In this example, we will show how to use lslx to conduct multi-group factor analysis. The example uses data HolzingerSwineford1939 in the package lavaan. Hence, lavaan must be installed.
In the following specification, x1 - x9 is assumed to be measurements of 3 latent factors: visual, textual, and speed.
The operator :=> means that the LHS latent factors is defined by the RHS observed variables. In this model, visual is mainly measured by x1 - x3, textual is mainly measured by x4 - x6, and speed is mainly measured by x7 - x9. Loadings of x1, x4, and x7 are fixed at 1 for scale setting. The above specification is valid for both groups. Details of model syntax can be found in the section of Model Syntax via ?lslx.
lslx is written as an R6 class. Everytime we conduct analysis with lslx, an lslx object must be initialized. The following code initializes an lslx object named lslx_mgfa.
library(lslx)
lslx_mgfa <- lslx$new(model = model_mgfa,
data = lavaan::HolzingerSwineford1939,
group_variable = "school",
reference_group = "Pasteur")An 'lslx' R6 class is initialized via 'data' argument.
Response Variables: x1 x2 x3 x4 x5 x6 x7 x8 x9
Latent Factors: visual textual speed
Groups: Grant-White Pasteur
Reference Group: Pasteur
NOTE: Because Pasteur is set as reference, coefficients in other groups actually represent increments from the reference.Here, lslx is the object generator for lslx object and new is the build-in method of lslx to generate a new lslx object. The initialization of lslx requires users to specify a model for model specification (argument model) and a data set to be fitted (argument sample_data). The data set must contain all the observed variables specified in the given model. Because in this example a multi-group analysis is considered, variable for group labeling (argument group_variable) must be specified. In lslx, two types of parameterization can be used in multi-group analysis. The first type is the same with the traditional multi-group SEM, which treats model parameters in each group separately. The second type sets one group as reference and treats model parameters in other groups as increments with respect to the reference. Under the second type of parameterization, the group heterogeneity can be efficiently explored if we treat the increments as penalized parameters. In this example, Pasteur is set as reference. Hence, the parameters in Grant-White now reflect differences from the reference.
After an lslx object is initialized, the heterogeneity of a multi-group model can be quickly respecified by $free_heterogeneity(), $fix_heterogeneity(), and $penalize_heterogeneity() methods. The following code sets x2<-visual, x3<-visual, x5<-textual, x6<-textual, x8<-speed, x9<-speed, and x2<-1, x3<-1, x5<-1, x6<-1, x8<-1, x9<-1 in Grant-White as penalized parameters. Note that parameters in Grant-White now reflect differences since Pasteur is set as reference.
The relation x1<-1 under Grant-White is set as PENALIZED with starting value = 0.
The relation x2<-1 under Grant-White is set as PENALIZED with starting value = 0.
The relation x3<-1 under Grant-White is set as PENALIZED with starting value = 0.
The relation x4<-1 under Grant-White is set as PENALIZED with starting value = 0.
The relation x5<-1 under Grant-White is set as PENALIZED with starting value = 0.
The relation x6<-1 under Grant-White is set as PENALIZED with starting value = 0.
The relation x7<-1 under Grant-White is set as PENALIZED with starting value = 0.
The relation x8<-1 under Grant-White is set as PENALIZED with starting value = 0.
The relation x9<-1 under Grant-White is set as PENALIZED with starting value = 0.
The relation x2<-visual under Grant-White is set as PENALIZED with starting value = 0.
The relation x3<-visual under Grant-White is set as PENALIZED with starting value = 0.
The relation x5<-textual under Grant-White is set as PENALIZED with starting value = 0.
The relation x6<-textual under Grant-White is set as PENALIZED with starting value = 0.
The relation x8<-speed under Grant-White is set as PENALIZED with starting value = 0.
The relation x9<-speed under Grant-White is set as PENALIZED with starting value = 0.
NOTE: Because Pasteur is set as reference, a relation under other group actually represents an increment.
NOTE: Please check whether the starting value for the increment represents a difference. Since the homogeneity of latent factor means may not be a reasonable assumption when examining measurement invariance, the following code relaxes this assumption
The relation visual<-1 under Grant-White is set as FREE with starting value = 0.
