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Non-Smooth Regularization for Structural Equation Models

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Definition-Variables-and-Multi-Group-SEM

Definition-Variables-and-Multi-Group-SEM
In lessSEM: Non-Smooth Regularization for Structural Equation Models

The basic SEM supported by lessSEM assumes that the data is independently and identically distributed. That is, each subject in the data set comes from the same population. This assumption may be unrealistic, however. Researchers may suspect that subgroups within the data set are more similar to one another than to other subgroups, for example. That is, they differ in their parameter vectors.

If possible groupings within the data set are known beforehand, multi-group models are a convenient way to allow for group-specific parameters. Setting up such models with lavaan is explained here. Unfortunately, lessSEM does not support the same syntax at the moment. "Throwing" a multi-group SEM into lessSEM will just result in errors. Instead, lessSEM follows a slightly different approach: You can pass multiple lavaan models at once that are then combined into a multi-group model.

In the following, we will look at a two-group model to better understand how multi-group models are implemented in lessSEM.

First Step: Setting up a Multi-Group Model

To set up a multi-group model in lessSEM, we first have to fit separate models for each of the groups in lavaan:

library(lavaan)

# For simplicity, we will use a subset of the Holzinger Swineford data set 
# that is also used at https://lavaan.ugent.be/tutorial/groups.html
# to demonstrate multi-group SEM

# To use mutli-group SEM in lessSEM, we have to set up a separate model
# for each of the groups: 
# - Pasteur: Children attending the Pasteur school
# - Grant_White: Children attending the Grant-White school
data(HolzingerSwineford1939)

## Pasteur ##
Pasteur <- subset(HolzingerSwineford1939, school == "Pasteur")

model_Pasteur <- paste0(' 
    visual  =~ l1_Pasteur*x1 + l2_Pasteur*x2 + l3_Pasteur*x3
    x1 ~~ v1*x1
    x2 ~~ v2*x2
    x3 ~~ v3*x3

    visual ~~ lv1*visual
    x1 ~ m1*1
    x2 ~ m2*1
    x3 ~ m3*1')
fit_Pasteur <- sem(model = model_Pasteur, 
                   data = Pasteur, 
                   std.lv = TRUE)

## Grant-White
Grant_White <- subset(HolzingerSwineford1939, school == "Grant-White")

model_Grant_White <- paste0(' 
    visual  =~ l1_Grant_White*x1 + l2_Grant_White*x2 + l3_Grant_White*x3
    x1 ~~ v1*x1
    x2 ~~ v2*x2
    x3 ~~ v3*x3

    visual ~~ lv1*visual
    x1 ~ m1*1
    x2 ~ m2*1
    x3 ~ m3*1')
fit_Grant_White <- sem(model = model_Grant_White, 
                       data = Grant_White, 
                       std.lv = TRUE)

Second Step: Pass the Model to lessSEM

Now that we have our group-specific models, we can pass them to lessSEM:

library(lessSEM)

# We will just estimate the parameters using the BFGS optimizer without any 
# regularization.
# Note that we pass the two models as a vector. lessSEM
# will then set up the multi-group model
fit <- bfgs(lavaanModel = c(fit_Pasteur, fit_Grant_White))

Let's have a look at the parameters:

coef(fit)
#>                                                                                                                           
#>   Tuning         ||--||  Estimates                                                                                        
#>  ------- ------- ||--|| ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ----------
#>   lambda   alpha ||--|| l1_Pasteur l2_Pasteur l3_Pasteur         v1         v2         v3         m1         m2         m3
#>  ======= ======= ||--|| ========== ========== ========== ========== ========== ========== ========== ========== ==========
#>   0.0000  0.0000 ||--||     0.7240     0.5610     0.8824     0.8449     1.0711     0.6108     4.9212     6.0770     2.2281
#>                                              
#>                                              
#>  -------------- -------------- --------------
#>  l1_Grant_White l2_Grant_White l3_Grant_White
#>  ============== ============== ==============
#>          0.7088         0.5536         0.7360

That's curious! There are group-specific parameters, but only for the parameters where we provided group-specific names!

