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FACTOR analysis (July 2014 updated)
 

FACTOR analysis (July 2014 updated)

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    FACTOR analysis (July 2014 updated) FACTOR analysis (July 2014 updated) Document Transcript

    • FACTOR ANALYSIS July 2014 updated Prepared by Michael Ling Page 1 QUANTITATIVE RESEARCH METHODS SAMPLE OF FACTOR ANALYSIS PROCEDURE Prepared by Michael Ling
    • FACTOR ANALYSIS July 2014 updated Prepared by Michael Ling Page 2 PROBLEM 1. Objectives of the tutorial for this week:- 1. Learn how to run a Factor Analysis. 2. Understand Varimax Rotation. 3. Application of Factor Analysis as a Data Reduction Technique. 2. Kay Sealey is the news director for KASI-TV, the local NBC affiliate for a large South-Western city. Sealey believes that the most important quality of an on-air newscaster is credibility in the eyes of the viewer. Accordingly, surveys are undertaken every six months that attempt to evaluate the credibility of the newscaster. One of the survey instruments used by the station is given below:- 3. Questionnaire Evaluate the anchorperson on the news broadcast that you reviewed by completing the following series of scales. Place a check mark on the scale position that most nearly matches your feelings about this anchorperson. For example, if you thought that in this anchorperson was extremely likeable, you would place a check mark in the blank nearest “likeable” (in this case, the far left blank). 1. likeable __ __ __ __ __ __ __ not likeable 2. knowledgeable __ __ __ __ __ __ __ not knowledgeable 3. unattractive __ __ __ __ __ __ __ attractive 4. intelligent __ __ __ __ __ __ __ not intelligent 5. good looking __ __ __ __ __ __ __ bad looking 6. not believable __ __ __ __ __ __ __ believable 4. This questionnaire was administered to 12 individuals. The following table contains the responses of the 12 people surveyed. The data was coded between 1 and 7. For example, a check mark closest to likeable would be coded as 1 and not likeable as 7. 1. likeable _1 _2 _3 _4 _5 _6 _7 not likeable ID Q1 Q2 Q3 Q4 Q5 Q6 1 1 2 5 3 3 6 2 1 2 5 3 3 6 3 5 6 5 5 5 5 4 2 2 5 2 3 5 5 2 2 5 2 3 5
    • FACTOR ANALYSIS July 2014 updated Prepared by Michael Ling Page 3 6 4 5 3 3 5 4 7 3 3 5 5 3 4 8 1 1 6 1 2 7 9 5 4 3 3 5 2 10 3 3 5 1 2 7 11 3 3 5 4 4 5 12 2 5 6 4 3 1 Step 1:-Produce a correlation matrix. Which variables are correlated? Does it appear that factor analysis would be appropriate for this data? Step 2:-Carry out a principal component factor analysis with unrotated factor analysis. How many factors? How many relevant factors? Use only Eigenvalue criterion to evaluate this. Also try a Scree plot using these Eigenvalues for different factors. How many factors should be retained (using Eigenvalue criterion)? Step 3:-Using Varimax rotation and the number of factors retained, run a factor model again. Are the unrotated factor loadings different from Varimax? Why or why not?? Try and interpret the results. Step 4:-Interpret the factors. Questions 1. Interpret the correlation matrix. (2Marks) 2. How many relevant factors are there? Use both Eigenvalue and Scree criteria to evaluate this. Provide a figure for the Scree plot. (3Marks) 3. How are the Varimax rotation factor loadings different from unrotated factor loadings? Which one makes more sense? Interpret the factors using Varimax rotation factor loadings. (2Marks) 4. Estimate Reliability for the factors identified in Question 3? (1Mark) 5. Why is factor analysis used for this data? Provide some tests to indicate the appropriateness of data for factor analysis. (2Marks) 6. Provide managerial recommendations to Kay Sealey. (5Marks)
    • FACTOR ANALYSIS July 2014 updated Prepared by Michael Ling Page 4 SOLUTON 1. Overall, the correlations were above .30, which indicated a fair amount of correlations among the variables and was also a recommended minimum for factor analysis. Some of the items were highly correlated such as Look/Likeable (r = .799**), Look/Know (r = .723**), Look/Attract (r = -.731*) and Likeable/Know (r = .774**). Some of the negative correlated items such as Likeable/Attract (r = -.653*) and Know/Believe (r = -.607*) were caused by the reverse coding of Attract and Believe scales compared to the other four scales (Table 1). (Note: * correlation is significant at .05 (2-taileed); ** correlation is significant at .01 (2 tailed)). 2. As there were two Eigenvalues whose values were greater than 1, being 3.751 and 1.107 and both accounted for 80.96% of total variance (Table 2), it suggested that it might be a 2- factor model. From the Scree Plot, 2 or 3 factors were possible (Figure 1). However, the Eigenvalues criterion was adopted as the Scree Plot was not very precise and did not reject a 2- factor model. From the unrotated component matrix, there were strong associations of Likeable, Know, Look and Believe with factor 1, and equally strong associations (cross-loadings) of Attract and Intell with both factors 1 and 2 (Table 3). With the presence of high loadings of 4 variables on one factor and high cross-loadings of 2 variables on two factors, interpretations would be difficult and hence rotation were required to redistribute the variances. The negative correlations were eliminated through variable transformations, which resulted in all positive correlations (Table 4). The component matrix of transformed variables showed that the loadings of variables on factor 1 were all positive but there were cross-loadings created by Attract and Intell (Table 5). The communalities values were high which meant that a large amount of the variances in the variables was extracted by the factoring structure (Table 6).
