Derek Ogle wrote: > > Members, > Can anyone provide me (a description or a reference will suffice) with a > convincing argument or demonstration of WHY the first eigenvector-eigenvalue > of the variance-covariance matrix represents the direction and magnitude of > the greatest variability in the "cloud of multivariate data"? I can convince > the students that this is what happens but I can't convince them of the why > this is what happens. Thank you in advance for your help. > A qualitative, visual answer: Imagine a spherical pin-cushion with pins sticking out in many different directions. Each pin represents a variable (test score or item score). Some pins are long (these represent variables with high communality); some are short. Pins sticking out in much the same direction represent variables which are highly correlated. A pin at right angles to another pin represents zero correlation. Imagine that there is a dominant cluster of long pins pointing in the one general direction, say eastwards. The vector sum of this set of pin directions (eastwards) represents the direction of the first principal component. It will have the largest eigenvalue. Eigenvalue is the sum of the squared loadings of each variable on that component. The loadings are large because many "pins" are correlated with each other. Hope this helps Paul Gardner
begin:vcard n:Gardner;Dr Paul tel;cell:0412 275 623 tel;fax:Int + 61 3 9905 2779 (Faculty office) tel;home:Int + 61 3 9578 4724 tel;work:Int + 61 3 9905 2854 x-mozilla-html:FALSE adr:;;;;;; version:2.1 email;internet:[EMAIL PROTECTED] x-mozilla-cpt:;-29488 fn:Dr Paul Gardner, Reader in Education and Director, Research Degrees, Faculty of Education, Monash University, Vic. Australia 3800 end:vcard
