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stats - lecture 12
Factor analysis
Purposes of factor analysis
Principal Components Analysis (PCA)
Common Factor Model
- Scientific method to uncover latent variables inductively (Exploratory Factor Analysis; EFA)
- Construct validation research (e.g., Confirmatory Factor Analysis; CFA) to address hypothesized construct structure (e.g., five or sixfactor model of personality traits)
- Psychometric test construction and validation (includes structural validation)
- Reducing a large set of variables to a smaller set of aggregate variables (technically the domain of principal components analysis which means you have a bunch of vari- ables and you want to create a smaller set of composite score)
Goal to reduce the observed variables to a smaller set of composite components(aggregates) that capture as much information Imagine if instead of using the current breakdown for your course evaluation we did a PCA of marks on the 5 assignments, mid-term and final and let the procedure determine the weights. It would essentially “look for” weights that would maximize the variance of the overall grades.
We focus on the variance in common between the variables We put an indication of how much variables have in common It's similar to R The diagonal represents the commonalties
Goal to uncover the latent structure (i.e., underlying common factors that explain the ob- served variables. The common factor model applies to both exploratory factor analysis (EFA) and confirmatory factor analysis (CFA).
The Common Factor Analysis Model Visually
Using Summary Data (i.e., Correlation Matrix)
Focuses on the shared or common variance and separates this variance from the unique variance.
This visual depiction shows an exploratory factor analysis (EFA). One main difference be- tween EFA and CFA is that EFA includes “crossloadings”. Each observed variable in the model (in a box) is influenced by all the factors (in circles). However there is usually a main “high” loading and secondary or cross loadings associated with each variable. This property is known as simple structure Simple structure: In your collection of factor loadings, you can identify the factors and you can link the variables with the factors easier You want variables that have high loadings on one factor and low loadings on other factors
Decathlon Example:
One Factor Model
You can use rank ordering to see if the athlete does well in one event, if they'll do good in another event
Decathlon Example in jamovi
Decathlon Example: Number of Factors to Extract?
Eigen values[SEARCH UP LATER]
Based on parallel analysis help you decide the number of factors
Plot of the eigenvalues There's are many factors as there are variables Scree plot - Like a mountain, and the scree is like fallen debris and you're trying to find important factors
Decathlon Example: Varimax vs. Oblique Rotation
Understanding Rotation (different example)
Factor 1 - 100m, long jump, 400m, 110H
Varimax is a type of independent factor rotation Orthogonal rotation - Independent factors Oblique - The angles aren't 90%
EFA vs. CFA
Visual Comparison of EFA and CFA Models
Similarities between Confirmatory Factor Analysis (CFA) and Exploratory Factor Analysis (EFA). Both use the common factor model in such a way that the factors/latent variables are hypothesized as explaining the manifestation or the scores on the observed variables. The observed variables for both can be individual questionnaire items or other quanti- tative variables Both can use the same maximum likelihood estimator (ML) which also provides a measure of model fit. If you were to draw the model for both (see next slide), they would look the same, except for the fact that for the EFA model, each factor would have paths/loadings go- ing to all the observed variables (also know as indicators), whereas in CFA, factors typically have paths/loadings going to a subset of the observed variables, thought as defining that factor.
