Factor Analysis

Question 1

What is factor analysis

Answer 1

a broad term surrounding a family of techniques that investigates clusters of variables - determines whether a larger number of variables can be reduced to a smaller number of variables (factors) by grouping together variables that are highly intercorrelated, while leaving out uncorrelated variables

Question 2

3 types of factor analysis

Answer 2

1. Principle Components Analysis
2. Exploratory Factor Analysis
3. Confirmatory Factor Analysis

Question 3

Differences between EFA and CFA

Answer 3

EFA:

• no pre-defined number of factors
• no pre-determined variable/factor relationships
• more common
• typically done via factor analysis (FA)

CFA

• pre-defined number of factors
• pre-determined variable/factor relationships
• less common

Question 4

what are the features of EFA

Answer 4

• used to determine appropriate scale/questions by identifying items that co-vary and load onto the construct which therefore comprise a factor
• Sometimes used in determine discriminate validity but not very robust in that regard

Question 5

observed correlation matrix

Answer 5

correlation matrix produced by the observed variables

Question 6

reproduced correlation matrix

Answer 6

correlation matrix produced from factors

Question 7

residual correlation matrix

Answer 7

difference between observed and reproduced correlation matrix. In good FA, correlations in resid matrix are small thus indicating a close fit between observed and reproduced matrix

Question 8

factor rotation

Answer 8

process by which the solution is made more interpretable without changing its underlying mathematical properties

Question 9

orthogonal rotation

Answer 9

all factors are uncorrelated with each other

Question 10

loading matrix

Answer 10

matrix of correlations between observed variables and factors. size of the loadings represent the relationships between each observed variable and each factor

Question 11

oblique rotation

Answer 11

factors themselves are correlated

Question 12

How can factor scores be combined

Answer 12

1. Weighted average - rarely used as too simplistic
2. Regression methods - more sophisticated but limits are imposed on way scores can relate to each other
3. Bartlett method - overcomes limitations of regression method by producing unbiased scores
4. Anderson-Rubin method - modification of Bartlett method that produces uncorrelated and standardised factor scores (recommended by Tabachnick & Fidell) if uncorrelated scores are required

Question 13

When to use factor analysis

Answer 13

1. To understand the structure of a set of variables - e.g. personality, mood, anxiety, culture, grief, well-being, and intelligence
2. To develop a questionnaire to measure a variable - to ensure that items themselves are in fact measuring what they say they are measuring
3. To reduce a data set to a more manageable size while retaining the data set’s essential qualities - reduces a large number of variables to a smaller number of factors which are then used in further analysis. Mean that instead of using a larger number of potentially related variables in a regression, can use a smaller number of more targeted variables and thus improving strength of the analysis

Question 14

When to use PCA or EFA

Answer 14

• PCA most commonly implemented in terms of scale development - default in SPSS
• Costello and Osborne indicate that its in fact a data reduction technique and not conducive to extracting factors from a particular dataset
• if you want to summarise a number of items, use PCA - PCA gives each item the same latent weight, therefore EFA much more robust in that regard (don’t predict latent variables to the same degree)

Question 15

Problems with PCA and EFA

Answer 15

• there are no external criterion such as group membership against which to test the solution
• after extraction, there is an infinite number of rotations available - all accounting for the same amount of variance in the original data, but with factors defined slightly differently
• FA is frequently used in an attempt to “save” poorly conceived research

Question 16

When not to do FA

Answer 16

Five key decisions according to Fabringer et al:

1. study design, particularly what variables are to be measured. Researchers need to consider the nature and number of common factors they wish to examine and ensure that such factors are represented in multiple measurements
2. determining whether EFA is appropriate
3. choice of model fitting procedure, specifically which factor extraction procedure is to be undertaken
4. number of factors - balancing parsimony with ability of the model to account for correlations between models (plausibility of model)
5. rotation method - specifically whether the researcher will allow for correlations between factors

Question 17

what does Field recommend to use as a cut-off point for multicollinearity

Answer 17

.3

Question 18

Additional tests for FA

Answer 18

Kaiser-Meyer test - covariance between items

• score close to 0 equals little covariance and therefore EFA not appropriate
• score closer to 1 suggests strong degree of common variance → Hair et al recommend that a score of .8 to .9 is excellent

