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  • 7126/6667 Survey Research & Design in Psychology Semester 1, 2010, University of Canberra, ACT, Australia James T. Neill Home page: http://ucspace.canberra.edu.au/display/7126 Lecture page: http://en.wikiversity.org/wiki/Exploratory_factor_analysis Notes page: http://en.wikiversity.org/wiki/Survey_research_methods_and_design_in_psychology Image source: http://www.flickr.com/photos/jimkster/832582262/ By jimkster: http://www.flickr.com/photos/jimkster License: CC-by-A 2.0 Description: This lecture is accompanied by a computer-lab based tutorial, the notes for which are available here: http://ucspace.canberra.edu.au/display/7126/Tutorial+-+Psychometrics Introduces and explains the use of exploratory factor analysis particularly for the purposes of psychometric instrument development.
  • Image source: http://commons.wikimedia.org/wiki/File:Information_icon4.svg License: Public domain
  • Image source: http://commons.wikimedia.org/wiki/File:Information_icon4.svg License: Public domain
  • Image source: http://www.flickr.com/photos/jimkster/832582262/ By jimkster: http://www.flickr.com/photos/jimkster License: CC-by-A 2.0
  • Image source: James Neill, Creative Commons Attribution 3.0, http://commons.wikimedia.org/wiki/Image:FactorAnalysis_ConceptualModel_DotsRings.png Factor analysis can be conceived of as a method for examining a matrix of correlations in search of clusters of highly correlated variables. Exploratory factor analysis is a tool to help a researcher ‘throw a hoop’ around clusters of related items (i.e., items that seem to share a central underlying theme), to distinguish between clusters, and to identify and eliminate irrelevant or indistinct (overlapping) items.
  • See also http://www.statsoft.com/textbook/stfacan.html A major purpose of factor analysis is data reduction, i.e., to reduce complexity in the data, by identifying underlying (latent) clusters of association. Commonly used in psychometric instrument development.
  • FA helps to eliminate: redundant variables (e.g, items which are highly correlated are unnecessary) unclear variables (e.g., items which don’t load cleanly on a single factor) irrelevant variables (e.g., variables which have low loadings)
  • FA helps to eliminate: redundant variables (e.g, items which are highly correlated are unnecessary) unclear variables (e.g., items which don’t load cleanly on a single factor) irrelevant variables (e.g., variables which have low loadings)
  • “ Introduced and established by Pearson in 1901 and Spearman three years thereafter, factor analysis is a process by which large clusters and grouping of data are replaced and represented by factors in the equation. As variables are reduced to factors, relationships between the factors begin to define the relationships in the variables they represent (Goldberg & Digman, 1994). In the early stages of the process' development, there was little widespread use due largely in part to the immense amount of hand calculations required to determine accurate results, often spanning periods of several months. Later on a mathematical foundation would be developed aiding in the process and contributing to the later popularity of the methodology. In present day, the power of super computers makes the use of factor analysis a simplistic process compared to the 1900's when only the devoted researchers could use it to accurately attain results (Goldberg & Digman, 1994).” From http://www.personalityresearch.org/papers/fehringer.html Terminology was introduced by Thurstone (1931) - http://www.statsoft.com/textbook/stfacan.html
  • CFA has become the professional standard now for testing and developing theoretical factor structures, but Exploratory factor analysis still has its place and is an important multivariate statistical tool. We will only be focusing on EFA, a more tradition method, which is, however, been gradually used less and CFA is becoming more common. CFA uses is more recent and powerful, usually covered at PhD level.
  • CFA has become the professional standard now for testing and developing theoretical factor structures, but Exploratory factor analysis still has its place and is an important multivariate statistical tool. We will only be focusing on EFA, a more tradition method, which is, however, been gradually used less and CFA is becoming more common. CFA uses is more recent and powerful, usually covered at PhD level.
  • Arrows represent single items (or variables) on a test/survey/questionnaire. The circles represent factors, i.e., the shared variance amongst different sets of related items.
  • Image source: http://www.siu.edu/~epse1/pohlmann/factglos/ This diagram represents the variance for a single factor. It also shows the variance for two items which correlate (r12) and which form part of the overall factor.
