The lavaan model syntax describes a latent variable model. The
function lavParTable turns it into a table that represents the full
model as specified by the user. We refer to this table as the parameter table.
lavaanify(model = NULL, meanstructure = FALSE, int_ov_free = FALSE,
int_lv_free = FALSE, marker_int_zero = FALSE,
orthogonal = FALSE, orthogonal_y = FALSE,
orthogonal_x = FALSE, orthogonal_efa = FALSE, std_lv = FALSE,
correlation = FALSE, composites = TRUE, composites_cov_free = FALSE,
effect_coding = "", conditional_x = FALSE,
fixed_x = FALSE, parameterization = "delta", constraints = NULL,
ceq_simple = FALSE, auto = FALSE, model_type = "sem",
auto_fix_first = FALSE, marker = NULL,
auto_fix_single = FALSE, auto_var = FALSE,
auto_cov_lv_x = FALSE, auto_cov_x = FALSE, auto_cov_y = FALSE,
auto_th = FALSE, auto_delta = FALSE, auto_efa = FALSE,
var_table = NULL, ngroups = 1L, nthresholds = NULL,
group_equal = NULL, group_partial = NULL, group_w_free = FALSE,
debug = FALSE, warn = TRUE, as_data_frame = TRUE, ...)lavParTable(model = NULL, meanstructure = FALSE, int_ov_free = FALSE,
int_lv_free = FALSE, marker_int_zero = FALSE,
orthogonal = FALSE, orthogonal_y = FALSE,
orthogonal_x = FALSE, orthogonal_efa = FALSE, std_lv = FALSE,
correlation = FALSE, composites = TRUE, composites_cov_free = FALSE,
effect_coding = "", conditional_x = FALSE,
fixed_x = FALSE, parameterization = "delta", constraints = NULL,
ceq_simple = FALSE, auto = FALSE, model_type = "sem",
auto_fix_first = FALSE, marker = NULL,
auto_fix_single = FALSE, auto_var = FALSE,
auto_cov_lv_x = FALSE, auto_cov_x = FALSE, auto_cov_y = FALSE,
auto_th = FALSE, auto_delta = FALSE, auto_efa = FALSE,
var_table = NULL, ngroups = 1L, nthresholds = NULL,
group_equal = NULL, group_partial = NULL, group_w_free = FALSE,
debug = FALSE, warn = TRUE, as_data_frame = TRUE, ...)
lavParseModelString(model_syntax = '', as_data_frame = FALSE,
parser = "open", warn = TRUE, debug = FALSE, ...)
A description of the user-specified model. Typically, the model
is described using the lavaan model syntax; see details for more
information. Alternatively, a parameter table (e.g., the output of
lavParseModelString) is also accepted.
The model syntax specifying the model. Must be a literal string.
If TRUE, intercepts/means will be added to
the model for both observed and latent variables.
If FALSE, the intercepts of the observed variables
are fixed to zero.
If FALSE, the intercepts of the latent variables
are fixed to zero.
Logical. Only relevant if the metric of each latent
variable is set by fixing the first factor loading to unity.
If TRUE, it implies meanstructure = TRUE and
std_lv = FALSE, and it fixes the intercepts of the marker
indicators to zero, while freeing the means/intercepts of the latent
variables. This only works correctly for single-group, single-level
models.
If TRUE, all covariances among
latent variables are set to zero.
If TRUE, all covariances among
endogenous latent variables only are set to zero.
If TRUE, all covariances among
exogenous latent variables only are set to zero.
If TRUE, all covariances among
latent variables involved in rotation only are set to zero.
If TRUE, the metric of each latent variable is
determined by fixing their variances to 1.0. If FALSE, the metric
of each latent variable is determined by fixing the factor loading of the
first indicator to 1.0. If there are multiple
groups, std_lv = TRUE and "loadings" is included in
the group.label argument, then only the latent variances
of the first group will be fixed to 1.0, while the latent
variances of the other groups are freely estimated.
If TRUE, a correlation structure is fitted. For
continuous data, this implies that the (residual) variances are no
longer parameters of the model.
