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Distribution Free Vs. Nondistribution Free Methods
1. Distribution Free vs.
Non-distribution Free Methods in
Factor Analysis
Nicola Ritter, M.Ed.
EPSY 643: Multivariate Methods
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2. Top 5 Take Away Points
1. Extracted factors from a covariance matrix are a
function of correlations and standard deviations.
2. Different factors may be extracted based on the
matrix of associations selected.
3. Correlational statistics represented in matrices
address different questions.
4. Factors are sensitive to the information available
in a given correlation statistic.
5. Factors are extracted from a matrix of
associations.
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3. 5. Factors are extracted from a matrix
of associations.
• Scores on measured variables are used to
compute matrices of bivariate associations.
• i.e. Covariance matrix or correlation matrix
Even given only a matrix of associations, all
steps in factor analysis can be completed
(except for calculating the factor scores).
VAR1 VAR2 VAR3 VAR4 VAR5 VAR6
VAR1 1
VAR2 1
VAR3 1
VAR4 1
VAR5 1
VAR6 1
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4. What are the different types of
correlation statistics?
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5. 4. Factors are sensitive to the information
available in a given correlation statistic.
Bivariate Correlation Coefficients
continuous rpb r
rank ρ
categorical Ф rpb
nominal ordinal interval
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6. Pearson r Correlation Matrix
• Most commonly used in EFA
• Default in most statistical packages
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7. Pearson’s r vs. Spearman’s rho
Pearson’s r Spearman’s ρ
• Variables are intervally • Variables are at least
scaled ordinally scaled.
If the data are intervally scaled, either correlation coefficient could be used.
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9. Pearson r Assumption
Participant X x̄ x Y Ybar y xy
1 3 4.0 -1.0 3 33.0 -30.0 30.0
2 4 4.0 0.0 4 33.0 -29.0 0.0
3 5 4.0 1.0 92 33.0 59.0 59.0
Sum 12.00 99.00 89.0
Mean 4.00 33.00
SD 1.00 51.10
COV 44.50
r 0.87
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10. Spearman rho Assumption
Participant X x̄ x Y Ybar y xy
1 1 2.0 -1.0 1 2.0 -1.0 1.0
2 2 2.0 0.0 2 2.0 0.0 0.0
3 3 2.0 1.0 3 2.0 1.0 1.0
Sum 6.00 6.00 2.0
Mean 2.00 2.00
SD 1.00 1.00
COV 2.00
r 1.00
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11. 3. Correlational statistics represented in
matrices address different questions.
Spearman’s rho Pearson r
1. Addresses the question, 1. Addresses the same
“How well do the two question AND
variables order the cases in 2. Addresses the question,
exactly the same (or the “To what extent do the two
opposite) order?” variables have the same
(Thompson, 2004, p. 130) shape?” (Thompson, 2006,
p. 130)
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12. 2. Different factors may be extracted based
on the matrix of associations selected.
I. Pearson r Correlation Matrix
II. Spearman’s rho Correlation Matrix
III. Covariance Matrix
Data from Thompson, 2004, Appendix A,
ID 001-007 & PER1-PER6
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19. 2. Different factors may be extracted based
on the matrix of associations selected.
Pearson’s r Spearman’s rho
Table 1 Table 2
Factor 1 Factor 2 h² Factor 1 Factor 2 h²
0.169 0.902 0.841 0.882 0.159 0.804
0.161 0.920 0.872 0.924 0.117 0.868
0.749 0.627 0.955 0.697 0.644 0.900
0.911 0.242 0.888 0.199 0.881 0.815
0.985 0.064 0.974 0.107 0.969 0.951
Notes. Principal components extraction, varimax-rotated factor matrix
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20. 1.Extracted factors from a covariance matrix are a
function of correlations and standard deviations.
• Matrix most commonly used in CFA
• Covariance is Pearson r with standard deviations
removed
rXY = COVXY / (SDX * SDY)
COVXY = rXY * SDX * SDY
• Jointly influenced by:
1. Correlation between the two variables
2. Variability of the first variable
3. Variability of the second variable
Thompson (2004)
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24. Matrices Sensitivity to Different
Aspects of the Data
FA with Pearson r FA with Spearman FA with Covariance
Matrix rho Matrix Matrix
Factor Factor Factor Factor Factor Factor
1 2 h² 1 2 h² 1 2 h²
0.169 0.902 0.841 0.882 0.159 0.804 0.163 0.923 0.878
0.161 0.920 0.872 0.924 0.117 0.868 0.174 0.898 0.836
0.749 0.627 0.955 0.697 0.644 0.900 0.760 0.615 0.955
0.911 0.242 0.888 0.199 0.881 0.815 0.900 0.250 0.873
0.985 0.064 0.974 0.107 0.969 0.951 0.990 0.055 0.982
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25. Top 5 Take Away Points
1. Extracted factors from a covariance matrix are a
function of correlations and standard deviations.
2. Different factors may be extracted based on the
matrix of associations selected.
3. Correlational statistics represented in matrices
address different questions.
4. Factors are sensitive to the information available
in a given correlation statistic.
5. Factors are extracted from a matrix of
associations.
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26. References
Gorsuch, R.L. (1983). Factor analysis (2nd ed.).
Hillsdale, NJ: Erlbaum.
Thompson, B. (2004). Exploratory and
confirmatory factor analysis: Understanding
concepts and applications. Washington, DC:
American Psychological Association.
Thompson, B. (2006). Foundations of behavioral
statistics: An insight-based approach. New
York, NY: Guilford.
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Editor's Notes
Thinking back to 640, what were some of the different types of bivariate relationships? Pearson r, covariance, etc.
Because the factors are extracted from a matrix of associations, the factors are sensitive to the information available in the given statistic measuring the bivariate relationship. For example, if a statistic only considers the rank of the measured variables, such as Spearman’s rho, then the factors will also be based on rank. On the other hand if a statistic considers both order and distance, such as the Pearson r, then the factors will also be based on order and distance.
If the data is intervally scaled, we could use either the Pearson r or the Spearman rho coefficients.
Principle Components extraction and varimax rotation (factors are orthogonal or factors are ‘uncorrelated’)
Now that the factors have been extracted using all three matrices, compare the three outputs. When we use the Spearman’s rho matrix, in general the values of the variables that do not contribute to the factors tend to attenuate. Difference between the two is due to the Pearson r matrix accounting for distance and order, while the Spearman rho matrix only accounts for order.
