1. EXPLORATORY FACTOR ANALYSIS (EFA)
Kalle Lyytinen & James Gaskin

2. Learning Objectives
Understand what is the factor analysis technique and its applications in research
Discuss exploratory factor analysis (EFA)
Run EFA with SPSS and interpret the resulted output
Estimate shortly reliability
Assess shortly construct validity

3. The whole works Analyzing the factor structure of the multi-item data
Theory
Constructs
Items linked to constructs
EFA
Collect data
Build/Run Structural
Model
Modify the
Measurement Model
Link items to constructs; Label constructs
Test structural hypotheses
Conduct CFA
Without CMB
With CMB
Conduct
Multi-group
CFA
Goodness of fit & psychometric properties filter
Data cleaning filter
Structural Model
Goodness of fit filter
Contribute to theory

4. Family Tree of SEM Source: PIRE
Multiple samples, multiple variables, over time, etc.
Is the difference between samples on a variable significant?
Is the correlation between different variables significant?
Multiple variables, overall model, measurement model, etc.
Source: PIRE

5. SCOPE of Factor Analysis today
Factor analysis and principal component analysis
Carrying out the analyses in SPSS
Deciding on the number of factors
Rotating factors
Producing factor and component scores
Assumptions and sample size
Exploratory and confirmatory FA

6. Types of Measurement Models
Exploratory (EFA)
Confirmatory (CFA)
Multitrait-Multimethod (MTMM)
Hierarchical CFA

7. EFA vs. CFA
Exploratory Factor Analysis is concerned with how many factors are necessary to explain the relations among a set of indicators and with estimation of factor loadings. It is associated with theory development.
Confirmatory Factor Analysis is concerned with determining if the number of factors “conform” to what is expected on the basis of pre-established theory. Do items load as predicted on the expected number of factors. Hypothesize beforehand the number of factors.

8. End-User Computing Satisfaction (EUCS)
EUCS: An instrument for measuring satisfaction with an information system
CONTENT:
Does the system provide the precise information you need?
Does the information content meet your needs?
Does the system provide reports that seem to be just about exactly what you need?
Does the system provide sufficient information?
ACCURACY:
Is the system accurate?
Are you satisfied with the accuracy of the system?
FORMAT:
Do you think the output is presented in a useful format?
Is the information clear?
EASE OF USE:
Is the system user friendly?
Is the system easy to use?
TIMELINESS:
Do you get the information you need in time?
Does the system provide up-to-date information?

9. Factor Analysis
Factor Analysis is a method for identifying a structure (or factors, or dimensions) that underlies the relations among a set of observed variables.
Factor analysis is a technique that transforms the correlations among a set of observed variables into smaller number of underlying factors, which contain all the essential information about the linear interrelationships among the original test scores.
Factor analysis is a statistical procedure that involves the relationship between observed variables (measurements) and the underlying latent factors.

10. Factor Analysis
Factor analysis is a fundamental component of Structural Equation modeling.
Factor analysis explores the inter-relationships among variables to discover if those variables can be grouped into a smaller set of underlying factors.
Many variables are “reduced” (grouped) into a smaller number of factors
These variables reflect the causal impact of the “latent” underlying factors
Statistical technique for dealing with multiple variables

11. Applications of Factor Analysis
Explore data for patterns.
Often a researcher is unclear if items or variables have a discernible patterns. Factor Analysis can be done in an Exploratory fashion to reveal
patterns among the inter-relationships of the items.
Data Reduction.
Factor analysis can be used to reduce a large number of variables into a smaller and more manageable number of factors. Factor analysis can create factor scores for each subject that represents these higher order variables.
Factor Analysis can be used to reduce a large number of variables into a parsimonious set of few factors that account better for the underlying variance (causal impact) in the measured phenomenon.
Confirm Hypothesis of Factor Structure.
Factor Analysis can be used to test whether a set of items designed to measure a certain variable(s) do, in fact, reveal the hypothesized factor structure (i.e. whether the underlying latent factor truly “causes” the variance in the observed variables and how “certain” we can be about it).
In measurement research when a researcher wishes to validate a scale with a given or hypothesized factor structure, Confirmatory Factor Analysis is used.
Theory Testing.
Factor Analysis can be used to test a priori hypotheses about the relations among a set of observed variables.

