Simulate Data

We’ll simulate a tri-variate data set, with two uncorrelated variables that are correlated to a third variable. To start, all variables will be centered at 0 (mu) and scaled to 1 (diagonal of Sigma):

library(tidyverse)
Sigma <- matrix(c(1.0, 0.6, 0.6,
                  0.6, 1.0, 0.0,
                  0.6, 0.0, 1.0),
                nrow = 3)
data <- MASS::mvrnorm(n = 1000,
                      mu = rep(0,3),
                      Sigma = Sigma,
                      empirical = T) %>% as.data.frame()

Let’s re scale V2 to increase it’s slope when predicting V1:

data <- data %>% 
  mutate(V2 = 5*V2+10)

Lets look at the correlation matrix (should be the same as Sigma):

knitr::kable(cor(data))

	V1	V2	V3
V1	1.0	0.6	0.6
V2	0.6	1.0	0.0
V3	0.6	0.0	1.0

and the covariance matrix (should only be different in the scale of V2):

knitr::kable(cov(data))

	V1	V2	V3
V1	1.0	3	0.6
V2	3.0	25	0.0
V3	0.6	0	1.0

Fit `lavaan` model

library(lavaan)
my_model <- '
V1 ~ a*V2 + b*V3
diff := a - b
'
fit <- sem(my_model, data = data)

If diff is computed on the standardized coefficients, we expect it to be 0.
If diff is computed on the unstandardized coefficients, we expect it to not 0.

summary(fit, standardized = T)

lavaan 0.6.13 ended normally after 1 iteration

  Estimator                                         ML
  Optimization method                           NLMINB
  Number of model parameters                         3

  Number of observations                          1000

Model Test User Model:
                                                      
  Test statistic                                 0.000
  Degrees of freedom                                 0

Parameter Estimates:

  Standard errors                             Standard
  Information                                 Expected
  Information saturated (h1) model          Structured

Regressions:
                   Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
  V1 ~                                                                  
    V2         (a)    0.120    0.003   35.857    0.000    0.120    0.600
    V3         (b)    0.600    0.017   35.857    0.000    0.600    0.600

Variances:
                   Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
   .V1                0.280    0.013   22.361    0.000    0.280    0.280

Defined Parameters:
                   Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
    diff             -0.480    0.017  -28.128    0.000   -0.480    0.000

We can see that the Estimate of diff is non-zero, implying that it was computed on the non-standardized coefficients. BUT we also see that the Std.all of diff is 0, implying that it was computed on the standardized coefficients.
What about the significance test?

parameterEstimates(fit)[7,] %>% knitr::kable()

	lhs	op	rhs	label	est	se	z	pvalue	ci.lower	ci.upper
7	diff	:=	a-b	diff	-0.48	0.0170646	-28.12843	0	-0.513446	-0.446554

standardizedSolution(fit)[7,] %>% knitr::kable()

	lhs	op	rhs	label	est.std	se	z	pvalue	ci.lower	ci.upper
7	diff	:=	a-b	diff	0	0.0236643	0	1	-0.0463812	0.0463812

We can see that we get two different \(z\)-tests, depending on the type of estimates we get. It seems that the test results returned from summary() are based on the standardizedSolution() function, and thus based on the unstandardized results.

Simulate Data

Fit lavaan model

Fit `lavaan` model