Exploratory Factor Analysis

Question 1

What techniques are utilised for identifying clusters of variables?

Answer 1

Factor analysis and Principle component analysis (PCA)

Question 2

What three uses do these techniques have?

Answer 2

1. Understanding the structure of a set of variables
2. Construct a questionairre to measure an underlying variable
3. Reduce a dataset to a more manageable size while retaining as much of the original information as possible

Question 3

How does factor analysis attempt to achieve parsimony?

Answer 3

By explaining the maximum amount of common variance in a correlation matrix using the smallest number of explanatory constructs (latent variables )

Question 4

How does this differ to PCA?

Answer 4

PCA attempts to explain the maximum amount of total variance in a correlation matrix by transforming the the original variables into linear components

Question 5

What is meant by the term factor loading?

Answer 5

A factor loading refers to the coordinate of a variable along a classification axis (e.g. Pearson correlation between factor and variable)

Question 6

What does a factor loading tell us

Answer 6

It tells us something about the relative contribution that a variable makes to a factor.

Question 7

How are scores on the measured variables predicted in factor analysis?

Answer 7

From the means of those variables plus the person’s scores on the common factors multiplied by their factor loadings, plus scores on any unique factors within the data

Question 8

What is meant by common factors?

Answer 8

factors that explain the correlations between variables

Question 9

How are components predicted in PCA?

Answer 9

In PCA, the components are predicted from the measured variables.

Question 10

Name a major assumption of factor analysis

Answer 10

One major assumption of factor analysis is that the algebraic factors represent real-world dimensions.

Question 11

How is the weighted average calculated after PCA

Answer 11

Multiplying their score by the factor loadings

Question 12

Why is the weighted average rarely used

Answer 12

as it is over simplistic and is influence by the measurement scores

Question 13

What is the simplest technique for calculating factor score coefficients?

Answer 13

A regression technique;

Question 14

What does the mean and variance look like using the regression technique?

Answer 14

the resulting actor scores have a mean of 0 and a variance equal to the squared multiple correlations between the estimated factor scores and the true factor values.

Question 15

What is a downside to the regression model?

Answer 15

A downside is that the scores can correlate with other factor scores from a different orthogonal factor

Question 16

What can be done to overcome this problem of the regression model? (2)

Answer 16

The Bartlett method and the Anderson-Rubin method can be used to overcome this problem. The Bartlett method produces factor scores that are unbiased and only produce correlations with their own factor. The Anderson-Rubin method is a method which produces factors which are uncorrelated and standardized.

Question 17

How is the matrix of factor score coefficients obtained?

Answer 17

By multiplying the factor loadings by the inverse of the original correlation or R matrix (dividing the factor loadings by the correlation coefficients)

Question 18

When is the Anderson-Rubin method best?

Answer 18

When uncorrelated scores are required

Question 19

What does the method for discovering factors depend on?

Answer 19

whether the results should be generalized from the sample to the population (1) and whether you are exploring your data or testing a specific hypothesis (2).

Question 20

What does random variance refer to?

Answer 20

Random variance refers to variance that is specific to one measure but not reliably so.

Question 21

What does communality refer to?

Answer 21

the proportion of common variance present in a variable.

Question 22

What is meant by extraction?

Answer 22

Extraction refers to the process of deciding how many factors to keep.

Question 23

What do Eigen values represent?

Answer 23

Eigenvalues associated with a variate indicate the substantive importance of that factor. Therefore, factors with large eigenvalues are retained. Eigenvalues represent the amount of variation explained by a factor.

Question 24

What is a scree plot?

Answer 24

A scree plot is a plot where each eigenvalue is plotted against the factor with which it is associated.

Question 25

What is meant by a point of inflection?

Answer 25

The point of inflexion is where the slope of the line changes dramatically. This point can be used as a cut-off point to retain factors

Question 26

What else can be used as a criterion?

Answer 26

The eigenvalues

Question 27

What is the difference between Kaiser’s and Joeliffe’s criterion?

Answer 27

Kaiser’s criterion is to retain factors with eigenvalues greater than 1. Joliffe’s criterion is to retain factors with eigenvalues greater than 0.7.

Question 28

With what method should you restrict your conclusions to the sample?

Answer 28

PCA

Question 29

What is meant by unique variance?

Answer 29

Variance that can be attributed to one measure

Question 30

What its the most common way of estimating communality?

Answer 30

SMC (squared multiple correlation)

Question 31

When can kaiser’s criterion be accurate?

Answer 31

When the number of variables is less than 30 and the resulting commonalities are all greater than 0.7 or when the sample size exceeds 250 and the commonalities are over 0.6

Question 32

What do commonalities mean for the validity of our results?

Answer 32

They represent a loss of information, the closer to one they are the better our factors are at explaining the data

Question 33

Once factors have been extracted how do we calculate the degree two which variables load onto these factors?

Answer 33

In order to make interpretation easier, rotation can be used where rotation rotates aces such that variables are loaded maximally to only one factor.

Question 34

What two different types of rotations are there?

Answer 34

Orthogonal rotation refers to rotation while keeping the factors uncorrelated. Oblique rotation allows factors to correlate.

Question 35

What does the choice of rotation depend on?

Answer 35

The choice of orthogonal or oblique rotation depends on whether there is a theoretical reason to suppose that the factors should correlate or should be uncorrelated and how the variables cluster on the factors before rotation. Orthogonal is used when its unrelated and arguably should not be used for human qualities

Question 36

Describe the three varieties of orthogonal rotation

Answer 36

Quartimax- more variables in less factors
Verimax- less variables in more factors (most common)
Equemax- erratic hybrid

Question 37

what is meant by direct quartimin rotation?

Answer 37

Oblique rotation when the delta is 0 and so doesn’t let the factors correlate too high

Question 38

What should happen if variable correlations are very high or low?

Answer 38

Remove the variables

Question 39

What does the determinant tell us?

Answer 39

The determinant tells us whether the correlation matrix is singular or if all variables are completely unrelated.

Question 40

What score should the determinant have?

Answer 40

The determinant should be larger than 0.00001.

Question 41

What is the likelihood of a non-positive definite matrix?

Answer 41

A non-positive definite matrix is not possible. The most likely reason for this is having too many variables and too few cases of data.

Question 42

What variables on the anti-image matrix should be removed? Why is this?

Answer 42

Variables in the diagonal line on the anti-image matrix with a score of less than 0.5 should be removed. These scores denote the Kaiser-Meyer-Olkin measure of sampling adequacy. The off-diagonal scores should be small.

Question 43

What is the first part of factor extraction?

Answer 43

The first part of factor extraction is to determine the linear components within the variables – the eigenvectors.

Question 44

_____ is a poor sample size
______ is good
______ is excellent

Answer 44

100; 300; 1,000

Question 45

What can decide what sample size is needed

Answer 45

more commonalities the less n needed and the more variables the more n needed

Question 46

What can be calculated to find the sampling adequacy?

Answer 46

KMO

Question 47

What test do we use to see if a correlation matrix is significantly different to an identity matrix?

Answer 47

Bartletts test

Question 48

What is meant by multicollinearity and singularity?

Answer 48

variables highly correlated; perfectly correlated

We try to avoid this

Question 49

When is an assumption of normality important?

Answer 49

When you’re generalising to the population