The relation textual<-1 under Grant-White is set as FREE with starting value = 0.
The relation speed<-1 under Grant-White is set as FREE with starting value = 0.
NOTE: Because Pasteur is set as reference, a relation under other group actually represents an increment.
NOTE: Please check whether the starting value for the increment represents a difference. To see more methods to modify a specified model, please check the section of Set-Related Method via ?lslx.
After an lslx object is initialized, method $fit_mcp() can be used to fit the specified model into the given data with MCP.
CONGRATS: Algorithm converges under EVERY specified penalty level.
Specified Tolerance for Convergence: 0.001
Specified Maximal Number of Iterations: 100 All the fitting result will be stored in the fitting field of lslx_mgfa.
Unlike traditional SEM analysis, lslx fits the model into data under all the penalty levels considered. To summarize the fitting result, a selector to determine an optimal penalty level must be specified. Available selectors can be found in the section of Penalty Level Selection via ?lslx. The following code summarize the fitting result under the penalty level selected by Haughton’s Bayesian information criterion (HBIC).
General Information
number of observations 301
number of complete observations 301
number of missing patterns none
number of groups 2
number of responses 9
number of factors 3
number of free coefficients 48
number of penalized coefficients 15
Numerical Conditions
selected lambda 0.115
selected delta 3.063
selected step none
objective value 0.470
objective gradient absolute maximum 0.001
objective Hessian convexity 0.188
number of iterations 9.000
loss value 0.430
number of non-zero coefficients 50.000
degrees of freedom 58.000
robust degrees of freedom 60.650
scaling factor 1.046
Fit Indices
root mean square error of approximation (rmsea) 0.090
comparative fit index (cfi) 0.919
non-normed fit index (nnfi) 0.900
standardized root mean of residual (srmr) 0.085
Likelihood Ratio Test
statistic df p-value
unadjusted 129.424 58.000 0.000
mean-adjusted 123.768 58.000 0.000
Root Mean Square Error of Approximation Test
estimate lower upper
unadjusted 0.090 0.065 0.115
mean-adjusted 0.089 0.063 0.114
Coefficient Test (Group = "Pasteur", Std.Error = "sandwich")
Factor Loading (reference component)
type estimate std.error z-value P(>|z|) lower upper
x1<-visual fixed 1.000 - - - - -
x2<-visual free 0.604 0.143 4.213 0.000 0.323 0.885
x3<-visual free 0.788 0.157 5.031 0.000 0.481 1.095
x4<-textual fixed 1.000 - - - - -
x5<-textual free 1.120 0.067 16.597 0.000 0.988 1.252
x6<-textual free 0.932 0.064 14.678 0.000 0.808 1.057
x7<-speed fixed 1.000 - - - - -
x8<-speed free 1.199 0.134 8.952 0.000 0.936 1.461
x9<-speed free 1.038 0.206 5.037 0.000 0.634 1.442
Covariance (reference component)
type estimate std.error z-value P(>|z|) lower upper
textual<->visual free 0.406 0.135 3.019 0.003 0.143 0.670
speed<->visual free 0.169 0.066 2.563 0.010 0.040 0.298
speed<->textual free 0.173 0.060 2.901 0.004 0.056 0.290
Variance (reference component)
type estimate std.error z-value P(>|z|) lower upper
visual<->visual free 0.801 0.229 3.492 0.000 0.352 1.251
textual<->textual free 0.880 0.135 6.532 0.000 0.616 1.144
speed<->speed free 0.306 0.083 3.699 0.000 0.144 0.467
x1<->x1 free 0.556 0.181 3.080 0.002 0.202 0.910
