Consider a simple one-factor model with 4 indicators. By default, lavaan will always fix the factor loading of the first indicator to 1. The other three factor loadings are free, and their values are estimated by the model. But suppose that you have good reasons to fix all the factor loadings to 1. The syntax below illustrates how this can be done:
f =~ y1 + 1*y2 + 1*y3 + 1*y4
In general, to fix a parameter in a lavaan formula, you need to pre-multiply
the corresponding variable in the formula by a numerical value. This is called
the pre-multiplication mechanism and will be used for many purposes. As another
example, consider again the three-factor Holzinger and Swineford CFA model.
Recall that, by default, all exogenous latent variables in a CFA model are
correlated. But if you wish to fix the correlation (or covariance) between a
pair of latent variables to zero, you need to explicity add a
covariance-formula for this pair, and fix the parameter to zero. In the syntax
below, we allow the covariance between the latent variables visual and
textual to be free, but the two other covariances are fixed to zero. In
addition, we fix the variance of the factor speed to unity. Therefore, there
is no need anymore to set the factor loading of its first indicator (x7)
equal to one. To force this factor loading to be free, we pre-multiply it with
NA, as a hint to lavaan that the value of this parameter is ‘missing’ and
therefore still unknown.
# three-factor model
visual =~ x1 + x2 + x3
textual =~ x4 + x5 + x6
speed =~ NA*x7 + x8 + x9
# orthogonal factors
visual ~~ 0*speed
textual ~~ 0*speed
# fix variance of speed factor
speed ~~ 1*speed
If you need to constrain all covariances of the latent variables in a CFA model
to be orthogonal, there is a shortcut. You can omit the covariance formulas in
the model syntax and simply add an argument orthogonal = TRUE to the function
call:
HS.model <- ' visual =~ x1 + x2 + x3
textual =~ x4 + x5 + x6
speed =~ x7 + x8 + x9 '
fit.HS.ortho <- cfa(HS.model,
data = HolzingerSwineford1939,
orthogonal = TRUE)
Similarly, if you want to fix the variances of all the latent
variables in a CFA model to unity, there is again a shortcut. Simply add
the argument std.lv = TRUE to the function call:
HS.model <- ' visual =~ x1 + x2 + x3
textual =~ x4 + x5 + x6
speed =~ x7 + x8 + x9 '
fit <- cfa(HS.model,
data = HolzingerSwineford1939,
std.lv = TRUE)
If the argument std.lv = TRUE is used, the factor loadings of the first
indicator of each latent variable will no longer be fixed to 1.
The lavaan package automatically generates starting values for all free
parameters. Normally, this works fine. But if you prefer to provide your own
starting values, you are free to do so. The way it works is based on the
pre-multiplication mechanism that we discussed before. But the numeric
constant is now the argument of a special function start().
An example will make this clear:
visual =~ x1 + start(0.8)*x2 + start(1.2)*x3
textual =~ x4 + start(0.5)*x5 + start(1.0)*x6
speed =~ x7 + start(0.7)*x8 + start(1.8)*x9
A nice property of the lavaan package is that all free parameters are automatically named according to a simple set of rules. This is convenient, for example, if equality constraints are needed (see the next subsection). To see how the naming mechanism works, we will use the model that we used for the Politcal Democracy data.
model <- '
# latent variable definitions
ind60 =~ x1 + x2 + x3
dem60 =~ y1 + y2 + y3 + y4
dem65 =~ y5 + y6 + y7 + y8
# regressions
dem60 ~ ind60
dem65 ~ ind60 + dem60
# residual (co)variances
y1 ~~ y5
y2 ~~ y4 + y6
y3 ~~ y7
y4 ~~ y8
y6 ~~ y8
'
fit <- sem(model,
data = PoliticalDemocracy)
coef(fit)
ind60=~x2 ind60=~x3 dem60=~y2 dem60=~y3 dem60=~y4 dem65=~y6
2.180 1.819 1.257 1.058 1.265 1.186
dem65=~y7 dem65=~y8 dem60~ind60 dem65~ind60 dem65~dem60 y1~~y5
1.280 1.266 1.483 0.572 0.837 0.624
y2~~y4 y2~~y6 y3~~y7 y4~~y8 y6~~y8 x1~~x1
1.313 2.153 0.795 0.348 1.356 0.082
x2~~x2 x3~~x3 y1~~y1 y2~~y2 y3~~y3 y4~~y4
0.120 0.467 1.891 7.373 5.067 3.148
y5~~y5 y6~~y6 y7~~y7 y8~~y8 ind60~~ind60 dem60~~dem60
2.351 4.954 3.431 3.254 0.448 3.956
dem65~~dem65
0.172
The function coef()
extracts the estimated values of the free parameters in the model,
together with their names. Each name consists of three parts and reflects the
part of the formula where the parameter was involved. The first part is the
variable name that appears on the left-hand side (lhs) of the formula.
