# PS Psychometrics

Shared by:
Categories
Tags
-
Stats
views:
1
posted:
9/5/2011
language:
English
pages:
28
Document Sample

```							    PS28C
Psychometrics

Lecture 6
Factor Analysis
What is Factor Analysis?

• Factor analysis is a group of similar
techniques used to simplify data
• Uses relationships between
variables to reduce a large number
of variables into a smaller number
of clusters
• Provides simplier method of
describing results
What is Factor Analysis?
• Goals of Factor Analysis are:
• To summarize patterns of correlations
among variables
• To reduce a large number of variables
to a smaller number of clusters
• To provide an operational definition of
a construct
• To test theory about an underlying
process
What is Factor Analysis?

Lone 1   Lone 2   Lone 3   Lsr 1   Lsr 2   Lsr 3

Lone 1   1.00
Lone 2   0.55     1.00
Lone 3   0.74     0.63     1.00
Lsr 1    0.01     0.01     0.01     1.00
Lsr 2    0.02     0.02     0.02     0.64    1.00
Lsr 3    0.03     0.03     0.03     0.72    0.54    1.00
What is Factor Analysis?
Leisure 3                 Leisure 2
Lonely 3

Lonely 2

Lonely 1                      Leisure 1

Loneliness Cluster                  Leisure Cluster
What is Factor Analysis?
• Combinations of variables are called
factors
• Factors represent a hypothetical
construct underlying a set of measures
• Factors are an error free or latent
measure of a construct
• Variables are manifest measure of a
construct
How is Factor Analysis
Used?
• Factor analysis is used in a
variety of ways:
• To simplify a large set of variables
• To select a few variables to represent
a larger construct
• To summarize a large number of
variables into a series of smaller
factors
• To develop a scale and establish its
validity
To Simplify a Large Set of
Variables

• Most common use of factor analysis
is to simplify a large number of
variables
• Goal is to identify a small number of
factors
• Easier to analyze and understand
the factors
To Select a Few Variables to
Represent a Larger Construct

• May not wish to use a number of
highly related variables in the same
data analysis
• Highly correlated variables can cause
problems with statistical analyses
• Factor analysis provides a method of
selecting one variable to represent a
cluster of related variables
Combine Many Variables into
a Few Factors

• Most common use of factor analysis
• Factor analysis can be used to
reduce several questions into a
single explanatory factor
• Factors account for much of
participants’ variability of response
To Create New Measures
• Factor analysis used to create new
measures
• Wide variety of items are written
• Groups of items are associated with
a specific aspect of the construct
• Each item is associated with only one
aspect of the construct
To Create New Measures
• Items administered to large number
of participants
• Those which are associated with
their hypothesized factor are kept
• Those which are not associated with
any factor are dropped
• Those associated with more than one
factor are dropped
How is Factor Analysis
Done?
• Factor analysis finds the best way
to combine clusters of variables
• Method maximizes the amount of
shared variability among the
variables
• Uses complex equation to weigh the
contribution of each variable to
predicting shared variability
How is Factor Analysis
Done?

y  a  bx  e
'

y  a  b1 x1  b2 x2  e
'
1
How is Factor Analysis
Done?
• In theory as many factors as
variables
• Practically only a small number of
factors are kept
• Factors are not equally important
• Factors are rank ordered in size
• First factor is always accounts for
greatest amount of variability
How is Factor Analysis
Done?
• Second factor formed from left
over variability
• First and second factors are
uncorrelated or orthogonal
• Factors summarize the pattern of
correlations in the correlation
matrix
How is Factor Analysis
Done?
• Each factor represents a hypothetical,
unobserved or latent variable
• Latent variable is error free, perfect
measure of a concept
• Weights from equation used to
calculate scores on latent variable
• Combination of the original variables is
called the manifest variable
Example of a Factor
Analysis
I1     I2    I3    I4     I5     I6     I7     I8     I9     I10
I1    1.00
I2    .55 1.00
I3    .74    .63 1.00
I4    .01    .01   .01   1.00
I5    .02    .02 .02     .65    1.00
I6    .03    .03 .03     .73    .55    1.00
I7    .04    .04   .04   .01    .01    .01    1.00
I8    .05    .05   .05   .02    .02    .02    .66    1.00
I9    .06    .06   .06   .03    .03    .03    .49    .63    1.00
I10   .07    .07   .07   .04    .04    .04    .72    .55    .63    1.00
Example of a Factor
Analysis

• End result of a factor analysis is the
factor matrix
• Columns of matrix are the new factors
• Rows of matrix are the original
variables
• Cells are correlations between original
variables and the new factors
Factor Matrix
Factor
1       2       3
ITEM10     .810   .049     .031
ITEM7      .802    .015    -.002
ITEM8      .769   .026     .009
ITEM9      .712   .040     .024
ITEM3      .030   .921     .012
ITEM1      .035   .797     .013
ITEM2      .041   .680     .013
ITEM4      .015   .000     .918
ITEM6      .016   .023     .778
ITEM5      .019    .013    .692
Extracting Factors
• Theoretically can extract as many
factors as variables
• Practice only extract a smaller
number of factors
• Decision on how many factors to
keep is complex and relies on many
criteria
Using Eigenvalues
• Eigenvalues express the amount of
variance accounted for by a cluster
of variables
• Each item contributes one unit of
variance
• Factors with eigenvalues with values
less than one account for less
variance than a single item
Analysis of Eigenvalues
% of      Cumulative
Factor   Eigenvalue
Variance       %
1          2.873      28.732       28.732
2          2.265      22.649        51.381
3           2.197     21.969       73.351
4            .537      5.372       78.723
5            .475      4.747       83.469
6            .472       4.715      88.185
7            .455      4.549       92.734
8            .266      2.660       95.394
9            .252       2.515      97.909
10            .209       2.091     100.000
Plot of Eigenvalues
Scree Plot
3.5

3.0

2.5

2.0

1.5

1.0

.5

0.0
1    2     3    4   5   6   7   8   9   10

Factor Number
Total Variability
Factor   Total     Cumulative % of
Variance
1        2.873       28.732
2        2.265       51.381
3        2.197       73.351
4         .537       78.723
5         .475       83.469
6         .472       88.185
7         .455       92.734
8         .266       95.394
9         .252       97.909
10         .209      100.000
Total Percentage of
Variance Accounted
• Final criterion for determining
number of factors
• Number kept should account for
30% or more of the total variability
in scores
• Three factors account for 79% of
the total variability in scores
Make Sense
• Comprehensibility of the factors is
the final important criteria
• If factor meets all other criteria
but the clusters of items can not be
interpreted than another factor
analytic solution should be tried
Rotation of Factors
very interpretable
matrix
the factors lie to maximize fit of
line to estimated scores on factors
• Does not alter amount of variance
accounted for

```
Related docs
Other docs by MikeJenny
South Moon Under