Important: If you set up a multi-group model with lessSEM, lessSEM will assume that all parameters with the same names should also have the same values. This includes parameters that you may have estimated, but for whom the names were provided by lavaan (e.g., variances).

Different Models with Shared Parameter Labels

All parameters that have the same labels in multiple models will be constrained to equality across models! If you are not careful, this can result in very annoying mistakes. To demonstrate this, we will use two very different models that may share some parameter names.

# Model from ?lavaan::sem
model <- ' 
  # latent variable definitions
     ind60 =~ x1 + x2 + x3
     dem60 =~ y1 + a*y2 + b*y3 + c*y4
     dem65 =~ y5 + a*y6 + b*y7 + c*y8

  # regressions
    dem60 ~ ind60
    dem65 ~ ind60 + dem60

  # residual correlations
    y1 ~~ y5
    y2 ~~ y4 + y6
    y3 ~~ y7
    y4 ~~ y8
    y6 ~~ y8
'

fitPolDem <- sem(model, 
                 data = PoliticalDemocracy,
                 meanstructure = TRUE)

# Model from ?lavaan::cfa
HS.model <- ' visual  =~ x1 + x2 + x3
              textual =~ x4 + x5 + x6
              speed   =~ x7 + x8 + x9 '

fitHS <- cfa(HS.model,
             data = HolzingerSwineford1939,
             meanstructure = TRUE)

## lessSEM does not care if the models passed to the function are similar 
# or even use different data sets. Of course, it probably does not make much sense
# to estimate two different models at the same time, but lessSEM won't stop you
# from trying...
fit <- bfgs(lavaanModel = c(fitPolDem, fitHS))

Let's first compare the fit of the separate models to that of the multi-group model. If lessSEM were to estimate the models truly separately we would expect the fit to be the same:

# fit for separate models
-2*logLik(fitPolDem) + (-2)*logLik(fitHS)
#> 'log Lik.' 10573.13 (df=39)

# fit for multi-group model:
fit@fits$m2LL
#> [1] 10813.96

Obviously, these two fits are not the same. What may have happened? Looking at the parameter estimates of the multi-group model shows that the two models, fitPolDem and fitHS did share some parameter labels! For instance, the intercepts of x1 is called x1~1 in both models. Therefore, lessSEM assumed that we wanted these parameters to have exactly the same value in both models.

coef(fit)
#>                                                                                                                              
#>   Tuning         ||--||  Estimates                                                                                           
#>  ------- ------- ||--|| ---------- ---------- ---------- ---------- ---------- ----------- ----------- ----------- ----------
#>   lambda   alpha ||--||  ind60=~x2  ind60=~x3          a          b          c dem60~ind60 dem65~ind60 dem65~dem60     y1~~y5
#>  ======= ======= ||--|| ========== ========== ========== ========== ========== =========== =========== =========== ==========
#>   0.0000  0.0000 ||--||     1.8164     1.5582     1.1879     1.1706     1.2511      1.4062      0.4696      0.8755     0.5389
#>                                                                                                                          
#>                                                                                                                          
#>  ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ----------
#>      y2~~y4     y2~~y6     y3~~y7     y4~~y8     y6~~y8     x1~~x1     x2~~x2     x3~~x3     y1~~y1     y2~~y2     y3~~y3
#>  ========== ========== ========== ========== ========== ========== ========== ========== ========== ========== ==========
#>      1.4270     2.2120     0.7425     0.3718     1.3734     0.0000     1.3895     1.2086     1.8544     7.5981     4.9592
#>                                                                                                                                
#>                                                                                                                                
#>  ---------- ---------- ---------- ---------- ---------- ------------ ------------ ------------ ---------- ---------- ----------
#>      y4~~y4     y5~~y5     y6~~y6     y7~~y7     y8~~y8 ind60~~ind60 dem60~~dem60 dem65~~dem65       x1~1       x2~1       x3~1
#>  ========== ========== ========== ========== ========== ============ ============ ============ ========== ========== ==========
#>      3.2006     2.2664     4.9911     3.6046     3.3135       0.5305       3.8021       0.2003     5.0409     5.8482     2.5445
#>                                                                                                                           
#>                                                                                                                           
#>  ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- -----------
#>        y1~1       y2~1       y3~1       y4~1       y5~1       y6~1       y7~1       y8~1 visual=~x2 visual=~x3 textual=~x5
#>  ========== ========== ========== ========== ========== ========== ========== ========== ========== ========== ===========
#>      5.4463     4.2330     6.5420     4.4295     5.1139     2.9502     6.1703     4.0150     0.2791     0.4461      1.1126
#>                                                                                                                    
#>                                                                                                                    
#>  ----------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- --------------
#>  textual=~x6  speed=~x8  speed=~x9     x4~~x4     x5~~x5     x6~~x6     x7~~x7     x8~~x8     x9~~x9 visual~~visual
#>  =========== ========== ========== ========== ========== ========== ========== ========== ========== ==============
#>       0.9221     1.1746     1.0056     0.3681     0.4434     0.3610     0.7750     0.4590     0.6025         1.3695
#>                                                                                                                        
#>                                                                                                                        
#>  ---------------- ------------ --------------- ------------- -------------- ---------- ---------- ---------- ----------
#>  textual~~textual speed~~speed visual~~textual visual~~speed textual~~speed       x4~1       x5~1       x6~1       x7~1
#>  ================ ============ =============== ============= ============== ========== ========== ========== ==========
#>            0.9839       0.4084          0.4666        0.2615         0.1746     3.0967     4.3804     2.2186     4.2060
#>                       
#>                       
#>  ---------- ----------
#>        x8~1       x9~1
#>  ========== ==========
#>      5.5507     5.3943

