Ch.4 - Dimensionality and Factor Analysis

Question 1

What is the dimensionality of a test?

Answer 1

The number of constructs a test measures

Question 2

What are the two main types of dimensionalities of tests?

Answer 2

• Unidimensional: measure one construct (e.g. algebra test in school)
  ~ All scores on items are combined into a total/composite score (total score reflects score on the single psychological attribute measured by the test)
• Multidimensional: measure multiple constructs (e.g. BIG 5, IQ tests and more)
  ~ If a multidimensional test has uncorrelated dimensions, each subtest has its own subtest score. In a sense, each subtest is in itself unidimensional. (e.g. BIG 5 has 5 scores on each personality dimension, but not a total score for personality in general)
  ~ if multidimensional test has correlated dimensions, subtests are often combined into a total test score.

Question 3

What are the 3 subtypes of multiple dimensionality based on how the constructs are correlated?

Answer 3

(See Picture 1)

Question 4

What are the 3 dimensionality questions and why are they important?

Answer 4

1) How many dimensions are reflected in the test? (Important because each dimension is likely to be scored differently, thus each dimension requires a different psychometric analysis)
2) If the test has more than one dimensions, are those dimensions correlated or not? (Important in determining if you should calculate a “total score” from all items of a test or not)
3) If a test has more than one dimensions, what are those dimensions? (If we want to score and interpret a dimension of a test effectively, then we must understand the score’s psychological meaning)

Question 5

Factor Analysis

Question 6

What is Factor Analysis?

Answer 6

Most common statistical method of studying a test’s dimensionality (addresses the 3rd question of dimensionality mentioned above)

Question 7

What are the 2 types of Factor Analysis?

Answer 7

• Exploratory Factor Analysis (EFA)
  ~ Most common method
  ~ used in early stages of psychometric analysis and development
  ~ easy to do using many software programs (e.g. SPSS)
• Confirmatory Factor Analysis (CFA)

Question 8

What are the 2 main differences between EFA and CFA?

Answer 8

1) EFA: There is no theory about the factor structure
1) CFA: There is a clear theory about the factor structure
(Just mentioned in the slides, but not that important, all the focus is on EFA)
2) EFA is used in situations were there are few, if any, ideas about the test’s dimensionality
2) CFA is used in situations were we have very clear knowledge about our test’s dimensionality
(Also, CFA uses completely different statistical and concepts than EFA)

Question 9

How do you conduct a Factor Analysis (general concept of factor analysis, not specifically EFA)

Answer 9

(See Method 2 for the whole process)
!!! Not applicable to binary items !!!
Eyeballing method - rarely works for real data

Question 10

What are the steps in conducting an EFA?

Answer 10

1) Choosing an extraction method
2) Identifying number of Factors and extracting them
- 2.a Correlation Matrix and Eigenvalues
- 2.b Select a number of Factors
3) Rotating the Factors
4) Examine Item-Factor Associations
5) Examine the Associations among Factors

Question 11

1) Choosing an extraction method

Answer 11

(extraction method for extracting Factors)
There are multiple techniques:
- Principal Axis Factoring (PAF)
- Principal Components Analysis (PCA)
(The above two are the most common, specifically PAF over PCA)
- maximum likelihood factor analysis

Question 12

2) Identifying number of factors and extracting them

Answer 12

Once you have chosen an extraction method (Step 1)), then you extract that number of factors.

Question 13

2.a., Eigenvalues

Answer 13

Eigenvalues are numbers that help us determine how many factors are in a test. They tell us how much variability does this item account for.
There have been two methods to determine how many factors there are by using Eigenvalues:
- Kaiser criterion: The amount of factors equals the number of Eigenvalues that have a value greater than 1
!!! Generally criticized and not used anymore !!! (because it is believed that through this method the actual number of factors will be overestimated)
- Scree plot (See Picture 3)

Question 14

3) Rotating the Factors

Answer 14

If a test is multidimensional then we rotate the factors to clarify the psychological meaning of the factors

Question 15

What are the 2 types of rotation?

Answer 15

• Orthogonal rotation: Generates factors that are uncorrelated (factors that are orthogonal to each other)
  ~ An example of orthogonal rotation is varimax rotation
• Oblique rotation: can generate correlated or uncorrelated factors (If factors want to be correlated they will be, or if they don’t then they won’t be) (Leads to most simple structure, see next flashcard)
  (See Picture 4 for both)

Question 16

4) Examining Item-Factor Correlations

Answer 16

Helpful to understand fully a scale’s dimension.
EFA presents correlations between items and factors through FACTOR LOADINGS.
- Range from -1 to +1 (if an item loads onto a factor with a negative value, that item is called contra-indicative item)
~ 0.3 to 0.4: reasonably strong correlations
~ 0.7 to 0.8: very strong correlations
- Two kinds of factor loadings:
~ Pattern coefficient: Correlation between an item and a factor
~ Structure coefficient: Correlation between person’s item responses and their levels of underlying factor
- If each item correlates to one and only one factor, then we have what’s called a SIMPLE STRUCTURE (See Picture 5)
~ In general, we sum a respondent’s response to the items that load together on a factor

Question 17

What are the 3 different type of matrices you get when applying factor loadings?

Answer 17

• (Before you apply factor loadings, you have the Factor Matrix - no rotation, factors are uncorrelated, correlations between item and the factor
• Pattern Matrix presents the pattern coefficients - factor loadings controlled for the correlation between the factors (factor loadings are accounted for), rotation (Usually has the most simple structure out of the 3 matrices, and is able to be interpreted the best)
• Structure Matrix presents the structure coefficients - correlation between item and factors, not cotrolling for the factor correlations
  !!! Pattern and Structure Matrices come about from Oblique rotation !!!

Question 18

If factors “want” to be correlated, what will happen if we apply orthogonal or oblique rotation?

Answer 18

• If we apply oblique rotation, then we will get high factor loadings, since this rotation allows for the factors to be correlated.
• If we apply orthogonal rotation, then we will get lower factor loadings, since this rotation only allows factors to be uncorrelated and this goes against the actual correlation of the factors.

Question 19

What should you do if there is factorial ambiguity (An item might not load strongly on any factor or an item loads strongly on many factors)?

Answer 19

• Revisit and reconsider the initial option of how many factors to extract
• Drop items with poor structure. If an item is not strongly associated with any factor, we might conclude it is not related to other items on the test. This could be because:
  ~ item reflects a psychological construct that differs from the one reflected by the other items on the scale
  ~ item is strongly affected by measurement error