Logical. If TRUE, use the new (0.6-20) approach
to handle composites.
Logical. Only relevant if the model contains
composites. If TRUE, the composite-indicator (co)variances (the
elements of the T matrix) are estimated as free parameters. If
FALSE (the default), they are fixed to their sample values.
Can be logical or character string. If
logical and TRUE, this implies
effect_coding = c("loadings", "intercepts"). If logical and
FALSE, it is set equal to the empty string.
If "loadings" is included, equality
constraints are used so that the average of the factor loadings (per
latent variable) equals 1. Note that this should not be used
together with std_lv = TRUE. If "intercepts" is
included, equality constraints are used so that the sum of the
intercepts (belonging to the indicators of a single latent variable)
equals zero.
As a result, the latent mean will be freely estimated and will usually
equal the average of the means of the indicators involved.
If TRUE, we set up the model conditional on
the exogenous `x' covariates; the model-implied sample statistics
only include the non-x variables. If FALSE, the exogenous `x'
variables are modeled jointly with the other variables, and the
model-implied statistics reflect both sets of variables.
If TRUE, the exogenous `x' covariates are considered
fixed variables and the means, variances and covariances of these variables
are fixed to their sample values. If FALSE, they are considered
random, and the means, variances and covariances are free parameters.
Currently only used if data is categorical. If
"delta", the delta parameterization is used. If "theta",
the theta parameterization is used.
Additional (in)equality constraints. See details for more information.
If TRUE, and no other general constraints are
used in the model, simple equality constraints are represented in the
parameter table as duplicated free parameters (instead of extra rows
with op = "==").
If TRUE, the default values are used for the auto.*
arguments, depending on the value of model_type.
Either "sem" or "growth"; only used if
auto=TRUE.
If TRUE, the factor loading of the first indicator
is set to 1.0 for every latent variable.
Optional named character vector mapping a latent variable
(name) to the observed indicator (name) whose loading should be fixed to
1.0 (instead of the first indicator) when auto_fix_first = TRUE.
Latent variables that are not named in this vector keep the first
indicator as their marker. This is used internally by the
bad.marker.crit mechanism.
If TRUE, the residual variance (if included)
of an observed indicator is set to zero if it is the only indicator of a
latent variable.
If TRUE, the (residual) variances of both observed
and latent variables are set free.
If TRUE, the covariances of exogenous latent
variables are included in the model and set free.
If TRUE, the covariances between exogenous latent
variables and observed exogenous covariates are also included in the
model and set free (this implies auto_cov_lv_x = TRUE). Ignored
(with a warning) if conditional_x = TRUE.
If TRUE, the covariances of dependent variables
(both observed and latent) are included in the model and set free.
If TRUE, thresholds for limited dependent variables
are included in the model and set free.
If TRUE, response scaling parameters for limited
dependent variables are included in the model and set free.
If TRUE, the necessary constraints are
imposed to make the (unrotated) exploratory factor analysis blocks
identifiable: for each block, factor variances are set to 1, factor
covariances are constrained to be zero, and factor loadings are
constrained to follow an echelon pattern.
The variable table containing information about the observed variables in the model.
The number of (independent) groups.
Either a single integer or a named vector of integers.
If nthresholds is a single integer, all endogenous
variables are assumed to be ordered with nthresholds indicating
the number of thresholds needed in the model. If nthresholds is a
named vector, it indicates the number of thresholds for these ordered
variables only. This argument should not be used in combination with
var_table.
A vector of character strings. Only used in
a multiple group analysis. Can be one or more of the following:
"loadings", "intercepts",
"means", "regressions", "residuals" or
"covariances", specifying the pattern of equality
constraints across multiple groups. When (in the model syntax) a vector
of labels is used as a modifier for a certain parameter, this will
override the group_equal setting if it applies to this parameter.
See also the Multiple groups section below for using modifiers in multiple
groups.
A vector of character strings containing the labels of the parameters which should be free in all groups (thereby overriding the group_equal argument for some specific parameters).
