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I was playing around in lavaan to bind together two simple models I previously tested on their own via simple regression analyses. I followed the tutorial provided on the following site: http://lavaan.ugent.be/tutorial/sem.html.

Basically, I have two latent variables ($lv1$ and $lv2$) one has three manifest indicators ($x1$, $x2$, $x3$), the other one has four ($x1$, $x2$, $x3$, $x4$). Both variables predict another variable $y$. Visually speaking:

sem model

The data used only contains positive values. I modeled this as following in R and lavaan:

semModel <- '
  # measurement models
    lv1 =~ x1 + x2 + x3
    lv2 =~ x1 + x2 + x3 + x4
  # regressions
    y ~ lv1
    y ~ lv2
  # residual correlations
    x1 ~~ x2
    x1 ~~ x3
    x1 ~~ x4
    x2 ~~ x3
    x2 ~~ x4
    x3 ~~ x4
'

Then I ran the following:

fit <- sem(semModel1, data = experimentalData)
summary(fit)

This returned the following errors:

1: In lav_model_vcov(lavmodel = lavmodel2, lavsamplestats = lavsamplestats,  :
  lavaan WARNING:
    Could not compute standard errors! The information matrix could
    not be inverted. This may be a symptom that the model is not
    identified.
2: In lav_object_post_check(object) :
  lavaan WARNING: some estimated lv variances are negative

I then added the option std.ov to standardise observed variables which still yields the error regarding the standard errors.

fit <- sem(semModel, data = experimentalData, std.ov = TRUE)

In lav_model_vcov(lavmodel = lavmodel2, lavsamplestats = lavsamplestats,  :
  lavaan WARNING:
    Could not compute standard errors! The information matrix could
    not be inverted. This may be a symptom that the model is not
    identified.

In the second case, output is as following:

lavaan 0.6-5 ended normally after 32 iterations

  Estimator                                         ML
  Optimization method                           NLMINB
  Number of free parameters                         21

  Number of observations                           583

Model Test User Model:

  Test statistic                                    NA
  Degrees of freedom                                -6
  P-value (Unknown)                                 NA

Parameter Estimates:

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

Latent Variables:
                   Estimate  Std.Err  z-value  P(>|z|)
  lv1 =~                                        
    x1                1.000                           
    x2                1.155       NA                  
    x3                1.691       NA                  
  lv2 =~                                       
    x1                1.000                           
    x2                2.006       NA                  
    x3                2.224       NA                  
    x4                1.422       NA                  

Regressions:
                   Estimate  Std.Err  z-value  P(>|z|)
  y ~                                        
    lv1               0.020       NA                  
    lv2               0.889       NA                  

Covariances:
                   Estimate  Std.Err  z-value  P(>|z|)
 .x1   ~~                                            
   .x2                0.033       NA                  
   .x3                0.134       NA                  
   .x4               -0.003       NA                  
 .x2   ~~                                            
   .x3               -0.104       NA                  
   .x4               -0.202       NA                  
 .x3   ~~                                            
   .x4                0.013       NA                  
  lv1  ~~                                        
    lv2              -0.159       NA                  

Variances:
                   Estimate  Std.Err  z-value  P(>|z|)
   .x1                0.918       NA                  
   .x2                0.415       NA                  
   .x3                0.444       NA                  
   .x4                0.405       NA                  
   .y                 0.772       NA                  
    lv1               0.105       NA                  
    lv2               0.294       NA                  

Where did I miss something? Are there (logical) errors in the definition of my model?

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The model is not identified, which means there is no unique solution to the estimation problem. Identification is a challenging topic, one that is often overlooked. First, your graphical model is incorrect. You have manifest variables pointing to the latent variables, when in your model, the manifest variables measure the latent variable. Second, the cause for the lack of identification is the residual covariances among all the indicators and the fact that all the indicators load on both latent variables. With so few indicators and most of them shared between the latent variables, you cannot supply any residual covariances. In general, each latent variable needs two unique indicators (i.e., unique to each latent variable), and residual covariances generally cannot be included without more than four indicators.

I highly recommend you look at the identification chapter in an SEM textbook, like Bollen (1989). There are specific rules to identification and ways to assess whether your model is identified.

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