ANADOLU UNivERSiTESi BiLiM VE TEKNOLOJi DERGiSi ANADOLU UNIVERSITY JOURNAL OF SCIENCE AND TECHNOLOGY Ci1Wol.:2 - Sayr/No: 1 : 67-72 (2001) ARA$TIRMA MAKALESi/RESEARCH ARTICLE MARDIA'S WRONG THEOREM Adil KORKMAZ1, SOIeyman GONAY2 ABSTRACT In this study it has been shown that Mardia's theorem about eigenvectors in correspondence analysis is wrong. Key Words: Correspondence Analysis, Standardised Eigenvector. MARDIA'NIN TEOREMi OZ Bu cahsmada uygunluk cozumlemesinde karsilasilan standart OZ vektorlere iliskin olmak iizere Mardia'nm bir teoreminin yanhs oldugu gosterilmistir, Anahtar Kelimeler: Uygunluk Cozumlemesi, Standart Oz Vektor, The technique of correspondence analysis effecti- vely takes both the above relationships simultaneously, and uses them to deduce scoring vectors rand s which rences of n species in p locations; that is, xij=1 if spe- cies i occurs in location j, and Xij=O otherwise. If ri is the wet-preference score allocated to the i th species, then the average wet-preference score of the species fo- und in location j is This is the estimate of wetness in location j produ- ced by the classical method of gradient analysis. One drawback of the above method is that the ri may be highly subjective. However, they themselves could be estimated by playing the same procedure in reverse -if Sj denotes the physical conditions in locati- on j, then ri could be estimated as the average score of the locations in which the i th species is found; that is 1. INTRODUCTION Mardia (1979, 1988 and 1989) has written that correspondence analysis is a way of interpreting con- tingency tables, which has several affinities with prin- cipal component analysis. In his referenced book he had introduced to the subject in the context of a botani- cal problem known as "gradient analysis". His senten- ces have been written below by not changing: «This concerns the quantification of the notion that certain species of flora prefer certain types of ha- bitat,and that their presence in a particular location can be taken as an indicator of the local conditions. Thus one species of grass might prefer wet conditions, whi- le another might prefer dry conditions. Other species may be indifferent. The classical approach to gradient analysis involves giving each species a "wet-preferen- ce score", according to its known preferences. Thus a wet-loving grass may score 10, and a dry-loving grass I, with a fickle or ambivalent grass perhaps receiving a score 5. The conditions in a given location may now be estimated by averaging the wet-preference scores of the species that are found here. To formalise this let X be the nxp one-zero matrix which represents the occur- Sj a L Xijr/x.j i ri a L XijS/Xi. i where where X.j = L Xij i Xi· = L Xij i 1 Akdeniz Universitesi Iktisadi ve Idari Bilimler Faktiltesi Iktisat Bolumu ANTALYA. 2 Hacettepe Universitesi Fen Bilimleri Enstitusu istatistik Bolumu Beytepe-ANKARA. Received: 10 February 2000; Accepted: 15 February 2001.