Education 795 Class Notes

1. Education 795 Class Notes Factor Analysis II Note set 7

2. Today’s Agenda • Announcements (ours and yours) • Revisiting factor analysis • Reliability • Very Brief Intro to Confirmatory Factor Analysis

3. Revisiting Factor Analysis

4. The Necessary Steps • Identify and gather data appropriate for factor analysis • Decide upon extraction approach and selection criteria • PCA vs. PAF • Eigenvalue => 1 • Scree Plot • Rotate extracted factors after deciding upon rotational approach • Varimax • Oblimin • Before naming factors, cycle through steps 2 and 3 until you have achieved a reasonable statistical and conceptual solution

5. Factor Selection Criteriaand Rotation • Tools to identify the appropriate number of factors: • In the interest of parsimony, n of factors should be less than the number of variables being analyzed • Scree plot • Specific theorized number /CRITERIA = FACTORS(n) • Amount of variance explained (Eigenvalue) /CRITERIA = MINEIGEN(1.0) • Varimax (Orthogonal) assumes factors will be uncorrelated. • Oblimin allows dependence between factors

6. Rotating Extracted Factors • Unrotated factor matrix is only one of many possible ones; transformations can clarify meaning without changing the underlying relationships amongst the variables • Rotation is used to ease interpretation but it should be tied to theory! • Desire to approach “simple structure” • Orthogonal (Varimax) or oblique (Oblimin)? • Is it cheating to rotate?

7. Interpreting and Naming Rotated Factors • Appropriate after cycling through various solutions and identifying the one that makes both statistical and conceptual sense • Naming should capture the essence of the variables that are most closely associated with each factor • Should take the relative strength of loading into account in naming factors

8. Technical Details • Coefficients associated with unrotated factors can be interpreted like regression betas. Specifically, the square of the coefficient in the factor matrix indicates the proportion of variance of a given indicator that is accounted for by the factor. • The Factor Pattern Matrix contains the coefficients for the regression of each indicator on the factors. • The Factor Structure Matrix consists of the correlations between indicators and factors. • When the factors are uncorrelated, the two matrices are equal. • The eigenvalue is equal to the sum of the squared loadings of the indicators on the factor with which it is associated.

9. Sampling and Sample Size • Probability sampling is necessary if one wants to generalize findings of EFA. • General Rule: at least 10 cases per variable in the factor analysis (Nunnally, 1978). • Many others disagree and just say, ‘Use large samples’!

10. A factor analysis, employing a principal components extraction using the Eigenvalue > 1.0 criterion, identified three interpretable factors, explaining 46.5 percent of the common variance After reviewing the results of the analysis, we named these three factors… Return to Our Example

11. Varimax Output: Naming Factor I: Factor II: Factor III:

12. Scree Plot

13. Creating Factor Scores • A straightforward scale • compute extrinsic =momoney+betterjb • compute intrinsic=gainege+moculture+ improve+moculture+prepgrad. • Or use an average • compute extrinsic=(momoney+betterjb)/2 • Or use the factor loadings • compute extrinsic=.84*momoney+.84*betterjb. • Be sure to represent ‘reversed’ items in creating scales: • If you have a negative sign in the factor group • recode Q4 (1=2) (2=1). • rerun the factor analysis.

14. Extending / Using FA Results • Validity • Whether a measurement instrument or technique measures what it is supposed to measure • Reliability • Reliability is a necessary but not sufficient condition for validity (a measure cannot be valid if it is not reliable but being reliable does not imply valid). • Reliability is the consistency or stability of a measure • Test-retest reliability -- consistency over time • Internal consistency reliability -- multiple items thought to measure the same construct should be correlated

15. Coefficient Alpha • A standard measure of internal consistency, developed by Cronbach • Expands the concepts of inter-item correlation averaging (add up all the correlations and divide by n), and split-half reliability (randomly divide the items measuring a single concept in half, compute total score for each half set of items, and then correlate them) • Mathematically equivalent to the average of all possible split-half estimates

16. Cronbach’s Alpha • Relatively low reliabilities OK and are tolerable in early phases of research. • Higher reliabilities are required when the measure is used to determine group differences (>.7) (Nunnally, 1978) • Very high reliabilities are needed for making important decisions about individuals (>.9) (Pedhazur, p. 109) • Ultimately it depends on how much error the researcher is willing to have

17. Reliability Example

18. Intro to Confirmatory Factor Analysis • Formulation of a model is a prerequesite for CFA—the aim is to “test” the model or assess the fit to the data • CFA is a submodel of Structural Equation Modeling • CFA is a measurement model of relations of indicators to factors as well as relations among factors

19. EFA vs. CFA • In EFA, all indicators have loadings; not necessarily so in CFA • Correlated factors are all or nothing in EFA. In CFA it is possible to specify that only some of the factors can be correlated. • In EFA, it is assumed that errors in indicators are not correlated. In CFA we can test this assumption.

20. CFA results from a study of college faculty

21. Some Last Details: Standardize First

22. For Next Week • Read Pedhazur Ch 6 p119-131 • Readings to be handed out in class on Affirmative Action Case Study