Author: Andreas M. Brandmaier
Date: 2025-11-26

# Constraints in semtree

#### Andreas M. Brandmaier

#### 2025-11-26

# Constraints

## Load the package

`semtree` allows different constraints on the split evaluation (`global.invariance`, `local.invariance`, and `focus.parameters`). These can be set in the following object and then passed to the `semtree` command:

    library(semtree)

    cnst <- semtree.constraints(local.invariance=NULL,
    global.invariance=NULL,
    focus.parameters=NULL)

    semtree(model.x, data=df, constraints=cnst)

## Simulation

To illustrate the invariance constraints, let’s set up a latent factor model with four indicators and simulate some data with a known predictor structure. Let’s create a dataset as follows:

- a latent construct, `lat`, measured by four indicators
- two predictors `p1` and `p2`
- only one predictor, `p1`, has an mean effect on the latent construct
- the other predictor, `p2`, has an effect on only one indicator (different loadings across groups; DIF)

    N <- 2000
    p1 <- sample(size = N,
                 x = c(0, 1),
                 replace = TRUE)
    p2 <- sample(size = N,
                 x = c(0, 1),
                 replace = TRUE)
    lat <- rnorm(N, mean = 0 + p1)
    loadings <- c(.5, .8, .7, .9)
    observed <- lat %*% t(loadings) + rnorm(N * length(loadings), sd = .1)
    observed[, 3] <- observed[, 3] + p2 * 0.5 * lat
    cfa.sim <- data.frame(observed, p1 = factor(p1), p2 = factor(p2))
    names(cfa.sim)[1:4] <- paste0("x", 1:4)

Here is the OpenMx model specification

    require("OpenMx");
    manifests<-c("x1","x2","x3","x4")
    latents<-c("F")
    model.cfa <- mxModel("CFA", type="RAM", manifestVars = manifests,
                         latentVars = latents,
    mxPath(from="F",to=c("x1","x2","x3","x4"),
           free=c(TRUE,TRUE,TRUE,TRUE), value=c(1.0,1.0,1.0,1.0) ,
           arrows=1, label=c("F__x1","F__x2","F__x3","F__x4") ),
    mxPath(from="one",to=c("x2","x3","x4"),
           free=c(TRUE,TRUE,TRUE), value=c(1.0,1.0,1.0) ,
           arrows=1, label=c("const__x2","const__x3","const__x4") ),
    mxPath(from="one",to=c("F"), free=c(TRUE),
           value=c(1.0) , arrows=1, label=c("const__F") ),
    mxPath(from="x1",to=c("x1"), free=c(TRUE),
           value=c(1.0) , arrows=2, label=c("VAR_x1") ),
    mxPath(from="x2",to=c("x2"), free=c(TRUE),
           value=c(1.0) , arrows=2, label=c("VAR_x2") ),
    mxPath(from="x3",to=c("x3"), free=c(TRUE),
           value=c(1.0) , arrows=2, label=c("VAR_x3") ),
    mxPath(from="x4",to=c("x4"), free=c(TRUE),
           value=c(1.0) , arrows=2, label=c("VAR_x4") ),
    mxPath(from="F",to=c("F"), free=c(FALSE),
           value=c(1.0) , arrows=2, label=c("VAR_F") ),
    mxPath(from="one",to=c("x1"), free=F, value=0, arrows=1),
    mxData(cfa.sim, type = "raw")
    );

## Global Invariance

Global invariance is a relatively naive way of estimating a tree, in which a chosen set of parameters is equal across all leafs. This is achieved by fixing one or more selected parameters to the sample estimate before the tree is grown. Global invariance assures that the chosen parameters don’t differ across the possible subgroups. However, if the parameters do differ, this leads to misspecification and jeopardizes the interpretation of the resulting tree structure. Using this approach is usually not recommended because it’s typically not safe to assume that parameters are identical across a wide range of splits of the data. There may be cases in which it may make sense to constrain loading structures or means but it seems particularly dangerous to constrain variance components with such an approach.

