PS Psychometrics

							    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
• Initial factor loading matrix not
  very interpretable
• Need to rotate initial factor loading
  matrix
• Loading changes the axes on which
  the factors lie to maximize fit of
  line to estimated scores on factors
• Does not alter amount of variance
  accounted for