--- title: "Definition-Variables-and-Multi-Group-SEM" output: rmarkdown::html_vignette vignette: > %\VignetteIndexEntry{Definition-Variables-and-Multi-Group-SEM} %\VignetteEngine{knitr::rmarkdown} %\VignetteEncoding{UTF-8} --- 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](https://lavaan.ugent.be/tutorial/groups.html). 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**: ```r 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**: ```r 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: ```r 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. ```r # 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: ```r # 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. ```r 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: ```r fit <- lasso(lavaanModel = c(fit_Pasteur, fit_Grant_White), regularized = c("l1_Pasteur"), nLambdas = 20) ``` The coefficients can be extracted as usual: ```r 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: ```r transformation <- " parameters: l1_Pasteur, l1_Grant_White, l1_delta l1_Grant_White = l1_Pasteur + l1_delta; " ``` Next, we pass this transformation to our model: ```r 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: ```r 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: ```r 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: ```r #### 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. ```r 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: ```r 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**: ```r fit <- bfgs(lavaanModel = models) ``` The parameters are given by: ```r 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 ``` ## Bibliography - Huang, P.-H. (2018). A penalized likelihood method for multi-group structural equation modelling. British Journal of Mathematical and Statistical Psychology, 71(3), 499–522. https://doi.org/10.1111/bmsp.12130 - Huang, P.-H. (2020). lslx: Semi-confirmatory structural equation modeling via penalized likelihood. Journal of Statistical Software, 93(7). https://doi.org/10.18637/jss.v093.i07