# 2. Linearity (in relationships among the variables--factors are linear constructions of the set of variables) F 2 X 4 U 4

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1 1 Neuendorf Factor Analysis Assumptions: 1. Metric (interval/ratio) data. Linearity (in relationships among the variables--factors are linear constructions of the set of variables) 3. Univariate and multivariate normal distributions 4. There are no dependent variables... Instead, the model is: X 1 U 1 (F=factor; U=unique var.) F 1 X U X 3 U 3 Decisions to make: 1. Intent: Exploratory vs. Confirmatory. Factor extraction (how factors represent variables): F X 4 U 4 Principal Component Factoring (the most common): -uses the "regular" correlation matrix. -factors derived on the basis of total variance for each variable (both the variance accounted for by Fs and Us (unique variances). -more widely used than common factoring. Common Factoring: -uses a correlation matrix with estimated communalities, not 1.0s, in diagonals. -factors derived on basis on common variance only (variance accounted for by Fs, but not the Us). There are several options for common factor extraction--principal axis factoring (most common), unweighted least squares, generalized least squares, maximum-likelihood, alpha, and image (see an SPSS manual or help for more info.). 3. Rotation:

3 3 Hair (p. 11) provides a chart for assessing the statistical significance of each loading. VAR1 = \$ 1 F1 + \$ F + \$ 3 F3 VAR = \$ 4 F1 + \$ 5 F + \$ 6 F3 etc. 3. Correlations between the variables and the factors--in orthogonal rotation, these are the same as loadings, in oblique they are not, and are contained in the structure matrix. 4. Communality (h )--the sum of all squared loadings for one variable. h = how much (%) of the variable's variance is explained by all the factors; 1-h = what is unique to the variable. e.g., \$ 1 + \$ + \$ 3 from # above would be the communality of VAR1. It is comparable to R VAR1.F1FF3. 5. Eigenvalue--the sum of all squared loadings for one factor. eigen = the amount (not expressed as a %) of the variables' total variance that is accounted for by one factor. e.g., \$ 1 + \$ 4 + \$ 7, etc. from # above would be the eigenvalue for F1. 6. Proportion of total variance accounted for by F1 = eigenvalue for F1 k where k=# of variables 7. Proportion of common variance accounted for by F1 = eigenvalue for F1 SUM of h 's 8. Factor score coefficients (betas predicting factors from variables \$ s below; these are not the same as loadings)--found in the component score coefficient matrix in SPSS. These values are used by SPSS to create "factor scores"--values on new scales created for each factor. These new scales must be saved in the FACTOR procedure if you want to use them for further analysis. There are three options--we typically use the regression method. SPSS will name the new scales automatically and place them at the end of the data set.

4 4 F1 = \$ 1 ZVAR1 + \$ ZVAR + \$ 3 ZVAR3 + \$ 4 ZVAR4 + etc. F = \$ 5 ZVAR1 + \$ 6 ZVAR + \$ 7 ZVAR3 + \$ 8 ZVAR4 + etc. F3 = \$ 9 ZVAR1 + \$ 10 ZVAR + \$ 11 ZVAR3 + \$ 1 ZVAR4 + etc. 9. Correlations among factors--for oblique rotation only! Found in the component correlation matrix, this indicates how much the various factors (and therefore their scales, if created) relate to one another in linear fashion.

5 5 Making Sense of Factor Loadings -Assume an orthogonal rotation. -Don't forget that they must be squared before adding to get communalities and eigenvalues. Factor Loadings: Communalities Communalities at 3 at 10 FACTOR 1 FACTOR FACTOR 3 Factors: Factors: Label: Q--News Q5--Westerns Q6--Action Q7--Weepies Q8--Game shows Q1--Soaps Q18--Talk shows Q--Comedies Q3--Horror Q4--Cop shows ? 1.0 Eigenvalues: sum=? sum=10.0 % of Total Variance: %? %? %? %? % of Common Variance: %? %? %? 100% /4/0

### 4. There are no dependent variables specified... Instead, the model is: VAR 1. Or, in terms of basic measurement theory, we could model it as:

1 Neuendorf Factor Analysis Assumptions: 1. Metric (interval/ratio) data 2. Linearity (in the relationships among the variables--factors are linear constructions of the set of variables; the critical source

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