Additivity Tests in the Two Way Anova with Single Sub-class Numbers

Implementation of the Tukey, Mandel, Johnson-Graybill, LBI, Tusell and modified Tukey non-additivity tests.

additivityTests is R package for additivity tests in the two way ANOVA with just one observation per cell.


In many applications of statistical methods, it is assumed that the response variable is a sum of several factors and a random noise. In a real world this may not be an appropriate model. For example, some patients may react differently to the same drug treatment or the effect of fertilizer may be influenced by the type of a soil. There might exist an interaction between factors.

If there is more than one observation per cell then standard ANOVA techniques may be applied. Unfortunately, in many cases it is infeasible to get more than one observation taken under the same conditions. For instance, it is not logical to ask the same student the same question twice.

Six tests of additivity hypothesis (under various alternatives) have been included in this package: Tukey test, modified Tukey test, Johnson-Graybill test, LBI test, Mandel test and Tusell test.


Let us generate 10 random subjects, 10 random treatmeants and combine them into a dataset (with no interaction):

    subjects = rnorm(10)
    treatments = rnorm(10)
    noise = rnorm(100)/100
    Y = matrix(rep(subjects,10), 10, 10) + matrix(rep(treatments, each=10), 10, 10) + noise

The tests should not reject the additive hypothesis:

    mtukey.test(Y, correction=2, Nboot=1000)

Now, the extra effect will be added to the last 5 subjects. The tests should reject the additive hypothesis:

    Y[1:5,] = Y[1:5,] + 10*rep(treatments, each=5)
    mtukey.test(Y, correction=2, Nboot=1000)


To install the additivityTests from Github, it's easiest to use the devtools package:

    # install.packages("devtools")

Or you can access the stable package version available on [CRAN]


Simecek, Petr and Simeckova, Marie. "Modification of Tukey's additivity test." Journal of Statistical Planning and Inference (2012). [ScienceDirect] [arXiv]

Rasch, Dieter, et al. "Tests of additivity in mixed and fixed effect two-way ANOVA models with single sub-class numbers." Statistical Papers 50.4 (2009): 905-916. [Springer]


additivityTests 1.1


  • "tusell" was previously misspelled as "tussel" (my apologies to F. Tusell)
  • bug in critical values quantile in lbi.test corrected


  • Vignette added
  • NEWS file added

Reference manual

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1.1-4 by Petr Simecek, 4 years ago

Browse source code at

Authors: Marie Simeckova [aut] , Thomas Rusch [aut] , Petr Simecek [cre]

Documentation:   PDF Manual  

GPL-3 license

Suggests knitr

Imported by biclust.

See at CRAN