    tree.gc <- semtree(model.cfa, data=cfa.sim, constraints=
                       semtree.constraints(global.invariance =
                                             c("F__x1","F__x2","F__x3","F__x4")))
    #> > Model was not run. Estimating parameters now.
    #> Beginning initial fit attemptFit attempt 0, fit=1318.29388440164, new current best! (was 23700.5750140635)                                                                             > Global Constraints:
    #> F__x1 F__x2 F__x3 F__x4
    #> > Freely Estimated Parameters:
    #> VAR_x1 VAR_x2 VAR_x3 VAR_x4 const__x2 const__x3 const__x4 const__F
    #> Beginning initial fit attemptFit attempt 0, fit=1318.2938844016, new current best! (was 1318.29388440164)                                                                            Beginning initial fit attemptFit attempt 0, fit=733.887584266354, new current best! (was 970.825458337159)                                                                             Beginning initial fit attemptFit attempt 0, fit=95.1677395282725, new current best! (was 363.032910748207)Beginning fit attempt 1 of at maximum 10 extra tries                         Fit attempt 1, fit=95.167739528257, new current best! (was 95.1677395282725)Beginning fit attempt 2 of at maximum 10 extra tries                        Fit attempt 2, fit=95.1677395281899, new current best! (was 95.167739528257)                                                                            Beginning initial fit attemptFit attempt 0, fit=90.9449345816668, new current best! (was 370.854673518056)Beginning fit attempt 1 of at maximum 10 extra tries                         Fit attempt 1, fit=90.9449345816006, new current best! (was 90.9449345816668)                                                                             Beginning initial fit attemptFit attempt 0, fit=112.496154030789, new current best! (was 347.468426064657)Beginning fit attempt 1 of at maximum 10 extra tries                         Fit attempt 1, fit=112.496154030731, new current best! (was 112.496154030789)Beginning fit attempt 2 of at maximum 10 extra tries                         Beginning fit attempt 3 of at maximum 10 extra triesBeginning fit attempt 4 of at maximum 10 extra triesBeginning fit attempt 5 of at maximum 10 extra triesBeginning fit attempt 6 of at maximum 10 extra triesBeginning fit attempt 7 of at maximum 10 extra triesBeginning fit attempt 8 of at maximum 10 extra triesBeginning fit attempt 9 of at maximum 10 extra triesBeginning fit attempt 10 of at maximum 10 extra tries                                                     ✔ Tree construction finished [took 3s].

    plot(tree.gc)

Even though we retrieve both predictors in the tree, the tree structure does not represent the data generating model (`p2` is found to be conditional only on `p1`>=.5) and the misspecification in the loadings leads to bias in the mean structure.

## Local Invariance

Local invariance builds a tree under which all parameters across the leafs of a tree may differ but the chosen parameters may not differ significantly from each other. If they differed significantly, the respective split is not considered a valid split and is not chosen. Local constraints are implemented by means of an additional test of measurement invariance. For each possible split, we fit an additional null model, in which the locally invariant parameters are constrained to be equal across the two resulting daughter nodes of a split. Only if we reject this null hypothesis, we believe that there is measurement non-invariance and disregard the split. A typical use-case is to the set of loadings of a factor model as `local.invariance` to allow a tree with weakly measurement-invariant leafs.

    tree.lc <- semtree(model.cfa, data=cfa.sim, constraints=
                       semtree.constraints(
                         local.invariance= c("F__x1","F__x2","F__x3","F__x4")))
    #> > Model was not run. Estimating parameters now.
    #> Beginning initial fit attemptFit attempt 0, fit=1318.29388440164, new current best! (was 23700.5750140635)                                                                             > No Invariance alpha selected. alpha.invariance set to:0.05
    #> Beginning initial fit attemptFit attempt 0, fit=1318.29388440161, new current best! (was 1318.29388440164)                                                                             Beginning initial fit attemptFit attempt 0, fit=717.155915399248, new current best! (was 970.825458996354)
    #> Beginning initial fit attemptFit attempt 0, fit=86.007086324807, new current best! (was 347.468425405495)Beginning fit attempt 1 of at maximum 10 extra tries                        Fit attempt 1, fit=86.0070863247715, new current best! (was 86.007086324807)Beginning fit attempt 2 of at maximum 10 extra tries                        Beginning fit attempt 3 of at maximum 10 extra triesBeginning fit attempt 4 of at maximum 10 extra triesBeginning fit attempt 5 of at maximum 10 extra triesBeginning fit attempt 6 of at maximum 10 extra triesBeginning fit attempt 7 of at maximum 10 extra triesBeginning fit attempt 8 of at maximum 10 extra triesBeginning fit attempt 9 of at maximum 10 extra triesBeginning fit attempt 10 of at maximum 10 extra tries
    #> ✔ Tree construction finished [took 3s].

