Multigroup models
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. We have to make sure that the column indicating the group (here called school) is a vector of Symbols, not strings - so we convert it.
dat = example_data("holzinger_swineford")
dat.school = Symbol.(replace.(dat.school, "-" => "_"))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, 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)
endYou 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)EnsembleParameterTable with groups: |Grant_White||Pasteur|
Grant_White:
--------- ---------- -------- ------- ------------- --------- ---------- ------
from relation to free value_fixed start estimate ⋯
Symbol Symbol Symbol Bool Float64 Float64 Float64 ⋯
--------- ---------- -------- ------- ------------- --------- ---------- ------
visual → x1 false 1.0 ⋯
visual → x2 true ⋯
visual → x3 true ⋯
textual → x4 false 1.0 ⋯
textual → x5 true ⋯
textual → x6 true ⋯
speed → x7 false 1.0 ⋯
speed → x8 true ⋯
speed → x9 true ⋯
x1 ↔ x1 true gGr ⋯
x2 ↔ x2 true gGr ⋯
x3 ↔ x3 true gGr ⋯
x4 ↔ x4 true gGr ⋯
x5 ↔ x5 true gGr ⋯
x6 ↔ x6 true gGr ⋯
⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋱
--------- ---------- -------- ------- ------------- --------- ---------- ------
1 column and 9 rows omitted
Latent Variables: [:visual, :textual, :speed]
Observed Variables: [:x1, :x2, :x3, :x4, :x5, :x6, :x7, :x8, :x9]
Pasteur:
--------- ---------- -------- ------- ------------- --------- ---------- ------
from relation to free value_fixed start estimate ⋯
Symbol Symbol Symbol Bool Float64 Float64 Float64 ⋯
--------- ---------- -------- ------- ------------- --------- ---------- ------
visual → x1 false 1.0 ⋯
visual → x2 true ⋯
visual → x3 true ⋯
textual → x4 false 1.0 ⋯
textual → x5 true ⋯
textual → x6 true ⋯
speed → x7 false 1.0 ⋯
speed → x8 true ⋯
speed → x9 true ⋯
x1 ↔ x1 true gPa ⋯
x2 ↔ x2 true gPa ⋯
x3 ↔ x3 true gPa ⋯
x4 ↔ x4 true gPa ⋯
x5 ↔ x5 true gPa ⋯
x6 ↔ x6 true gPa ⋯
⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋱
--------- ---------- -------- ------- ------------- --------- ---------- ------
1 column and 9 rows omitted
Latent Variables: [:visual, :textual, :speed]
Observed Variables: [:x1, :x2, :x3, :x4, :x5, :x6, :x7, :x8, :x9]
The parameter table can be used to create a multigroup Sem model:
model_ml_multigroup = Sem(
specification = partable,
data = dat,
semterm_column = :school,
groups = groups)Structural Equation Model- Loss Functions
> SemML
- id: Grant_White
- observed: SemObservedData
- implied: RAM
- weight: 0.482
> SemML
- id: Pasteur
- observed: SemObservedData
- implied: RAM
- weight: 0.518
Instead of choosing the workflow "Graph -> EnsembleParameterTable -> model", you may also directly specify RAMMatrices for each group (for an example see this test).
We now fit the model and inspect the parameter estimates:
sem_fit = fit(model_ml_multigroup)
update_estimate!(partable, sem_fit)
details(partable)
--------------------------------- Variables ---------------------------------
Latent variables: visual textual speed
Observed variables: x1 x2 x3 x4 x5 x6 x7 x8 x9
Group: Grant_White
---------------------------- Parameter Estimates -----------------------------
Loadings:
visual
to estimate value_fixed start free from label relation
x1 0.0 1.0 0.0 visual const →
x2 0.6 1.0 visual λ₂ →
x3 0.78 1.0 visual λ₃ →
textual
to estimate value_fixed start free from label relation
x4 0.0 1.0 0.0 textual const →
x5 1.08 1.0 textual λ₅ →
x6 0.91 1.0 textual λ₆ →
speed
to estimate value_fixed start free from label relation
x7 0.0 1.0 0.0 speed const →
x8 1.2 1.0 speed λ₈ →
x9 1.04 1.0 speed λ₉ →
Directed Effects:
from to estimate value_fixed start free label
Variances:
from to estimate value_fixed start free label ⋯
x1 ↔ x1 0.65 1.0 gGrant_White ⋯
x2 ↔ x2 0.94 1.0 gGrant_White ⋯
x3 ↔ x3 0.61 1.0 gGrant_White ⋯
x4 ↔ x4 0.33 1.0 gGrant_White ⋯
x5 ↔ x5 0.39 1.0 gGrant_White ⋯
x6 ↔ x6 0.44 1.0 gGrant_White ⋯
x7 ↔ x7 0.6 1.0 gGrant_White ⋯
x8 ↔ x8 0.41 1.0 gGrant_White ⋯
x9 ↔ x9 0.54 1.0 gGrant_White ⋯
visual ↔ visual 0.73 1.0 gGrant_White ⋯
textual ↔ textual 0.91 1.0 gGrant_White ⋯
speed ↔ speed 0.48 1.0 gGrant_White ⋯
1 column omitted
Covariances:
from to estimate value_fixed start free label ⋯
textual ↔ visual 0.44 1.0 gGrant_White ⋯
speed ↔ visual 0.32 1.0 gGrant_White ⋯
speed ↔ textual 0.23 1.0 gGrant_White ⋯
1 column omitted
Group: Pasteur
---------------------------- Parameter Estimates -----------------------------
Loadings:
visual
to estimate value_fixed start free from label relation
x1 0.0 1.0 0.0 visual const →
x2 0.6 1.0 visual λ₂ →
x3 0.78 1.0 visual λ₃ →
textual
to estimate value_fixed start free from label relation
x4 0.0 1.0 0.0 textual const →
x5 1.08 1.0 textual λ₅ →
x6 0.91 1.0 textual λ₆ →
speed
to estimate value_fixed start free from label relation
x7 0.0 1.0 0.0 speed const →
x8 1.2 1.0 speed λ₈ →
x9 1.04 1.0 speed λ₉ →
Directed Effects:
from to estimate value_fixed start free label
Variances:
from to estimate value_fixed start free label
x1 ↔ x1 0.55 1.0 gPasteur_1
x2 ↔ x2 1.27 1.0 gPasteur_2
x3 ↔ x3 0.89 1.0 gPasteur_3
x4 ↔ x4 0.44 1.0 gPasteur_4
x5 ↔ x5 0.51 1.0 gPasteur_5
x6 ↔ x6 0.27 1.0 gPasteur_6
x7 ↔ x7 0.85 1.0 gPasteur_7
x8 ↔ x8 0.52 1.0 gPasteur_8
x9 ↔ x9 0.66 1.0 gPasteur_9
visual ↔ visual 0.81 1.0 gPasteur_10
textual ↔ textual 0.92 1.0 gPasteur_13
speed ↔ speed 0.31 1.0 gPasteur_15
Covariances:
from to estimate value_fixed start free label
textual ↔ visual 0.42 1.0 gPasteur_11
speed ↔ visual 0.17 1.0 gPasteur_12
speed ↔ textual 0.18 1.0 gPasteur_14Other 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.