12. How would you group these Items?

13. Exploratory Factor Analysis
In EFA, the researcher
is attempting to explore
the relationships among items to determine if the
items can be grouped
into a smaller number of underlying factors.
In this analysis, all items are assumed to be related to all factors. V1 ε
Factor 1 V2 ε V3 ε
Factor 1 V4 ε

14. Factorial Solution
Factor
Loading
Item
Cross-Loading ?

15. Exploratory Factor Analysis
Measured Variables or
Indicators:
These variables are those that the researcher has observed or measured.
In this example, they are the four items on the scale.
Note, they are drawn as rectangles or squares. V1 ε
Factor 1 V2 ε V3 ε
Factor 1 V4 ε

16. Exploratory Factor Analysis
Unmeasured or Latent Variables:
These variables are not directly measurable, rather the researcher only
has indicators of these measures.
These variables are more often the more interesting, but more difficult variables
to measure (e.g., self-efficacy).
In this example, the latent variables are the two factors.
Note, they are drawn as elipses V1 ε
Factor 1 V2 ε V3 ε
Factor 1 V4 ε

17. Exploratory Factor Analysis
Factor Loadings:
Measure the relationship between the items and the factors.
Factor loadings can be interpreted like correlation coefficients;
ranging between -1.0 and +1.0.
The closer the value is to 1.0,
positive or negative, the stronger the relationship between the factor and the item.
Loadings can be both positive
or negative. V1 ε
Factor 1 V2 ε V3 ε
Factor 1 V4 ε

18. Exploratory Factor AnalysisV1 ε
Factor Loadings:
Note the direction of the arrows;
the factors are thought to influence the indicators, not vice versa.
Each item is being predicted by the factors.
Factor 1 V2 ε V3 ε
Factor 1 V4 ε

19. Exploratory Factor Analysis
Errors in Measurement:
Each of the indicator variables has some error in measurement.
The small circles with the ε indicate the error.
The error is composed of 'we know not what' or are not measured directly.
These errors in measurement are considered the reliability estimates for each indicator variable. V1 ε
Factor 1 V2 ε V3 ε
Factor 1 V4 ε

20. Multi-Indicator Approach
A multiple-indicator approach reduces the overall effect of measurement error of any individual observed variable on the accuracy of the results
A distinction is made between observed variables (indicators) and underlying latent variables or factors (constructs)
Together the observed variables and the latent variables make up the measurement model

21. Conceptual Model
This model holds that there are two uncorrelated factors that explain the relationships among the six emotion variables
Awe
Joy
Happiness
Guilt
Fear
Sadness
Positive
Affect
Negative
Affect
Variables Factor
(Observed) (Latent) 10

22. Measurement Model Items Positive Affect (Factor 1) Negative Affect
Joy
Loading*
Awe
Loading
Happiness
Fear
Guilt
Sadness
*The loading is a data-driven parameter that estimates the relationships
(correlation) between an observed item and a latent factor.

23. Assumptions of Factor Analysis
Data Matrix must have sufficient number of correlations
Variables must be inter-related in some way since factor analysis seeks the underlying common dimensions among the variables. If the variables are not related each variable will be its own factor!!
Rule of thumb: substantial number of correlations greater than .30
Metric variables are assumed, although dummy variables may be used (coded 0,1).
The factors or unobserved variables are assumed to be independent of one another. All variables in a factor analysis must consist of at least an ordinal scale. Nominal data are not appropriate for factor analysis.

24. Quick Quips about Factor Analysis
How many cases? Rule of 10—10 cases for every item; rule of 100– number of respondents should be the larger of (1) 5 times number of variables or (2) 100.
How many variables do I need to FA? More the better (at least 3)
Is normality of data required? Nope
Is it necessary to standardize one variables before FA? Nope
Can you pool data from two samples together in a FA? Yep, but must show they have same factor structure.

