Exploratory factor analysis

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  1. Lecture 5 Survey Research & Design in Psychology James Neill, 2010 Exploratory Factor Analysis
  2. Overview
    • What is factor analysis?
    • Assumptions
    • Steps / Process
    • Examples
    • Summary
  3. What is factor analysis?
    • What is factor analysis?
    • Purpose
    • History
    • Types
    • Models
  4. Universe : Galaxy All variables : Factor
  5. Conceptual model of factor analysis FA uses correlations among many items to search for common clusters.
  6. Factor analysis...
    • is used to identify clusters of inter-correlated variables (called ' factors ').
    • is a family of multivariate statistical techniques for examining correlations amongst variables.
    • empirically tests theoretical data structures .
    • is commonly used in psychometric instrument development .
  7. Purposes There are two main applications of factor analytic techniques:
      • Data reduction : Reduce the number of variables to a smaller number of factors.
      • Theory development : Detect structure in the relationships between variables, that is, to classify variables.
  8. Purposes: Data reduction
    • Simplifies data structure by revealing a smaller number of underlying factors
    • Helps to eliminate or identify items for improvement :
      • redundant variables
      • unclear variables
      • irrelevant variables
    • Leads to calculating factor scores
  9. Purposes: Theory development
    • Investigates the underlying correlational pattern shared by the variables in order to test theoretical models e.g., how many personality factors are there?
    • The goal is to address a theoretical question as opposed to calculating factor scores.
  10. History of factor analysis
    • Invented by Charles Spearman (1904)
    • Usage hampered by onerousness of hand calculation
    • Since the advent of computers, usage has thrived, esp. to develop:
      • Theory e.g., determining the structure of personality
      • Practice e.g., development of 10,000s+ of psychological screening & measurement tests
  11. EFA = Exploratory Factor Analysis
      • explores & summarises underlying correlational structure for a data set
    CFA = Confirmatory Factor Analysis
      • tests the correlational structure of a data set against a hypothesised structure and rates the “goodness of fit”
    Two main types of FA: Exploratory vs. confirmatory factor analysis
  12. This (introductory) lecture focuses on Exploratory Factor Analysis (recommended for undergraduate level). However, note that Confirmatory Factor Analysis (and Structural Equation Modeling) is generally preferred but is more advanced and recommended for graduate level. This lecture focuses on exploratory factor analysis
  13. Conceptual model - Simple model
    • e.g., 12 items testing might actually tap only 3 underlying factors
    • Factors consist of relatively homogeneous variables.
    Factor 1 Factor 2 Factor 3
  14. Eysenck’s 3 personality factors e.g., 12 items testing three underlying dimensions of personality Extraversion/ introversion Neuroticism Psychoticism talkative shy sociable fun anxious gloomy relaxed tense unconventional nurturing harsh loner
  15. Question 1 Conceptual model - Simple model Question 2 Question 3 Question 4 Question 5 Factor 1 Factor 2 Factor 3 Each question loads onto one factor
  16. Question 1 Conceptual model - Complex model Question 2 Question 3 Question 4 Question 5 Factor 1 Factor 2 Factor 3 Questions may load onto more than one factor
  17. Conceptual model – Area plot Correlation between X1 and X2 A theoretical factor which is partly measured by the common aspects of X1 and X2
  18. How many factors? One factor? Three factors? Nine factors? (independent items)
  19. Does personality consist of 2, 3, or 5, 16, etc. factors? e.g., the “Big 5”?
    • Neuroticism
    • Extraversion
    • Agreeableness
    • Openness
    • Conscientiousness
    Example: Personality
  20. Does intelligence consist of separate factors, e.g,.
    • Verbal
    • Mathematical
    • Interpersonal, etc.?
