Model Implied Covariance Matrix From a Lavaan-Specified SEM

Given a structural equation model written in lavaan model syntax with all of its parameters fixed to their population values, compute the model implied population covariance matrix \(\Sigma(\theta)\) of the observed variables and, when the model has a mean structure, the model implied population mean vector \(\mu(\theta)\). This function requires lavaan to be installed.

This is the helper that drives the population side of the sample size planning workflow for SEM: it lets the user state a population model, obtain the \(\Sigma(\theta)\) (and \(\mu(\theta)\)) those fixed values imply, and then pass that population to ss_aipe_sem_path, ss_aipe_sem_path_sensitivity, ss_aipe_rmsea_sensitivity, ss_power_composite_sem, or ss_aipe_composite_sem.

Usage

cov_sem(model)

Arguments

model: A single character string giving a structural equation model in lavaan model syntax (see model.syntax), with every parameter fixed to its population value. A factor loading, a structural path, a variance, or a covariance is fixed by prefixing the numeric value to the variable with the * operator, for example "f1 =~ 1*y1 + 0.8*y2 + 0.8*y3" for the loadings, "f2 ~ 0.5*f1" for a structural path, and "y1 ~~ 0.5*y1" for a residual variance. A model with a mean structure (for example a latent growth curve model) also fixes every intercept and latent mean, for example "t1 ~ 0*1" and "s ~ 0.3*1". lavaan syntax embeds the values and knows which variables are latent, so no separate parameter vector or list of latent variables is needed.

Value

A list with components:

sigma_theta: The model implied population covariance matrix of the observed variables, with rows and columns named.
mu_theta: The model implied population mean vector of the observed variables, named, in the row order of sigma_theta. A vector of zeros when the model has no mean structure.
observed_vars: Character vector of observed variable names in the row/column order of sigma_theta.

Details

The function builds a non-fitted lavaan object from model with all parameters held at the population values written into the syntax, and reads back the model implied covariance matrix of the observed variables. Because the object is created with do.fit = FALSE, no estimation is performed and the placeholder sample covariance lavaan needs to construct the object is never used; the returned \(\Sigma(\theta)\) comes entirely from the fixed parameter values. The observed-variable names are taken from the model syntax and fix the row and column order of the returned matrix.

References

Lai, K., & Kelley, K. (2011). Accuracy in parameter estimation for targeted effects in structural equation modeling: Sample size planning for narrow confidence intervals. Psychological Methods, 16(2), 127–148. doi:10.1037/a0021764

Rosseel, Y. (2012). lavaan: An R package for structural equation modeling. Journal of Statistical Software, 48(2), 1–36. doi:10.18637/jss.v048.i02

Author

Ken Kelley kkelley@nd.edu

Examples

# Population model with all parameters fixed to their values: two factors,
# three indicators each, and a structural path f2 ~ f1 of 0.5.
pop_model <- "
  f1 =~ 1*y1 + 0.8*y2 + 0.8*y3
  f2 =~ 1*y4 + 0.8*y5 + 0.8*y6
  f2 ~ 0.5*f1
  f1 ~~ 1*f1
  f2 ~~ 0.75*f2
  y1 ~~ 0.5*y1; y2 ~~ 0.5*y2; y3 ~~ 0.5*y3
  y4 ~~ 0.5*y4; y5 ~~ 0.5*y5; y6 ~~ 0.5*y6
"
cov_sem(pop_model)$sigma_theta
#>     y1   y2   y3  y4   y5   y6
#> y1 1.5 0.80 0.80 0.5 0.40 0.40
#> y2 0.8 1.14 0.64 0.4 0.32 0.32
#> y3 0.8 0.64 1.14 0.4 0.32 0.32
#> y4 0.5 0.40 0.40 1.5 0.80 0.80
#> y5 0.4 0.32 0.32 0.8 1.14 0.64
#> y6 0.4 0.32 0.32 0.8 0.64 1.14

# A population model with a mean structure: a linear latent growth curve
# over four waves. The intercepts and latent means are fixed too, and
# mu_theta carries the model implied means (5.0, 5.3, 5.6, 5.9).
pop_lgm <- "
  i =~ 1*t1 + 1*t2 + 1*t3 + 1*t4
  s =~ 0*t1 + 1*t2 + 2*t3 + 3*t4
  i ~~ 1*i
  s ~~ 0.2*s
  i ~~ -0.15*s
  t1 ~~ 0.5*t1; t2 ~~ 0.5*t2; t3 ~~ 0.5*t3; t4 ~~ 0.5*t4
  t1 ~ 0*1; t2 ~ 0*1; t3 ~ 0*1; t4 ~ 0*1
  i ~ 5*1
  s ~ 0.3*1
"
cov_sem(pop_lgm)$mu_theta
#>  t1  t2  t3  t4 
#> 5.0 5.3 5.6 5.9