25. Tests for Basic Assumptions
Two statistics on the SPSS output allow you to look at some of the basic assumptions.
Kaiser-Meyer-Olkin (KMO) Measure of Sampling Adequacy, and Bartlett's Test of Sphericity
Kaiser-Meyer-Olkin Measure of Sampling Adequacy generally indicates whether or not the variables are able to be grouped into a smaller set of underlying factors. That is, will data factor well???
KMO varies from 0 to 1 and should be .60 or higher to proceed (can us .50 more lenient cut-off)
High values (close to 1.0) generally indicate that a factor analysis may be useful with your data.
If the value is less than .50, the results of the factor analysis probably won't be very useful.

26. Kaiser-Meyer-Olkin (KMO)
Marvelous s
Meritorious s
Middling s
Mediocre s
Miserable s
Unacceptable below .50

27. KMO Statistics: Interpreting the Output
In this example, the data support the use of factor analysis and suggest that the data may be grouped into a smaller set of underlying factors.
What does Bartlett’s Test of Sphericity explore?

28. Correlation Matrix Bartlett's Test of Sphericity
Tests hypothesis that correlation matrix is an identity matrix.
Diagonals are ones
Off-diagonals are zeros
Significant result indicates matrix is not an identity matrix.

29. Bartlett’s Test of Sphericity
Bartlett’s Test of Sphericity compares your correlation matrix to an identity matrix’
An identity matrix is a correlation matrix with 1.0 on the principal diagonal and zeros in all other correlations. So clearly you want your Bartlett value to be significant as you are expecting relationships between your variables, if a factor analysis is going to be appropriate!
Problem with Bartlett’s test occurs with large n’s as small correlations tend to be statistically significant – so test may not mean much!

30. Two Extraction Methods
Principal Component Analysis
Considers all of the available variance (common + unique) (places 1’s on diagonal of correlation matrix).
Seeks a linear combination of variables such that maximum variance is extracted—repeats this step.
Use when there is concern with prediction, parsimony and knows specific and error variance are small.
Results in orthogonal (uncorrelated factors)
Principal Axis Factoring (PFA) or Common Factor Analysis
Considers only common variance (places communality estimates on diagonal of correlation matrix).
Seeks least number of factors that can account for the common variance (correlation) of a set of variables.
PAF is only analyzing common factor variability; removing the uniqueness or unexplained variability from the model.
Called Principal Axis Factoring (PFA).
PFA preferred in SEM cause it accounts for co-variation, whereas PCS accounts for total variance

31. Methods of Factor Extraction
Principal-axis factoring (PAF)
diagonals replaced by estimates of communalities
iterative process
continues until negligible changes in communalities

32. What is a Common Factor?
It is an abstraction, a hypothetical construct that affects at least two of our measurement variables.
We want to estimate the common factors that contribute to the variance in our variables.
Is this an act of discovery or an act of invention?

33. What is a Unique Factor?
It is a factor that contributes to the variance in only one variable.
There is one unique factor for each variable.
The unique factors are unrelated to one another and unrelated to the common factors.
We want to exclude these unique factors from our solution.

34. Comparison of Extraction Models
PCA vs. PAF
Factor loadings and eigenvalues are a little larger with Principal Components
One may always obtain a solution with Principal Components
Often little practical difference
FYI—Other less-used Extraction Methods (Image, alpha, ML ULS, GLS factoring)

35. Principal Components Extraction
A communality (C) is the extent to which an item correlates with all other items.
Thus, in PCA extraction method when the initial communalities are set to 1.0, then all of the variability of each item is accounted for in the analysis.
Of course some of the variability is explained and some is unexplained.
In PCA with these initial communalities set to 1.0, you are trying to find both the common factor variance and the unique or error variance.

36. Principal Components Extraction
Statisticians have indicated that assuming that all of the variability of the items whether explained or unique can be accounted for in the analysis is flawed and definitely should not be used in an exploratory factor model.
Some researchers suggest PAF as the appropriate method for factor extraction using EFA.
In PAF extraction, the amount of variability each item shares with all other items is determined and this value is inserted into the correlation matrix replacing the 1.0 on the diagonals. As a result, PAF is only analyzing common factor variability; removing the uniqueness or unexplained variability from the model.

