Lecture 5 practical_guidelines_assignments

Introduction to Applied Statistics and Applied Statistical Methods Practical guidelines
Prof. Dr. Chang Zhu page 1
Table of Contents
LECTURE 5................................................................................................................................................ 2
RECAP ...................................................................................................................................................... 2
EXPLORATORY FACTOR ANALYSIS (EFA).................................................................................................. 2
OUTPUT INTEPRETATION .................................................................................................................... 4
PRELIMINARY ANALYSIS ...................................................................................................................... 4
FACTOR EXTRACTION .......................................................................................................................... 5
HOW MANY FACTORS TO EXTRACT?................................................................................................... 5
RELIABILITY ANALYSIS (Cronbach’s α) ..................................................................................................... 7
REPORTING THE RESULTS OF THE EFA .................................................................................................... 8
ASSIGNMENT 5........................................................................................................................................ 9

LECTURE 5
EXPLORATORY FACTOR ANALYSIS (EFA)
A researcher has generated a new questionnaire which is designed to measure happiness. The
questionnaire that she has generated has 10 items on it and she has collected responses for this from
200 respondents. The items on the questionnaire are as follows:
- I feel enthusiastic
- I feel full of energy
- I have lots of interesting things to do
- I have lots of friends
- I have a lot to look forward to
- I feel excited at the start of each day
- I love meeting people
- I want to contact friends and family
- I want to go out and party
- The people at work inspire me
The questionnaire is measured on a five point scale where 1 = strongly disagree and 5 = strongly
agree.
The data file is named Happy_measure.sav
This example is taken from:
http://wps.pearsoned.co.uk/ema_uk_he_dancey_statsmath_4/84/21627/5536653.cw/content/index.html
RECAP
Before we proceed to a worked example, it is useful to look at some criteria that we are to apply in exploratory
factor analysis (EFA):
 To decide the sample size, the rule of thumb is that for one variable, we should have 5 to 10 participants
(Field, 2009).
 The Kaiser-Meyer-Olkin of sampling adequacy (KMO) should be greater than .5 to be acceptable.
 Barlett’s test should be significant to indicate variables are relatively independent from one another.
 Rotation option: choose Varimax if factors are theoretically independent; Direct oblimin or Promax if
factors might correlate.
 The community is the proportion of common variance within a variable, and should be greater than .3.
 Look at the table labelled Reproduced Correlations: the percentage of “nonresidual with absolute
values > .05” should be less than 50%.
 The factor loadings should be greater than .512 with a sample size of 100, and .364 with one of 200
(Stevens, 2002).
After EFA, it’s suggested that we will conduct reliability analysis for each of the factor.

In SPSS, choose Analyse > Dimension Reduction > Factor
Move all the variables to the Variables area.
Click on the Descriptive button to access the dialog box. We can select all of the options to get to
know all the preliminary analysis of our data. Click Continue to proceed to the next step.
Access the Extraction dialog box, for Method: choose Principal components. It is advised that we
choose Correlation matrix as it’s the standardized version of the Covariance matrix to overcome
differences in measurement scales.
Select Unrotated factor solution to see if the rotation can yield better intepretation of the factors.
The Scree plot and Eigenvalues greater than 1 are the two important options we should select to
come up with a decision of how many factors are retained. Click Continue to proceed to the next step.
Access the Rotation option: choose Varimax (as our factors are supposed to be independent from
each other). Under the Display: select Rotated solution (the result of the analysis) and Loading plot(s)
(a graphical display of the cluster of variables to their corresponding factor).
As default, SPSS sets the Maximum Iterations for Convergence at 25 times. However, if we have huge
data set, we can reset it to 30. Click Continue to proceed to the next step.
e

Access the Scores button, we can calculate the factor scores for each participant by selecting Save as
variables with the Anderson-Rubin method as our factor scores are independent. For correlated
factor scores, select the Regression method. Click Continue to proceed to the next step.
Click on the Options button, under the Missing Values, select Exclused cases listwise. For the
Coefficient Display Format, select Sorted by size and Suppress small coefficients with Absolute value
below: .4 (this value is set by us) so that we can have a clean table of results with factor loadings up
to .4.
Click Continue to return to the main dialog box, then select OK to run the anaysis.
OUTPUT INTEPRETATION
There are many output tables, but we will choose the most important tables to provide the basis for
our decision making of which variables will belong to which factors.
PRELIMINARY ANALYSIS
The first table we should look at is labeled KMO and Barlett’s Test. The KMO value is .79 (above .05)
and the Barlett’s test is significant (p < .001), which indicates that the sample is adequate for factor
analysis.
KMO and Bartlett’s Test
Kaiser-Meyer-Olkin Measure of Sampling Adequacy. .790
Bartlett's Test of Sphericity Approx. Chi-Square 819.746
Df 45
Sig. .000
The KMO values for each variable can be found in the table labeled Anti-image Matrices, by looking
at Anti-image Correlation. The values, as can be found, are well-above .5.

