By Ravindra Khattree

Real-world difficulties and knowledge units are the spine of this publication, which gives a distinct method of the subject, integrating statistical tools, info research, and functions. Now widely revised, the publication comprises new information regarding combined results versions, purposes of the combined process, regression diagnostics with the corresponding IML strategy code, and covariance constructions. The authors' method of the data will relief professors, researchers, and scholars in quite a few disciplines and industries. broad SAS code and the corresponding high-resolution output accompany pattern difficulties, and transparent causes of SAS systems are incorporated. Emphasis is on right interpretation of the output to attract significant conclusions. that includes either the theoretical and the sensible, themes lined contain multivariate research of experimental information and repeated measures information, graphical illustration of knowledge together with biplots, and multivariate regression. additionally, a brief creation to the IML method with distinctive connection with multivariate facts comes in an appendix. SAS courses and output built-in with the textual content make it effortless to learn and persist with the examples.

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R2 respectively represent the portion of the total variation accounted for by the dimensions 1, . . , r . 2) then we get an approximate biplot for Y and the corresponding goodness of fit is measured by η= λ21 + λ22 . r 2 i=1 λi If the actual dimension of Y is 2, then η = 1. If r ≥ 3, η < 1. Thus if η is near one, the two-dimensional biplot will give a good visual approximation of the data matrix Y which has dimension r . Gabriel (1971) suggested these choices for the coordinates of the biplot.

5’; /* Generate n Wishart matrices of order p by p with degrees of freedom f */ proc iml; n = 4 ; f = 7 ; Chapter 1 Multivariate Analysis Concepts 19 seed = 4509049 ; sigma = {4 2 1, 2 3 1, 1 1 5 } ; g = root(sigma); p = nrow(sigma) ; print ’Wishart Random Matrix’; do i = 1 to n ; t = normal(repeat(seed,f,p)) ; x = t*g ; w = x‘*x ; print w ; end ; These programs can be easily modified to generate the Beta matrices of either Type 1 or Type 2, as the generation of such matrices essentially amounts to generating the pairs of Wishart matrices with appropriate degrees of freedom and then combining them as per their definitions.

5 Then save these four vectors as the rows of 4 by 3 matrix Y. It is easy to see that   µ µ   E(Y) =   µ  = M. µ 18 Applied Multivariate Statistics Also, let G be a matrix such that Σ = G G. This matrix is obtained using the ROOT function which performs the Cholesky decomposition of a symmetric matrix. 4’; /* Generate n random vector from a p dimensional population with mean mu and the variance covariance matrix sigma */ proc iml ; seed = 549065467 ; n = 4 ; sigma = { 4 2 1, 2 3 1, 1 1 5 }; mu = {1, 3, 0}; p = nrow(sigma); m = repeat(mu‘,n,1) ; g =root(sigma); z =normal(repeat(seed,n,p)) ; y = z*g + m ; print ’Multivariate Normal Sample’; print y; We first generate a 4 by 3 random matrix Z, with all its entries distributed as N (0, 1).

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