  • Image sources: http://www.creative-wisdom.com/teaching/assessment/subscales.html
  • Image source: http://people.cs.uct.ac.za/~dburford/masters/SATNAC/2000/index.html Image license: Unknown Rv
  • 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. http://hdl.handle.net/1947/2580 Information Technology Interfaces, Proceedings of the 25th International Conference on ?, Volume , Issue , 16-19 June 2003 Page(s): 187 - 191 Summary: We present the results of an exploratory factor analysis to determine the minimum number of facial features required for recognition. In the introduction, we provide an overview of research we have done in the area of the evaluation of facial recognition systems. In the next section, we provide a description of the facial recognition process (including signal processing operations such as face capture, normalisation, feature extraction, database comparison, decision and action). The following section provides description of the variables. We follow with our main result, where we have used a sample of 80 face images from a test database. We have preselected 20 normalized facial features and extracted six essential orthogonal factors, representing 76.5 % of the total facial feature variability. They are (in order of importance): upper-lip, eyebrow-position, nose-width, eye-position, eye/eyebrow-length and face-width. http://ieeexplore.ieee.org/Xplore/login.jsp?url=/iel5/8683/27508/01225343.pdf
  • Image source: http://commons.wikimedia.org/wiki/File:Information_icon4.svg License: Public domain
  • Image sources: ClipArt (Unknown) Garbage In –> Garbage Out i.e., Put Crap (i.e., data with no logical relationship) into Factor Analysis –> Get Crap Out of Factor Analysis i.e., check your assumptions
  • Image source: Table 7 Fabrigar et al (1999).
  • Example: Someone who works full-time, studies full-time, averages an HD, and has 2 children.
  • Based on Francis Section 5.6, which is based on the Victorian Quality Schools Project. Francis, G. (2007). Introduction to SPSS for Windows: v. 15.0 and 14.0 with Notes for Studentware (5th ed.). Sydney: Pearson Education.
  • Image source: http://www.acer.edu.au/research/programs/documents/RoweITUC_2004Paper.pdf This is from the questionnaire used in Francis 5.6, pp. 157-164.
  • Image source: Francis 5.6, pp. 157-164?
  • Image source: Francis 5.6, pp. 157-164 Be careful to examine the anti-image correlation matrix (not the anti-image covariance matrix)
  • Large KMO values indicate that a factor analysis of the variables is a good idea. Should be > .5 for a satisfactory factor analysis
  • Image source: This is from Francis 5.6, pp. 157-164
  • Image source: http://commons.wikimedia.org/wiki/File:Information_icon4.svg License: Public domain See also http://en.wikiversity.org/wiki/Exploratory_factor_analysis/Data_analysis_tutorial See also http://en.wikiversity.org/wiki/Exploratory_factor_analysis/Data_analysis_tutorial
  • Image source: http://www.siu.edu/~epse1/pohlmann/factglos/
  • Image source: Francis 5.6, pp. 157-164
  • Image: Francis 5.6, pp. 157-164
  • Image: Francis 5.6, pp. 157-164
  • If all the EVs are added together, it will equal the number of items. In other words, each item contributes a variance of 1 to be explained.
  • Image: Francis 5.6, pp. 157-164 Y-axis shows variance (eigen variables) for each factor. Look for “elbow” in the curve - point where additional factors don’t add much to explained variance.
  • Image source: James Neill, Creative Commons Attribution 2.5 Australia, http://creativecommons.org/licenses/by/2.5/au/
  • The max. no. of possible factors = no. of variables Its important to refer back to original wording of questionnaire items – get the item labels to show on the SPSS output
  • Image source: http://www.siu.edu/~epse1/pohlmann/factglos/
  • Image source (left): http://www.xs-stock.co.uk/shopimages/products/normal/Kerplunk.jpg Image source (right): http://mayhem-chaos.net/photoblog/images/super_sized_kerplunk.jpg
  • Orthogonal - Vectors (lines of best fit or factors) are perpendicular Oblimin - Vectors angles are permitted to be acute, i.e., correlation allowed between factors
  • There is also: Quartimax maximises variance of loadings on each variable Equamax compromise between quartimax and varimax
  • Image source: Unknown
  • Image source: Francis 5.6, pp. 157-164
  • Image source: Francis 5.6, pp. 157-164 ‘ Compliant’ (6) and ‘Self-controlled’ (7) load onto 2 factors - run analysis without these two variables.