Regularized Multi-Group Models

All multi-group models can be regularized similar to the standard SEM: Instead of using the bfgs-function, we use (for instance), the lasso-function:

fit <- lasso(lavaanModel = c(fit_Pasteur, fit_Grant_White),
             regularized = c("l1_Pasteur"), 
             nLambdas = 20)

The coefficients can be extracted as usual:

coef(fit, criterion = "AIC")
#>                                                                                                                           
#>   Tuning         ||--||  Estimates                                                                                        
#>  ------- ------- ||--|| ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ----------
#>   lambda   alpha ||--|| l1_Pasteur l2_Pasteur l3_Pasteur         v1         v2         v3         m1         m2         m3
#>  ======= ======= ||--|| ========== ========== ========== ========== ========== ========== ========== ========== ==========
#>   0.0000  1.0000 ||--||     0.7239     0.5609     0.8826     0.8450     1.0712     0.6107     4.9212     6.0769     2.2280
#>                                              
#>                                              
#>  -------------- -------------- --------------
#>  l1_Grant_White l2_Grant_White l3_Grant_White
#>  ============== ============== ==============
#>          0.7087         0.5536         0.7361

Regularizing Differences Between Parameters using lessSEM

Where regularized multi-group models shine is when automatically testing for group-differences. This was proposed by Huang (2018) and provides a convenient way to decide which of the parameters should be group-specific. To this end, differences between parameters must be regularized. Say, we are interested in the loading l1 and wonder if we do indeed need separate loadings for students attending the Pasteur school (l1_Pasteur) and the Grant-White school (l1_Grant_White). Using the Pasteur school as baseline group (see Huang, 2018, for more details), we can define l1_Grant_White = l1_Pasteur + l1_delta, where l1_delta is the difference between the two schools. If l1_delta is zero, then both schools have the same loading (i.e., we have measurement invariance). Within lessSEM, we can regularize such differences using transformations (see vignette(topic = "Parameter-transformations", package = "lessSEM") for more details). Therefore, the first step is to define the transformation:

transformation <- "
parameters: l1_Pasteur, l1_Grant_White, l1_delta
l1_Grant_White = l1_Pasteur + l1_delta;
"

Next, we pass this transformation to our model:

fit <- lasso(lavaanModel = c(fit_Pasteur, fit_Grant_White),
             regularized = c("l1_delta"), # we want to regularize the difference!
             nLambdas = 20,
             modifyModel = modifyModel(transformations = transformation))

