1 / 60

Factor Analysis 2006

Factor Analysis 2006. Lecturer: Timothy Bates Tim.bates@ed.ac.uk Lecture Notes based on Austin 2005 Bring your hand out to the tutorial Please read prior to the tutorial. FACTOR ANALYSIS.

alaina
Download Presentation

Factor Analysis 2006

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Factor Analysis 2006 Lecturer: Timothy Bates Tim.bates@ed.ac.uk Lecture Notes based on Austin 2005 Bring your hand out to the tutorial Please read prior to the tutorial

  2. FACTOR ANALYSIS • A statistical tool to account for variability in observed traits in terms of a smaller number of factors • Factor = "unobserved random variable" • Measured item = Observed random variable • Values for an observation are recovered (with some error) from a linear combination of (usually much smaller set of) extracted factors.

  3. Visually…

  4. FA as a Data reduction technique • Simplify complex multivariate datasets by finding “natural groupings” within the data • May correspond to underlying ‘dimensions’. • Subsets of variables that correlate strongly with each other and weakly with other variables in the dataset. • Natural groupings (factors) can assist the theoretical interpretation of complex datasets • Theoretical linkage of factors to underlying (latent) constructs, e.g. “extraversion”, liberal attitudes, interest in ideas, ability

  5. EXAMPLE DATASET 210 students produced self-ratings on a list of trait adjectives. Correlations above 0.2 marked in bold • 1. ASSERTIVE, 2. TALKATIVE, 3.EXTRAVERTED, 4. BOLD • 5. ORGANIZED • 6. EFFICIENT, 7. THOROUGH, 8. SYSTEMATIC • 9. INSECURE • 10. SELF-PITYING, 11 NERVOUS, 12. IRRITABLE • Clear structure in this sorted matrix • How easy would this be to see in a larger matrix?

  6. THE THREE FACTORS FROM THE EXAMPLE DATA • The numbers are factor loadings = correlation of each variable with the underlying factor. • Loadings less than 0.1 omitted.) • Can construct factor score (multiplied factor loadings) • N =(0.75*Nervous) + (.73*Irritable) + (.73*Insecure) + (.72*Self-pity) – (.10*Extraverted) –(.21*Assertive) • Main loadings are large and highly significant. • Smaller (cross-)loadings may be informative. • Factors are close to simple structure.

  7. OBJECTIVES AND OUTCOMES OF FACTOR ANALYSIS • Aim of factor analysis is to objectively detect natural groupings of variables (factors) • Can deal with large matrices, uses (reasonably) objective statistical criteria. • Can obtain quantitative information • e.g. factor scores. • Factors are (should be) of theoretical interest. • In the example the factors correspond to the personality traits of Extraversion, Neuroticism and Conscientiousness • Exploratory method, uncovering structure in data • Confirmatory factor analysis (model testing) is also possible.

  8. SOME TECHNICAL REQUIREMENTS FOR A FACTOR ANALYSIS TO BE VALID AND USEFUL • Simple structure • Each item loads highly on one factor and close to zero on all others • Factors have a meaningful theoretical interpretation • Rotation • Factors retain most of the variance in the raw data • Parsimony compared to starting variables achieved without loss of explanatory power • Factors are Replicable

  9. Assumptions • Large enough sample • So that the correlations are reliable • Somewhat normal variables, No outliers • No variables uncorrelated with any other • No variables correlated 1.0 with each other • Remove one of each problematic pair, or use sum if appropriate.

  10. DATA QUALITY • Sample Size • Rough rule is that 300 is OK, smaller numbers may be OK. • Subjects/variables ratio • Much discussion (less agreement) • Values between 2:1 and 10:1 have been proposed as a minimum. • Simulations suggest that overall sample size is more important. • Well-defined factors (large loadings) will replicate in smaller samples than poorly-defined ones (small loadings)

  11. STAGES OF ANALYSIS • Examine data for outliers and correlations • Choose number of factors • Scree plot • Rotate factors if necessary • Interpret factors • Obtain scores • Check reliability of scales defining factors • Further experiments to validate factors