The middle part is the operator type (op) of the formula, and the third part
is the variable in the right-hand side (rhs) of the formula that
corresponds with the parameter.
Often, it is convenient to choose your own labels for specific parameters. The
way this works is similar to fixing a parameter. But instead of pre-multiplying
with a numerical constant, we use a character string (the
label) instead. In the example below, we ‘label’ the factor loading of the
x3 indicator with the label myLabel:
model <- '
# latent variable definitions
ind60 =~ x1 + x2 + myLabel*x3
dem60 =~ y1 + y2 + y3 + y4
dem65 =~ y5 + y6 + y7 + y8
# regressions
dem60 ~ ind60
dem65 ~ ind60 + dem60
# residual (co)variances
y1 ~~ y5
y2 ~~ y4 + y6
y3 ~~ y7
y4 ~~ y8
y6 ~~ y8
'
It is important that labels start with a letter (a-zA-Z), and certainly not
with a digit. For example ‘13bis’ is not a valid label, and will confuse the
lavaan syntax parser. (Note: before version 0.4-8, it was necessary to use the
modifier label() to specify a custom label. Although it is still supported,
it is not recommended anymore. The only reason why it should be used in new
syntax is if the label contains an operator like “=~”.)
We have seen the use of the pre-multiplication mechanism (using the *
operator) a number of times: to fix a parameter, to provide a starting value,
and to label a parameter. We refer to these operations as modifiers, because
they modify some properties of certain model parameters. More modifiers will be
introduced later.
Each term on the right hand side in a formula can have one modifier only. If you want to specify more modifiers for the same parameter, you need to list the term multiple times in the same formula. For example:
f =~ y1 + y2 + myLabel*y3 + start(0.5)*y3 + y4
The indicator y3 was listed twice, each time with a different modifier. The
parser will accumulate all the different modifiers, but still treat y3 as a
single indicator.
In some applications, it is useful to impose equality constraints on one or
more otherwise free parameters. Consider again the three-factor H&S CFA model.
Suppose a user has a priori reasons to believe that the factor loadings of the
x2 and x3 indicators are equal to each other. Instead of estimating two free
parameters, lavaan should only estimate a single free parameter, and use that
value for both factor loadings. The main mechanism to specify this type of
(simple) equality constraints is by using labels: if two parameters have the
same label, they will be considered to be the same, and only one value will be
computed for them. This is illustrated in the following syntax:
visual =~ x1 + v2*x2 + v2*x3
textual =~ x4 + x5 + x6
speed =~ x7 + x8 + x9
Remember: all parameters having the same label will be constrained to be equal.
An alternative approach is to use the equal() modifier. This is useful if no
custom label has been specified, and one needs to refer to the automatically
generated label. For example:
visual =~ x1 + x2 + equal("visual=~x2")*x3
textual =~ x4 + x5 + x6
speed =~ x7 + x8 + x9
Consider the following regression:
y ~ b1*x1 + b2*x2 + b3*x3
where we have explicitly labeled the regression coefficients as b1, b2
and b3. We create a toy dataset containing these four variables and fit
the regression model:
set.seed(1234)
Data <- data.frame(y = rnorm(100),
x1 = rnorm(100),
x2 = rnorm(100),
x3 = rnorm(100))
model <- ' y ~ b1*x1 + b2*x2 + b3*x3 '
fit <- sem(model, data = Data)
coef(fit)
b1 b2 b3 y~~y
-0.052 0.084 0.139 0.970
Suppose that we need to impose the following two (nonlinear) constraints on $b_1$: $b_1 = (b_2+b_3)^2$ and $b_1 \geq \exp(b_2 + b_3)$. The first constraint is an equality constraint. The second is an inequality constraint. To specify these constraints, you can use the following syntax:
model.constr <- ' # model with labeled parameters
y ~ b1*x1 + b2*x2 + b3*x3
# constraints
b1 == (b2 + b3)^2
b1 > exp(b2 + b3) '
To see the effect of the constraints, we refit the model:
model.constr <- ' # model with labeled parameters
y ~ b1*x1 + b2*x2 + b3*x3
# constraints
b1 == (b2 + b3)^2
b1 > exp(b2 + b3) '
fit <- sem(model.constr, data = Data)
coef(fit)
b1 b2 b3 y~~y
0.495 -0.405 -0.299 1.610
The reader can verify that the constraints are indeed respected. The equality constraint holds exactly. The inequality constraint has resulted in an equality between the left-hand side ($b_1$) and the right-hand side ($\exp(b_2 + b_3)$).