This example demonstrates a full workflow for estimating factor
complexity in an Exploratory Structural Equation Modeling (ESEM) context
using the facomplex package. We use a two-factor target
rotation on a set of 12 items. This is a real data, about motivations
for research in university teachers
We start by loading the dataset fullclean which should
contain the observed variables for the ESEM model, 12 items, 2
factors.
We define the hypothesized factor structure using a target matrix. This matrix guides the target rotation by specifying which items are expected to load on each factor.
Using lavaan, we define an ESEM model with exploratory
factors f1 and f2. All items load on both
factors through exploratory syntax.
We fit the model using the sem() function from
lavaan, applying a target rotation based on our predefined
matrix.
INV.esem.fit <- sem(INV.esem.model,
data = fullclean,
ordered = FALSE,
estimator = "ulsmv",
rotation = "target",
rotation.args = list(target = INV.target,
geomin.epsilon = 0.01,
rstarts = 30,
algorithm = "gpa",
std.ov = TRUE))We examine the model fit and standardized solution:
summary(INV.esem.fit, standardized = TRUE, fit.measures = TRUE)
#> lavaan 0.6-21 ended normally after 32 iterations
#>
#> Estimator ULS
#> Optimization method NLMINB
#> Number of model parameters 37
#> Row rank of the constraints matrix 2
#>
#> Rotation method PST OBLIQUE
#> Rotation algorithm (rstarts) GPA (30)
#> Standardized metric TRUE
#> Row weights None
#>
#> Used Total
#> Number of observations 588 589
#>
#> Model Test User Model:
#> Standard Scaled
#> Test Statistic 118.607 152.535
#> Degrees of freedom 43 43
#> P-value (Unknown) NA 0.000
#> Scaling correction factor 0.846
#> Shift parameter 12.312
#> simple second-order correction
#>
#> Model Test Baseline Model:
#>
#> Test statistic 2868.367 1173.406
#> Degrees of freedom 66 66
#> P-value NA 0.000
#> Scaling correction factor 2.511
#>
#> User Model versus Baseline Model:
#>
#> Comparative Fit Index (CFI) 0.973 0.901
#> Tucker-Lewis Index (TLI) 0.959 0.848
#>
#> Robust Comparative Fit Index (CFI) 0.967
#> Robust Tucker-Lewis Index (TLI) 0.949
#>
#> Root Mean Square Error of Approximation:
#>
#> RMSEA 0.055 0.066
#> 90 Percent confidence interval - lower 0.043 0.055
#> 90 Percent confidence interval - upper 0.067 0.077
#> P-value H_0: RMSEA <= 0.050 0.240 0.010
#> P-value H_0: RMSEA >= 0.080 0.000 0.021
#>
#> Robust RMSEA 0.061
#> 90 Percent confidence interval - lower 0.050
#> 90 Percent confidence interval - upper 0.071
#> P-value H_0: Robust RMSEA <= 0.050 0.045
#> P-value H_0: Robust RMSEA >= 0.080 0.001
#>
#> Standardized Root Mean Square Residual:
#>
#> SRMR 0.054 0.054
#>
#> Parameter Estimates:
#>
#> Standard errors Robust.sem
#> Information Expected
#> Information saturated (h1) model Unstructured
#>
#> Latent Variables:
#> Estimate Std.Err z-value P(>|z|) Std.lv Std.all
#> f1 =~ efa1