  • Image source: (Based on?) Francis 5.6, pp. 157-164
  • Tip: The rotated factor loading results can be printed out and color-coded with a highlighter for careful cross-comparison
  • Factor analysis of the Condom Use Self-Efficacy Scale among multicultural college students Thomas W. Barkley, Jr and Jason L. Burns (2000) http://her.oxfordjournals.org/cgi/content/full/15/4/485
  • Image source: http://commons.wikimedia.org/wiki/File:Information_icon4.svg License: Public domain
  • Image source: http://commons.wikimedia.org/wiki/File:Information_icon4.svg License: Public domain See also http://en.wikiversity.org/wiki/Exploratory_factor_analysis/Data_analysis_tutorial See also http://en.wikiversity.org/wiki/Exploratory_factor_analysis/Data_analysis_tutorial

Transcript

  • 1. Lecture 5 Survey Research & Design in Psychology James Neill, 2010 Exploratory Factor Analysis
  • 2. Overview
    • What is factor analysis?
    • 3. Assumptions
    • 4. Steps / Process
    • 5. Examples
    • 6. Summary
  • 7. What is factor analysis?
  • 12. Universe : Galaxy All variables : Factor
  • 13. Conceptual model of factor analysis FA uses correlations among many items to search for common clusters.
  • 14. 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 .
    • 15. is commonly used in psychometric instrument development .
  • 16. Purposes There are two main applications of factor analytic techniques:
      • Data reduction : Reduce the number of variables to a smaller number of factors.
      • 17. Theory development : Detect structure in the relationships between variables, that is, to classify variables.
  • 18. Purposes: Data reduction
    • Simplifies data structure by revealing a smaller number of underlying factors
    • 19. Helps to eliminate or identify items for improvement :
      • redundant variables
      • 20. unclear variables
      • 21. irrelevant variables
    • Leads to calculating factor scores
  • 22. 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?
    • 23. The goal is to address a theoretical question as opposed to calculating factor scores.
  • 24. History of factor analysis
    • Invented by Charles Spearman (1904)
    • 25. Usage hampered by onerousness of hand calculation
    • 26. Since the advent of computers, usage has thrived, esp. to develop:
      • Theory e.g., determining the structure of personality
      • 27. Practice e.g., development of 10,000s+ of psychological screening & measurement tests
  • 28. 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
  • 29. 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
  • 30. Conceptual model - Simple model
    • e.g., 12 items testing might actually tap only 3 underlying factors
    • 31. Factors consist of relatively homogeneous variables.
    Factor 1 Factor 2 Factor 3
  • 32. 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
  • 33. 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
  • 34. 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
  • 35. Conceptual model – Area plot Correlation between X1 and X2 A theoretical factor which is partly measured by the common aspects of X1 and X2
  • 36. How many factors? One factor? Three factors? Nine factors? (independent items)
  • 37. Does personality consist of 2, 3, or 5, 16, etc. factors? e.g., the “Big 5”? Example: Personality
  • 42. Does intelligence consist of separate factors, e.g,. ...or is it one global factor (g)? ...or is it hierarchically structured? Example: Intelligence
  • 45. Example: Essential facial features ( Ivancevic, 2003)
  • 46. Six orthogonal factors, represent 76.5% of the total variability in facial recognition (in order of importance): Example: Essential facial features ( Ivancevic, 2003)
  • 52. Assumptions
  • 59. G arbage . I n . G arbage . O ut
    • Screen the data
    • 60. Use variables that theoretically “go together”
  • 61. 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
  • 62. Assumption testing: Sample size Comrey and Lee (1992): 50 = very poor, 100 = poor, 200 = fair, 300 = good, 500 = very good 1000+ = excellent
  • 63. Assumption testing: Sample size
  • 64. 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.