Now we find `p1` as the only predictor that yields subgroups that pass the measurement invariance test. Even though we have chosen the four factor loadings as `local.invariance` constraint, they are allowed to differ numerically but there was no significant difference between them.

    plot(tree.lc)

## Focus Parameters

First, we create some data. Here, we generate 1,000 bivariate observations `x1` and `x2` gathered in a data frame called `obs`. Also, there are two predictors `grp1` and `grp2`. Predictor `grp1` predicts a mean difference in `x1` whereas `grp2` predicts differences only in `x2`.

    set.seed(123)
    N <- 1000
    grp1 <- sample(x = c(0,1), size=N, replace=TRUE)
    grp2 <- sample(x = c(0,1), size=N, replace=TRUE)
    Sigma <- matrix(byrow=TRUE,
                            nrow=2,c(2,0.2,
                                    0.2,1))
    obs <- MASS::mvrnorm(N,mu=c(0,0),
                          Sigma=Sigma)
    obs[,1] <- obs[,1] + ifelse(grp1,3,0)
    obs[,2] <- obs[,2] + ifelse(grp2,3,0)
    df.biv <- data.frame(obs, grp1=factor(grp1), grp2=factor(grp2))
    names(df.biv)[1:2] <- paste0("x",1:2)

A tree without constraints should recover both parameters (given large enough sample and effect sizes) because it explores differences in both means and covariance structure of `x1` and `x2`. Here is an OpenMx model specification for a saturated bivariate model:

    manifests<-c("x1","x2")
    model.biv <- mxModel("Bivariate_Model",
    type="RAM",
    manifestVars = manifests,
    latentVars = c(),
    mxPath(from="x1",to=c("x1","x2"),
           free=c(TRUE,TRUE), value=c(1.0,.2) ,
           arrows=2, label=c("VAR_x1","COV_x1_x2") ),
    mxPath(from="x2",to=c("x2"), free=c(TRUE),
           value=c(1.0) , arrows=2, label=c("VAR_x2") ),
    mxPath(from="one",to=c("x1","x2"), label=c("mu1","mu2"),
           free=TRUE, value=0, arrows=1),
    mxData(df.biv, type = "raw")
    );

Let’s run this model in OpenMx:

    result <- mxRun(model.biv)
    #> Running Bivariate_Model with 5 parameters
    summary(result)
    #> Summary of Bivariate_Model
    #>
    #> free parameters:
    #>        name matrix row col  Estimate  Std.Error A
    #> 1    VAR_x1      S  x1  x1 4.0283800 0.18015402
    #> 2 COV_x1_x2      S  x1  x2 0.3039196 0.11443875
    #> 3    VAR_x2      S  x2  x2 3.2282345 0.14437018
    #> 4       mu1      M   1  x1 1.4187341 0.06346925
    #> 5       mu2      M   1  x2 1.4628999 0.05681735
    #>
    #> Model Statistics:
    #>                |  Parameters  |  Degrees of Freedom  |  Fit (-2lnL units)
    #>        Model:              5                   1995              8233.926
    #>    Saturated:              5                   1995                    NA
    #> Independence:              4                   1996                    NA
    #> Number of observations/statistics: 1000/2000
    #>
    #> Information Criteria:
    #>       |  df Penalty  |  Parameters Penalty  |  Sample-Size Adjusted
    #> AIC:       4243.926               8243.926                 8243.986
    #> BIC:      -5547.046               8268.465                 8252.584
    #> To get additional fit indices, see help(mxRefModels)
    #> timestamp: 2025-11-26 16:54:27
    #> Wall clock time: 0.02205181 secs
    #> optimizer:  SLSQP
    #> OpenMx version number: 2.22.9
    #> Need help?  See help(mxSummary)

Now, we grow a tree without constraints:

    tree.biv <- semtree(model.biv, data=df.biv)
    #> > Model was not run. Estimating parameters now.
    #> Beginning initial fit attemptFit attempt 0, fit=8233.92582585158, new current best! (was 14528.4141425595)                                                                             Beginning initial fit attemptFit attempt 0, fit=8233.92582585143, new current best! (was 8233.92582585158)                                                                             Beginning initial fit attemptFit attempt 0, fit=3454.12434636158, new current best! (was 4066.88531956853)                                                                             Beginning initial fit attemptFit attempt 0, fit=1555.54412300078, new current best! (was 1720.05414323717)                                                                             Beginning initial fit attemptFit attempt 0, fit=1569.26472590267, new current best! (was 1734.07020312441)                                                                             Beginning initial fit attemptFit attempt 0, fit=3566.5692080098, new current best! (was 4167.0405062829)                                                                           Beginning initial fit attemptFit attempt 0, fit=1593.91684303245, new current best! (was 1780.60715329424)                                                                             Beginning initial fit attemptFit attempt 0, fit=1576.27862642528, new current best! (was 1785.96205471556)                                                                             ✔ Tree construction finished [took 2s].

