Exploratory Factor Analysis with R
Annotated Syntax and Output
The following document is supplemental material to Sakaluk and Short (in press). Below
is selected annotated syntax and output for conducting exploratory factor analysis in R using the
psych package (Revelle, 2015) from the supplementary file “R_EFA_syntax.R”.
One Factor Model
One factor EFA model with confidence intervals using the following code :
> singfac <- fa(dat, nfactors = 1, rotate = "none", SMC = TRUE, fm =
"ml", alpha = .05, n.iter=1000)
> print(singfac)
Output for one factor model
Factor Analysis with confidence intervals using method = fa(r = dat,
nfactors = 1, n.iter = 1000, rotate = "none", SMC = TRUE,
fm = "ml", alpha = 0.05)
Factor Analysis using method = ml
Call: fa(r = dat, nfactors = 1, n.iter = 1000, rotate = "none", SMC =
TRUE,
fm = "ml", alpha = 0.05)
Standardized loadings (pattern matrix) based upon correlation matrix
ML1
h2
u2 com
y1 0.57 0.327 0.67
1
y2 0.34 0.114 0.89
1
y3 0.43 0.182 0.82
1
y4 0.41 0.172 0.83
1
y5 0.31 0.098 0.90
1
y6 0.48 0.229 0.77
1
ML1
SS loadings
1.12
Proportion Var 0.19
Mean item complexity = 1
Test of the hypothesis that 1 factor is sufficient.
The degrees of freedom for the null model are 15 and the objective
function was 0.47 with Chi Square of 91.9
The degrees of freedom for the model are 9 and the objective function
was 0.1
The root mean square of the residuals (RMSR) is
0.07
The df corrected root mean square of the residuals is
0.09
The harmonic number of observations is 200 with the empirical chi
square 28.38 with prob < 0.00082
The total number of observations was 200 with MLE Chi Square = 18.6
with prob < 0.029
Tucker Lewis Index of factoring reliability = 0.791
RMSEA index = 0.074 and the 95 % confidence intervals are
BIC = -29.09
Fit based upon off diagonal values = 0.88
Measures of factor score adequacy
ML1
Correlation of scores with factors
0.77
Multiple R square of scores with factors
0.59
Minimum correlation of possible factor scores 0.18
NA 0.128
Coefficients and bootstrapped confidence intervals
low ML1 upper
y1 0.37 0.57 0.77
y2 0.13 0.34 0.54
y3 0.24 0.43 0.61
y4 0.21 0.41 0.62
y5 0.11 0.31 0.51
y6 0.28 0.48 0.67
Factor Analysis using method = ml
Two Factor Model
Two factor EFA model with confidence intervals using the following code:
> twofac <- fa(dat, nfactors = 2, rotate = "promax", SMC = TRUE,
fm = "ml", alpha = .05, n.iter = 1000)
> print(twofac)
Output for two factor model
Factor Analysis with confidence intervals using method = fa(r = dat,
nfactors = 2, n.iter = 1000, rotate = "promax", SMC = TRUE,
fm = "ml", alpha = 0.05)
Factor Analysis using method = ml
Call: fa(r = dat, nfactors = 2, n.iter = 1000, rotate = "promax", SMC
= TRUE,
fm = "ml", alpha = 0.05)
Standardized loadings (pattern matrix) based upon correlation matrix
ML1
ML2
h2
u2 com
y1 0.52 0.15 0.37 0.63 1.2
y2 0.62 -0.20 0.31 0.69 1.2
y3 0.36
y4 -0.02
y5 -0.11
y6 0.20
0.12
0.57
0.52
0.34
0.18
0.32
0.23
0.22
0.82
0.68
0.77
0.78
SS loadings
Proportion Var
Cumulative Var
Proportion Explained
Cumulative Proportion
1.2
1.0
1.1
1.6
ML1
0.83
0.14
0.14
0.51
0.51
ML2
0.79
0.13
0.27
0.49
1.00
With factor correlations of
ML1 ML2
ML1 1.00 0.47
ML2 0.47 1.00
Mean item complexity = 1.2
Test of the hypothesis that 2 factors are sufficient.
The degrees of freedom for the null model are 15 and the objective
function was 0.47 with Chi Square of 91.9
The degrees of freedom for the model are 4 and the objective function
was 0.02
The root mean square of the residuals (RMSR) is 0.02
The df corrected root mean square of the residuals is
0.05
The harmonic number of observations is 200 with the empirical chi
square 3.63 with prob < 0.46
The total number of observations was 200 with MLE Chi Square = 3.12
with prob < 0.54
Tucker Lewis Index of factoring reliability = 1.043
RMSEA index = 0 and the 95 % confidence intervals are NA 0.108
BIC = -18.07
Fit based upon off diagonal values = 0.98
Measures of factor score adequacy
ML1 ML2
Correlation of scores with factors
0.76 0.74
Multiple R square of scores with factors
0.57 0.55
Minimum correlation of possible factor scores 0.14 0.10
Coefficients and bootstrapped confidence intervals
low
ML1 upper
low
ML2 upper
y1 -0.35 0.52 1.57 -0.68 0.15 1.23
y2 0.11 0.62 1.11 -0.69 -0.20 0.52
y3 -0.28 0.36 1.13 -0.49 0.12 0.91
y4 -0.49 -0.02 0.64 0.01 0.57 1.22
y5 -0.69 -0.11 0.65 -0.09 0.52 1.17
y6 -0.36 0.20 0.96 -0.31 0.34 1.11
Interfactor correlations and bootstrapped confidence intervals
lower estimate upper
ML1-ML2 0.16
0.47 0.59
Conduct a Parallel Analysis and Produce a Scree Plot
The psych package also allows for parallel analysis using the fa.parallel( ) function. A
parallel analysis can be conducted using the below code.
> parallel <- fa.parallel(dat, fm = "ml", fa = "fa", n.iter = 50, SMC = TRUE, show.legend =
TRUE)
This syntax will produce a scree plot that can be used to examine the eigenvalues from
the actual dataset versus a simulated or resampled dataset. However, this plot is not quite APA
publication standards. R has a variety of plotting packages, and we provide an example with the
ggplot2 package because this package is incredibly powerful and flexible. The example syntax is
omitted, but below is the final plot.