#> INV1 0.648 0.066 9.771 0.000 0.648 0.652
#> INV4 0.640 0.062 10.287 0.000 0.640 0.652
#> INV5 0.534 0.062 8.599 0.000 0.534 0.615
#> INV7 0.656 0.062 10.532 0.000 0.656 0.680
#> INV11 0.522 0.061 8.508 0.000 0.522 0.593
#> INV12 0.385 0.063 6.081 0.000 0.385 0.415
#> INV3 0.033 0.055 0.587 0.557 0.033 0.035
#> INV6 0.183 0.067 2.710 0.007 0.183 0.187
#> INV8 0.087 0.052 1.666 0.096 0.087 0.092
#> INV9 0.091 0.056 1.629 0.103 0.091 0.093
#> INV13 -0.110 0.051 -2.139 0.032 -0.110 -0.106
#> INV14 -0.101 0.064 -1.570 0.117 -0.101 -0.090
#> f2 =~ efa1
#> INV1 -0.037 0.062 -0.592 0.554 -0.037 -0.037
#> INV4 -0.108 0.052 -2.088 0.037 -0.108 -0.110
#> INV5 -0.036 0.058 -0.621 0.535 -0.036 -0.042
#> INV7 0.002 0.052 0.048 0.962 0.002 0.003
#> INV11 0.050 0.055 0.901 0.368 0.050 0.056
#> INV12 0.289 0.053 5.418 0.000 0.289 0.311
#> INV3 0.405 0.056 7.171 0.000 0.405 0.438
#> INV6 0.295 0.063 4.696 0.000 0.295 0.302
#> INV8 0.573 0.062 9.249 0.000 0.573 0.606
#> INV9 0.590 0.061 9.630 0.000 0.590 0.604
#> INV13 0.765 0.056 13.728 0.000 0.765 0.737
#> INV14 0.473 0.072 6.618 0.000 0.473 0.421
#>
#> Covariances:
#> Estimate Std.Err z-value P(>|z|) Std.lv Std.all
#> f1 ~~
#> f2 0.595 0.051 11.763 0.000 0.595 0.595
#>
#> Variances:
#> Estimate Std.Err z-value P(>|z|) Std.lv Std.all
#> .INV1 0.595 0.054 10.964 0.000 0.595 0.602
#> .INV4 0.623 0.056 11.060 0.000 0.623 0.647
#> .INV5 0.491 0.053 9.318 0.000 0.491 0.651
#> .INV7 0.499 0.046 10.747 0.000 0.499 0.535
#> .INV11 0.468 0.042 11.146 0.000 0.468 0.605
#> .INV12 0.497 0.040 12.481 0.000 0.497 0.577
#> .INV3 0.674 0.053 12.759 0.000 0.674 0.789
#> .INV6 0.765 0.058 13.197 0.000 0.765 0.806
#> .INV8 0.498 0.047 10.699 0.000 0.498 0.557
#> .INV9 0.533 0.042 12.846 0.000 0.533 0.559
#> .INV13 0.580 0.058 9.990 0.000 0.580 0.538
#> .INV14 1.088 0.072 15.061 0.000 1.088 0.860
#> f1 1.000 1.000 1.000
#> f2 1.000 1.000 1.000We now use the facomplex package to calculate various
indices of factor complexity.
The items grouped in both lists, within the items_target
argument, are the expected items in their factors. The use of FSI to
interpret its results at the factor level requires this prior knowledge
of the items in their expected factors.
simload(data = lavInspect(INV.esem.fit, what = "std")$lambda,
items_target = list(f1 = c(1,2,3,4,5,6),
f2 = c(7,8,9,10,11,12)))
#> $TSFI
#> [1] 0.955
#>
#> $SFI
#> f1 f2
#> 0.968 0.938
#>
#> $IFS
#> Items IFS
#> 1 INV1 0.997
#> 2 INV4 0.972
#> 3 INV5 0.995
#> 4 INV7 1.000
#> 5 INV11 0.991
#> 6 INV12 0.439
#> 7 INV3 0.994
#> 8 INV6 0.616
#> 9 INV8 0.977
#> 10 INV9 0.976
#> 11 INV13 0.979
#> 12 INV14 0.954Hofmann(data = lavInspect(INV.esem.fit, what = "std")$lambda)
#> CHof CHof_R
#> INV1 1.006 0.994
#> INV4 1.057 0.946
#> INV5 1.009 0.991
#> INV7 1.000 1.000
#> INV11 1.018 0.982
#> INV12 1.853 0.540
#> INV3 1.013 0.987
#> INV6 1.670 0.599
#> INV8 1.046 0.956
#> INV9 1.047 0.955
#> INV13 1.041 0.960
#> INV14 1.091 0.916