    ...or is it one global factor (g)? ...or is it hierarchically structured? Example: Intelligence
  21. Example: Essential facial features ( Ivancevic, 2003)
  22. Six orthogonal factors, represent 76.5% of the total variability in facial recognition (in order of importance):
    • upper-lip
    • eyebrow-position
    • nose-width
    • eye-position
    • eye/eyebrow-length
    • face-width
    Example: Essential facial features ( Ivancevic, 2003)
  23. Assumptions
    • GIGO
    • Sample size
    • Levels of measurement
    • Normality
    • Linearity
    • Outliers
    • Factorability
  24. G arbage . I n . G arbage . O ut
    • Screen the data
    • Use variables that theoretically “go together”
  25. Assumption testing: Sample size Some guidelines:
    • Min. : N > 5 cases per variable
      • e.g., 20 variables, should have > 100 cases (1:5)
    • Ideal : N > 20 cases per variable
      • e.g., 20 variables, ideally have > 400 cases (1:20)
    • Total N > 200 preferable
  26. Assumption testing: Sample size Comrey and Lee (1992): 50 = very poor, 100 = poor, 200 = fair, 300 = good, 500 = very good 1000+ = excellent
  27. Assumption testing: Sample size
  28. Assumption testing: Level of measurement
    • All variables must be suitable for correlational analysis, i.e., they should be ratio/metric data or at least Likert data with several interval levels.
  29. Assumption testing: Normality
    • FA is robust to violation of assumptions of normality
    • If the variables are normally distributed then the solution is enhanced
  30. Assumption Testing: Linearity
    • Because FA is based on correlations between variables, it is important to check there are linear relations amongst the variables (i.e., check scatterplots)
  31. Assumption testing: Outliers
    • FA is sensitive to outlying cases
      • Bivariate outliers (e.g., check scatterplots)
      • Multivariate outliers (e.g., Mahalanobis’ distance)
    • Identify outliers, then remove or transform
    • 15 classroom behaviours of high-school children were rated by teachers using a 5-point scale.
    • Task : Identify groups of variables (behaviours) that are strongly inter-related & represent underlying factors.
    Example factor analysis: Classroom behaviour
  32. Classroom behaviour items
    • Cannot concentrate ? can concentrate
    • Curious & enquiring ? little curiousity
    • Perseveres ? lacks perseverance
    • Irritable ? even-tempered
    • Easily excited ? not easily excited
    • Patient ? demanding
    • Easily upset ? contented
    • Control ? no control
    • Relates warmly to others ? disruptive
    • Persistent ? frustrated
    • Difficult ? easy
    • Restless ? relaxed
    • Lively ? settled
    • Purposeful ? aimless
    • Cooperative ? disputes
    Classroom behaviour items
  33. Classroom behaviour items
  34. Assumption testing: Factorability Check the factorability of the correlation matrix (i.e., how suitable is the data for factor analysis?) by one or more of the following methods:
    • Correlation matrix correlations > .3?
    • Anti-image matrix diagonals > .5?
    • Measures of sampling adequacy (MSAs)?
      • Bartlett’s sig.?
      • KMO > .5 or .6?
  35. Assumption testing: Factorability (Correlations) Are there SOME correlations over .3? If so, proceed with FA Takes some effort with a large number of variables, but accurate
    • Examine the diagonals on the anti-image correlation matrix
    • Consider variables with correlations less that .5 for exclusion from the analysis – they lack sufficient correlation with other variables
    • Medium effort, reasonably accurate
    Assumption testing: Factorability: Anti-image correlation matrix
  36. Anti-Image correlation matrix Make sure to look at the anti-image CORRELATION matrix
    • Global diagnostic indicators - correlation matrix is factorable if:
      • Bartlett’s test of sphericity is significant and/or
      • Kaiser-Mayer Olkin (KMO) measure of sampling adequacy > .5 or .6
    • Quickest method, but least reliable
    Assumption testing: Factorability: Measures of sampling adequacy
  37. Assumption testing: Factorability
  38. Summary: Measures of sampling adequacy Draw on one or more of the following to help determine the factorability of a correlation matrix:
    • Several correlations > .3?
    • Anti-image matrix diagonals > .5?