  • 65. Assumption testing: Normality
    • FA is robust to violation of assumptions of normality
    • 66. If the variables are normally distributed then the solution is enhanced
  • 67. 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)
  • 68. Assumption testing: Outliers
    • FA is sensitive to outlying cases
      • Bivariate outliers (e.g., check scatterplots)
      • 69. Multivariate outliers (e.g., Mahalanobis’ distance)
    • Identify outliers, then remove or transform
  • 70.
    • 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
  • 71. Classroom behaviour items
    • Cannot concentrate ↔ can concentrate
    • 72. Curious & enquiring ↔ little curiousity
    • 73. Perseveres ↔ lacks perseverance
    • 74. Irritable ↔ even-tempered
    • 75. Easily excited ↔ not easily excited
    • 76. Patient ↔ demanding
    • 77. Easily upset ↔ contented
  • 78.
    • Control ↔ no control
    • 79. Relates warmly to others ↔ disruptive
    • 80. Persistent ↔ frustrated
    • 81. Difficult ↔ easy
    • 82. Restless ↔ relaxed
    • 83. Lively ↔ settled
    • 84. Purposeful ↔ aimless
    • 85. Cooperative ↔ disputes
    Classroom behaviour items
  • 86. Classroom behaviour items
  • 87. 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?
    • 88. Anti-image matrix diagonals > .5?
    • 89. Measures of sampling adequacy (MSAs)?
      • Bartlett’s sig.?
      • 90. KMO > .5 or .6?
  • 91. 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
  • 92.
    • Examine the diagonals on the anti-image correlation matrix
    • 93. Consider variables with correlations less that .5 for exclusion from the analysis – they lack sufficient correlation with other variables
    • 94. Medium effort, reasonably accurate
    Assumption testing: Factorability: Anti-image correlation matrix
  • 95. Anti-Image correlation matrix Make sure to look at the anti-image CORRELATION matrix
  • 96.
    • Global diagnostic indicators - correlation matrix is factorable if:
      • Bartlett’s test of sphericity is significant and/or
      • 97. Kaiser-Mayer Olkin (KMO) measure of sampling adequacy > .5 or .6
    • Quickest method, but least reliable
    Assumption testing: Factorability: Measures of sampling adequacy
  • 98. Assumption testing: Factorability
  • 99. 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?
    • 100. Anti-image matrix diagonals > .5?
    • 101. Bartlett’s test significant?
    • 102. KMO > .5 to .6? (depends on whose rule of thumb)
  • 103. Steps / Process
    • Test assumptions
    • 104. Select type of analysis
    • 105. Determine no. of factors (Eigen Values, Scree plot, % variance explained)
    • 106. Select items (check factor loadings to identify which items belong in which factor; drop items one by one; repeat)
    • 107. Name and define factors
    • 108. Examine correlations amongst factors
    • 109. Analyse internal reliability
    • 110. Compute composite scores
  • 111. Type of EFA: Extraction method: PC vs. PAF Two main approaches to EFA:
    • Analyses all variance: Principle Components (PC)
    • 112. Analyses shared variance: Principle Axis Factoring (PAF)
  • 113. Principal components (PC)
    • More common
    • 114. More practical
    • 115. Used to reduce data to a set of factor scores for use in other analyses
    • 116. Analyses all the variance in each variable
  • 117. Principal axis factoring (PAF)
    • Used to uncover the structure of an underlying set of p original variables
    • 118. More theoretical
    • 119. Analyses only shared variance (i.e. leaves out unique variance)
  • 120. Total variance of a variable Principal Components (PC) Principal Axis Factoring (PAF)
  • 121.
    • Often there is little difference in the solutions for the two procedures.
    • 122. If unsure, check your data using both techniques
    • 123. 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
  • 124. Communalities
    • Each variable has a communality =
      • the proportion of its variance explained by the extracted factors
      • 125. sum of the squared loadings for the variable on each of the factors
    • Ranges between 0 and 1
    • 126. If communality for a variable is low (e.g., < .5, consider extracting more factors or removing the variable)
  • 127.