  12. Partitioning item variance • Variance of each item can be thought of in three partitions: 1. Shared variance • Common variance, explained by factors + Unique variance Not explained by other factors • 2. Specific variance • 3. Error variance • Communality • The proportion of common variance for a given variable • Sum of squares of item factor loadings • Large communalities are required for a valid and useful factor solution

  13. Computing a Factor Analysis • Two main approaches • Differ in estimating communalities • Principal components • Simplest computationally • Assumes all variance is common variance (implausible) but gives similar results to more sophisticated methods. • SPSS default. • Principal factor analysis • Estimates communalities first

  14. How many Factors? • Initially unknown • Needs to be specified by the investigator on the basis of preliminary analysis • No 100% foolproof statistical test for number of factors • Similar problems with other multivariate methods

  15. How many factors? • There are potentially as many factors as items • We don’t want to retain factors which account for little variance. • Most commonly-used method to decide the number of factors is the “scree” plot of the “eigenvalues” • Variance explained by each factor. • A point of inflection or kink or in the scree plot is a good method of making a cut-off

  16. EQ Scree

  17. Goldberg Scree

  18. Food and health Scree

  19. IQ Scree

  20. OTHER METHODS FOR FACTOR NUMBERS • Eigenvalues > 1 • Eigenvalues sum to the number of items, so an eigenvalue of >1 = more informative than a single average item • Not a useful guide in practice • Parallel Analysis • Repeatedly randomise the correlation matrix and determine how large an eigenvalue appears by chance in many thousands of trials. • Excellent method • Theory-driven • Extract a number of factors based on theoretical considerations • Hard to justify

  21. How to align the factors? • The initial solution is “un-rotated” • Two undesirable features make it hard to interpret: • Designed to maximise the loadings of all items on the first factor • Most items have large loadings on more than one factor • Hides groupings in the data

  22. UNROTATED FACTORS FOR THE EXAMPLE DATA

  23. ROTATION – DETAIL (1) • Rotation shows up the groups of items in the data. • Orthogonal rotation • Factors remain independent • Oblique rotation • Factors allowed to correlate • Theoretical reasons to choose a type of rotation • (e.g. for intelligence test scores); • May explore both types • Choose oblique if there are large correlations between factors, orthogonal otherwise.

  24. Item loadings on the first 2 factors +1 N X X X X -1 +1 X X X X C -1

  25. Lack of Simple Structure +1 N X X X X X -1 +1 C X X -1

  26. Rotation Defines New Axes WhichReveal the Item Groups X X X X X X

  27. Oblique Rotation X X X X X X

  28. ROTATION -DETAIL (2) • Rotated and un-rotated solutions are mathematically equivalent • Rotation is performed for purposes of interpretation. • Most common types: • Oblique • Direct oblimin • Orthogonal • Varimax (maximzes squared colun variance) • Most common • Quartimax (maximises row variance) • Equamax simplifies rows and columns of a factor matrix

  29. INTERPRETING FACTORS • Done on the basis of ‘large’ loadings • Often taken to be above 0.3. • Size of loading which should be considered substantive is sample-size dependent. • For large samples loadings of 0.1 or below may be significant but do not explain much variance. • Well-defined factor should have at least three high-loading variables • Existence of factors with only one or two large loadings indicates factors over-extracted, or multi-colinearity problems. • Assigning meaning to factors.

  30. FACTOR SCORES • Factor scores • Estimate of each subject’s score on the underlying latent variable • Calculated from the factor loadings of each item. • Simple scoring methods • Often used for, e.g., personality questionnaires is to sum the individual item scores (reverse-keying where necessary). • This method is reasonable when all variables are measured on the same scale; • What if you have a mix of items measured on different scales? • (e.g. farmer’s extraversion score, farm annual profit, farm area).