    • Bartlett’s test significant?
    • KMO > .5 to .6? (depends on whose rule of thumb)
  39. Steps / Process
    • Test assumptions
    • Select type of analysis
    • Determine no. of factors (Eigen Values, Scree plot, % variance explained)
    • Select items (check factor loadings to identify which items belong in which factor; drop items one by one; repeat)
    • Name and define factors
    • Examine correlations amongst factors
    • Analyse internal reliability
    • Compute composite scores
  40. Type of EFA: Extraction method: PC vs. PAF Two main approaches to EFA:
    • Analyses all variance: Principle Components (PC)
    • Analyses shared variance: Principle Axis Factoring (PAF)
  41. Principal components (PC)
    • More common
    • More practical
    • Used to reduce data to a set of factor scores for use in other analyses
    • Analyses all the variance in each variable
  42. Principal axis factoring (PAF)
    • Used to uncover the structure of an underlying set of p original variables
    • More theoretical
    • Analyses only shared variance (i.e. leaves out unique variance)
  43. Total variance of a variable Principal Components (PC) Principal Axis Factoring (PAF)
    • Often there is little difference in the solutions for the two procedures.
    • If unsure, check your data using both techniques
    • If you get different solutions for the two methods, try to work out why and decide on which solution is more appropriate
    PC vs. PAF
  44. Communalities
    • Each variable has a communality =
      • the proportion of its variance explained by the extracted factors
      • sum of the squared loadings for the variable on each of the factors
    • Ranges between 0 and 1
    • If communality for a variable is low (e.g., < .5, consider extracting more factors or removing the variable)
    • High communalities (> .5): Extracted factors explain most of the variance in the variables being analysed
    • Low communalities (< .5): A variable has considerable variance unexplained by the extracted factors
      • May then need to extract MORE factors to explain the variance
  45. Communalities - 2
  46. 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
  47. Explained variance 3 factors explain 73.5% of the variance in the items
  48. Eigen values
    • Each factor has an eigen value
    • Indicates overall strength of relationship between a factor and the variables
    • Sum of squared correlations
    • Successive EVs have lower values
    • Rule of thumb: Eigen values over 1 are ‘stable’ (Kaiser's criterion)
  49. Explained variance The eigen values ranged between .16 and 9.35. Two factors satisfied Kaiser's criterion (EVs > 1) but the third EV is .93 and appears to be a useful factor.
  50. Scree plot
    • A line graph of Eigen Values
    • Depicts amount of variance explained by each factor
    • Cut-off: Look for 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
  51. Scree plot
  52. Scree plot Scree plot
  53. Scree plot Scree plot
  54. How many factors? A subjective process ... Seek to explain maximum variance using fewest factors, considering:
    • Theory – what is predicted/expected?
    • Eigen Values > 1? (Kaiser’s criterion)
    • Scree Plot – where does it drop off?
    • Interpretability of last factor ?
    • Try several different solutions ? (consider FA type, rotation, # of factors)
    • Factors must be able to be meaningfully interpreted & make theoretical sense?
  55. How many factors?
    • Aim for 50-75% of variance explained with 1/4 to 1/3 as many factors as variables/items.
    • Stop extracting factors when they no longer represent useful/meaningful clusters of variables
    • Keep checking/clarifying the meaning of each factor – make sure you are reading the full wording of each item.
    • Factor loadings (FLs) indicate relative importance of each item to each factor.
      • In the initial solution , each factor tries “selfishly” to grab maximum unexplained variance.
      • All variables will tend to load strongly on the 1st factor
    Initial solution: Unrotated factor structure
  56. Initial solution - Unrotated factor structure
    • Factors are weighted combinations of variables
    • A factor matrix shows variables in rows and factors in columns
  57. 1st factor extracted:
    • Best possible line of best fit through the original variables
    • Seeks to explain lion's share of all variance
    • A single factor, best summary of the variance in the whole set of items
    Initial solution - Unrotated factor structure
    • Each subsequent factor tries to explain the maximim possible amount of remaining unexplained variance.
