# SEM Instead of MANOVA

I do not have sufficient reputation to comment on a question, so I hope this post is acceptable.

Regarding the accepted answer to this question:

How to do Simple Confirmatory Factory Analysis/SEM in R?

Let's say we have a simple SEM that would normally be analyzed via MANOVA:

$$y_{1} \sim a + b \\ y_{2} \sim a + b$$

where $$y_{i} \sim \mathcal{N}(0, \sigma^{2})$$. However, heteroscedasticity is present in both models, so MANOVA may not be appropriate. Would a likelihood ratio test between this SEM and the SEM orthogonal to it be an acceptable substitute to MANOVA?

UPDATE: Example data and analysis with multivariate $$p$$-value (thank you, @JeremyMiles!)

library(lavaan)

close(offspring)

# You should now have a data frame called "OM.full"
# Two "treatment" levels: cues, nocues
# Two response variables: dispersed, total.weight

# Scale response variables to z-scores

OM.full$$clutch.size <- scale(OM.full$$dispersed)
OM.full$$clutch.weight <- scale(OM.full$$total.weight)

# Desaturate the model to obtain a multivariate p-value
OM.sem <- "clutch.size ~ 0 * treatment
clutch.weight ~ 0 * treatment"

fit <- sem(OM.sem,
estimator = "MLMVS",
data = OM.full)

summary(fit)

lavaan 0.6-7 ended normally after 16 iterations

Estimator                                         ML
Optimization method                           NLMINB
Number of free parameters                          3

Number of observations                           128

Model Test User Model:
Standard      Robust
Test Statistic                                 2.085       1.984
Degrees of freedom                                 2       1.993
P-value (Chi-square)                           0.352       0.369
Scaling correction factor                                  1.051
mean and variance adjusted correction

Parameter Estimates:

Standard errors                           Robust.sem
Information                                 Expected
Information saturated (h1) model          Structured

Regressions:
Estimate  Std.Err  z-value  P(>|z|)
clutch.size ~
treatment         0.000
clutch.weight ~
treatment         0.000

Covariances:
Estimate  Std.Err  z-value  P(>|z|)
.clutch.size ~~
.clutch.weight     0.848    0.091    9.293    0.000

Variances:
Estimate  Std.Err  z-value  P(>|z|)
.clutch.size       0.992    0.099   10.006    0.000
.clutch.weight     0.992    0.097   10.180    0.000


• Thank you, @JeremyMiles. I was referring to the accepted answer to the question in the link above. @dmartin set up a likelihood ratio test via anova() in R. I was wondering if his procedure would produce the SEM equivalent of a parametric MANOVA, i.e. test of significance with corresponding $p$-values. – Tavaro Evanis Nov 25 '20 at 0:26