In Best et al. (2009) we looked at nonparametric tests in randomized block designs, with a particular focus on ties and ordered alternatives. Formulae were given for the Page, umbrella and Friedman test statistics. It was also noted that orthogonal trend contrasts can be used to partition the Friedman into the Page, umbrella and a residual.

In Thas et al. (2012) this work was extended to cover the completely randomized, randomized block and balanced incomplete block designs. Again there was an emphasis on ties and ordered alternatives and partitioning the Kruskal-Wallis, Friedman and Durbin test statistics. The interpretation of these tests was also discussed. Without an assessment of the location shift (LS) model these tests cannot be interpreted in terms of location (mean/median) shift. The tests are consistent under the stochastic ordering model (SOM). Since SOM É LS conclusions other than location shift may be valid.

Recently we have extended an idea of Conover (1999) who, for randomized blocks and balanced incomplete blocks, suggested carrying out an analysis of variance (ANOVA) on the ranks and using the F test for treatment differences. Use of general linear model (GLM) routines permits the handling of ties and missing values. Empirical evidence demonstrates that the relevant F tests give test sizes generally at least as close to nominal as the competitor tests and power generally at least as good as that of the competitor tests.

References
BEST, D.J., RAYNER, J.C.W. and THAS, O. (2009). Nonparametric tests for randomized block data with ties and ordered alternatives. Proceedings of the Third Annual Applied Statistics Education and Research Collaboration (ASEARC) Research Conference, 7-8 December 2009: Newcastle, Australia.
CONOVER, W. (1999), Practical non-parametric statistics (3rd edn). New York: Wiley.
THAS, O., BEST, D.J. and RAYNER, J.C.W. (2012). Using orthogonal trend contrasts for testing ranked data with ordered alternatives. Statisticia Neerlandica, 66(4), 452-471.