Metrika (2002) 55: 233–245 > Springer-Verlag 2002 Measures of nonlinearity for a biadditive ANOVA model A.Pa ´zman1, J.-B. Denis2 1 Dept. of Probability and Statistics, Faculty of Mathematics, Physics and Informatics, Commenius University, Mlynska ´ dolina, 842 48 Bratislava, Slovakia (e-mail: pazman@fmph.uniba.sk) 2 Unite ´ BIA, INRA, domaine de Vilvert, F-78352 Jouy-en-Josas, France (e-mail: jbdenis@jouy.inra.fr) Abstract. The biadditive models are ANOVA models extended by quadratic (bilinear) terms which are frequently used in Agronomy and Plant Breeding in- terpretation of experiments. In a biadditive model, we derived explicit expres- sions for the intrinsic and parameter-e¤ect measures of nonlinearity. Biadditive models are specific in the sense that they are singular (overparametrized), and regularized by nonlinear parameter constraints, so the approach from Pa ´zman (1999) has been applied, however using advantageously the orthogonal struc- ture of the model under a full design. Two competitive parametrizations of the same model are compared for their influence on the measures of nonlinearity. A review of numerical values obtained from already published results is com- mented. Key words: Nonlinear regression, parameter constraints, two-way ANOVA 1 Introduction When considering the e¤ects of factors on a variable of interest, one often uses bilinear models called ‘‘biadditive’’ following Denis & Gower (1994). The ear- liest paper we know on that point dates back to 1921 when Mooers proposed by means of graphical representations what is called nowadays a joint regres- sion analysis. Two years later, a more statistical paper was published by Fisher and Mackenzie. They compared an additive model ½a i þ b j  with a multipli- cative model ½g i d j , where subscripts i and j denotes the levels of two crossed factors, and Greek letters designate unknown parameters. A model combin- ing featuresoffactoranalytic ½g i d j  andanalysisofvariance ½a i þ b j  techniques was develped much later by Gollob (1968). Models combining both linear and For this work, the first author was supported by the Slovak VEGA grant No. 1/7295/20.