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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)

offspring <- url("https://drive.google.com/uc?export=download&id=1yXXlcHUZSMZ3QGtxnmuqvrFy6g0o2QeN")

load(offspring)

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
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Yes, if I have understood correctly, use of SEM can relax the homoscedasticity assumption - using a robust estimator (e.g. Satorra-Bentler) does not make the homoscedasticity assumption (it's a sandwich estimator). (But I'm not sure what you mean by "this SEM and the SEM orthogonal to it").

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  • $\begingroup$ 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. $\endgroup$ – Tavaro Evanis Nov 25 '20 at 0:26
  • $\begingroup$ You ge that without the anova test (but you need to allow the residuals of y1 and y2 to correlate). Quite some time ago, I wrote about the equivalence of SEM and Manova (in terms of power, but it works for this). $\endgroup$ – Jeremy Miles Nov 25 '20 at 0:39
  • $\begingroup$ Orthogonal in the previous answer refers to the correlations of the latent variables - you don't have any latent variables. $\endgroup$ – Jeremy Miles Nov 25 '20 at 0:39
  • $\begingroup$ I believe your post on MANOVA is here: stats.stackexchange.com/questions/331848/…. However, I get almost the same p-values from SEM as I do with independent ANOVAs. How do I obtain a single p-value? $\endgroup$ – Tavaro Evanis Nov 25 '20 at 1:58
  • $\begingroup$ It's easiest to explain if you post some code. $\endgroup$ – Jeremy Miles Nov 25 '20 at 18:55

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