Lecture 9 - Exploratory factor analysis

Question 1

Describe the steps for developing a construct

Answer 1

Steps:

Conceptualization:
1. Develop conceptual construct def.

Develop measures:
2. Generate items for construct: Questions
3. Asses content validity of items

Model specification:
4. Formally specify measurement model. Decide on measure

Scale evaluation & refinement:
5. Collect data to conduct pretest
6. Scale purification & refinement: EFA

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From slide:
1. Select appropriate questions/items
2. Define what to measure
3. Generate questions/items
4. Decide which measure it is
5. Collect data
6. EFA

Question 2

(?) Describe measurement dimensionality

Answer 2

Construct dimensionality:
- Often more dim. to capture
- Sub-dim. can be sub-construct

Decision:
- Do sub-dim. manifest construct or define characteristics of it?
- Do construct exist separate on deeper level than sub-dimensions?
- Change in construct associated with change in one, few or all sub-dimensions?

Question 3

(!) Describe EFA in general & considerations

Answer 3

General:
- Mathematic factor
- Linear combination of obs. variables
- Explorative technique: No one best way
- Group variables w. most correlation
- Researcher decide def. & interpretation of factor: Final rotation
- Quality of final analysis = Factors interpretability & stability
- Correlation matrix by eigenvalue/eigenvector to factor loading
- Find factors & its association with each variables factor loading

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Considerations:

Within cluster:
- Seek high correlation

Between clusters:
- Seek low correlation
- Separate them

No clear clusters:
- Unsuitable data
- Survey done wrong

Correlated factors:
- Should differ

Plan for # factors:
- Bad planning if new one appear

Question 4

(!) Describe the components in factor analysis

Answer 4

Eigenvalue:
- Scalar multiple
- Used w. eigenvector
- Give scale-number telling spread on the line
- Amount of variance explained by given factor of variables
- A factor with an eigenvalue ≥1 explain more variance than single variable
- Collapsed variance under diagonal in correlation matrix
- Factor eigenvector gets scaled w. when transformed by matrix

Eigenvector:
- Direction of line

Items:
- Questions
- Dots at line

Factor:
- Sum of underlying variables
- The wanted & hidden variable
- Factor line try position so strong relationship to cluster of items
- Visualized at x & y-axis with dots: Items

Factor loading:
- Correlation between variable & factor
- Factor loading = Eigenvalue * Eigenvector
- Presented in the matrix
- Parameter to help define factor
- Connect invisible factors to visible questions
- Should be above 0,8

Question 5

(!) Describe the procedure of EFA

Answer 5

Get data & check assumptions:

Extract factors:
- Initially: # factors = # variables
- Second: Find interesting factor
- Later: Force items to be expressed by interesting factor

Rotate factors:
- Reorient for simpler interpretation
- To make a simple structure
- Minimize # factors to explain variables
- Increase interpretability
- Which items to which factor
- Locate highest factor loading
- Delete variable with low association to factors
- Should become clear structure: No factor overlap

Interpretation:
- Interpret results
- Save factor score for future analysis
- Name underlying factor pattern

Question 6

(!) How to perform EFA: 1. Prepare data

Answer 6

Sample size:
- Above 3-5 Items pr. factor
- 5-10 obs. per Item
- 100 & less obs: Well-determined factor
- 100-200 obs: Well-determined factors with low common correlation between variables
- 300 obs. Medium common correlation
- 500 obs. Low level of correlation + poor defined factors

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Check assumptions & factorability of R:

Normality/case & variable outliers:
- Normality enhance solution
- Normality sensitive for large samples
- Skewness & kurtosis should approach 0

Outliers:
- Histograms, q plots, box plots
- Remove variables with low squared multiple correlation & low correlation to factors

Linearity, multicollinearity:
- Avoid singularity: Extreme multicollinearity

Factorability of R:

General:
- Check correlations above 0,3
- Matrix singular if determinant is 0 in linear algebra
- Check partial correlation matrices for pattern of correlation: There should be

Bartletts test of sphericity:
- Check if correlation is too sensitive: Zero

Kaisers measure of sampling adequacy:
- Sum of squared correlations
- Value above 0,6 accepted

Question 7

(?) How to perform EFA: 1. Extract data

Answer 7

Factor-extracting procedure:
- Methods to estimate parameters of factor model, factor loadings & unique variances
- Principal components: Try explain all variances not separating them
- Principal axis factoring
- Estimate communalities squared multiple correlation of the items
- Try to explain variance with factors

Question 8

PCA versus EFA

Answer 8

PCA:

General:
- Reduce variables
- Principal component analysis
- Principal component = Eigenvectors
- Capture all variance in data
- Try explain same variance w. less variables
- Try find variable best describing factor
- Identify variables that are composite of observed variables
- Each variable add equally to variance
- If all components retained = Poss. perfect reprod. backward correlation matrix

Internal communality:
- Not calculated
- Just set to 1: 100%

Extracted communality:
- Sum of squared loadings for variable across all extracted factors

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EFA:

General:
- Reduce factors
- Capture variance shared by variables
- Latent factors

Internal communality:
- Squared multiple correlation of each variable

Extracted communality:
- Sum of squared loadings for variable across all extracted factors

Question 9

(!) Describe considerations on selecting # factors & on interpretation

Answer 9

Selecting factors:

General:
- Parsimony vs. accuracy
- Overfactoring > Underfactoring: Stability

Rules of thumb:

Kaiser criterion:
- Keep factors w. eigenvalues above 1. Explain more than variable

Scree plot:
- Look for turning point at eigenvalues plotted against factors

Variance explained:
- Total factors retained should explain min. 60% of variance

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Interpretation:
- Look at highest factor loading
- Mark items above 0,8
- Interpret result: Give factor a name
- Decide structure

Question 10

(?) Describe how to perform rotation

Answer 10

General:
- Change factor loading to fulfill simple structure criteria
- Create new matrices

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Types:

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Orthogonal:

General:
- Interpret rotated matrix

Varimax:
- Simplify columns of loading matrix: Maxing loading variance on each factor
- Dont use: Too much rotation
- Used before computers

Quartimax:
- Simplify rows of loading matrix: Maxing loading variance within variables

Equimax:
- Combination. Try simplify both rows & columns of loading matrix

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Oblique rotation:

General:
- Structure or pattern matrix
- Clean association between factors & items
- Choose pattern & consider variables w. higher loading than 0.32: Or even 0.71

Oblimin:
- Minimize cross-product of loadings
- Choose level of corr. between factors by choice of delta-level
- Choose this since constraining factors to be uncorrelated make no sense

Promax:
- Orthogonal rotated again
- Allow corr. among factors
- Orthogonal loadings raised to powers: 2,3,6
- Not guarantee that factors are correlated: Then same as orthogonal