Bartlett’s test - correlation between items

• needs to be significant in order to proceed with EFA. significance means large degree of overlap amongst items
• measured using chi square rather than pearson’s correlation

Question 19

what is the proportion of common variance called

Answer 19

communality

Question 20

what is the purpose of extraction and how is it done

Answer 20

After factors have been discovered, decision have to made about how many and which factors to keep, this is called extraction - one method of extracting is with EIGENVALUES (represents the proportion of variance accounted for by a factor)

→ higher the eigenvalue, greater the proportion of explained variance

Question 21

What is an alternative way to determine which factors to keep in analysis

Answer 21

Kaiser (1960) recommended retaining all factors with eigenvalues greater than one because one represents a substantial amount of variance as explained. Others have suggested this is too rigid and can over-extract. recommend retaining all factors with eigenvalues greater than 0.6 or 0.7.

Question 22

What is the monte carlo parallel analysis

Answer 22

• most robust method of determining factors, but very under-utilised as it requires syntax
• compares observed eigenvalues taken from correlation matrix with eigenvalues extracted from the simulation of a number of parallel datasets
• determines the expected eigenvalues by averaging the resulting randomly generated eigenvalues for each factor
• any eigenvalue in the original dataset that exceeds those randomly generated is considered significant

Question 23

What is the purpose of rotation

Answer 23

to maximise high correlations between factors and variables and to minimise low correlations. Rotation makes it easier to accurately discriminate between factors, thus improving the interpretability and scientific utility of your model. 2 types:
→ Orthogonal (unrelated, perpendicular) rotation vs oblique rotation

Question 24

what is orthogonal rotation

Answer 24

easier to interpret, describe, and report results; more suitable if factors are almost independent - costello and osborne suggest it is counter-intuitive because rarely are two factors uncorrelated (since both are predicting same factor)

Question 25

oblique rotation

Answer 25

more suitable if factors are correlated.

Question 26

different types of oblique rotation

Answer 26

→ direct quartimin: direct rotation of data - recommended by Howard unless you have solid reason determining degree of relationship between factors

→ direct oblimin: same as quartimin but includes extra parameter that allows to control for degree of obliqueness

→ promax: uses orthogonal techniques as far as mathematically possible where it rotates obliquely

Field (2018) recommends that atfirst factor analysis, start with the orthogonal technique of varimax rotation (varimax meaning variance maximising) to simplify the interpretation of factors.

Question 27

inter-rater reliability

Answer 27

decisions from different raters are compared to each other to see how consistent the rater’s decisions are

Question 28

test-retest reliability

Answer 28

same survey given to same group of people at different times

Question 29

parallel forms

Answer 29

when same group of people complete two similar versions of the survey. each version of the survey is trying to measure the same thing. then, the results from the two versions are compared in order to determine the consistency of the results between the similar versions of the survey

Question 30

internal consistency

Answer 30

when different survey items that are trying to measure same construct are compared to see how they produce similar results. 2 types:

→ typically measured through alpha but research now suggesting to use omega (uses regression weights and error of items in order to calculate internal consistency)

Question 31

average inter-item correlation

Answer 31

when group of people complete a survey and then all items on the survey that they are measuring the same construct are compared with each other. then the items are compared overall to create an average of those comparisons

Question 32

split half correlation

Answer 32

when group of people complete survey then all items on survey that are measuring same construct are split in half to form two sets of items. then the two sets of items are compared to each other to see if they all consistently measure the construct

Question 33

face validity

Answer 33

when researchers simply look at the items on the survey and give their opinion if the items appears to accurately measure what they are trying to measure

Question 34

content validity

Answer 34

how well a survey covers the range of meanings included within a concept that is being measured

Question 35

construct validity

Answer 35

when we are able to generalise out construct of interest (how truthful we are labelling out construct)

Question 36

2 types of construct validity

Answer 36

• convergent: demonstrated through having strong (positive or negative) correlations with very similar or dissimilar variables
• divergent: indicated through no associations with variables presumed to be completely unrelated with the construct

Question 37

criterion validity

Answer 37

when results from survey accurately relate to some kind of external variable

• predictive: where scale is able to predict future performance
• concurrent: where scale strongly correlates with other scales purporting to measure the same construct