2. Factor analysis
Factor analysis is a class of procedures used for data reduction and
summarization.
It is an interdependence technique: no distinction between
dependent and independent variables.
Factor analysis is used:
To identify underlying dimensions, or factors, that explain the correlations among a set
of variables.
To identify a new, smaller, set of uncorrelated variables to replace the original set of
correlated variables.
3. Types of factor analysis
Exploratory Factor Analysis :- Researchers are not aware of how many
underlying dimensions( Factors) can be found from the variables that are under
study. Depending upon the co linearity between the variables limited number of
factors are derived.
Confirmatory Factor Analysis :- In this analysis researchers test the hypothesis
whether the variables under study based on theoretical support actually conform
to the factor structure or not.
4. Methods used for factor analysis in
SPSS
In Principal components analysis(mostly preferred) the total variance in the data is
considered.
-Used to determine the min number of factors that will account for max variance in the
data.
It is the most common method which the researchers use. Also, it extracts the maximum
variance and put them into the first factor. Subsequently, it removes the variance explained
by the first factor and extracts the second factor. Moreover, it goes on until the last factor.In
Common factor analysis, the factors are estimated based only on the common variance.
6. Factor Analysis Model
Each variable is expressed as a linear combination of factors. The factors are some
common factors plus a unique factor. The factor model is represented as:
Xi = Ai 1F1 + Ai 2F2 + Ai 3F3 + . . . + AimFm + ViUi
where
Xi = i th standardized variable
Aij = standardized mult reg coeff of var i on common factor j
Fj = common factor j
Vi = standardized reg coeff of var i on unique factor i
Ui = the unique factor for variable i
m = number of common factors
7. Statistics in factor analysis
Bartlett's test of sphericity. Bartlett's test of sphericity is used to test the
hypothesis that the variables are uncorrelated in the population (i.e., the population
corr matrix is an identity matrix)
Correlation matrix. A correlation matrix is a lower triangle matrix showing the
simple correlations, r, between all possible pairs of variables included in the
analysis. The diagonal elements are all 1.
8. Communality. Amount of variance a variable shares with all the other variables.
This is the proportion of variance explained by the common factors.
Eigenvalue. Represents the total variance explained by each factor.
Factor loadings. Correlations between the variables and the factors.
Factor matrix. A factor matrix contains the factor loadings of all the variables on
all the factors
9. Factor scores. Factor scores are composite scores estimated for each respondent
on the derived factors.
Kaiser-Meyer-Olkin (KMO) measure of sampling adequacy. Used to examine
the appropriateness of factor analysis. High values (between 0.5 and 1.0) indicate
appropriateness. Values below 0.5 imply not.
Percentage of variance. The percentage of the total variance attributed to each
factor.
Scree plot. A scree plot is a plot of the Eigenvalues against the number of factors
in order of extraction.
10. Steps to run factor analysis in SPSS
Analyze/Dimension Reduction/Factor
Mention variables(in our example vehicle type…fuel efficiency)
Descriptives (initial solution, coefficients, KMO & Bartlett Test) CONTINUE
Extraction (method: principal components, correlation matrix, unrotated
factor solution, Scree Plot) CONTINUE
Rotation (method: Varimax, display rotated solution) CONTINUE
Scores (Save As Variables, regression) CONTINUE)
OK
11. Interpretations of factor analysis in
SPSS
• The next output from the analysis is the correlation coefficient. A correlation matrix is simply a
rectangular array of numbers that gives the correlation coefficients between a single variable and
every other variable in the investigation.
• The correlation coefficient between a variable and itself is always 1, hence the principal diagonal of
the correlation matrix contains 1s. The correlation coefficients above and below the principal
diagonal are the same.
12. KMO and Barlett’s Test
Kaiser-Meyer-Olkin Measure of Sampling Adequacy – It is an index used to examine
appropriateness of factor analysis.This measure varies between 0 and 1, and values closer to 1
are better.
Values equal to or greater than 0.5 indicate that factor analysis is appropriate.
Bartlett’s Test of Sphericity – This tests the null hypothesis that the correlation matrix is an
identity matrix. An identity matrix is matrix in which all of the diagonal elements are 1 and all
off diagonal elements are 0 that is the variables are uncorrelated in the population
You want to reject this null hypothesis.
the significance value should be is less than 0.05 to reject null hypothesis and conclude
that the variables are correlated in the population
13. Communalities
Communality is the amount of variance a variable shares with all the other variables being
considered. Small values indicate variables that do not fit well with the factor solution,
Extraction – The values in this column indicate the proportion of each variable’s variance that can be
explained by the retained factors. Variables with high values are well represented in the common factor
space, while variables with low values are not well represented. Here we can see that all extraction values
are high.
14. Total variance explained
The next item shows all the factors extractable from the analysis along with their eigenvalues,
the percent of variance attributable to each factor, and the cumulative variance of the factor
and the previous factors. Here we can three factors are selected because three factors have
eigen values greater than one 5.994,1.654 and 1.123
Eigen Value: The eigenvalue represents the total variance explained by each factor.
Factors having eigenvalues over 1 are selected for further study.
15. Scree plot
The scree plot is a graph of the eigenvalues against all the factors. The graph is useful for
determining how many factors to retain.
One rule is to consider only those points with eigenvalues over 1.
16. Component matrix
The elements of the Component Matrix are correlations of the item with each component.
• Summing the squared component loadings(correlations) across the components (columns)
gives you the communality estimates for each item,
• and summing each squared loading down the items (rows) gives you the eigenvalue for each
component.
17. Rotations
ORTHOGONAL ROTATION- Rotations that assume the factors are not
correlated are called orthogonal rotations(Axes maintained at right
angles(varimax is a popular choice)
OBLIQUE ROTATION- Rotations that allow for correlation between
factors are called oblique rotations(. Axes not maintained at right angles)
18. Rotated component matrix
The idea of rotation is to reduce the number factors on which the variables under
investigation have high loadings.
The maximum of each row(excluding the sign) shows that the particular variable belongs
to the respective component
Example: vehicle type has the maximum value 0.954 under factor 3 so it belongs to factor
three
19. Try it yourself:
Example-2(Toothpaste data file)
To determine benefits from toothpaste
Responses were obtained on 6 variables:
V1: It is imp to buy toothpaste to prevent cavities
V2: I like a toothpaste that gives shiny teeth
V3: A toothpaste should strengthen your gums
V4: I prefer a toothpaste that freshens breath
V5: Prevention of tooth decay is not imp
V6: The most imp consideration is attractive teeth
Responses on a 7-pt scale (1=strongly disagree; 7=strongly agree)