Now, let's look at the parameter estimates:

coef(fit, criterion = "AIC")@estimates[,c("l1_Pasteur", "l1_delta")]
#> l1_Pasteur   l1_delta 
#>   0.716718   0.000000

As the l1_delta parameter has been set to zero, we can assume measurement invariance. Note that you won't find l1_Grant_White in the parameters of the model. This is because l1_Grant_White is a deterministic function of the actual parameters l1_Pasteur and l1_delta. If you want to find the value for l1_Grant_White, have a look at:

fit@transformations
#>          lambda alpha l1_Grant_White
#> 1  0.0056517504     1      0.7167576
#> 2  0.0053542898     1      0.7167180
#> 3  0.0050568293     1      0.7166009
#> 4  0.0047593687     1      0.7161368
#> 5  0.0044619082     1      0.7155761
#> 6  0.0041644476     1      0.7151688
#> 7  0.0038669871     1      0.7147807
#> 8  0.0035695265     1      0.7141878
#> 9  0.0032720660     1      0.7137695
#> 10 0.0029746054     1      0.7132645
#> 11 0.0026771449     1      0.7128818
#> 12 0.0023796844     1      0.7125411
#> 13 0.0020822238     1      0.7120493
#> 14 0.0017847633     1      0.7115889
#> 15 0.0014873027     1      0.7110626
#> 16 0.0011898422     1      0.7107554
#> 17 0.0008923816     1      0.7102373
#> 18 0.0005949211     1      0.7097089
#> 19 0.0002974605     1      0.7094042
#> 20 0.0000000000     1      0.7088852

Note that lslx (Huang, 2020) supports different penalties for the delta parameter (l1_delta) and the baseline parameter (l1_Pasteur). This is currently not supported by lessSEM.

Cross-Validation

Automatic cross-validation for multi-group models with, for instance, cvLasso is not yet implemented. This is because it can be difficult to decide how to split up the data set in each submodel. If you want to use cross-validation, you will (unfortunately) have to set up the procedure manually.

Definition Variables

Models with definition variables are basically the same as multi-group models, with the sole exception that the group-specific parameters are not estimated but fixed to specific values.

If your main interest is in setting up a multi-group SEM with lessSEM and you don't care about the details, the lessTemplates package (https://github.com/jhorzek/lessTemplates) provides means to easily set up such models (see SEMWithDefinitionVariables function in lessTemplates).

In the following, we will look in detail at how definition variables can be used in lessSEM

The details ...

Unfortunately, lavaan does not allow us to set up models for $N=1$, however. This is required for many definition variable applications, such as latent growth curve models with subject-specific measurement occasions. In the following, we will use a workaround.

Let's first simulate some data:

#### Population parameters ####
intercept_mu <- 0
intercept_sigma <- 1
slope_mu <- .3
slope_sigma <- 1

#### data set ####
N <- 50
intercepts <- rnorm(n = N, 
                    mean = intercept_mu, 
                    sd = intercept_sigma)
slopes <- rnorm(n = N, 
                mean = slope_mu, 
                sd = slope_sigma)
times <- matrix(seq(0,5,1),
                nrow = N,
                ncol = 6,
                byrow = TRUE) +
  cbind(0,matrix(round(runif(n = N*5, min = -.2,max = .2),2),
                 nrow = N,
                 ncol = 5,
                 byrow = TRUE)) # we add some jitter to make the times person-specific

lgcData <- matrix(NA, nrow = N, ncol = ncol(times), dimnames = list(NULL, paste0("x", 0:5)))

for(i in 1:N){
  lgcData[i,] <- intercepts[i] + times[i,]* slopes[i] + rnorm(ncol(lgcData),0,.3)
}
lgcData <- as.data.frame(lgcData)

