= R lavann =

'''lavann''' is a framework for fitting a [[Statistics/StructuralEquationModeling|SEM]].

<<TableOfContents>>

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== Installation ==

{{{
install.package('lavaan')
}}}

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== Example ==

{{{
"
Y =~ y1 + y2 + y3
X =~ x1 + x2 + x3
Z =~ z1 + z2 + z3
Y ~ X + Z
" |>
  sem(data = df, estimator = "ML", std.lv = TRUE) |>
  lavaanPlot(coefs = TRUE)
}}}

Note that [[Statistics/Variance|variances]] of a term prefixed by a dot (like `.Y`) are [[Statistics/Residuals|error residuals]].



=== Model Specification ===

A model is specified in a domain specific syntax.

{{{
mod <- '
# Measurement paths
X =~ x1 + x2 + x3
Y =~ y1 + y2 + y3
Z =~ z1 + z2 + z3

# Regression paths
X ~ Y

# Declare covariance structure
x1 ~~ x2 + x3            #shorthand for `x1 ~~ x2` and `x1 ~~ x3`
x2 ~~ x3

# Constrain coefficient
Z ~ 1*Y

# Constrain intercept
Y ~ 0

# Constrain covariance
X ~~ 0*x1 + 0*x2 + 0*x3  #shorthand for `X ~~ 0*x1`, `X ~~ 0*x2`, and `X ~~ 0*x3`
'
}}}

Paths, like regressions in R, are indicated with a tilde (`~`).

Measurement paths use the compound `=~` symbol, and should be read as ' ''X'' is measured by ''x1'', ''x2'', and ''x3'' '. All latent variables must be declared like this, with the name of the variable on the left-hand side. Right-hand side only (i.e., strictly exogenous latent variables) are not supported.

Covariances are indicated with double tilde (`~~`). A variance is indicated by covariance of a variable with itself.



=== Estimators ===

Available estimators are:
 * Maximum likelihood (ML) (default)
 * ML with [[CorrectionsToTestStatisticsAndStandardErrorsInCovarianceStructureAnalysis|Satorra-Bentler adjustment]] (MLM)
   * Only works with complete data
 * [[StructuralEquationModelingWithRobustCovariances|Yuan-Bentler robust]] (MLR)

To use incomplete data, pass the `missing="ML"` option. Note that `"FIML"` is an alias for `"ML"`, and some documents prefer that naming.



=== Constraints ===

To constrain a loading/coefficient in a path, insert a constant and an asterisk (`*`) next to a variable on the right-hand side, as in `Z ~ 1*Y`. (Co)variances are constrained in the same way, as in `Y ~~ 1*Y`.

There is a shorthand for declaring all latent variables to be orthogonal/independent: try `sem(orthogonal = TRUE)`.

Intercepts are constrained using a syntax similar to path declarations. The concept is that the constant measurement of a variable is its expected value.



=== Default Behaviors ===

Regarding (co)variances, note the following default behaviors:
 * All latent variable (co)variances are allowed to vary freely, but are not necessarily estimated. To force estimation, declare it without a constraint.
 * All observed variable (co)variances are calculated and then constrained as given.
 * When a (co)variance declaration or constraint is made on an outcome/endogenous variable, it is automatically interpreted as the residual variance.

By default, intercepts are not estimated. To force estimation, declare it without a constraint or use the `meanstructure=TRUE` option. Note however that the `missing="ML"` option flips this default, because means are used in the incomplete data procedure.

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== Tips ==

=== Growth Curves ===

Specify the model like:

{{{
mod <- '
intercept = 1*t1 + 1*t2 + 1*t3 + 1*t4
slope     = 0*t1 + 1*t2 + 2*t3 + 3*t4
'
}}}



=== Multilevel ===

There is limited support for multilevel models. Try:

{{{
mod <- '
# Within cluster
level: 1
       x ~ x1 + x2 + x3 + Y

# Across clusters
level: 2
       Y =~ y1 + y2 + y3
'

mod.fit <- sem(model = mod, data = df, estimator = "ML", cluster = "clusterid")
}}}



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