Determination of Minimum and Maximum Experimental Size in One-, Two- and Three-way ANOVA with Fixed and Mixed Models by
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Determination of Minimum and Maximum Experimental Size in One‑, Two‑ and Three‑way ANOVA with Fixed and Mixed Models by R Dieter Rasch1 · Rob Verdooren2
© Grace Scientific Publishing 2020
Abstract In the paper for all classifications, the models of analysis of variance (ANOVA) have up to three factors and all have the fixed factor A which has to be tested. R-programs of the package OPDOE for the determination of the experimental size are given. Keywords One-way ANOVA · Two-way ANOVA · Three-way ANOVA · Noncentrality parameter λ of F(df1, df2, λ) · Minimal experimental size
1 Introduction We consider in this paper all classifications and models with at least one fixed factor of ANOVA with up to three factors. If the fixed factor has at least three levels, the minimum experimental size depends on the values of its factor levels. As described in [3, 6], we calculate the minimum experimental size for the most favorable (minimin size) and the least favorable (maximin size) case of the allocation of the values of the factor levels of A. What we understand by the minimin and the maximin size is described in section 5.2.2 of [3]. In ( ) ( ) F df1 , df2 , 0|1 − 𝛼 = F df1 , df2 , 𝜆|𝛽 , (1) ( ) with the P-quantiles F df1 , df2 , 𝜆|P of the non-central F-distribution with df1 and df2 degrees of freedom of the numerator and the denominator of the F-distribution, respectively, and the risk of the first kind α and the risk of the second kind Part of special issue guest edited by Dieter Rasch, Jürgen Pilz, and Subir Ghosh—Advances in Statistical and Simulation Methods. * Dieter Rasch d_rasch@t‑online.de 1
Institute of Statistics, University of Natural Resources and Life Sciences, Vienna, Austria
2
Division Data Sciences, Danone Nutrica Research, Utrecht, The Netherlands
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Vol.:(0123456789)
57
Page 2 of 25
Journal of Statistical Theory and Practice
(2020) 14:57
β of the F-test, and the non-centrality parameter λ, the basic equation for all other sections of this paper is given. Further, the difference δ between the largest and the smallest effects (main effect or in the following sections also interaction effect) of the fixed factor A, to be tested against null, belongs to the precision requirement. The solution λ in (1) is denoted by ( ) 𝜆 = 𝜆 𝛼, 𝛽, df1 , df2 . (2) Let δ = amax − amin be the difference between the maximum and the minimum of a real effects a1 , … , aa of the fixed factor A (or of interaction effects). We consider two extreme configuration cases for ai: (1) a1 = amin, aa = amax and the other a − 2 ai equal ā = �2 ( )2 ∑a � 𝛿2 ai − ā = 2 𝛿2 ; hence, λmin = 2𝜎 . 2 i=1
amax +amin , 2
which results in
(2) For even a = 2m if m values ai are equal to amin and the other m values ai are equal �2 ( )2 ∑a � a +a 𝛿2 to amax; hence, ā = max 2 min and i=1 ai − ā = a 𝛿2 ; hence, λmax = a 4𝜎 . 2 For odd a = 2m + 1, we consider the same configuration as given above but the remaining ai is size equal to either amin or amax. Case (1) results in the maximum sample size maximin nmax and Case (2) in the
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