1.
FACTOR ANALYSIS July 2014 updated
Prepared by Michael Ling Page 1
QUANTITATIVE RESEARCH METHODS
SAMPLE OF
FACTOR ANALYSIS PROCEDURE
Prepared by
Michael Ling
2.
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
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FACTOR ANALYSIS July 2014 updated
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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)
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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).
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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.
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FACTOR ANALYSIS July 2014 updated
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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.
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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.
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FACTOR ANALYSIS July 2014 updated
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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.
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FACTOR ANALYSIS July 2014 updated
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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.
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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.
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