  31. EXAMPLE 1 – FACTOR STRUCTURE OF DIETARY BEHAVIOUR • Research question: Is there a dimension of healthy vs. unhealthy diet preferences? • (Mac Nicol et al 2003) • 451 schoolchildren completed a 35-item questionnaire mainly on food items regularly consumed (also some general health behaviour items) • Subjects:variables 12.9. Population not representative for SES. • Scree suggested three factors, two diet related • F1: Unhealthy foods (chips, fizzy drinks etc) • F2 Healthy foods (fruit, veg etc) • Validation • Higher SES and better nutrition knowledge associated with healthier eating patterns. • Factor reliabilities low • Problem of yes/no items • Sample in-homogeneity.

  32. EXAMPLE 2 –FACTOR STRUCTURE OF THE AQ(Austin, 2005) • Does the AQ have the factor structure that its original author thinks it has? • The AQ is a 50-item questionnaire designed to assess autistic traits in the general population and at the high-functioning end of the clinical range. • Designed produce a general factor and to have subscales assessing well-known clinical characteristics of autism: • Poor social skills • Strong focus of attention • Attention to detail • Poor communication • Poor imagination/play • Completed by 201 undergraduates. • Subjects: variables 4:1. • Scree suggested a general factor + three sub-factors • Poor social skills, attention to detail and poor communication. • Reliabilities OK, some validation (males vs. females, arts vs. science)

  33. EXAMPLE 3 –FACTOR STRUCTURE OF AN EI SCALE • How many factors in a published emotional intelligence scale, and can it be improved by adding more items? • (Saklofske, etal. 2003; Austin et al., 2004). • 354 undergraduates completed a 33-item EI scale for which previous findings on the factor structure had given contradictory results. • Scree plot (and some confirmatory modelling) suggested four factors, one with poor reliability. • The factor structure has been replicated although other factor structures have been reported. • A longer 41-item version of the same scale was constructed with more reverse-keyed items than the original scale, and also with additional items targeted on the low-reliability factor (utilisation of emotions). • Completed by 500 students and was found to have a three-factor structure. • Reliability of utilisation subscale increased, but still below 0.7.

  34. EXAMPLE 4 – ABNORMAL PERSONALITY • How does personality disorder relate to normal personality? • Deary et al. (1998). • Scale-level analysis of DSM-III-R personality disorders & EPQ-R • Sample = 400 students • Joint analysis gives four factors: • N+ Borderline, Self-defeating, Paranoid • P+ Antisocial, Passive-aggressive, Narcissistic • E+ avoidant(-), histrionic • P(-) Obsessive-compulsive, Narcissistic

  35. EXAMPLE 5 - THE ATTITUDES TO CHOCOLATE QUESTIONNAIRE • 80 items on attitudes to chocolate were constructed using interviews and related literature. • Aspects assessed included • difficulty controlling consumption, positive attitudes, negative attitudes, craving. • Self-report chocolate consumption was obtained; participants also performed a bar-pressing task with chocolate button reinforcements delivered on a progressive ratio schedule. • Factor analysis gave three factors (eigenvalue 1 criterion) • 33.2%, 14.1% & 6.1% of the variance. • Third scale had low reliability • Probably over-factored. • Follow up paper (Cramer & Hartleib, 2001) has confirmed the first two factors.

  36. Factors Found • Craving • I like to indulge in chocolate • I often go into a shop for something else and end up buying chocolate), • Guilt • I feel guilty after eating chocolate • Functional approach • I eat chocolate to keep my energy levels up when doing physical exercise. • High-craving individuals reported • Consuming more bars per month • Were prepared to work harder to get chocolate buttons

  37. Example 6: Criterion based FA(Kline, Easy Guide, Ch 9) • Two groups: long-term tranquilliser users and matched controls • Measured • Personality • Psychological distress • Life events • Health data • Visits to GP • Ratings by GP • etc. etc. • What factor(s) predict group membership? • High loadings for the group membership variable • In this study the best factor loaded • Anxiety • Few friends • High GP contact • High repeat prescriptions • Some variables unrelated (life events, job satisfaction, church attendance…) • Alternative approaches • Regression • Cluster analysis

  38. End of Lecture I • See you next week :-)

  39. INTERPRETING FACTORS • Done on the basis of ‘large’ loadings • Often taken to be above 0.3. • Size of loading which should be considered substantive is sample-size dependent. • For large samples loadings of 0.1 or below may be significant but do not explain much variance. • Well-defined factor should have at least three high-loading variables • Existence of factors with only one or two large loadings indicates factors over-extracted, or multi-colinearity problems. • Assigning meaning to factors.