37. Factor Rotation: Orthogonal
Varimax (most common)
minimizes number of variables with high loadings (or low) on a factor—makes it possible to identify a variable with a factor
Quartimax
minimizes the number of factors needed to explain each variable. Tend to generate a general factor on which most variables load with med to high vales—not helpful for research
Equimax
combination of Varimax and Quartimax
Q&A:
Why use rotation method? Rotation causes factor loading to be more clearly differentiated—necessary to facilitate interpretation

38. Non-orthogonal (oblique)
The real issue is you don’t have a basis for knowing how many factors there are or what they are much less whether they are correlated! Researchers assume variables are indicators of two or more factors, a measurement model which implies orthogonal rotation.
Direct oblimin (DO)
Factors are allowed to be correlated. Diminished interpretability
Promax
Computationally faster than DO
Used for large datasets

39. Oblique Rotation
The variables are assessed for the unique relationship between each factor and the variables (removing relationships that are shared by multiple factors)
The matrix of unique relationships is called the pattern matrix.
The pattern matrix is treated like the loading matrix in orthogonal rotation.

40. Decisions to be made EXTRACTION: ROTATION: PCA vs PAF
Orthogonal or Oblique (non-orthogonal)

41. Procedures for Factor Analysis
Multiple different statistical procedures exist by which the number of appropriate number of factors can be identified.
These procedures are called "Extraction Methods."
By default SPSS does PCA extraction
This Principal Components Method is simpler and until more recently was considered the appropriate method for Exploratory Factor Analysis.
Statisticians now advocate for a different extraction method due to a flaw in the approach that Principal Components utilizes for extraction.

42. What else? How many factors do you extract?
One convention is to extract all factors with eigenvalues greater than 1 (e.g. PCA)
Another is to extract all factors with non-negative eigenvalues
Yet another is to look at the scree plot
Number based on theory
Try multiple numbers and see what gives best interpretation.

43. Eigenvalues greater than 1

44. Scree Plot
Three Factor Solution

45. Criteria For Retention Of Factors
Eigenvalue greater than 1
Single variable has variance equal to 1
Plot of total variance - Scree plot
Gradual trailing off of variance accounted for is called the scree.
Note cumulative % of variance of rotated factors

46. Interpretation of Rotated Matrix
Loadings of .40 or higher
Name each factor based on 3 or 4 variables with highest loadings.
Do not expect perfect conceptual fit of all variables.

47. Loading size based on sample (from Hair et al 2010 Table 3-2)
Significant Factor Loadings based on Sample Size
Sample Size
Sufficient Factor Loading 50
0.75 60
0.70 70
0.65 85
0.60
100
0.55
120
0.50
150
0.45
200
0.40
250
0.35
350
0.30

48. What else? How do you know when the factor structure is good?
When it makes sense and has a (relatively) simple and clean structure.
Total Variance Explained > .60
How do you interpret factors?
Good question, that is where the true art of this comes in.

49. Why EFA?

50. Why EFA? ?

51. Reflective versus Formative
Diet (Reflective)
Health (Formative)
R1. I eat healthy food.
R2. I do not each much junk food.
R3. I have a balanced diet.
F1. I have a balanced diet
F2. I exercise regularly
F3. I get sufficient sleep each night e3
Diet
Health R1 R2 R3 F1 F2 F3 e1 e2 e3
EDM 643

52. Diet (Reflective) Health (Formative)
Direction of causality is from construct to measure
Measures expected to be correlated
Indicators are interchangeable
Direction of causality is from measure to construct
No reason to expect the measures are correlated
Indicators are not interchangeable
EDM 643
*From Jarvis et al 2003

53. Adequacy Residuals ≤ 5% KMO ≥ 0.8 is better
Communalities ≥ 0.5 is better

54. Validity Face Validity (do they make sense?) Pattern Matrix
Convergent (high loadings)
Discriminant (no cross-loadings)
Factor Correlations
≤.7 is better
EDM 643

55. Reliability Split data and do two EFAs
Cronbach’s Alpha (>.70) for each factor
SPSS: Scale  Reliability Analysis
EDM 643