FACTOR
ANALYSIS
Factor Analysis is the Data reduction tool that removes
redundancy or duplication from a set of correlated variables.It represents
correlated variables with a smaller set of “derived” variables. Factors are
formed that are relatively independent of one another.
Two types of “variables”
–latent variables: factors
–observed variables
Applications of Factor Analysis
Identification of Underlying Factors:
·
clusters variables into homogeneous sets
·
creates new variables (i.e. factors)
·
allows us to gain insight to categories
Screening of Variables:
·
identifies groupings to allow us to select one variable to
represent many
·
useful in regression (recall collinearity)
Commonalities - This is the proportion of each variable's variance that can
be explained by the factors. It can be defined as the sum of squared factor
loadings for the variables.
Initial - With principal factor axis factoring, the initial values on
the diagonal of the correlation matrix are determined by the squared multiple
correlation of the variable with the other variables. For example, if you
regressed items 14 through 24 on item 13, the squared multiple correlation
coefficient would be .564.
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.
Factor - The initial number of factors is the same as the number of
variables used in the factor analysis.
Initial Eigenvalues - Eigenvalues
are the variances of the factors.
% of Variance - This column contains the percentage of total variance
accounted for by each factor.
Rotation is a method used to
simplify interpretation of a factor analysis.
• In principal components, the first
factor describes most of variability. Uses “ambiguity” or non-uniqueness of
solution to make interpretation more simple
• After choosing number of factors
to retain, we want to spread variability more evenly among factors.
• To do this we “rotate” factors:
– redefine factors such that
loadings on various factors tend to be very high (-1 or 1) or very low (0)
– Intuitively, it makes sharper
distinctions in the meanings of the factors
Extraction
Method – specifies the method of
extraction which are principal components, unweighted least squares,
generalized least squares, maximum
likelihood, principal axis factoring and image factoring.
Under Analyze, either the
Correlation matrix or Covariance matrix may be selected. Non rotated factor
solution – prints out the non-rotated pattern matrix Scree plot – plots the
eigenvalues in descending order. The number of factors extracted may be based
on either Eigenvalues over a set number or a specified Number of Factors.
How
to do factor analysis in SPSS
Go to: Analyse---Data
Reduction—Factor—Select the factors
·
Select
Extraction tab, ant then check the box against scree plot
·
Select
Descriptive tab, and then initial solution
·
Select
Rotation, and then click the radio button against Verimax
·
Then a
solution will be generated, which contains the tables, and graphs which can
assist in factor analysis.
The
tables, and the graphs generated are
·
Communalities
·
Component
matrix
·
Rotated
component matrix
·
Scree Plot
Some of
the components are shown below:
Rotated Component Matrix(a)-Critical
Element for Factor analysis
|
Component
|
||
|
1
|
2
|
|
|
Price in thousands
|
-.005
|
.924
|
|
Engine size
|
.498
|
.768
|
|
Horsepower
|
.221
|
.911
|
|
Wheelbase
|
.931
|
.040
|
|
Width
|
.779
|
.371
|
|
Length
|
.887
|
.104
|
|
Curb weight
|
.716
|
.585
|
|
Fuel capacity
|
.725
|
.486
|
|
Fuel efficiency
|
-.562
|
-.641
|
·
Extraction Method: Principal
Component Analysis.
·
Rotation Method: Varimax with Kaiser
Normalization.
·
Rotation converged in 3 iterations.
·
Rotation:
Tries to equalize the variance/Cumulative variance should remain same 3
Factors, on 3 axis of a cube, Variables mapped inside the box
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