FACTOR EXTRACTION
If we look at the Communalities table, we can see the proportion of common variance within a
variable before and after extraction. The table shows that all values are above the cut-off value of .3.
Communalities
Initial Extraction
I feel enthusiastic 1.000 .566
I have lots of friends 1.000 .569
I love meeting people 1.000 .482
I feel full of energy 1.000 .646
I have lots of interesting things to
do
1.000 .421
I have a lot to look forward to 1.000 .682
I want to contact friends & family 1.000 .701
I want to go out and party 1.000 .796
The people at work inspire me 1.000 .608
I feel excited at the start of each
day
1.000 .645
Extraction Method: Principal Component Analysis.
HOW MANY FACTORS TO EXTRACT?
To decide how many factors to extract, we should look at 3 outputs:
The Total Variance Explained table shows that there are 2 components with eigen values bigger than
1, namely 3.18 and 2.92 respectively. The first component accounts for 31.86% of variance, and the
second factor 29.28%.
Total Variance Explained
Component
Initial Eigenvalues Extraction Sums of Squared Loadings Rotation Sums of Squared Loadings
Total % of Variance Cumulative % Total % of Variance Cumulative % Total % of Variance Cumulative %
1 3.186 31.862 31.862 3.186 31.862 31.862 3.170 31.699 31.699
2 2.928 29.279 61.140 2.928 29.279 61.140 2.944 29.442 61.140
3 .757 7.569 68.710
4 .658 6.583 75.293
5 .637 6.369 81.662
6 .522 5.220 86.882
7 .429 4.290 91.171
8 .380 3.801 94.973
9 .316 3.155 98.128
10 .187 1.872 100.000

The scree plot illustrates that the inflexion point where the two lines that represent the vertical and
horizontal parts met is at 3, meaning that we can extract 2 factors (because we do not include the
factor at the inflexion point).
The information given at the end of the Reproduced Correlations table indicates that there are 20
residuals (44%) with absolute values greater than .05. Though the percentage is quite big, it does not
exceeding the limit of 50%.
The Component Matrix table also indicates a two-factor solution, but our final decision will be based
on the table named Rotated Component Matrix. As indicated by the factor loadings, we can decide
that 2 factors will be extracted.
After making the decision, we will examine the underlying theme of each factor. The first factor
(including items in blue) can be described as a measure of sociability (How sociable a person is) and
the second factor positive feeling (or feeling good about the world).
Reproduced Correlations
I feel enthusiastic I have lots of friends I love meeting people I feel full of energy
I have lots of
interesting things to do
I have a lot to look
forward to
I want to contact
friends & family
I want to go out and
party
The people at work
inspire me
I feel excited at the
start of each day
Reproduced Correlation I feel enthusiastic .566a
.045 .041 .594 .480 .620 .080 .083 .070 .603
I have lots of friends .045 .569a
.523 -.067 -.047 .018 .631 .672 .587 .019
I love meeting people .041 .523 .482a
-.063 -.044 .016 .581 .618 .540 .017
I feel full of energy .594 -.067 -.063 .646a
.521 .657 -.042 -.047 -.044 .639
I have lots of interesting things to do .480 -.047 -.044 .521 .421a
.531 -.026 -.029 -.028 .517
I have a lot to look forward to .620 .018 .016 .657 .531 .682a
.053 .054 .045 .663
I want to contact friends & family .080 .631 .581 -.042 -.026 .053 .701a
.747 .653 .054
I want to go out and party .083 .672 .618 -.047 -.029 .054 .747 .796a
.695 .055
The people at work inspire me .070 .587 .540 -.044 -.028 .045 .653 .695 .608a
.046
I feel excited at the start of each day .603 .019 .017 .639 .517 .663 .054 .055 .046 .645a
Residualb
I feel enthusiastic -.023 .006 -.083 -.054 -.162 -.006 -.059 .053 -.112
I have lots of friends -.023 -.095 .013 -.005 .041 -.161 .001 -.159 -.016
I love meeting people .006 -.095 .061 -.003 .008 -.141 -.115 -.079 -.056
I feel full of energy -.083 .013 .061 -.170 -.022 -.017 .019 -.033 -.114
I have lots of interesting things to do -.054 -.005 -.003 -.170 -.131 .012 .010 .021 -.113
I have a lot to look forward to -.162 .041 .008 -.022 -.131 -.043 .040 -.056 -.047
I want to contact friends & family -.006 -.161 -.141 -.017 .012 -.043 -.006 -.033 .043
I want to go out and party -.059 .001 -.115 .019 .010 .040 -.006 -.120 -.018
The people at work inspire me .053 -.159 -.079 -.033 .021 -.056 -.033 -.120 .015
I feel excited at the start of each day -.112 -.016 -.056 -.114 -.113 -.047 .043 -.018 .015
a. Reproduced communalities
b. Residuals are computed between observed and reproduced correlations. There are 20 (44.0%) nonredundant residuals with absolute values greater than 0.05.

