Application of correspondences analysis method to burt tables

In this study, the method of the correspondences analysis is investi gated from the point of view of tensor algebra and the method is applied to a medical problem. First in 1930's H.Hotelling, C. Spearman and K.Pearson took into ac count this method and then it is improved in 1960 by J.P.Benzecri. Let I and J be two sets with finite number of elements. We consider the following Ku rectangular table Ku = {k(i,j) | iel, jeJ} we obtain from Kjj the frequency table Fij: Fij = {fij |.?/, jeJ] ; /*; = ^ k = £}{k(», i) | i e I, j ? J] Here dividing every fij(i G I) by the mass /_,- which is the sum of jth column, it yields This table can be considered as a mixed tensor of type (1,1). By assigning the mass fj to each fj, we obtain the set of points that we represent by N(J) W) = {(fl, fj) I jeJ}c.Rj this is the set of variables.In the same manner the Fj table and the set of individuals N(I) can be obtained from Fjj Hi) = {(/>, fd I ieijcRj In the space Ri we define a metric by the quadratic form ci' mn={mu' | i,i' Gl}, m"' = -Ş- Ji The matrix M which represents the above quadratic form defines an isometry of R1 on R1. If uf is the unit vector in the space Ri according to the metric m11 with matrix M, the projection of fj on the axis Aa« is represented by V>£ and is defined in tensorial form by ft = M{uJ, //) = *£(/*') Here o is the transition defined between the measure fj and the func tion iff, which is the orthogonal projection on the axes Aa«. At the same time this is the density function of uf with respect to fj. The center of gravity gi of set of points obtained as follows *' = £{/;// I ;eJ} = £tf/; I jeJ} = fr and it can be written as a tensor gr = // o fj. The quadratic inertia of the set of points about the subspace which is orthogonal to M~x(Sj) can be written as a tensor *n = (ffo-ff)ofj Let Z be the matrix corresponding to Ff, V be the matrix corre sponding to the tensor an and Dp be quadratic diagonal weights matrix whose general element is djj = fj. It can be written as follows * V = Z.DP.Z' where V is a variance-covariance matrix. The aim of correspondances analysis is to search the nearest Hi sub- space to the set of N(J). This is {An? | a?[l,*]} The principal component of any point of the set which is on the a th axis is found as M(// ) - £) = Vİ o // = M(«?)// = M(«?, //) = 0 iv ) Let V be variance-covariance matrix, then Vfri, A) = A«<£' The principal components satisfy therefore the following properties; i) Ver Sil, *İ = ff = Z'fâ) ii ) i\)Ja are eigendirections of Z'MZDPiii ) Mean values of ^ are zero. iv ) ^(a£ A) are orthogonal to Dv. To each principal axis of the set N(J) represented by uj (a 6 A), corresponds a triple ((pI1iJ}J, Xa). In fact, to every principal axis which is represented by Uj(a E A) of the set N(I), corresponds another triple (¥>J,^J,Aa). Hence every element of N(I) can be obtained by means of the elements of N( J) in the following way i ) «? = £V(y£) = ^ZD^i) ii) uf = M(u?) = j-ZD,M) iii) ri,£ = Z'M(uJ) = Z'tpIa In the same way, every factor in a triple can be written in terms of the factor of the other triple as follows These equations can be written in tensor form as follows Keyword: Burt tabloları = Burt tables ; Karşılaştırmalı analiz = Comparative analysis