    • High communalities (> .5): Extracted factors explain most of the variance in the variables being analysed
    • 128. Low communalities (< .5): A variable has considerable variance unexplained by the extracted factors
      • May then need to extract MORE factors to explain the variance
    Communalities
  • 129. Communalities - 2
  • 130. Explained variance
    • A good factor solution is one that explains the most variance with the fewest factors
    • 131. Realistically, happy with 50-75% of the variance explained
  • 132. Explained variance 3 factors explain 73.5% of the variance in the items
  • 133. Eigen values
    • Each factor has an eigen value
    • 134. Indicates overall strength of relationship between a factor and the variables
    • 135. Sum of squared correlations
    • 136. Successive EVs have lower values
    • 137. Rule of thumb: Eigen values over 1 are ‘stable’ (Kaiser's criterion)
  • 138. 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.
  • 139. Scree plot
    • A line graph of Eigen Values
    • 140. Depicts amount of variance explained by each factor
    • 141. Cut-off: Look for where additional factors fail to add appreciably to the cumulative explained variance
    • 142. 1st factor explains the most variance
    • 143. Last factor explains the least amount of variance
  • 144. Scree plot
  • 145. Scree plot Scree plot
  • 146. Scree plot Scree plot
  • 147. How many factors? A subjective process ... Seek to explain maximum variance using fewest factors, considering:
    • Theory – what is predicted/expected?
    • 148. Eigen Values > 1? (Kaiser’s criterion)
    • 149. Scree Plot – where does it drop off?
    • 150. Interpretability of last factor ?
    • 151. Try several different solutions ? (consider FA type, rotation, # of factors)
    • 152. Factors must be able to be meaningfully interpreted & make theoretical sense?
  • 153. How many factors?
    • Aim for 50-75% of variance explained with 1/4 to 1/3 as many factors as variables/items.
    • 154. Stop extracting factors when they no longer represent useful/meaningful clusters of variables
    • 155. Keep checking/clarifying the meaning of each factor – make sure you are reading the full wording of each item.
  • 156.
    • 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.
      • 157. All variables will tend to load strongly on the 1st factor
    Initial solution: Unrotated factor structure
  • 158. Initial solution - Unrotated factor structure
    • Factors are weighted combinations of variables
    • 159. A factor matrix shows variables in rows and factors in columns
  • 160. 1st factor extracted:
    • Best possible line of best fit through the original variables
    • 161. Seeks to explain lion's share of all variance
    • 162. A single factor, best summary of the variance in the whole set of items
    Initial solution - Unrotated factor structure
  • 163.
    • 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
  • 164. Vectors (Lines of best fit)
  • 165. Initial solution: Unrotated factor structure
    • Seldom see a simple unrotated factor structure
    • 166. Many variables load on 2 or more factors
    • 167. Some variables may not load highly on any factors (check: low communality)
    • 168. Until the FLs are rotated, they are difficult to interpret.
    • 169. Rotation of the FL matrix helps to find a more interpretable factor structure.
  • 170. Two basic types of factor rotation Orthogonal (Varimax) Oblique (Oblimin)
  • 171. Two basic types of factor rotation
    • Orthogonal minimises factor covariation, produces factors which are uncorrelated
    • 172. Oblimin allows factors to covary, allows correlations between factors
  • 173. Orthogonal rotation
  • 174. Why rotate a factor loading matrix?
    • After rotation, the vectors (lines of best fit) are rearranged to optimally go through clusters of shared variance
    • 175. Then the FLs and the factor they represent can be more readily interpreted
  • 176. Why rotate a factor loading matrix?
    • A rotated factor structure is simpler & more easily interpretable
      • each variable loads strongly on only one factor
      • 177. each factor shows at least 3 strong loadings
      • 178. all loading are either strong or weak, no intermediate loadings
  • 179. Orthogonal vs. oblique rotations
    • Consider purpose of factor analysis
    • 180. If in doubt, try both
    • 181. Consider interpretability
    • 182. Look at correlations between factors in oblique solution
      • if >.3 then go with oblique rotation (>10% shared variance between factors)
  • 183. Interpretability
    • It is dangerous to be driven by factor loadings only – think carefully - be guided by theory and common sense in selecting factor structure.
