Notes on Preliminary Notation and Terminology | EDMS 657, Study notes of Statistics

Download Notes on Preliminary Notation and Terminology | EDMS 657 and more Study notes Statistics in PDF only on Docsity! Preliminary notation and terminology Diagram symbols an observed, measured V (X, Y) variable for which you have data a latent variable ("con- F (ξ, η) struct," "factor") not meas- ured directly error or residual in a E (δ, ε) measured variable not explained F1 V1 theory says F1 might influence V1, but not vice versa F1 F2 no causal relation between F1 and F2 is hypothesized, but they might correlate Independent (Exogenous) thing: One whose variation is assumed to be determined by causes outside the hypothesized model. There are no causal inputs. In factor analysis these are factors or errors. Dependent (Endogenous) thing: One whose variation is explained by exogenous or other endogenous things in the model. There is at least one causal input. In factor analysis these are measured variables. Preliminary remarks regarding factor analysis Factor analysis -- a collection of procedures for inferring, or testing, the existence of underlying "latent" variables as an explanation for observed relations among measured variables. Exploratory Confirmatory principal components analysis Maximum likelihood factor analysis principal axis factoring image factoring alpha factoring "I wonder what factors underlie these data" "I hypothesize that the following factors underlie these data. Let me test how well the data confirm (fail to disconfirm) my theory" If the theory expressed in the diagram is correct, then elements in each column (exclusive of the diagonal) should be roughly proportional to those in all other columns. VC VF VE VM VP VT VC (c2) cf ce cm cp ct VF fc (f2) fe fm fp ft VE ec ef (e2) em ep et VM mc mf me (m2) mp mt VP pc pf pe pm (p2) pt VT tc tf te tm tp (t2) VC VF VE VM VP VT VC 1.00 .83 .78 .70 .66 .63 VF .83 1.00 .67 .67 .65 .57 VE .78 .67 1.00 .64 .54 .51 VM .70 .67 .64 1.00 .45 .51 VP .66 .65 .54 .45 1.00 .40 VT .63 .57 .51 .51 .40 1.00 Check VC and VE: Convinced of the existence of a general intelligence factor ("g"), Spearman needed a method to solve for the paths from the factor to each variable. Spearman's method of triads: (cf)(ce)/(fe) = c2 = (.83)(.78)/(.67) = .966 (cf)(cm)/(fm) = c2 = (.83)(.70)/(.67) = .867 (cf)(cp)/(fp) = c2 = (.83)(.66)/(.65) = .843 (cf)(ct)/(ft) = c2 = (.83)(.63)/(.57) = .917 (ce)(cm)/(em) = c2 = (.78)(.70)/(.64) = .853 (ce)(cp)/(ep) = c2 = (.78)(.66)/(.54) = .953 (ce)(ct)/(et) = c2 = (.78)(.63)/(.51) = .964 (cm)(cp)/(mp) = c2 = (.70)(.66)/(.45) = 1.027 (cm)(ct)/(mt) = c2 = (.70)(.63)/(.51) = .865 (cp)(ct)/(pt) = c2 = (.66)(.63)/(.40) = 1.040 Could average all these and take a square root, but a slightly more stable way is: Estimate of c2 = sum of all triad numerators = 5.1701 sum of all triad denominators 5.6100 = .9216 Estimate of c = 9216. = .9600 This process would then be repeated for all paths: c, f, e, m, p, t. This is a very cumbersome method, particularly as the number of variables increases. For a system with p variables, each estimate would be based on (p-1)(p-2)/2 equations! Using the method of triads, we would get: c = .9600 m = .7495 f = .8870 p = .6662 e = .8037 t = .6438 Original observed correlations VC VF VE VM VP VT VC 1.00 VF .83 1.00 VE .78 .67 1.00 VM .70 .67 .64 1.00 VP .66 .65 .54 .45 1.00 VT .63 .57 .51 .51 .40 1.00 Reproduced correlations (after plugging solutions for c, f, e, m, p, t back into the model-implied correlation matrix): VC VF VE VM VP VT VC .92 VF .85 .79 VE .77 .71 .65 VM .72 .66 .60 .57 VP .64 .59 .54 .50 .44 VT .62 .57 .52 .48 .43 .41 Residual correlations (original minus reproduced) VC VF VE VM VP VT VC .08 VF -.02 .21 VE .01 -.04 .35 VM -.02 .01 .04 .43 VP .02 .06 .00 -.05 .56 VT .01 .00 -.01 .03 -.03 .59