANOVA.

Leys and Schumann (2010) present a worked example which tests for an interaction in a 2x2 between subjects factor ANOVA (see [http://www.sciencedirect.com/science/article/pii/S002210311000034X here) using an adjusted rank transform with a working [attachment:nonint.xls input spreadsheet.]

More recently Zimmerman (2011) has shown that using t-tests on ranked outcomes, even when there are a large proportion of equal scores (having tied ranks), have greater power and smaller type I error than the conventional t-test on raw scores when the outcome is not normally distributed. He also claims using ranked outcomes works well with one-way ANOVAs.

References

Akritas, MG (1991) Limitations of the rank transform procedure: a study of repeated measures designs, part 1. Journal of the Americal Statistical Association 86(414) 457-460.

Blair, RC, Sawilowsky, SS and Higgins, JJ (1987). Limitations of the rank transform statistic in tests for itneractions. Communications in Statistics - Simulation and Computation 16 1133-1145.

Conover, WJ and Iman, RL (1981). Rank transformations as a bridge between parametric and nonparametric statistics. American Statistician 35 124-129. doi:10.2307/2683975

Huitema, BE (1980). The analysis of covariance and alternatives. 2nd Edition. Wiley: New York.

Leys, C. and Schumann, S (2010). A nonparametric method to analysze interactions: the adjusted rank transform test. Journal of Experimental Psychology 46 684-688.

Quade, D (1967). Rank analysis of covariance. Journal of the American Statistical Association 62 1187-1200.

Shirley, EAC (1981). A distribution-free method for analysis of covariance based on ranked data. Applied Statistics 30 158-162.

Shirley, EAC (1987). Applications of ranking methods to multiple comparison procedures and factorial experiments. Applied Statistics 36(2) 205-213.

Wilcox, RR (2003, p.604). Applying contemporary statistical techniques. Academic Press:New York.

Zimmerman, DW (2011). Power comparisons of significance tests of location using scores, ranks, and modular ranks. British Journal of Mathematical and Statistical Psychology 64 233-243.