Logical. If TRUE, the group frequencies are
considered to be free parameters in the model. In this case, a
Poisson model is fitted to estimate the group frequencies. If
FALSE (the default), the group frequencies are fixed to their
observed values.
If TRUE, return the list of model parameters
as a data.frame.
Character. If "old", use the original/classic parser.
If "new", use the new/ldw parser.
If "open", use the newest, extensible parser.
The default (as of version 0.7-1) is "open".
If TRUE, some (possibly harmless) warnings are printed
out.
If TRUE, debugging information is printed out.
To accept old argument names with dots. No other arguments are accepted.
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 a modifier resulting in a numerical constant. Here are some examples:
Fixing the regression coefficient of the predictor
x2:
y ~ x1 + 2.4*x2 + x3Specifying an orthogonal (zero) covariance between two latent variables:
f1 ~~ 0*f2Specifying 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*y14Instead 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.
Additionally, the == operator can be used to set a labeled parameter
equal to a specific numeric value. This will be demonstrated in the section below
about (In)equality constraints.
User-provided starting values can be given by using the special function
start(), containing a numeric constant. For example:
y ~ x1 + start(1.0)*x2 + x3Note that if a starting value is provided, the parameter is not automatically considered to be free.
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 their own labels for specific parameters simply by pre-multiplying
the corresponding term (on the right hand side of the operator only) by
a character string (starting with a letter).
For example, in the formula f1 =~ x1 + x2 + mylabel*x3, the parameter
corresponding with the factor loading of
x3 will be named "mylabel".
An alternative way to specify the label is as follows:
f1 =~ x1 + x2 + label("mylabel")*x3,
where the label is the argument of special function label();
this can be useful if the label contains a space, or an operator (like "~").
There are two ways to constrain a parameter
to be equal to another target parameter. If you
have specified your own labels, you can use the fact that
equal labels imply equal parameter values.
If you rely on automatic parameter labels, you
can use 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 ~ int2*1
y2b ~ int2*1
y2c ~ int2*1
'
In a multiple group analysis, modifiers that contain a single element should be replaced by a vector, having the same length as the number of groups. If you provide a single element, it will be recycled for all the groups. This may be dangerous, in particular when the modifier is a label. In that case, the (same) label is copied across all groups, and this would imply an equality constraint across groups. Therefore, when using modifiers in a multiple group setting, it is always safer (and cleaner) to specify the same number of elements as the number of groups. Consider this example with 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 + c(x9.group1, x9.group2)*x9 '
In this example, the factor loading of the `x2' indicator is fixed to the value 0.5 for both groups. However, the factor loadings of the `x3' indicator are fixed to 0.6 and 0.8 for group 1 and group 2 respectively. The same logic is used for all modifiers. Note that character vectors can contain unquoted strings.
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 + x3.loading*x3In a two-level model (with a cluster= argument), the rv()
modifier can be used in the level-1 (within) part of the model to turn a
regression coefficient into a random slope: a latent variable at
the second (between) level. The (quoted) argument of rv() is the
name of this latent variable. At the between level, the random slope can
then be given a (residual) variance, covariances, regressions on
between-level covariates, and an intercept (which is the `fixed effect'
of the slope). For example (cfr. ex9.8 in the Mplus User's Guide):
model <- '
level: 1
fw =~ y1 + y2 + y3 + y4
fw ~ rv("s1")*x1 + rv("s2")*x2
level: 2
fb =~ y1 + y2 + y3 + y4
s1 + s2 + fb ~ w
'
Here, the within-level regression coefficients of fw on the
observed covariates x1 and x2 vary across clusters; the
random slopes s1 and s2 are regressed on the between-level
covariate w.
The covariate that carries a random slope may also be a latent (within-level) variable, as in
model <- '
level: 1
fxw =~ x1 + x2 + x3
fyw =~ y1 + y2 + y3
fyw ~ rv("s1")*fxw
level: 2
fxb =~ x1 + x2 + x3
fyb =~ y1 + y2 + y3
fyb ~ fxb
fyb ~~ s1
'
Such a slope multiplies two random variables, and the marginal
loglikelihood is no longer available in closed form: it is computed by
(Gauss-Hermite) quadrature over the latent-covariate slopes only,
while all other random effects are still integrated out analytically
(Rockwood, 2020). The number of quadrature points (per dimension) can
be set via the integration.ngh option (default: 21).