x2<->x2 free 1.269 0.172 7.370 0.000 0.931 1.606
x3<->x3 free 0.881 0.131 6.747 0.000 0.625 1.136
x4<->x4 free 0.446 0.070 6.328 0.000 0.308 0.584
x5<->x5 free 0.502 0.083 6.019 0.000 0.339 0.666
x6<->x6 free 0.263 0.058 4.518 0.000 0.149 0.377
x7<->x7 free 0.849 0.113 7.520 0.000 0.628 1.070
x8<->x8 free 0.516 0.094 5.477 0.000 0.331 0.701
x9<->x9 free 0.657 0.118 5.587 0.000 0.426 0.887
Intercept (reference component)
type estimate std.error z-value P(>|z|) lower upper
x1<-1 free 4.914 0.095 51.570 0.000 4.728 5.101
x2<-1 free 6.088 0.080 75.898 0.000 5.930 6.245
x3<-1 free 2.487 0.093 26.781 0.000 2.305 2.669
x4<-1 free 2.778 0.087 31.915 0.000 2.608 2.949
x5<-1 free 4.035 0.103 39.172 0.000 3.833 4.237
x6<-1 free 1.926 0.075 25.776 0.000 1.779 2.072
x7<-1 free 4.432 0.087 51.183 0.000 4.263 4.602
x8<-1 free 5.569 0.074 75.563 0.000 5.425 5.714
x9<-1 free 5.409 0.070 77.118 0.000 5.272 5.547
Coefficient Test (Group = "Grant-White", Std.Error = "sandwich")
Factor Loading (increment component)
type estimate std.error z-value P(>|z|) lower upper
x1<-visual fixed 0.000 - - - - -
x2<-visual pen 0.000 - - - - -
x3<-visual pen 0.000 - - - - -
x4<-textual fixed 0.000 - - - - -
x5<-textual pen 0.000 - - - - -
x6<-textual pen 0.000 - - - - -
x7<-speed fixed 0.000 - - - - -
x8<-speed pen 0.000 - - - - -
x9<-speed pen 0.000 - - - - -
Covariance (increment component)
type estimate std.error z-value P(>|z|) lower upper
textual<->visual free 0.020 0.144 0.139 0.890 -0.263 0.303
speed<->visual free 0.144 0.106 1.364 0.173 -0.063 0.351
speed<->textual free 0.050 0.109 0.461 0.645 -0.163 0.263
Variance (increment component)
type estimate std.error z-value P(>|z|) lower upper
visual<->visual free -0.084 0.198 -0.424 0.672 -0.473 0.305
textual<->textual free -0.010 0.167 -0.060 0.952 -0.337 0.317
speed<->speed free 0.170 0.094 1.802 0.072 -0.015 0.355
x1<->x1 free 0.094 0.178 0.527 0.598 -0.254 0.442
x2<->x2 free -0.329 0.221 -1.491 0.136 -0.761 0.103
x3<->x3 free -0.277 0.138 -2.001 0.045 -0.548 -0.006
x4<->x4 free -0.103 0.094 -1.100 0.271 -0.286 0.080
x5<->x5 free -0.126 0.103 -1.220 0.222 -0.327 0.076
x6<->x6 free 0.174 0.093 1.874 0.061 -0.008 0.356
x7<->x7 free -0.250 0.132 -1.887 0.059 -0.510 0.010
x8<->x8 free -0.109 0.141 -0.772 0.440 -0.386 0.168
x9<->x9 free -0.125 0.142 -0.883 0.377 -0.404 0.153
Intercept (increment component)
type estimate std.error z-value P(>|z|) lower upper
visual<-1 free 0.050 0.132 0.377 0.706 -0.209 0.309
textual<-1 free 0.576 0.120 4.788 0.000 0.340 0.812
speed<-1 free -0.072 0.089 -0.808 0.419 -0.246 0.102
x1<-1 pen 0.000 - - - - -
x2<-1 pen 0.000 - - - - -
x3<-1 pen -0.531 0.117 -4.521 0.000 -0.761 -0.301
x4<-1 pen 0.000 - - - - -
x5<-1 pen 0.000 - - - - -
x6<-1 pen 0.000 - - - - -
x7<-1 pen -0.440 0.108 -4.063 0.000 -0.652 -0.228
x8<-1 pen 0.000 - - - - -
x9<-1 pen 0.000 - - - - - In this example, we can see that all of the loadings are invariant across the two groups. However, the intercepts of x3 and x7 seem to be not invariant. The $summarize() method also shows the result of significance tests for the coefficients. In lslx, the default standard errors are calculated based on sandwich formula whenever raw data is available. It is generally valid even when the model is misspecified and the data is not normal. However, it may not be valid after selecting an optimal penalty level.