      • Second factor is orthogonal to first factor - seeks to maximise its own eigen value (i.e., tries to gobble up as much of the remaining unexplained variance as possible)
    Initial solution - Unrotated factor structure
  58. Vectors (Lines of best fit)
  59. Initial solution: Unrotated factor structure
    • Seldom see a simple unrotated factor structure
    • Many variables load on 2 or more factors
    • Some variables may not load highly on any factors (check: low communality)
    • Until the FLs are rotated, they are difficult to interpret.
    • Rotation of the FL matrix helps to find a more interpretable factor structure.
  60. Two basic types of factor rotation Orthogonal (Varimax) Oblique (Oblimin)
  61. Two basic types of factor rotation
    • Orthogonal minimises factor covariation, produces factors which are uncorrelated
    • Oblimin allows factors to covary, allows correlations between factors
  62. Orthogonal rotation
  63. 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
  64. Why rotate a factor loading matrix?
    • 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
  65. Orthogonal vs. oblique rotations
    • Consider purpose of factor analysis
    • If in doubt, try both
    • Consider interpretability
    • Look at correlations between factors in oblique solution
      • if >.3 then go with oblique rotation (>10% shared variance between factors)
  66. Interpretability
    • It is dangerous to be driven by factor loadings only – think carefully - be guided by theory and common sense in selecting factor structure.
    • You must be able to understand and interpret a factor if you’re going to extract it.
  67. Interpretability
    • However, watch out for ‘seeing what you want to see’ when evidence might suggest a different, better solution.
    • There may be more than one good solution! e.g., in personality
      • 2 factor model
      • 5 factor model
      • 16 factor model
  68. Factor loadings & item selection A factor structure is most interpretable when: 1. Each variable loads strongly (> + .40) on only one factor 2. Each factor shows 3 or more strong loadings; more loadings = greater reliability 3. Most loadings are either high or low, few intermediate values. 4. These elements give a ‘simple’ factor structure.
  69. Initial solution – Unrotated factor structure 4
  70. Rotated factor matrix Task Orientation Sociability Settledness
  71. 3-d plot
    • Bare min. = 2
    • Recommended min. = 3
    • Max. = unlimited
    • More items:
    -> ? reliability -> ? 'roundedness' -> Law of diminishing returns
    • Typically = 4 to 10 is reasonable
    How many items per factor?
  72. How do I eliminate items? A subjective process; consider:
    • Size of main loading (min = .4)
    • Size of cross loadings (max = .3?)
    • Meaning of item (face validity)
    • Contribution it makes to the factor
    • Eliminate 1 variable at a time, then re-run, before deciding which/if any items to eliminate next
    • Number of items already in the factor
  73. Factor loadings & item selection Comrey & Lee (1992): loadings > .70 - excellent > .63 - very good > .55 - good > .45 - fair > .32 - poor
  74. Factor loadings & item selection Cut-off for acceptable loadings:
    • Look for gap in loadings (e.g., .8, .7, .6 , .3, .2 )
    • Choose cut-off because factors can be interpreted above but not below cut-off
  75. Other considerations: Normality of items Check the item descriptives . e.g. if two items have similar Factor Loadings and Reliability analysis, consider selecting items which will have the least skew and kurtosis. The more normally distributed the item scores, the better the distribution of the composite scores.
  76. Factor analysis in practice
    • To find a good solution, consider each combination of:
      • PC-varimax
      • PC-oblimin
      • PAF-varimax
      • PAF-oblimin
    • Apply the above methods to a range of possible/likely factors, e.g., for 2, 3, 4, 5, 6, and 7 factors
    • Eliminate poor items one at a time, retesting the possible solutions
    • Check factor structure across sub-groups (e.g., gender) if there is sufficient data
    • You will probably come up with a different solution from someone else!