The same route is taken when the covariate is a `split' both-level observed variable (an observed variable that appears in both the level-1 and the level-2 part of the model): the random slope then multiplies its latent within-cluster component (latent centering), and the covariate is jointly modeled:
model <- '
level: 1
y ~ rv("s1")*x
level: 2
y ~ x
y ~~ s1
'
This is the latent-covariate contextual model of Rockwood (2020, section 4). Note the contrast with a within-only covariate (a variable that does not appear in the level-2 part): there, the slope multiplies the observed values, which are conditioned upon.
The current implementation is limited to: two-level models, a single
group, continuous data, estimator = "ML" and fixed.x =
TRUE; the observed covariates that carry a random slope must be
within-only (level-1) variables. Missing data can be handled by
full-information maximum likelihood (missing = "ml"); the
exogenous covariates themselves must be complete (cases with missing
values on exogenous variables are removed). The (observed-data)
loglikelihood is conditional on all exogenous
covariates: the covariates carrying random slopes, any other
within-only exogenous covariates (as in Mplus
TYPE = TWOLEVEL RANDOM), and the between-level exogenous
covariates. Between-only endogenous observed variables
(level-2 outcomes, e.g., indicators of a between-level factor) are
supported; they must be pure indicators (not involved in
regressions), with a free (or fixed to a nonzero value) residual
variance, and without missing values. No
chi-square test statistic (or fit indices) are available for models
with random slopes; fitMeasures() provides the
loglikelihood-based measures only (npar, ntotal,
logl, aic, bic, bic2), plus (with
estimator = "MLR") the H0 scaling correction factor
(scaling.factor.h0), which can be used for scaled
(loglikelihood) difference tests. Standard errors can be
"standard" (observed
information) or "robust.huber.white" (sandwich; this is also
what estimator = "MLR" implies). Both optim.method =
"nlminb" and optim.method = "em" are supported; with complete
data, "nlminb" is the default, while with missing =
"ml", the EM algorithm is the default optimizer.
lavPredict() provides empirical Bayes (posterior mean)
predictions: with level = 2, the
between-level latent variables including the random slopes (and, with
se = "standard", their posterior standard deviations); with
level = 1 (the default), the within-level factor scores. For
latent-covariate models, these are EAP (expected a posteriori)
estimates: the posterior is a mixture over the quadrature nodes.
For random slopes with latent covariates, some additional restrictions
apply (for now): a random-slope label may not be attached to both an
observed and a latent covariate, and the optimizer is always
"nlminb". The latent-covariate slopes may be regressed on
between-level covariates (e.g., s1 ~ w in the level-2 part),
as in the PISA example of Rockwood (2020).
The == operator can be used either to fix a parameter to a specific value,
or to set an estimated parameter equal to another parameter. Adapting the
example in the Parameter labels and equality constraints section, we
could have used different labels for the second factor's intercepts:
y2a ~ int1*1
y2b ~ int2*1
y2c ~ int3*1
Then, we could fix the first intercept to zero by including in the syntax an operation that indicates the parameter's label equals that value:
int1 == 0
Whereas we could still estimate the other two intercepts under an equality constraint by setting their different labels equal to each other:
int2 == int3
Optimization can be less efficient when constraining parameters this way (see
the documentation linked under See also for more information). But the
flexibility might be advantageous. For example, the constraints could be
specified in a separate character-string object, which can be passed to the
lavaan(..., constraints=) argument, enabling users to compare results
with(out) the constraints.
