FACTOR analysis (July 2014 updated)

FACTOR ANALYSIS July 2014 updated

Prepared by Michael Ling Page 1

QUANTITATIVE RESEARCH METHODS

SAMPLE OF

FACTOR ANALYSIS PROCEDURE

Prepared by

Michael Ling



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



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)



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).



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.



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.



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.



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.



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


Component Analysis.

a. 2 components extracted.



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


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


Component Analysis.

Rotation Method: Varimax with

Kaiser Normalization.

a. Rotation converged in 3 iterations.



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)

Education

FACTOR analysis (July 2014 updated)