Department of Industrial and Systems Engineering, IIT Kharagpur Subject: Applied Multivariate Statistical Modeling I (IM60061)

Home Assignment X on Exploratory Factor Analysis (EFA)

Prepared by Prof J Maiti of ISE, IIT Kharagpur

1. What is exploratory factor analysis (EFA)? Is it different from PCA? Explain. 2. State the assumptions of EFA. 3. Prove that in the population factor model, TLL = + [the notations represent their

usual meaning]. 4. What is commonality? If jj is the variance of the j th original variable, then show that,

th thcommonality specific variancejj j j = + . 5. Let X = [X1, X2, ., Xp]T and = Cov (X). Further, m (< p) oblique factors are extracted. Show

that TL L = + [the notations represent their usual meaning]. 6. What is exploratory factor analysis (EFA)? What way it is different from confirmatory factor

analysis? 7. What is factor rotation? Why it is used? 8. Show that the variance explained by a m-factor model remains unchanged even after

rotation [Hint: * *TL L = + ]. 9. Name three methods of orthogonal rotation of factors. Compare them. 10. What is factor score? Show that using weighted least square method, the estimated i-th

factor score is 1 1 1 ( ) ( )T Ti if L L L x x = for I = 1,2,.,n [Hint: See section 9.5 (factor

scores) from Johnson and Wichern, page 515]. 11. What is confirmatory factor analysis (CFA)? What way CFA is different from EFA? 12. The correlation matrix for X = [x1, x2, x3, x4, x5, x6]T is given below.

The following factor loadings are extracted by the maximum likelihood procedure:

Variable Estimated factor loadings F1 F2 x1 x2 x3 x4 x5 x6

0.602 0.467 0.926 1.000 0.874 0.894

0.200 0.154 0.143 0.000 0.476 0.327

Obtain the maximum likelihood estimates of the following:

a. The specific variances, b. The communalities, and c. The proportion of variance explained by each factor.

1.000.505 1.000.569 .422 1.000.602 .467 .926 1.000.621 .482 .877 .874 1.000.603 .450 .878 .894 .937 1.000

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13. A large manufacturing company is successful in implementing lean engineering. The factors (hidden) that determine leanness are just-in-time (JIT) practices, workers involvement, and autonomation. 30 officials responsible for lean production at the different levels of organizational hierarchy were asked on six questions shown in Table below. The exploratory factor model retains two factors (based on Kaisers rule) as shown in the Table. Correlation matrix was used.

Variable Estimated factor loadings F1 F2 x1 = Use of pull system x2 = Reduction of set up time x3 = On-time delivery x4 = Continuous improvement x5 = Quality circles x6 = Work standardization

0.50 0.60 0.70 0.40 0.50 0.60

-0.50 -0.60 -0.70 0.40 0.60 0.60

a) Obtain the specific variances, communalities, and proportion of variance explained by

each factor. b) Does rotation help in interpreting the factors? If so, what are the rotated factor

loadings? [Hint: Use graph paper]. c) Comment on the results.

14. In an exploratory study involving ten variables (X1 to X10), two factors are extracted using

sample correlation matrix. The factor loadings are given below. Variables X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 Factor 1 0.20 -0.30 0.40 0.50 -0.30 0.20 0.40 0.40 0.50 0.20 Factor 2 0.20 -0.30 0.30 0.40 0.30 -0.20 -0.30 -0.70 -0.80 -0.90

(a) Obtain the explained variance for each of the 10 variables. (b) What is the variability of X explained by factor 1 and factor 2, respectively? (c) Can the factor loadings be improved? If yes, compute the improved factor loadings [Hint:

Use graph paper].

15. The un-rotated factor loadings for a study are shown below.

Variables Factor (F1) Factor (F2) 1X 0.50 0.80

2X 0.60 0.70

3X 0.90 -0.25

4X 0.80 -0.30

5X 0.60 -0.50

Use varimax rotation and obtain the rotated factor loadings [Hint: Use graph paper].

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