The lavaan1 syntax since version 0.6.4 allows for exploratory blocks for latent variables. The EFAST package builds on this functionality to combine exploratory latent variable models (such as EFA) with structural parameters (in EFAST the residual covariance structure). Here is how to create a basic 3-factor EFA model in lavaan using the x1 - x9 variables from the famous Holzinger and Swineford data included in the lavaan package:
data("HolzingerSwineford1939")
efa_model <- "
efa('block1')*F1 =~ x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9
efa('block1')*F2 =~ x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9
efa('block1')*F3 =~ x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9
"The efa('block1') modifier indicates that the factors F1, F2, and F3 comprise an exploratory block (with the name block1) and thus that we do not exactly know the loading structure of the variables x1 - x9 on these three factors.
The EFA model can then be estimated as follows, (releasing the residual variances and implying the EFA constraints):
efa_hs39 <- lavaan(
model = efa_model,
data = HolzingerSwineford1939,
auto.var = TRUE,
auto.efa = TRUE
)In the results, among the parameter estimates we will see the pattern of factor loadings resulting from the oblique geomin rotation:
## lavaan 0.6-5 ended normally after 53 iterations
##
## Estimator ML
## Optimization method NLMINB
## Number of free parameters 36
## Number of equality constraints 3
## Row rank of the constraints matrix 3
##
## Rotation method GEOMIN OBLIQUE
## Geomin epsilon 0.01
## Rotation algorithm (rstarts) GPA (100)
## Standardized metric FALSE
## Row weights None
##
## Number of observations 301
##
## Model Test User Model:
##
## Test statistic 22.897
## Degrees of freedom 12
## P-value (Chi-square) 0.029
##
## Parameter Estimates:
##
## Information Expected
## Information saturated (h1) model Structured
## Standard errors Standard
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|)
## F1 =~ block1
## x1 0.715 0.091 7.876 0.000
## x2 0.588 0.095 6.205 0.000
## x3 0.787 0.086 9.149 0.000
## x4 0.041 0.038 1.061 0.289
## x5 -0.070 0.041 -1.707 0.088
## x6 0.098 0.056 1.759 0.079
## x7 -0.096 0.097 -0.992 0.321
## x8 0.169 0.140 1.203 0.229
## x9 0.414 0.114 3.636 0.000
## F2 =~ block1
## x1 0.209 0.091 2.303 0.021
## x2 0.042 0.053 0.787 0.432
## x3 -0.092 0.072 -1.275 0.202
## x4 0.971 0.061 15.891 0.000
## x5 1.140 0.067 16.988 0.000
## x6 0.877 0.061 14.475 0.000
## x7 0.042 0.051 0.818 0.413
## x8 -0.042 0.045 -0.939 0.348
## x9 0.023 0.039 0.604 0.546
## F3 =~ block1
## x1 0.023 0.045 0.520 0.603
## x2 -0.145 0.085 -1.702 0.089
## x3 0.012 0.050 0.240 0.810
## x4 0.005 0.041 0.132 0.895
## x5 0.011 0.034 0.319 0.750
## x6 -0.013 0.040 -0.330 0.741
## x7 0.774 0.076 10.214 0.000
## x8 0.693 0.094 7.376 0.000
## x9 0.451 0.076 5.909 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .x1 0.696 0.087 8.038 0.000
## .x2 1.035 0.102 10.151 0.000
## .x3 0.692 0.097 7.134 0.000
## .x4 0.377 0.048 7.902 0.000
## .x5 0.403 0.061 6.590 0.000
## .x6 0.365 0.042 8.613 0.000
## .x7 0.594 0.106 5.624 0.000
## .x8 0.479 0.080 5.958 0.000
## .x9 0.551 0.060 9.132 0.000
## F1 1.000
## F2 1.000
## F3 1.000In this summary, we can see that the geomin rotation has produced the structure we expect for this data: the first factor is most associated with the first three variables, the second with x4 - x6, and the last three variables load mostly on the third factor.
This is also the same structure we get when we run an oblique geomin-rotated exploratory factor analysis using the same data in the psych package (except for the ordering of the factors):
efa_psych <- psych::fa(r = HolzingerSwineford1939[, 7:15], nfactors = 3, rotate = "geominQ")
loadings(efa_psych)##
## Loadings:
## MR1 MR3 MR2
## x1 0.193 0.595
## x2 0.507 -0.125
## x3 0.689
## x4 0.844
## x5 0.884
## x6 0.804
## x7 -0.128 0.733
## x8 0.148 0.682
## x9 0.399 0.451
##
## MR1 MR3 MR2
## SS loadings 2.187 1.294 1.223
## Proportion Var 0.243 0.144 0.136
## Cumulative Var 0.243 0.387 0.523The EFAST package implements this exact pattern, and allows for extracting the loadings similar to the psych package:
##
## Loadings:
## F1 F2 F3
## x1 0.603 0.187
## x2 0.506 -0.119
## x3 0.690
## x4 0.838
## x5 0.886
## x6 0.805
## x7 -0.150 0.725
## x8 0.106 0.702
## x9 0.368 0.462
##
## F1 F2 F3
## SS loadings 1.275 2.180 1.248
## Proportion Var 0.142 0.242 0.139
## Cumulative Var 0.142 0.384 0.523Bonus: a shorter way to write the same syntax would be:
efa('block1')*F1 +
efa('block1')*F2 +
efa('block1')*F3 =~ x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9which is how the EFAST package works (see Extending EFAST models).
Yves Rosseel (2012). lavaan: An R Package for Structural Equation Modeling. Journal of Statistical Software, 48(2), 1-36. URL http://www.jstatsoft.org/v48/i02/.↩︎