EPSY 546: LECTURE 4 EXPLORATORY AND CONFIRMATORY FACTOR ANALYSIS

1. EPSY 546: LECTURE 4EXPLORATORY AND CONFIRMATORYFACTOR ANALYSIS George Karabatsos

2. TRUE SCORERELIABILITY ANALYSIS • A reliability analysis provides a (an approximate) test of the hypothesis that the set of test items are unidimensional. • Items that have non-positive correlations with the rest of the items may be removed. • Such items measure a different dimension, i.e., latent trait.

3. TRUE SCORERELIABILITY ANALYSIS • The interpretation of correlation coefficients in decisions to retain/remove items involves some degree of subjectivity. • Also, it may be desirable to determine whether subsets of items of the test each consist of correlated clusters of items. Perhaps each subset tap different measurement dimensions (dims being uncorrelated or uncorrelated).

4. ITEM CORRELATION ANALYSIS: DIFFERENT VIEW

5. ITEM CORRELATION ANALYSIS: DIFFERENT VIEW • Regression of items on the total score.

6. REGRESSION OF ITEMS E(X+n)= a + b1Xn1 + b2Xn2 +…+ bjXnj +…+ bjXnj

7. REGRESSION OF ITEMS • E(X+n)= a + b1Xn1 + b2Xn2 +…+ bjXnj +…+ bjXnj • Over all subjects n = 1,…,N : • Correlate the E(X+n)with the Xnj • for each item j = 1,…,J. • (“loading” of item j in Factor 1)

8. REGRESSION OF ITEMS • E(X+n)= a + b1Xn1 + b2Xn2 +…+ bjXnj +…+ bjXnj • Over all subjects n = 1,…,N : • Correlate the E(X+n)with the Xnj • for each item j = 1,…,J. • (“loading” of item j in Factor 1) • Calculate the residual: • Res2n= X+ E(X+n)

9. REGRESSION OF ITEMS E(Res2n)= a + b1Xn1 + b2Xn2 +…+ bjXnj +…+ bjXnj

10. REGRESSION OF ITEMS • E(Res2n)= a + b1Xn1 + b2Xn2 +…+ bjXnj +…+ bjXnj • Over all subjects n = 1,…,N : • Correlate the E(Res2n)with the Xnj • for each item j = 1,…,J. • (“loading” of item j in Factor 1)

11. REGRESSION OF ITEMS • E(Res2n)= a + b1Xn1 + b2Xn2 +…+ bjXnj +…+ bjXnj • Over all subjects n = 1,…,N : • Correlate the E(Res2n)with the Xnj • for each item j = 1,…,J. • (“loading” of item j in Factor 2) • Calculate the residual: • Res3n= Res2n  E(Res2n)

12. REGRESSION OF ITEMS • E(Res3n)= a + b1Xn1 + b2Xn2 +…+ bjXnj +…+ bjXnj • Over all subjects n = 1,…,N : • Correlate the E(Res3n)with the Xnj • for each item j = 1,…,J. • (“loading” of item j in Factor 3) • Calculate the residual: • Res4n= Res3n  E(Res3n)

13. REGRESSION OF ITEMS E(Res4n)= a + b1Xn1 + b2Xn2 +…+ bjXnj +…+ bjXnj

14. REGRESSION OF ITEMS • E(Res4n)= a + b1Xn1 + b2Xn2 +…+ bjXnj +…+ bjXnj • …and repeat until the residuals equal 0. (The number of factors never exceeds the number of items)

15. EXPLORATORY FACTOR ANALYSIS • Confirmatory factor analysis (CFA) is • the reverse of exploratory factor analysis. • Whereas EFA seeks to explore and extract factors from the data, CFA enables the analyst to test hypotheses that items fit pre-specified factors.

16. CONFIRMATORY FACTOR ANALYSIS • E(Res4n)= a + b1Xn1 + b2Xn2 +…+ bjXnj +…+ bjXnj • …and repeat until the residuals equal 0. (The number of factors never exceeds the number of items) • This is the basic idea of exploratory factor analysis (in this case, principal components analysis).