The Cochran–Mantel–Haenszel (CMH) methodology is a suite of tests applicable to tables of count data. The data are counts of the number of times the ith of t treatments results in an outcome in the hth of c categories on the jth of b blocks or strata. Inference is conditional on the treatment and outcome totals on each stratum being known before sighting the data. One important application of the CMH tests is to randomized blocks data when the responses are categorical rather than continuous. Software exists for analyzing CMH data but data entry can be tedious if there are many strata.

I will outline the CMH tests and discuss three projects:

• the convenient calculation of the CMH test statistics
• extensions of the CMH mean scores and correlation tests and
• the development of unconditional analogous to the CMH tests.

Another project I have recently been involved in can be downloaded at http://bookboon.com/en/introductory-nonparametrics-ebook.

I will explain!

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.

(John Rayner, John Best & Olivier Thas)

For multifactor experimental designs in which the levels of at least one of the factors are ordered we demonstrate the use of components that provide a deep nonparametric scrutiny of the data. The components assess generalized correlations and the resulting tests include and extend the Page and umbrella tests.

(Thomas Seusse, John Rayner & Olivier Thas)

Smooth tests were developed to test for a finite mixture distribution using two smooth models. Each has its own strengths and weaknesses. These tests are demonstrated by testing for a mixture of two normal distributions. Some sizes and powers are given, as is an example.

(John Rayner, John Best & Olivier Thas)

For multifactor experimental designs in which the levels of at least one of the factors are ordered we demonstrate the use of components that provide a deep nonparametric scrutiny of the data. The components assess generalized correlations and the resulting tests include and extend the Page and umbrella tests.

(Thomas Seusse, John Rayner & Olivier Thas)

Smooth tests were developed to test for a finite mixture distribution using two smooth models. Each has its own strengths and weaknesses. These tests are demonstrated by testing for a mixture of two normal distributions. Some sizes and powers are given, as is an example.

(John Rayner, John Best & Olivier Thas)

For multifactor experimental designs in which the levels of at least one of the factors are ordered we demonstrate the use of components that provide a deep nonparametric scrutiny of the data. The components assess generalized correlations and the resulting tests include and extend the Page and umbrella tests.

(Thomas Seusse, John Rayner & Olivier Thas)

Smooth tests were developed to test for a finite mixture distribution using two smooth models. Each has its own strengths and weaknesses. These tests are demonstrated by testing for a mixture of two normal distributions. Some sizes and powers are given, as is an example.

(John Rayner, John Best & Olivier Thas)

For multifactor experimental designs in which the levels of at least one of the factors are ordered we demonstrate the use of components that provide a deep nonparametric scrutiny of the data. The components assess generalized correlations and the resulting tests include and extend the Page and umbrella tests.

(Thomas Seusse, John Rayner & Olivier Thas)

Smooth tests were developed to test for a finite mixture distribution using two smooth models. Each has its own strengths and weaknesses. These tests are demonstrated by testing for a mixture of two normal distributions. Some sizes and powers are given, as is an example.

The rank transform procedure is often used in the ANOVA when observations are not consistent with normality. The data are ranked and the ANOVA is applied to the ranked data. Often the rank residuals will be consistent with normality and a valid analysis results even though the residuals of the raw data were not consistent with normality. Here we observe that the rank transform procedure is equivalent to applying the intended ANOVA analysis to what we call scaled first order orthonormal polynomials. Using higher order scaled orthonormal polynomials extends the analysis to higher order effects, detecting, roughly, dispersion, skewness etc. differences between treatment ranks. We discuss using unscaled orthonormal polynomials.

AUTHORS: J.C.W. Rayner and D.J. Best