    • 184. You must be able to understand and interpret a factor if you’re going to extract it.
  • 185. Interpretability
    • However, watch out for ‘seeing what you want to see’ when evidence might suggest a different, better solution.
    • 186. There may be more than one good solution! e.g., in personality
  • 189. 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.
  • 190. Initial solution – Unrotated factor structure 4
  • 191. Rotated factor matrix Task Orientation Sociability Settledness
  • 192. 3-d plot
  • 193.
    • Bare min. = 2
    • 194. Recommended min. = 3
    • 195. Max. = unlimited
    • More items:
    -> ↑ reliability -> ↑ 'roundedness' -> Law of diminishing returns
    • Typically = 4 to 10 is reasonable
    How many items per factor?
  • 196. How do I eliminate items? A subjective process; consider:
    • Size of main loading (min = .4)
    • 197. Size of cross loadings (max = .3?)
    • 198. Meaning of item (face validity)
    • 199. Contribution it makes to the factor
    • 200. Eliminate 1 variable at a time, then re-run, before deciding which/if any items to eliminate next
    • 201. Number of items already in the factor
  • 202. Factor loadings & item selection Comrey & Lee (1992): loadings > .70 - excellent > .63 - very good > .55 - good > .45 - fair > .32 - poor
  • 203. Factor loadings & item selection Cut-off for acceptable loadings:
    • Look for gap in loadings (e.g., .8, .7, .6 , .3, .2 )
    • 204. Choose cut-off because factors can be interpreted above but not below cut-off
  • 205. 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.
  • 206. Factor analysis in practice
    • To find a good solution, consider each combination of:
    • Apply the above methods to a range of possible/likely factors, e.g., for 2, 3, 4, 5, 6, and 7 factors
  • 210.
    • 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
    • 211. You will probably come up with a different solution from someone else!
    • Check/consider reliability analysis
    Factor analysis in practice
  • 212. Example: Condom use
    • The Condom Use Self-Efficacy Scale (CUSES) was administered to 447 multicultural college students.
    • 213. PC FA with a varimax rotation.
    • 214. Three distinct factors were extracted:
      • Appropriation
      • 215. Sexually Transmitted Diseases
      • 216. Partners' Disapproval
  • 217. 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
  • 218. 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
  • 219. 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
  • 220. Summary
  • 223. Introduction: Summary
    • Factor analysis is a family of multivariate correlational data analysis methods for summarising clusters of covariance .
    • 224. FA summarises correlations amongst items.
    • 225. The common clusters (called factors) are summary indicators of underlying fuzzy constructs.
  • 226. Assumptions: Summary
    • Sample size
      • 5+ cases per variables (ideally 20+ cases per variable)
      • 227. N > 200
    • Bivariate & multivariate outliers
    • 228. Factorability of correlation matrix (Measures of Sampling Adequacy)
    • 229. Normality enhances the solution
  • 230. Summary: Steps / Process
    • Test assumptions
    • 231. Select type of analysis
    • 232. Determine no. of factors (Eigen Values, Scree plot, % variance explained)
    • 233. Select items (check factor loadings to identify which items belong in which factor; drop items one by one; repeat)
    • 234. Name and define factors
    • 235. Examine correlations amongst factors
    • 236. Analyse internal reliability
    • 237. Compute composite scores
      Next lecture
  • 238. Summary: Types of FA
    • PC: Data reduction
      • uses all variance
    • PAF: Theoretical data exploration
      • uses shared variance
    • Try both ways
      • Are solutions different? Why?
  • 239. Summary: Rotation
    • Orthogonal (varimax)
      • perpendicular vectors
    • Oblique (oblimin)
      • angled vectors
    • Try both ways
      • Are solutions different? Why?
  • 240. 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
  • 241.
    • Comrey, A. L., & Lee, H. B. (1992). A first course in factor analysis (2nd ed.). Hillsdale, NJ: Lawrence Erlbaum.
    • 242. 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.
    • 243. 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.
    References
  • 244. Open Office Impress
    • This presentation was made using Open Office Impress.
    • 245. Free and open source software.
    • 246. http://www.openoffice.org/product/impress.html