#
library(tidyverse) # This gives us access to dplyr.
library(psych) # This allows us to perform factor analysis.
library(DescTools) # This will let us compute Cronbach's alpha.
library(magrittr) # This allows us to use the double pipe.
TidyData <- read.csv('TidyData.csv', header = TRUE,
stringsAsFactors = FALSE) # Here's the data we'll use.PSY460: Advanced Quantitative Methods
Week #8: Factor Analysis
Today, we’ll discuss the conceptual underpinnings of exploratory factor analysis, and we’ll run a factor analysis on a new dataset. We will also check whether our factors have sufficient internal consistency. Finally, we’ll discuss creating new variables and other critical steps in the data cleaning process.
“Quiz”
- Write two sentences describing what you learned about factor analysis from the assigned reading and video.
- Write down two questions you have about factor analysis.
- Reflect on your performance in the course since our one-on-one meeting. What have you done well, and what you could be working to improve?
What is factor analysis?
- A “factor” is an unobserved (or “latent”) variable (e.g., IQ) that is indicated by a set of observed measures (e.g., scores on Woodcock-Johnson tests).
- Factors are identified by a common, underlying source of variance across multiple measures, thus ignoring variance from idiosyncratic aspects of each measure (including measurement error).
- Typically, exploratory factor analysis (EFA) is conducted prior to confirmatory factor analysis (CFA). For your purposes, you will only use EFA.
Factor extraction
- There are different ways of “extracting” factor solutions from the data.
- Maximum likelihood (ML) extraction is a common way of estimating how well the factor solution from the EFA is able to account for the actual relationships among the indicator variables.
Factor rotation
- There are also different mathematical transformations that can be applied to the factor loadings to “rotate” the solution in multidimensional space for improving the interpretability of results.
- Oblique rotations, which allow for correlations between factors, are more realistic.
- Promax rotation is a common oblique rotation method.
- Oblique rotations, which allow for correlations between factors, are more realistic.
Factor rotation
Factor selection
- EFA does not always produce a clear-cut indication of how many latent factors exist for a given set of variables.
- Various techniques, such as parallel analysis, can provide a rough estimation of how many factors exist in an optimal solution.
- However, subjective considerations of interpretability can be used to override more objective considerations.
Internal consistency
- In general, the items in each factor should have a substantial amount of shared variance. This indicates a form of reliability called “internal consistency”.
- This form of reliability is typically indexed by a measure called Cronbach’s alpha.
- As a general rule of thumb, the alpha coefficient should be above .70.
- This form of reliability is typically indexed by a measure called Cronbach’s alpha.
Questions posed to participants about targets’ moral character
- Would you consider [name] to have strong moral integrity?
- Do you think that [name] is trustworthy?
- How interested would you be in pursuing a friendship or business partnership with [name]?
- How warm do you feel toward [name]?
- Do you expect [name] to act in [respectful, humane, etc.] ways in the future?
Packages needed for today’s analyses
Determining the number of factors
Determining the number of factors
Parallel analysis suggests that the number of factors = 2 and the number of components = NA Performing a factor analysis: Two factors
Performing a factor analysis: Two factors
Factor Analysis using method = ml
Call: fa(r = FA_variables, nfactors = 2, rotate = "promax", residuals = TRUE,
fm = "ml")
Standardized loadings (pattern matrix) based upon correlation matrix
item ML2 ML1 h2 u2 com
IntegrityChange 1 0.83 0.04 0.76 0.24 1.0
TrustworthyChange 2 0.62 0.26 0.73 0.27 1.4
FutureBehChange 5 0.47 0.20 0.42 0.58 1.3
PartnerChange 3 0.06 0.87 0.84 0.16 1.0
WarmChange 4 0.28 0.64 0.78 0.22 1.4
ML2 ML1
SS loadings 1.83 1.71
Proportion Var 0.37 0.34
Cumulative Var 0.37 0.71
Proportion Explained 0.52 0.48
Cumulative Proportion 0.52 1.00
With factor correlations of
ML2 ML1
ML2 1.00 0.84
ML1 0.84 1.00
Mean item complexity = 1.2
Test of the hypothesis that 2 factors are sufficient.
df null model = 10 with the objective function = 3.42 with Chi Square = 6974.54
df of the model are 1 and the objective function was 0
The root mean square of the residuals (RMSR) is 0
The df corrected root mean square of the residuals is 0
The harmonic n.obs is 2044 with the empirical chi square 0.01 with prob < 0.92
The total n.obs was 2044 with Likelihood Chi Square = 0.05 with prob < 0.82
Tucker Lewis Index of factoring reliability = 1.001
RMSEA index = 0 and the 90 % confidence intervals are 0 0.036
BIC = -7.57
Fit based upon off diagonal values = 1
Measures of factor score adequacy
ML2 ML1
Correlation of (regression) scores with factors 0.94 0.95
Multiple R square of scores with factors 0.88 0.90
Minimum correlation of possible factor scores 0.76 0.81Performing a factor analysis: One factor
Performing a factor analysis: One factor
Factor Analysis using method = ml
Call: fa(r = FA_variables, nfactors = 1, rotate = "promax", residuals = TRUE,
fm = "ml")
Standardized loadings (pattern matrix) based upon correlation matrix
V ML1 h2 u2 com
WarmChange 4 0.89 0.79 0.21 1
PartnerChange 3 0.88 0.77 0.23 1
TrustworthyChange 2 0.85 0.72 0.28 1
IntegrityChange 1 0.82 0.68 0.32 1
FutureBehChange 5 0.65 0.42 0.58 1
ML1
SS loadings 3.37
Proportion Var 0.67
Mean item complexity = 1
Test of the hypothesis that 1 factor is sufficient.
df null model = 10 with the objective function = 3.42 with Chi Square = 6974.54
df of the model are 5 and the objective function was 0.05
The root mean square of the residuals (RMSR) is 0.02
The df corrected root mean square of the residuals is 0.03
The harmonic n.obs is 2044 with the empirical chi square 19.04 with prob < 0.0019
The total n.obs was 2044 with Likelihood Chi Square = 105.22 with prob < 4.2e-21
Tucker Lewis Index of factoring reliability = 0.971
RMSEA index = 0.099 and the 90 % confidence intervals are 0.083 0.116
BIC = 67.11
Fit based upon off diagonal values = 1
Measures of factor score adequacy
ML1
Correlation of (regression) scores with factors 0.96
Multiple R square of scores with factors 0.93
Minimum correlation of possible factor scores 0.85Identifying potential factors
Internal consistency
Creating new variables
Creating other variables
- Every dataset must be cleaned in unique ways. However, one common feature is that datasets are almost never set up how you would like them to look for analysis.
- For example, you may have data spread across different similar but mutually exclusive variables (e.g., different columns for single individuals and for families).
- In cases this like, you can use the function “coalesce” from dplyr to combine your data across variables, as “coalesce” will create a new variable with the first non-missing value across an array of variables.
- For example, you may have data spread across different similar but mutually exclusive variables (e.g., different columns for single individuals and for families).
Creating other variables
- In many cases, you will want to create new variables that are conditional upon values of other variables. One option for doing this is to specify a value for each condition, like this:
Creating other variables
- A more streamlined option for creating conditional variables is to make use of functions such as “if_else” or “case_when” from dplyr, like this: