By B. S. Everitt (auth.)
Latent variable types are utilized in many components of the social and behavioural sciences, and the expanding availability of desktop programs for becoming such types is probably going to extend their attractiveness. This e-book makes an attempt to introduce such types to utilized statisticians and learn staff attracted to exploring the constitution of covari ance and correlation matrices when it comes to a small variety of unob servable constructs. The emphasis is at the functional program of the techniques instead of on certain dialogue in their mathe matical and statistical homes. it really is assumed that the reader understands the main conventional statistical suggestions and strategies, rather regression, and in addition has a good wisdom of matrix algebra. My thank you are as a result of my colleagues Dr David Hand and Dr Graham Dunn for worthwhile reviews at the ebook, to Mrs Bertha Lakey for her cautious typing of a tricky manuscript and to Peter Cuttance for advice with the LlSREL package deal. moreover the textual content truly owes very much to the paintings on structural equation versions released via Karl Joreskog, Dag Sorbom, Peter Bentler, Michael Browne and others.
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Extra resources for An Introduction to Latent Variable Models
The elements of B represent direct causal effects of 'I variables on other 'I variables (so that the diagonal elements of B are zero), and the elements of r represent direct causal effects of ~ variables on 'I variables. We assume that' is uncorrelated with ~. Corresponding to the two sets of latent variables are two sets of observed, manifest variables, y'=[Y1'''''Yq ] and x'=[xto ... 5) where Il and (j are vectors, the elements of which represent errors of measurement in y and x, and the matrices Ay(q x I) and AAp x m) contain regression weights of yon 'I and x on ~.
1 _ - 0-1 ~ 0·2 ........ 0·3 ......... Math - 0-2 0·4 ern Ot - 0·5 .... ICOI 0'6 0·7 0-8 -6 ........ _5.... -4 - 0-3 - 0-4 Figure 2_1 Graphical rotation offactors interpreted very simply in terms of two oblique or correlated factors, one 'verbal' in content and the other 'mathematical'. The 'general factor' suggested by the original loadings is now submerged in these. When dealing with a two-factor model this graphical method of rotation is generally sufficient to find a new position for the axes which eases interpretation.
The elements of ~xx' ~yy and ~xy are functions of the elements of Ay, Ax, B, r, (f), '1', (Ja and (J6' In any particular application some of these elements will be fixed (usually to zero or one), some will be unknown but constrained to equal other elements and some will be unknown and unconstrained. Thus, the elements of the eight parameter matrices are of three kinds: (1) fixed parameters which have been assigned given values; (2) constrained parameters whose values are unknown but equal to one or more other parameters; (3) free parameters which are unknown and not constrained to be equal to any other parameter.
An Introduction to Latent Variable Models by B. S. Everitt (auth.)