    • Check/consider reliability analysis
    Factor analysis in practice
  77. Example: Condom use
    • The Condom Use Self-Efficacy Scale (CUSES) was administered to 447 multicultural college students.
    • PC FA with a varimax rotation.
    • Three distinct factors were extracted:
      • Appropriation
      • Sexually Transmitted Diseases
      • Partners' Disapproval
  78. Factor loadings & item selection .56   I feel confident I could gracefully remove and dispose of a condom after sexual intercourse .61   I feel confident I could remember to carry a condom with me should I need one .65   I feel confident I could purchase condoms without feeling embarrassed .75   I feel confident in my ability to put a condom on myself or my partner   FL Factor 1: Appropriation
  79. Factor loadings & item selection .80   I would not feel confident suggesting using condoms with a new partner because I would be afraid he or she would think I thought they had a sexually transmitted disease .86   I would not feel confident suggesting using condoms with a new partner because I would be afraid he or she would think I have a sexually transmitted disease .72   I would not feel confident suggesting using condoms with a new partner because I would be afraid he or she would think I've had a past homosexual experience   FL Factor 2: STDs
  80. Factor loadings & item selection .58   If my partner and I were to try to use a condom and did not succeed, I would feel embarrassed to try to use one again (e.g. not being able to unroll condom, putting it on backwards or awkwardness) .65   If I were unsure of my partner's feelings about using condoms I would not suggest using one .73   If I were to suggest using a condom to a partner, I would feel afraid that he or she would reject me   FL Factor 3: Partner's reaction
  81. Summary
    • Introduction
    • Assumptions
    • Steps/Process
  82. Introduction: Summary
    • Factor analysis is a family of multivariate correlational data analysis methods for summarising clusters of covariance .
    • FA summarises correlations amongst items.
    • The common clusters (called factors) are summary indicators of underlying fuzzy constructs.
  83. Assumptions: Summary
    • Sample size
      • 5+ cases per variables (ideally 20+ cases per variable)
      • N > 200
    • Bivariate & multivariate outliers
    • Factorability of correlation matrix (Measures of Sampling Adequacy)
    • Normality enhances the solution
  84. Summary: Steps / Process
    • Test assumptions
    • Select type of analysis
    • Determine no. of factors (Eigen Values, Scree plot, % variance explained)
    • Select items (check factor loadings to identify which items belong in which factor; drop items one by one; repeat)
    • Name and define factors
    • Examine correlations amongst factors
    • Analyse internal reliability
    • Compute composite scores
      Next lecture
  85. Summary: Types of FA
    • PC: Data reduction
      • uses all variance
    • PAF: Theoretical data exploration
      • uses shared variance
    • Try both ways
      • Are solutions different? Why?
  86. Summary: Rotation
    • Orthogonal (varimax)
      • perpendicular vectors
    • Oblique (oblimin)
      • angled vectors
    • Try both ways
      • Are solutions different? Why?
  87. No. of factors to extract?
    • Inspect Evs
      • look for > 1 or sudden drop (Inspect scree plot)
    • % of variance explained
      • aim for 50 to 75%)
    • Interpretability / theory
    Summary: Factor extraction
    • Comrey, A. L., & Lee, H. B. (1992). A first course in factor analysis (2nd ed.). Hillsdale, NJ: Lawrence Erlbaum.
    • Ivancevic, V., Kaine, A.K., MCLindin, B.A, & Sunde, J. (2003). Factor analysis of essential facial features . In the proceedings of the 25th International Conference on Information Technology Interfaces (ITI), pp. 187-191, Cavtat, Croatia.
    • Fabrigar, L. R., Wegener, D. T., MacCallum, R. C., & Strahan, E. J. (1999). Evaluating the use of exploratory factor analysis in psychological research. Psychological Methods , 4 (3), 272-299.
    • Tabachnick, B. G. & Fidell, L. S. (2001). Principal components and factor analysis. In Using multivariate statistics . (4th ed., pp. 582 - 633). Needham Heights, MA: Allyn & Bacon.
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