head(lgcData)
#>            x0         x1         x2        x3        x4        x5
#> 1  0.26801876 -0.5353028 -1.0262295 -2.533564 -2.890246 -4.655061
#> 2  0.26260181 -0.4802635 -2.0790754 -2.686983 -3.677022 -4.540501
#> 3 -0.71054447 -0.5477048 -0.5747618 -1.231821 -1.229388 -1.418401
#> 4  0.14564354  1.0690979  1.8919916  2.842025  2.953471  4.654879
#> 5  0.09501816  1.6350899  2.6331440  4.729328  5.423381  6.799296
#> 6  2.51013193  3.0917904  4.1193617  5.749542  6.222599  7.716919

head(times)
#>      [,1] [,2] [,3] [,4] [,5] [,6]
#> [1,]    0 0.94 2.13 3.01 3.91 5.12
#> [2,]    0 0.84 2.13 2.91 4.10 5.19
#> [3,]    0 0.83 2.14 3.12 3.95 4.93
#> [4,]    0 0.88 2.03 3.16 4.01 5.03
#> [5,]    0 1.07 2.01 3.00 3.81 4.82
#> [6,]    0 1.17 2.11 2.88 4.06 5.06

Note that the times are random and subject-specific. We need a separate model for each subject. Because lavaan won't let us set up such models, we will instead set up models using the entire data set and replace the data post-hoc.

models <- c()
for(i in 1:N){
model_i <- paste0(
  "
int =~ 1*x0 + 1*x1 + 1*x2 + 1*x3 + 1*x4 + 1*x5
slope =~ ",times[i,1],"*x0 + 
         ",times[i,2],"*x1 + 
         ",times[i,3],"*x2 +  
         ",times[i,4],"*x3 + 
         ",times[i,5],"*x4 + 
         ",times[i,6],"*x5

         int ~ intMean*1
         slope ~ slopeMean*1

         int ~~ intVar*int + 0*slope
         slope ~~ slopeVar*slope

x0 ~~ v*x0
x1 ~~ v*x1
x2 ~~ v*x2
x3 ~~ v*x3
x4 ~~ v*x4
x5 ~~ v*x5

x0 ~ 0*1
x1 ~ 0*1
x2 ~ 0*1
x3 ~ 0*1
x4 ~ 0*1
x5 ~ 0*1
"  
)

fit_i <- sem(model = model_i, 
             data = lgcData, 
             do.fit = FALSE)
internalData <- lavInspect(fit_i, "data")
# replace the data set
fit_i@Data@X[[1]] <- as.matrix(lgcData[i,colnames(internalData),drop = FALSE])

models <- c(models, 
            fit_i)
}

Exemplarily, it makes sense to look at one of the models:

cat(model_i)
#> 
#> int =~ 1*x0 + 1*x1 + 1*x2 + 1*x3 + 1*x4 + 1*x5
#> slope =~ 0*x0 + 
#>          1.13*x1 + 
#>          1.81*x2 +  
#>          3.06*x3 + 
#>          4.12*x4 + 
#>          4.98*x5
#>          
#>          int ~ intMean*1
#>          slope ~ slopeMean*1
#>          
#>          int ~~ intVar*int + 0*slope
#>          slope ~~ slopeVar*slope
#> 
#> x0 ~~ v*x0
#> x1 ~~ v*x1
#> x2 ~~ v*x2
#> x3 ~~ v*x3
#> x4 ~~ v*x4
#> x5 ~~ v*x5
#> 
#> x0 ~ 0*1
#> x1 ~ 0*1
#> x2 ~ 0*1
#> x3 ~ 0*1
#> x4 ~ 0*1
#> x5 ~ 0*1

Note that the loadings of the slope are fixed to the time points at which person i provided data.

Now we can pass the models to lessSEM:

fit <- bfgs(lavaanModel = models)

The parameters are given by:

coef(fit)
#>                                                                               
#>   Tuning         ||--||  Estimates                                            
#>  ------- ------- ||--|| ---------- ---------- ---------- ---------- ----------
#>   lambda   alpha ||--||    intMean  slopeMean     intVar   slopeVar          v
#>  ======= ======= ||--|| ========== ========== ========== ========== ==========
#>   0.0000  0.0000 ||--||    -0.0152     0.2921     0.7703     0.9411     0.0851

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