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Multigroup models

using StructuralEquationModels

As an example, we will fit the model from the lavaan tutorial with loadings constrained to equality across groups.

We first load the example data and split it between groups:

dat = example_data("holzinger_swineford")

dat_g1 = dat[dat.school .== "Pasteur", :]
dat_g2 = dat[dat.school .== "Grant-White", :]

dat = example_data("holzinger_swineford")

dat_g1 = dat[dat.school .== "Pasteur", :]
dat_g2 = dat[dat.school .== "Grant-White", :]

We then specify our model via the graph interface:

latent_vars = [:visual, :textual, :speed]
observed_vars = Symbol.(:x, 1:9)

graph = @StenoGraph begin
    # measurement model
    visual  → fixed(1.0, 1.0)*x1 + label(:λ₂, :λ₂)*x2 + label(:λ₃, :λ₃)*x3
    textual → fixed(1.0, 1.0)*x4 + label(:λ₅, :λ₅)*x5 + label(:λ₆, :λ₆)*x6
    speed   → fixed(1.0, 1.0)*x7 + label(:λ₈, :λ₈)*x8 + label(:λ₉, :λ₉)*x9
    # variances and covariances
    _(observed_vars) ↔ _(observed_vars)
    _(latent_vars)   ⇔ _(latent_vars)
end

latent_vars = [:visual, :textual, :speed]
observed_vars = Symbol.(:x, 1:9)

graph = @StenoGraph begin
    # measurement model
    visual  → fixed(1, 1)*x1 + label(:λ₂, :λ₂)*x2 + label(:λ₃, :λ₃)*x3
    textual → fixed(1, 1)*x4 + label(:λ₅, :λ₅)*x5 + label(:λ₆, :λ₆)*x6
    speed   → fixed(1, 1)*x7 + label(:λ₈, :λ₈)*x8 + label(:λ₉, :λ₉)*x9
    # variances and covariances
    _(observed_vars) ↔ _(observed_vars)
    _(latent_vars)   ⇔ _(latent_vars)
end

You can pass multiple arguments to fix() and label() for each group. Parameters with the same label (within and across groups) are constrained to be equal. To fix a parameter in one group, but estimate it freely in the other, you may write fix(NaN, 4.3).

You can then use the resulting graph to specify an EnsembleParameterTable

groups = [:Pasteur, :Grant_White]

partable = EnsembleParameterTable(
    graph, 
    observed_vars = observed_vars,
    latent_vars = latent_vars,
    groups = groups)

The parameter table can be used to create a Dict of RAMMatrices with keys equal to the group names and parameter tables as values:

specification = convert(Dict{Symbol, RAMMatrices}, partable)

That is, you can asses the group-specific RAMMatrices as specification[:group_name].

!!! note "A different way to specify" Instead of choosing the workflow "Graph -> EnsembleParameterTable -> RAMMatrices", you may also directly specify RAMMatrices for each group (for an example see this test).

The next step is to construct the model:

model_g1 = Sem(
    specification = specification[:Pasteur],
    data = dat_g1
)

model_g2 = Sem(
    specification = specification[:Grant_White],
    data = dat_g2
)

model_ml_multigroup = SemEnsemble(model_g1, model_g2)

We now fit the model and inspect the parameter estimates:

solution = sem_fit(model_ml_multigroup)
update_estimate!(partable, solution)
sem_summary(partable)

Other things you can query about your fitted model (fit measures, standard errors, etc.) are described in the section Model inspection and work the same way for multigroup models.