The Lavaan Model Syntax

Description

The lavaan model syntax describes a latent variable model. The function lavaanify turns it into a list that represents the full model as specified by the user.

Usage

lavaanify(model.syntax = NULL, meanstructure = FALSE, int.ov.free = FALSE, 
    int.lv.free = FALSE, orthogonal = FALSE, std.lv = FALSE, fixed.x = TRUE, 
    constraints = NULL, auto = FALSE, model.type = "sem",
    auto.fix.first = FALSE, auto.fix.single = FALSE,  auto.var = FALSE, 
    auto.cov.lv.x = FALSE, auto.cov.y = FALSE, ngroups = 1L, group.equal = NULL,
    group.partial = NULL, debug = FALSE, warn = TRUE, as.data.frame. = TRUE)

lavaanNames(object, type = "ov", group = NULL)

Arguments

Details

The model syntax consists of one or more formula-like expressions, each one describing a specific part of the model. The model syntax can be read from a file (using readLines), or can be specified as a literal string enclosed by single quotes as in the example below.

myModel <- '
  # latent variable definitions
    f1 =~ y1 + y2 + y3
    f2 =~ y4 + y5 + y6
    f3 =~ y7 + y8 + 
          y9 + y10 
    f4 =~ y11 + y12 + y13

  ! this is also a comment
 
  # regressions
    f1 ~ f3 + f4
    f2 ~ f4
    y1 + y2 ~ x1 + x2 + x3

  # (co)variances
    y1 ~~ y1
    y2 ~~ y4 + y5 
    f1 ~~ f2

  # intercepts
    f1 ~ 1; y5 ~ 1
'

1. Latent variable definitions: The "=~" operator can be used to define (continuous) latent variables. The name of the latent variable is on the left of the "=~" operator, while the terms on the right, separated by "+" operators, are the indicators of the latent variable.

   The operator "=~" can be read as “is manifested by”.

2. Regressions: The "~" operator specifies a regression. The dependent variable is on the left of a "~" operator and the independent variables, separated by "+" operators, are on the right. These regression formulas are similar to the way ordinary linear regression formulas are used in R, but they may include latent variables. Interaction terms are currently not supported.

3. Variance-covariances: The "~~" (‘double tilde’) operator specifies (residual) variances of an observed or latent variable, or a set of covariances between one variable, and several other variables (either observed or latent). Several variables, separated by "+" operators can appear on the right. This way, several pairwise (co)variances involving the same left-hand variable can be expressed in a single expression. The distinction between variances and residual variances is made automatically.

4. Intercepts: A special case of a regression formula can be used to specify an intercept (or a mean) of either an observed or a latent variable. The variable name is on the left of a "~" operator. On the right is only the number "1" representing the intercept. Including an intercept formula in the model automatically implies meanstructure = TRUE. The distinction between intercepts and means is made automatically.

Usually, only a single variable name appears on the left side of an operator. However, if multiple variable names are specified, separated by the "+" operator, the formula is repeated for each element on the left side (as for example in the third regression formula in the example above).

In the right-hand side of these formula-like expressions, each element can be multiplied (using the "*" operator) by either a numeric constant, an expression resulting in a numeric constant, or one of three special functions: start(), label() and equal(). This provides the user with a mechanism to fix parameters, to provide alternative starting values, to label the parameters, and to define equality constraints among model parameters. This is explained in more detail in the following sections.

Fixing parameters

It is often desirable to fix a model parameter that is otherwise (by default) free. Any parameter in a model can be fixed by using the pre-multiplication mechanism. Here are some examples:

• Fixing the regression coefficient of the predictor x2:

  y ~ x1 + 2.4*x2 + x3

• Specifying an orthogonal (zero) covariance between two latent variables:

  f1 ~~ 0*f2

• Specifying an intercept and a linear slope in a growth model:

  i =~ 1*y11 + 1*y12 + 1*y13 + 1*y14
  s =~ 0*y11 + 1*y12 + 2*y13 + 3*y14

Instead of a numeric constant, one can use a mathematical function that returns a numeric constant, for example sqrt(10). Multiplying with NA will force the corresponding parameter to be free.

Starting values

User-provided starting values can be given by using the same pre-multiplication mechanism. But the numeric constant is now the argument of a special function start(). For example:

y ~ x1 + start(1.0)*x2 + x3

Parameter labels and equality constraints

Each free parameter in a model is automatically given a name (or label). The name given to a model parameter consists of three parts, coerced to a single character vector. The first part is the name of the variable in the left-hand side of the formula where the parameter was implied. The middle part is based on the special ‘operator’ used in the formula. This can be either one of "=~", "~" or "~~". The third part is the name of the variable in the right-hand side of the formula where the parameter was implied, or "1" if it is an intercept. The three parts are pasted together in a single string. For example, the name of the fixed regression coefficient in the regression formula y ~ x1 + 2.4*x2 + x3 is the string "y~x2". The name of the parameter corresponding to the covariance between two latent variables in the formula f1 ~~ f2 is the string "f1~~f2".

Although this automatic labeling of parameters is convenient, the user may specify its own labels for specific parameters by using the label() modifier in a formula. For example, in the formula f1 =~ x1 + x2 + label("mylabel")*x3, the parameter corresponding with the factor loading of x3 will be named "mylabel" instead of the default name "f1=~x3".

To constrain a parameter to be equal to another target parameter, the pre-multiplication mechanism can be used using the special function equal(). The argument of equal() is the (automatic or user-specified) name of the target parameter. For example, in the confirmatory factor analysis example below, the intercepts of the three indicators of each latent variable are constrained to be equal to each other. For the first three, we have used the default names. For the last three, we have provided a custom label for the y2a intercept.

model <- '
  # two latent variables with fixed loadings
    f1 =~ 1*y1a + 1*y1b + 1*y1c
    f2 =~ 1*y2a + 1*y2b + 1*y2c

  # intercepts constrained to be equal
  # using the default names
    y1a ~ 1
    y1b ~ equal("y1a~1") * 1
    y1c ~ equal("y1a~1") * 1

  # intercepts constrained to be equal
  # using a custom label
    y2a ~ label("int2") * 1
    y2b ~ equal("int2") * 1
    y2c ~ equal("int2") * 1
'

Multiple groups

In a multiple group analysis, you can still use the pre-multiplication techniques, but the single argument is now replaced by a vector of arguments, one for each group. If it is a single value, the same value is used for all groups. For example, if there are two groups:

HS.model <- ' visual  =~             x1 +
                                 0.5*x2 +
                         c(0.6, 0.8)*x3

              textual =~                    x4 +
                         start(c(1.2, 0.6))*x5 +
                                            x6

              speed   =~                        x7 +
                                                x8 +
                         label(c("x9.group1",
                                 "x9.group2"))*x9 '

Multiple modifiers

In the model syntax, you can specify a variable more than once on the right hand side of an operator; therefore, several ‘modifiers’ can be applied simultaneously; for example, if you want to fix the value of a parameter and also label that parameter, you can use something like:

 f1 =~ x1 + x2 + 4*x3 + label("x3.loading")*x3