As expected, we obtain a tree structure that has both `p1` and `p2` (here we use the viridis colors to give each leaf node a different frame color, which we’ll use later again):

    # default white color for all nodes
    cols <- rep("black", semtree:::getNumNodes(tree.biv))
    cols[as.numeric(row.names(semtree:::getTerminalNodes(tree.biv)))] <- viridis:::viridis_pal()(4)

    plot(tree.biv, border.col=cols)

Let us visualize what this tree is doing. Here is the empirical joint distribution of both `x1` and `x2`:

    require("ggplot2")
    ggplot(data = df.biv, aes(x=x1, y=x2))+
      geom_density_2d()+
      theme_classic()

And here is the partition of the observed two-dimensional space implied by the leafs of the tree (compare the density colors to the colors of the leaf nodes in the earlier dendrogram):

    df.biv.pred <- data.frame(df.biv,
      leaf=factor(getLeafs(tree=tree.biv, data = df.biv)))
    ggplot(data = df.biv.pred, aes(x=x1, y=x2))+
      geom_density_2d(aes(colour=leaf))+
      viridis::scale_color_viridis(discrete=TRUE)+
      theme_classic()

What if we were interested only in splits with respect to one of the two dimensions? In this case, we can set a `focus.parameter`. Focus parameters change the split evaluation such that only splits are evaluated that maximize the misfit between a model in which all parameters are free and a model in which all parameters are free but the parameters given in `focus.parameter`.

Let us first set `mu1` as focus parameter:

    tree.biv2 <- semtree(model.biv, df.biv, constraints=
                          semtree.constraints(focus.parameters = "mu1"))
    #> > Model was not run. Estimating parameters now.
    #> Beginning initial fit attemptFit attempt 0, fit=8233.92582585158, new current best! (was 14528.4141425595)                                                                             Beginning initial fit attemptFit attempt 0, fit=8233.92582585143, new current best! (was 8233.92582585158)                                                                             Beginning initial fit attemptFit attempt 0, fit=3740.92185296731, new current best! (was 4086.36288912548)                                                                             Beginning initial fit attemptFit attempt 0, fit=3795.16307144921, new current best! (was 4147.56293672595)                                                                             ✔ Tree construction finished [took 1s].

    plot(tree.biv2)

As expected, the resulting tree structure only has `grp1` as predictor because `grp1` predicts differences in `mu1`. Predictor `grp2` did not come up anymore. Now, if we set `mu2`, we should see the exact opposite picture:

    tree.biv3 <- semtree(model.biv, df.biv, constraints=
                          semtree.constraints(focus.parameters = "mu2"))
    #> > Model was not run. Estimating parameters now.
    #> Beginning initial fit attemptFit attempt 0, fit=8233.92582585158, new current best! (was 14528.4141425595)                                                                             Beginning initial fit attemptFit attempt 0, fit=8233.92582585143, new current best! (was 8233.92582585158)                                                                             Beginning initial fit attemptFit attempt 0, fit=3454.12434636158, new current best! (was 4066.88531956853)                                                                             Beginning initial fit attemptFit attempt 0, fit=3566.5692080098, new current best! (was 4167.0405062829)                                                                           ✔ Tree construction finished [took 1s].

And, indeed, we see only `grp2` as predictor whereas `grp1` was not selected this time.

    plot(tree.biv3)

Finally, we set the `focus.parameter` to one of the variance parameters. None of the predictors predicts differences in the variance, so we expect an empty tree (only the root node; no predictors selected):

    tree.biv4 <- semtree(model.biv, df.biv, constraints=
                          semtree.constraints(focus.parameters = "VAR_x2"))
    #> > Model was not run. Estimating parameters now.
    #> Beginning initial fit attemptFit attempt 0, fit=8233.92582585158, new current best! (was 14528.4141425595)                                                                             Beginning initial fit attemptFit attempt 0, fit=8233.92582585143, new current best! (was 8233.92582585158)                                                                             ✔ Tree construction finished [took less than a second].

    plot(tree.biv4)