    • FACTOR ANALYSIS July 2014 updated Prepared by Michael Ling Page 5 3. The Varimax rotated factor loadings indicated that the associations of Like (.796), Look (.790) and Attract (.948) with factor 1were stronger than factor 2, and the associations of Know (.769), Intell (.896) and Believe (.753) with factor 2 were stronger than factor 1 (Table 7). More importantly, the substantial cross-loadings in Attract and Intell were eliminated and the loadings of each variable on one factor were maximized. As a result, the Varimax rotation factor loadings made more sense. Thus, factor 1 could be interpreted as the Resourcefulness (Knowledgeable, Intelligent and Believable) of the newscasters and factor 2 could be interpreted as the External Appearances (Likeable, Good Looking and Attractive) of the newscasters. 4. Cronbach’s alpha was .863, which showed good reliability as it was greater than the generally agreed upon lower limit of .70 (Table 8). 5. Firstly, it is important that the data matrix has sufficient correlations to justify the use of factor analysis. A visual inspection of the correlation matrix revealed the presence of a large number of significant correlations greater than .30 (Table 1). Secondly, the Bartlett test of sphericity indicated that there was statistical significance, Sig < .001, that the correlation matrix had significant correlations among its variables (Table 9). Thirdly, the KMO statistic test for measure of sampling adequacy was .694, which was towards the middling value of .60 (Table 9). Fourthly, the Measure of Sampling Adequacy (MSA) index (Table 10) indicated that most of the variables had indices over .70 but only Intell (.645) and Attract (.521) had lower indices. As MSA increases with sample size and the current sample size was only 12, a larger sample size would likely to improve the individual MSA values. The overall MSA value was .69, which was greater than the recommended .50 value.
    • FACTOR ANALYSIS July 2014 updated Prepared by Michael Ling Page 6 6. Based on the results of factor analysis, the six items that are originally used by Kay Sealey as measures of credibility for the newscasters can now be reduced or simplified to two broader underlying evaluating dimensions – the Resourcefulness and the External Appearances of the newscasters. These two dimensions (or factors) are composites of the six specific variables, which in turn allow the dimensions to be interpreted and described. The Resourcefulness factor accounts for the variances of knowledge and intelligence of the newscasters as well as whether they are ‘believable’. The External Appearances factor accounts for the variances of attractiveness, good looking of the newscasters as well as whether they are likeable. The results of the factor analysis provides for Sealey a smaller set of dimensions (two of them) to consider in its strategic or operational plans, while still providing insight into what constitutes each dimension (i.e. the individual variables defining each factor). In terms of building and enhancing the quality of its newscasters, KASI-TV can now use the two broader dimensions – Resourcefulness and External Appearances - to develop its hiring and recruitment strategies, training and development programs.
    • FACTOR ANALYSIS July 2014 updated Prepared by Michael Ling Page 7 Appendix Table 1: Correlations Likeable Know Attract intell Look Believe Likeable Pearson Correlation 1 .774** -.653* .423 .799** -.419 Sig. (2-tailed) .003 .021 .171 .002 .176 N 12 12 12 12 12 12 Know Pearson Correlation .774** 1 -.360 .618* .723** -.607* Sig. (2-tailed) .003 .251 .032 .008 .037 N 12 12 12 12 12 12 Attract Pearson Correlation -.653* -.360 1 -.072 -.731** .294 Sig. (2-tailed) .021 .251 .824 .007 .354 N 12 12 12 12 12 12 intell Pearson Correlation .423 .618* -.072 1 .560 -.520 Sig. (2-tailed) .171 .032 .824 .058 .083 N 12 12 12 12 12 12 Look Pearson Correlation .799** .723** -.731** .560 1 -.497 Sig. (2-tailed) .002 .008 .007 .058 .100 N 12 12 12 12 12 12 Believe Pearson Correlation -.419 -.607* .294 -.520 -.497 1 Sig. (2-tailed) .176 .037 .354 .083 .100 N 12 12 12 12 12 12 **. Correlation is significant at the 0.01 level (2-tailed). *. Correlation is significant at the 0.05 level (2-tailed). Table 2: Total Variance Explained Component Initial Eigenvalues Extraction Sums of Squared Loadings Total % of Variance Cumulative % Total % of Variance Cumulative % 1 3.751 62.515 62.515 3.751 62.515 62.515 2 1.107 18.445 80.960 1.107 18.445 80.960 3 .535 8.916 89.876 4 .379 6.315 96.191 5 .139 2.322 98.513 6 .089 1.487 100.000 Extraction Method: Principal Component Analysis.