  40. FACTOR SCORES • Factor scores • Estimate of each subject’s score on the underlying latent variable • Calculated from the factor loadings of each item. • Simple scoring methods • Often used for, e.g., personality questionnaires is to sum the individual item scores (reverse-keying where necessary). • This method is reasonable when all variables are measured on the same scale; • What if you have a mix of items measured on different scales? • (e.g. farmer’s extraversion score, farm annual profit, farm area).

  41. STATISTICAL TESTS FOR DATA QUALITY • Examine KMO statistic. • Kaiser-Meyer-Olkin test of sampling adequacy • Should be 0.5 or more. • Low values indicate diffuse correlations with no substantive groupings. • KMO statistics for each item • Item values below 0.5 indicate item does not belong to a group and may be removed • Bartlett’s test of sphericity. • Tests that the correlation matrix is significantly different from an identity matrix. • p-value should be significant • Tests that there are not duplicate items in the matrix

  42. SPSS ASPECTS • Path to follow is analyse, data reduction, factor. • EXTRACTION • Select scree plot for initial run. • Choose number of factors. • ROTATION • Select rotation method • Increase number of iterations for rotation if necessary (default 25) • DESCRIPTIVES • KMO and Bartlett tests • Reproduced correlations and residuals • Anti-Image matrix • SCORES • Save as variables • Method

  43. Sort coefficients by size Suppress small loadings OPTIONS

  44. SCORING ETC. • Factor scores constructed as above or by related methods can be used in further analyses • e.g. are there M/F differences in scores on N, E, C? • Do the factor scores correlate with other measures (exam anxiety, subjective reports of life quality, number of friends, exam success…)

  45. OTHER ASPECTS OF FACTOR ANALYSIS • Discussion so far has been in terms of questionnaire items, but factor analysis is possible with any set of measures for which correlations can be calculated. • Hypothetical example: personality traits, socio-economic status, salary, life satisfaction, number of serious illnesses etc in the last five years • Datasets of this type raise issues of factor analysis vs. regression modelling. • Scale-level analysis can be very useful in the study of personality/individual differences. • Hierarchical factor structures. • Best-known example is intelligence test scores. • Scores on a diverse range of tests are usually all positively intercorrelated (positive manifold). • Can extract either • A general ability (g) factor (positive loadings from all tests) • or • Examine clustering of tests in more detail giving correlated (oblique) lower-level factors. • Choice of level of description; both descriptions are equally ‘correct’.

  46. Nested Analysis g gs d gf gr gc Specific tests

  47. USING FACTORS • Naming – use content of high-loading items as a guide • Assess internal reliability for each factor • Scores – ‘unit weighting’ best for comparison between samples • Validation – do factor scores correlate as expected with other variables? Issues of convergent/divergent validity with other tests if relevant.

  48. Scale Reliability • Factor Derived Scales can be assessed as with any other scale • For instance using Cronbach’s Alpha • Check alpha if item deleted to identify poorly-functioning items • Adequate reliability is defined as 0.7 or above

  49. CONFIRMATORY FACTOR ANALYSIS • Hypothesis testing • Test the “fit” of a pre-specified model • Compare different Models • Available in several packages • AMOS, Mx, Mplus • Not covered in this course

  50. How to assess FA • Sample size • To things matter: • ratio of subjects to Items • Total sample size • Item to subject ratio is important • Can get away with smaller numbers when communalities are high (i.e. factors well-defined) • Restriction of range (subject too similar) • reduces correlations • Items per factor. • Need at least three per factor, four is better. Some published analyses discuss factors with only one item loading! • Use of eigenvalue-1. • Often seen in papers where factor number comes out implausibly high. • Rotation. • Orthogonal used when oblique should have been tried first. • Generally safest to assume by default that factors will correlate. • Scores. • SPSS and other packages give scores which are sample-dependent. • Use of unit weighting of items is better practice.

More Related