EFA and PCA

what reliability measures are used to evaluate factors

internal reliability and external (test-retest) reliability

internal reliability

Cronbach’s alpha / Tau-equivalent reliability or Kuder-Richardson-20 (KR20)

External (test-retest) reliability

correlation r

Varimax rotation

most common orthogonal rotation technique, maximises the variance of factor loading within each factor, keeps factors uncorrelated

when to use Varimax rotation

when factors are theoretically independent

PCA

extracts component that maximise variance and are orthogonal, without regard to interpretability

Factor Analysis

goes further than PCA by rotating components to achieve simple structure - distinct, interpretable factor loadings

why is factor rotation used

rotation makes factor loading more distinct (simple structure) so that each variable loads strongly on one factor and near-zero on others

what does factor rotation achieve

makes factors easier to interpret

Bartlett test of sphericity

assesses whether the correlation matrix is significantly different from a diagonal (spherical) matrix

what does a significant result mean for Bartlett’s test of sphericity

means variables are sufficiently intercorrelated for factor analysis to be worthwhile

cross-loading

occurs when a variable has a factor loading >0.3 on more than one factor

when is cross-loading a problem

not considered a problem if the difference between the two loadings is greater than 0.2

Kaiser-Meyer-Olkin test

measures sampling adequacy - the proportion of variance shared across variables relative to total variance

what do values mean in the Kaiser-Meyer-Olkin test

marvelous - >0/9, meritorious - 0.8-0.89, middling - 0.7-0.79, unacceptable - under 0.5

K1 rule in factor retention (Kaiser’s criterion)

retain only factors with an eigenvalue greater than 1

when is the K1 rule used

often used alongside the scree plot, retaining factors before the ‘elbow’

eigenvalue

scalar (lambda) that represents how much an eigenvector is stretched or compressed during a linear transformation

eigenvector

special vector (direction) associated with a linear transformation (matrix) that does not change its direction when that transformation is applied

factor loadings

the correlations between the original observed variables and the extracted factors (or principal components, for orthogonal solutions)

what are factor loadings used for

used to interpret the meaning of each factor

core idea behind latent variables in factor analysis

observed variables are assumed to be caused by underlying latent variables. factor analysis estimates these latent variables from the pattern of correlations in the data

what is an eigenvalue in PCA

reports the proportion of total variance explained by each principle component

when is an eigenvalue used in PCA

alongside the scree plot and K1 rule to decide how many factors to retain

factor scores

an individual participant’s value on each extracted factor, can themselves be used in further analyses

what are the Big 5 personality factors as measured by the IPIP-BFF (Goldberg, 1992)

openness, conscientiousness, extraversion, agreeableness and neuroticism, the scale uses 50 items (10 per factor) with postively and negatively keyed items requiring reverse scoring

communality (h²) in factor analysis

the proportion of a variables variance explained by all extracted factors together. high communalities (>0.6) require fewer participants (-100). low communalities (<0.5) with many factors may need N>500

orthogonal rotation

keeps factors uncorrelated at 90 degrees

oblique rotation

allow factors to correlate, which may better reflect reality when constructs are related

what are the recommended sample size ratios for EFA

N/P (participants per item): 5:1 to 10:1, minimum 100 participants. P/M (items per factor): 4:1 N/M (participants per factor): 6:1

two key constraints that define Principal Component Analysis (PCA)

1. each successive component explains the maximum remaining variance, 2. all components are mutually orthogonal (uncorrelated), providing non-redundant information