    • FACTOR ANALYSIS July 2014 updated Prepared by Michael Ling Page 8 Figure 1: Scree Plot Table 3: Component Matrixa Component 1 2 Likeable .879 -.246 Know .878 .211 Attract -.660 .682 intell .668 .599 Look .922 -.193 Believe -.690 -.375 Extraction Method: Principal Component Analysis. a. 2 components extracted.
    • FACTOR ANALYSIS July 2014 updated Prepared by Michael Ling Page 9 Table 4: Correlations likeable_2 know_2 intell_2 Look_2 Attract Believe likeable_2 Pearson Correlation 1 .774** .423 .799** .653* .419 Sig. (2-tailed) .003 .171 .002 .021 .176 N 12 12 12 12 12 12 know_2 Pearson Correlation .774** 1 .618* .723** .360 .607* Sig. (2-tailed) .003 .032 .008 .251 .037 N 12 12 12 12 12 12 intell_2 Pearson Correlation .423 .618* 1 .560 .072 .520 Sig. (2-tailed) .171 .032 .058 .824 .083 N 12 12 12 12 12 12 Look_2 Pearson Correlation .799** .723** .560 1 .731** .497 Sig. (2-tailed) .002 .008 .058 .007 .100 N 12 12 12 12 12 12 Attract Pearson Correlation .653* .360 .072 .731** 1 .294 Sig. (2-tailed) .021 .251 .824 .007 .354 N 12 12 12 12 12 12 Believe Pearson Correlation .419 .607* .520 .497 .294 1 Sig. (2-tailed) .176 .037 .083 .100 .354 N 12 12 12 12 12 12 **. Correlation is significant at the 0.01 level (2-tailed). *. Correlation is significant at the 0.05 level (2-tailed). Table 5: Component Matrixa Component 1 2 likeable_2 .879 -.246 know_2 .878 .211 intell_2 .668 .599 Look_2 .922 -.193 Attract .660 -.682 Believe .690 .375 Extraction Method: Principal Component Analysis. a. 2 components extracted.
    • FACTOR ANALYSIS July 2014 updated Prepared by Michael Ling Page 10 Table 6: Communalities Initial Extraction likeable_2 1.000 .833 know_2 1.000 .815 intell_2 1.000 .805 Look_2 1.000 .887 Attract 1.000 .900 Believe 1.000 .617 Extraction Method: Principal Component Analysis. Table 7. Rotated Component Matrixa Component 1 2 likeable_2 .796 .446 know_2 .474 .769 intell_2 .050 .896 Look_2 .790 .514 Attract .948 -.018 Believe .225 .753 Extraction Method: Principal Component Analysis. Rotation Method: Varimax with Kaiser Normalization. a. Rotation converged in 3 iterations.
    • FACTOR ANALYSIS July 2014 updated Prepared by Michael Ling Page 11 Table 8. Reliability Statistics Cronbach's Alpha N of Items .863 6 Table 9: KMO and Bartlett's Test Kaiser-Meyer-Olkin Measure of Sampling Adequacy. .694 Bartlett's Test of Sphericity Approx. Chi-Square 37.242 df 15 Sig. .001 Table 10: Anti-image Matrices likeable_2 know_2 intell_2 Look_2 Attract Believe Anti-image Covariance likeable_2 .222 -.134 .003 -.028 -.084 .075 know_2 -.134 .220 -.023 -.054 .091 -.142 intell_2 .003 -.023 .390 -.131 .159 -.125 Look_2 -.028 -.054 -.131 .153 -.134 .023 Attract -.084 .091 .159 -.134 .244 -.090 Believe .075 -.142 -.125 .023 -.090 .549 Anti-image Correlation likeable_2 .782a -.604 .010 -.151 -.359 .214 know_2 -.604 .721a -.078 -.294 .391 -.407 intell_2 .010 -.078 .645a -.538 .515 -.270 Look_2 -.151 -.294 -.538 .718a -.693 .078 Attract -.359 .391 .515 -.693 .521a -.245 Believe .214 -.407 -.270 .078 -.245 .766a a. Measures of Sampling Adequacy(MSA)