Inequality constraints work in much the same way, using the < or >
operator to indicate which estimated parameter is hypothesized to be greater or
less than either a specific value or another estimated parameter. For example, a
variance can be constrained to be nonnegative:
y1a ~~ var1a*y1a
## hypothesized constraint:
var1a > 0
Or the factor loading of a particular indicator might be expected to exceed other indicators' loadings:
f1 =~ L1*y1a + L2*y1b + L3*y1c
## hypothesized constraints:
L1 > L2
L3 < L1
Functions of parameters can be useful to test particular hypotheses. Following
from the Multiple groups example, we might be interested in which group's
factor loading is larger (i.e., an estimate of differential item functioning
(DIF) when the latent scales are linked by anchor items with equal loadings).
speed =~ c(L7, L7)*x7 + c(L8, L8)*x8 + c(L9.group1, L9.group2)*x9 '
## user-defined parameter:
DIF_L9 := L9.group1 - L9.group2
Note that this hypothesis is easily tested without a user-defined parameter by
using the lavTestWald() function. However, a user-defined parameter
additionally provides an estimate of the parameter being tested.
User-defined parameters are particularly useful for specifying indirect effects in models of mediation. For example:
model <- ' # direct effect
Y ~ c*X
# mediator
M ~ a*X
Y ~ b*M# user defined parameters:
# indirect effect (a*b)
ab := a*b
# total effect (defined using another user-defined parameter)
total := ab + c
'
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 <- '
# 1. 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
# 2. regressions
f1 ~ f3 + f4
f2 ~ f4
y1 + y2 ~ x1 + x2 + x3
# 3. (co)variances
y1 ~~ y1
y2 ~~ y4 + y5
f1 ~~ f2
# 4. intercepts
f1 ~ 1; y5 ~ 1
# 5. thresholds
y11 | t1 + t2 + t3
y12 | t1
y13 | t1 + t2
# 6. scaling factors
y11 ~*~ y11
y12 ~*~ y12
y13 ~*~ y13
# 7. composites
f5 <~ z1 + z2 + z3 + z4
'
Blank lines and comments can be used in between the formulas, and formulas can be split over multiple lines. Both the sharp (#) and the exclamation (!) characters can be used to start a comment. Multiple formulas can be placed on a single line if they are separated by a semicolon (;).
There can be seven types of formula-like expressions in the model syntax:
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''.
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.
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. In 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.
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, and the only
term on the right is 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.
Thresholds: The "|" operator can be used to define the
thresholds of categorical endogenous variables (on the left-hand side
of the operator). By convention, the
thresholds (on the right-hand side, separated by the "+" operator)
are named "t1", "t2", and so on.
Scaling factors: The "~*~" operator defines a scale factor.
The variable name on the left hand side must be the same as the variable
name on the right hand side. Scale factors are used in the Delta
parameterization, in a multiple group analysis when factor indicators
are categorical.
Composites: The "<~" operator can be used to define
a composite (on the right hand side of the operator).
A composite is a weighted linear combination of its composite indicators.
The name of the composite variable is on the left of the "<~"
operator, while the terms on the right, separated by "+"
operators, are the indicators of the composite variable.
There are 4 additional operators, also with left- and right-hand sides, that can
be included in model syntax. Three of them are used to specify (in)equality
constraints on estimated parameters (==, >, and <), and
those are demonstrated in a later section about
(In)equality constraints.
The final additional operator (:=) can be used to define ``new'' parameters
that are functions of one or more other estimated parameters. The :=
operator is demonstrated in a section about User-defined parameters.
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). The only exception is scaling factors, where
only a single element is allowed on the left-hand side.
In the right-hand side of these formula-like expressions, each element can be
modified (using the "*" operator) by either a numeric constant,
an expression resulting in a numeric constant, an expression resulting
in a character vector, 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. All "*" expressions are
referred to as modifiers. They are explained in more detail in the
following sections.
Rockwood, N. J. (2020). Maximum likelihood estimation of multilevel structural equation models with random slopes for latent covariates. Psychometrika, 85(2), 275--300. tools:::Rd_expr_doi("10.1007/s11336-020-09702-9")
Rosseel, Y. (2012). lavaan: An R package for structural equation
modeling. Journal of Statistical Software, 48(2), 1--36.
tools:::Rd_expr_doi("10.18637/jss.v048.i02")