Syntax

Loadings, Regressions, and Intercepts

The following specifies loadings of a latent variable eta on manifest variables y1-y4:

eta =~ y1 + y2 + y3

Regressions are specified with ~:

xi  =~ x1 + x2 + x3
eta =~ y1 + y2 + y3
# predict eta with xi:
eta ~  xi

Add covariances with ~~

xi  =~ x1 + x2 + x3
eta =~ y1 + y2 + y3
# predict eta with xi:
eta ~  xi
x1 ~~ x2

Intercepts are specified with ~1

xi  =~ x1 + x2 + x3
eta =~ y1 + y2 + y3
# predict eta with xi:
eta ~  xi
x1 ~~ x2

eta ~ 1

Note: In lavaan’s sem-function, the loading on the first item of each latent variable is constrained to one by default. Estimating this loading freely requires replacing xi =~ x1 + x2 + x3 with xi =~ NA*x1 + x2 + x3. In mxsem, a different approach is used. When calling the mxsem-function set the argument scale_loadings to FALSE to freely estimate all loadings.

Parameter labels and constraints

Add labels to parameters as follows:

xi  =~ l1*x1 + l2*x2 + l3*x3
eta =~ l4*y1 + l5*y2 + l6*y3
# predict eta with xi:
eta ~  b*xi

Fix parameters by using numeric values instead of labels:

xi  =~ 1*x1 + l2*x2 + l3*x3
eta =~ 1*y1 + l5*y2 + l6*y3
# predict eta with xi:
eta ~  b*xi

Bounds

Lower and upper bounds allow for constraints on parameters. For instance, a lower bound can prevent negative variances.

xi  =~ 1*x1 + l2*x2 + l3*x3
eta =~ 1*y1 + l5*y2 + l6*y3
# predict eta with xi:
eta ~  b*xi
# residual variance for x1
x1 ~~ v*x1
# bound:
v > 0

Upper bounds are specified with v < 10. Note that the parameter label must always come first. The following is not allowed: 0 < v or 10 > v.

(Non-)linear constraints

Assume that latent construct eta was observed twice, where eta1 is the first observation and eta2 the second. We want to define the loadings of eta2 on its observations as l_1 + delta_l1. If delta_l1 is zero, we have measurement invariance.

eta1 =~ l1*y1 + l2*y2 + l3*y3
eta2 =~ l4*y4 + l5*y5 + l6*y6
# define new delta-parameter
!delta_1; !delta_2; !delta_3
# redefine l4-l6
l4 := l1 + delta_1
l5 := l2 + delta_2
l6 := l3 + delta_3

Alternatively, implicit transformations can be used as follows:

eta1 =~ l1*y1 + l2*y2 + l3*y3
eta2 =~ {l1 + delta_1} * y4 + {l2 + delta_2} * y5 + {l3 + delta_3} * y6

This is inspired by the approach in metaSEM (Cheung, 2015).

Definition variables

Definition variables allow for person-specific parameter constraints. Use the data.-prefix to specify definition variables.

I =~ 1*y1 + 1*y2 + 1*y3 + 1*y4 + 1*y5
S =~ data.t_1 * y1 + data.t_2 * y2 + data.t_3 * y3 + data.t_4 * y4 + data.t_5 * y5

I ~ int*1
S ~ slp*1

Model name

You can specify a model name using the following syntax:

# start with at least three equal signs:
=== model_name ===
I =~ 1*y1 + 1*y2 + 1*y3 + 1*y4 + 1*y5
S =~ data.t_1 * y1 + data.t_2 * y2 + data.t_3 * y3 + data.t_4 * y4 + data.t_5 * y5

I ~ int*1
S ~ slp*1

Note that mxsem will ignore everything above the three (or more) equal signs! That is, the following will result in problems:

# the following two lines will be ignored:

I =~ 1*y1 + 1*y2 + 1*y3 + 1*y4 + 1*y5
S =~ data.t_1 * y1 + data.t_2 * y2 + data.t_3 * y3 + data.t_4 * y4 + data.t_5 * y5

# start with at least three equal signs:
=== model_name ===

I ~ int*1
S ~ slp*1

Starting Values

mxsem differs from lavaan in the specification of starting values. Instead of providing starting values in the model syntax, the set_starting_values function is used.

References

• Boker, S. M., Neale, M., Maes, H., Wilde, M., Spiegel, M., Brick, T., Spies, J., Estabrook, R., Kenny, S., Bates, T., Mehta, P., & Fox, J. (2011). OpenMx: An Open Source Extended Structural Equation Modeling Framework. Psychometrika, 76(2), 306–317. https://doi.org/10.1007/s11336-010-9200-6
• Rosseel, Y. (2012). lavaan: An R package for structural equation modeling. Journal of Statistical Software, 48(2), 1–36. https://doi.org/10.18637/jss.v048.i02