Factor Analysis

Question 1

What is Factor Analysis?

Answer 1

• Factor Analysis is a family of techniques used to investigate the underlying structure of variables.
  
  • Determining whether many variables can be reduced to fewer, higher order (latent) variables (called factors)
  • Groups together highly correlated variables
• Three main types:
  
  • Principle Components Analysis; data reduction technique (doesn’t extract factors. All items given same regression weight, creates components (not factors).
  • Exploratory Factor Analysis; Extract factors predicting the latent construct (items which covary together stay together), creates factors
  • Confirmatory Factor Analysis; when you already have a theory

Question 2

What are the differences between EFA and CFA?

Answer 2

Question 3

What are the three main uses of factor analysis according to Field?

Answer 3

1. To understand the structure of a set of variables; such as investigating intelligence, wellbeing, personality and other complex constructs
2. To develop a questionnaire to measure a variable; Factor analysis is often used to test the validity and internal structure of questionnaires
3. To reduce a data set to a more manageable size while retaining the data set’s essential qualities; Especially when there are issues with multicollinearity, highly correlated variables can be condensed into factors

Question 4

According to Fabringer et al. what 5 decisions need to be made when using factor analysis?

Answer 4

1. Study Design and What variables are to be measured: Consider nature and number of common factors they wish to examine, ensure those factors are represented in multiple measurements
2. Determine whether EFA is appropriate; goal of EFA is to produce a more parsimonious model of factors
3. Choice of model fitting procedure (which factor extraction procedure to use)
4. Numbers of factors: balancing parsimony with ability of model to account for correlations between variables (plausibility)
5. Rotation Methods: whether to allow for correlations between factors

Question 5

What are three ways of viewing relationships in factor analysis?

Answer 5

• R-Matrix; a correlation table used to eyeball the data (look for groups of high correlations)
• Plot; Presenting the proposed factors as axis on a graph, plot the correlation of each variable with the factor on the graph to reveal clusters.
  
  • Factor/Component Loading = coordinate of a variable along a classification axis (given by pearson correlation)
• Mathematically; A matrix representation of the factor loadings
  
  • Column per factor, Row per variable
    *

Question 6

What are the four methods of combining factor scores?

Answer 6

• Weighted Average; rarely used because it is too simplistic.
  
  • Uses a linear model formula, sub in the individual’s scores
  • Cannot compare results across different measurements
• Regression Method; more sophisticated but there are limits imposed on how scores can relate to each other
  
  • Generates coefficients adjusted for intial correlations between variables
  • Recommended for most circumstances
• Bartlett Method; overcomes limits of regression by producing unbiased scores. Factor scores can still correlate with each other though.
• Anderson-Rubin Method; modification of Bartlett method which produces uncorrelated and standardised factor scores
  
  • recommended when requiring standardised scores

Question 7

What is Communality?

Answer 7

• Communality is the proportion of common variance present in a variable
  
  • Empasse: FA requires knowledge of communality but you need to conduct FA to discover communality.
• Determining Shared Variance:
  
  • PCA: Assume a value of 1 (no unique variance) for all variables
    
    • Problem; assumes no measurement error
  • Estimating Communality; many methods available eg
    
    • Squared Multiple Correlation (SMC); Run a multiple regression using one measure as outcome and others as predictors. R2 is used as a communality estimate for that factor

Question 8

What are some different ways of extracting factors?

Answer 8

• Eigenvalues; Represent the proportion of variance accounted for by a factor
• Kaisers Criterion/Jiffy: Retain all factors with eigenvalues > 1
  
  • Overly simplistic- tends to over or under extract
  • Joliffe recommended .7 cutoff
• Catells’ Scree Plot: Eigenvalues on Y axis, factor associated on x; look for the ‘elbow’ of the curve
  
  • Very subjective, up to researchers discretion
• Monte Carlo Parallel Analysis; Similar process to bootstrapping - compares observed eigenvalues to average of many simulated data sets. Retain factors that exceed those expected
  
  • Underutilised since requires SPSS syntax

Question 9

What is rotation and what are the different types?

Answer 9

• Rotation is a technique which maximises an item’s loading on its primary factor while minimising loading on other factors
• Orthogonal Rotation; Assume factors to be uncorrelated. Counter-intuitive for Psyc.
  
  • Varimax Rotation; Good starting point. Maximises dispersion to aid interpretation.
• Oblique Rotation; Assumes factors to be correlated. All techniques produce similar results
  
  • Direct Quartimin; direct rotation
  • Direct Oblimin; direct rotation + predetermining degree of correlation
  • Promax; faster, uses orthogonal up to a point

Question 10

What assumptions need to be met when conducting factor analysis?

Answer 10

• Data Cleaning; inaccurate entries and missing data are important.
  
  • Missing data; cannot use listwise deletion when participants are answering 120 questions. Remember to check for patterns
• Normality; Not strictly required but enhances factor structure
• Linearity; Need to examine scatterplots for each pair of variables. Can be done more quickly using a scatterplot matrix.
• Correlations; Create a correlation matrix for all the variables
  
  • Too Low: Consider excluding variables with several low (.3) correlations
    
    • Bartletts test; should be significant (otherwise it is essentially an identity matrix = disaster)
  • Too High; multicollinearity
    
    • Determinant of R-Matrix should be > .00001

Question 11

What sample size is required for factor analysis?

Answer 11

• Sample Size; Aim for 300 - 1000+ participants
  
  • Bare minimum is 5 participants per variable, more common recommendation is 10 - 15 per variable
• Sample size can be less for factors with stronger loadings; if a factor has 4+ loadings > .6 it is reliable regardless of sample size
• Kaiser Meyer Olken Measure of sampling adequacy (KMO)
  
  • Measures ratio of squared correlations between variables to squared partial correlations
  • Values below .5 = Merde, values >.9 = marvellous

Question 12

How is reliability assessed in factor analysis?

Answer 12

• Internal consistency is measured using Chronbach’s Alpha (measure of split half internal consistency)
• Chronbach’s a; most commonly reported statistic, with values above .8 generally considered reliable. But:
  
  • a is biased by sample size
  • a is affected by reverse phrasing
  • a is not a measure of uni-dimensionality
  • Assumes uncorrelated errors multivariate normality, etc
• SPSS: Check overall, corrected item column, if item deleted column.
  
  • Run separate analyses for each subscale