An introduction to factor analysis

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An introduction to factor analysis
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The Big Five - history

• Allport and Odbert (1936)
• lexical hypothesismost important individual
  differences are encoded into language
• compiled a list of 4504 personality-describing
  adjectives from English dictionaries
• Cattel (1943)
• reduced these 4504 terms to 171
• 100 persons rated by two friends on these items
• factor analysis leads to 35 factors
• 200 persons rate themselves on these factors
• factor analysis leads to 16 factors (16PF), five
  second-order factors

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The Big Five - history

• Fiske (1949) constructed simplified descriptions
  from Catells variables
• Tupes and Christal (1961) used these and found
  five relatively strong and recurrent factors
  (through factor analysis)
• Norman (1963) replicated and labelled these
  factors
• OCEAN Opennes to Experience, Conscientiousness,
  Extraversion, Agreeableness and Neuroticism

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The Big Five - history

• Big Five Factors Goldberg (1981)
• Since then factor structures resembling the Big
  Five were identified in numerous sets of
  variables
• Neo-PI-R most used Big-Five questionnaire
• Importance of Big Five derives from their ability
  to bring a common language to personality
  psychology

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Components of the Big Five Hogan and Hogan (2007)
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What is factor analysis?

• Introduction to factor analysis largely based on
• Bortz Statistik für Sozialwissenschaftler
• Presentation by James Neill

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What is factor analysis?

• A family of techniques to examine correlations
  amongst variables.
• Uses correlations among many items to search for
  common clusters.
• Aim is to identify groups of variables which are
  relatively homogeneous.
• Groups of related variables are called factors.

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Purpose of a factor analysis

• The main applications of factor analytic
  techniques are
• to reduce the number of variables and
• to detect structure in the relationships between
  variables, that is to classify variables.

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Factor analysis - motivation

• Imagine we have scores from two tests which are
  correlated.
• Correlation could be explained e.g. if both tests
  measure intelligence or motivation.
• We can check this by looking at the correlation
  between test scores and variables measuring
  intelligence and motivation.
• Problem with many variables involved we will end
  up with a mess of correlations

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Factor analysis - motivation

• Need a method that structures a large number of
  variables into several groups based on the
  correlations to determine which variables contain
  joint or separate information.
• A factor analysis creates a synthetic variable (a
  factor) which correlates as highly as possible
  with all variables
• If we partial out the influence of this factor we
  can try to explain the remaining variance between
  the variables with another factor and so on...
• We will end up with a number of independent
  factors which explain the relations between the
  variables.

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Conceptual Model for Factor Analysis
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Conceptual Model for Factor Analysis
One factor
Independent items
Three factors
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What is a factor?

• Example
• Questionnaire with the following items
• I blush easily
• I am often self-conscious
• I like to sit by the sea and listen to the sound
  of the waves
• I like to take a walk in the woods
• Leads to the following table of correlations

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What is a factor?

• Given the structure of the correlations we are
  likely to get two factors (one for items 1 and 2,
  one for 3 and 4)
• Factor analysis is a data reducing procedure.

Factor 1
Factor 2
Blush easily
Sound of waves
Walk in the woods
Self-conscious
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What is a factor?

• What do these factors mean?
• One interpretation Inner restlessness (factor 1)
  and love for nature (factor 2)

Factor 1
Factor 2
Blush easily
Sound of waves
Walk in the woods
Self-conscious
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Steps to Factor Analysis

• Step 1 correlational matrix correlations of all
  variables with each other
• Step 2 clusters - computer identifies the
  patterns of correlationsfactors
• Step 3 loading - correlation between each
  factor each original variable
• Step 4 labeling naming and interpreting the
  factors

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Principal Component Analysis (PCA)

• Most widely used way to do factor analysis
• Introductory example Test with the following
  tasks
• Rebus (Bilderrätsel)
• Math problem
• Puzzle
• Memory task
• Crossword
• Each participant is asked to solve these tasks as
  quickly as possible. For each task and person the
  time needed is recorded.

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Principal Component Analysis (PCA)

• Important terms
• Factor loading Correlation between a variable i
  and a factor j.
• Communality The proportion of variance in each
  variable which can be explained by the factors
• High communalities (gt .5) show that the factors
  extracted explain most of the variance in the
  variables being analysed
• Eigenvalue of a factor measures how much of the
  total variance of all variables is explained by
  this factor

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Eigen Values

• EV sum of squared correlations for each factor
• EV overall strength of relationship between a
  factor and the variables
• Successive EVs have lower values
• Eigen values over 1 are stable

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Explained Variance

• A good factor solution is one that explains the
  most variance with the fewest factors
• Realistically happy with 50-75 of the variance
  explained

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Scree Plot

• A bar graph of Eigen Values
• Depicts the amount of variance explained by each
  factor.
• Look for point where additional factors fail to
  add appreciably to the cumulative explained
  variance.
• 1st factor explains the most variance
• Last factor explains the least amount of variance

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Why Rotate a Factor Loading Matrix?

• After rotation, the vectors (lines of best fit)
  are rearranged to optimally go through clusters
  of shared variance
• Then the FLs and the factor they represent can be
  more readily interpreted
• A rotated factor structure is simpler more
  easily interpretable
• each variable loads strongly on only one factor
• each factor shows at least 3 strong loadings
• all loading are either strong or weak, no
  intermediate loadings

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Factor Analysis (continued)
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Back to the Big Five

• Extensive questionnaires such as the 16 PF or the
  NEO-PI-R are too long to be used in surveys
• Shorter versions have been developped e.g. for
  the SOEP (3 items for each of the five
  dimensions)
• These mini-versions have to be validated i.e. do
  they measure the same as the usual Big Five
  inventories
• SOEP data is available to researchers

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Back to the Big Five

• Extensive questionnaires such as the 16 PF or the
  NEO-PI-R are too long to be used in surveys
• Shorter versions have been developped e.g. for
  the SOEP (3 items for each of the five
  dimensions)
• These mini-versions have to be validated i.e. do
  they measure the same as the usual Big Five
  inventories
• SOEP data is available to researchers

From Gerlitz/Schupp 2005
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Back to the Big Five

• Extensive questionnaires such as the 16 PF or the
  NEO-PI-R are too long to be used in surveys
• Shorter versions have been developped e.g. for
  the SOEP (3 items for each of the five
  dimensions)
• These mini-versions have to be validated i.e. do
  they measure the same as the usual Big Five
  inventories
• SOEP data is available to researchers

From Gerlitz/Schupp 2005