Rotated Component Matrixa
Component
1
sociability
2
positive feeling
Q8_I want to go out and party .892
Q7_I want to contact friends & family .837
Q9_The people at work inspire me .779
Q2_I have lots of friends .754
Q3_I love meeting people .694
Q6_I have a lot to look forward to .825
Q10_I feel excited at the start of each day .802
Q4_I feel full of energy .801
Q1_I feel enthusiastic .748
Q5_I have lots of interesting things to do .647
Rotation Method: Varimax with Kaiser Normalization.
a. Rotation converged in 3 iterations.
RELIABILITY ANALYSIS (Cronbach’s α)
When we develop our own questionnaire, after factor analysis, we should check the reliability of the
sub-scales.
Based on the factor analysis, we have 2 factors extracted or 2 sub-scales and the respective items as
below:
 Sub-scale 1 (sociability): Q2, 3, 7, 8, and 9
 Sub-scale 2 (positive feeling): Q1, 4, 5, 6, 10
We will calculate the Cronbach’s α for sub-scale 1 first.
In SPSS, choose Analyse > Scale > Reliability Analysis
Move the variables Q2, 3, 7, 8, and 9 to the Items area. The default model is set as Alpha, so we do
not need to change it.
Click on the Statistics button to access the Statistics dialog box. The important option we should
select is Scale if item deleted.

Click Continue to return to the main dialog box and OK to run the analysis.
In the output, the table Reliability Statistics tells us that the internal consistency of the 5 items is
measured with α = .851 (which is high).
Reliability Statistics
Cronbach's Alpha N of Items
.851 5
If we look at the final column in the table named Item-Total Statistics, we can see that if we remove
one of the items, the values of α are not any bigger than .851. Hence, the 5 items display strong
internal consistency to one another.
Item-Total Statistics
Scale Mean if Item
Deleted
Scale Variance if
Item Deleted
Corrected Item-
Total Correlation
Cronbach's Alpha if
Item Deleted
Q2_I have lots of friends 13.3850 12.188 .612 .832
Q3_I love meeting people 13.2250 11.914 .552 .850
Q7_I want to contact friends & family 13.1800 11.173 .712 .806
Q8_I want to go out and party 13.3550 10.773 .799 .782
Q9_The people at work inspire me 13.3150 11.825 .645 .824
We can report the reliability of the first sub-scale following APA style:
The sociability subscale of the Happiness questionnaire had high reliability with Cronbach’s α = .85.
Or:
The reliability of the sociability scale of the Happiness questionnaire was satisfactory (Cronbach’s α
= .85).
Repeat the same analysis for sub-scale 2 (positive feeling), we can obtain an α equal to .812.
REPORTING THE RESULTS OF THE EFA
Field (2009) suggests that we should report 2 things: a description of the analysis and a table of
factor loadings, eigenvalues, varianced explained, and Cronbach’s α as follows.
A principal component analysis (PCA) was conducted on the 10 items with orthogonal rotation
(varimax). The Kaiser-Meyer-Olkin measure verified the sampling adequacy for the analysis: KMO
= .79 which is good according to Field (2009). All KMO values for individual items are well above
the acceptable limit of .50 (Field, 2009). Bartlett’s test of spherity χ² (45) = 819.746, p < .001,
indicated that correlations between items were sufficiently large for PCA. Two components had
eigenvalues over Kaiser’s criterion of 1 and in combination explained 61.14% of the variance. The
scree plot also supports a two-factor structure. Table 1 shows the factor loadings after rotation.
The items that cluster on the same factors suggest that factor 1 represents sociability and factor 2
positive feeling.

Table 1
Exploratory Factor Analysis Results For The Happiness Meaure Questionnaire (N = 200)
Items
Rotated Factor Loadings
sociability positive feeling
Q8_I want to go out and party .892
Q7_I want to contact friends & family .837
Q9_The people at work inspire me .779
Q2_I have lots of friends .754
Q3_I love meeting people .694
Q6_I have a lot to look forward to .825
Q10_I feel excited at the start of each day .802
Q4_I feel full of energy .801
Q1_I feel enthusiastic .748
Q5_I have lots of interesting things to do .647
Eigenvalues 3.18 2.92
% of variance 31.86 29.28
𝜶 .85 .81
ASSIGNMENT 5
(You can work alone or in group for this assignment. If you work in group, please stay in the same
group of previous assignments and indicate the group members in the submission document).
The questionnaire is designed by Dr. Dian Williams to measure organizational ability. She predicted
a five-factor structure measuring different dimensions of organizational ability, which are
theoretically independent. The items are completed by 239 people, using a 7-point Likert scale (1 =
strongly disagree, 4 = neither, 7 = strongly agree).
The data file is named organization_measure.sav
Conduct factor and reliability analyses, and report the result in APA style. To come up with the
appropriate analysis option, and the final decision making on the number of factors that should be
extracted and the report, think about these questions to help you:
1. Are the constructs independent or dependent from each other? What is the appropriate
rotation method?
2. How many components/factors with eigenvalues bigger than 1? How many factors can be
extracted as suggested by the scree plot and the Rotated Component Matrix?
3. What dimension (name of factor, e.g. sociability) can we attribute to each of the
component?
4. Do the results from the reliability analysis suggest to remove any items that help to
increase the Cronbach’s alpha?

Lecture 5 practical_guidelines_assignments

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