PS28C Psychometrics

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PS28CPsychometrics

• Lecture 6
• Factor Analysis

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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

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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

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What is Factor Analysis?
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What is Factor Analysis?
Leisure 3
Leisure 2
Lonely 3
Lonely 2
Leisure 1
Lonely 1
Loneliness Cluster
Leisure Cluster
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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

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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

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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

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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

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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

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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

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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

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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

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How is Factor Analysis Done?
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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

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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

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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

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Example of a Factor Analysis
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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

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Factor Matrix
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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

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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

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Analysis of Eigenvalues
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Plot of Eigenvalues
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Total Variability
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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

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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

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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