[alessio lombini]

Dear Christopher Bratt,

I followed your advice and I performed a CFA to test measurement invariance. This is the code and the output I obtained:

. sem ($xlist <- GOVERNANCE_QUALITY), method(ml)

Endogenous variables

Measurement:  indg_accountability indg_control_corruption indg_govt_effect indg_political_stability indg_regulatory_quality indg_rule_law

Exogenous variables

Latent:       GOVERNANCE_QUALITY

Fitting target model:

Iteration 0:   log likelihood = -2333.1374  (not concave)
Iteration 1:   log likelihood =  -2255.416  
Iteration 2:   log likelihood =  -2120.146  
Iteration 3:   log likelihood = -1956.0777  
Iteration 4:   log likelihood = -1952.7927  
Iteration 5:   log likelihood = -1952.6845  
Iteration 6:   log likelihood = -1952.6844  

Structural equation model                       Number of obs     =        649
Estimation method  = ml
Log likelihood     = -1952.6844

 ( 1)  [indg_accountability]GOVERNANCE_QUALITY = 1
------------------------------------------------------------------------------------------------
                               |                 OIM
                               |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------------------------+----------------------------------------------------------------
Measurement                    |
  indg_accountability          |
            GOVERNANCE_QUALITY |          1  (constrained)
                         _cons |   .5579661   .0356381    15.66   0.000     .4881167    .6278155
  -----------------------------+----------------------------------------------------------------
  indg_control_corruption      |
            GOVERNANCE_QUALITY |    1.37988   .0450685    30.62   0.000     1.291548    1.468213
                         _cons |   .6728875   .0395549    17.01   0.000     .5953612    .7504138
  -----------------------------+----------------------------------------------------------------
  indg_govt_effect             |
            GOVERNANCE_QUALITY |   1.098787   .0363079    30.26   0.000     1.027625    1.169949
                         _cons |   .8366163   .0317185    26.38   0.000     .7744491    .8987835
  -----------------------------+----------------------------------------------------------------
  indg_political_stability     |
            GOVERNANCE_QUALITY |   .8816656    .040445    21.80   0.000     .8023948    .9609364
                         _cons |   .3306148   .0323192    10.23   0.000     .2672702    .3939594
  -----------------------------+----------------------------------------------------------------
  indg_regulatory_quality      |
            GOVERNANCE_QUALITY |   1.016093   .0344509    29.49   0.000     .9485705    1.083615
                         _cons |   .8468197   .0298789    28.34   0.000     .7882581    .9053814
  -----------------------------+----------------------------------------------------------------
  indg_rule_law                |
            GOVERNANCE_QUALITY |    1.26069   .0400029    31.51   0.000     1.182286    1.339094
                         _cons |    .726265   .0355271    20.44   0.000     .6566332    .7958968
-------------------------------+----------------------------------------------------------------
     var(e.indg_accountability)|   .3181308   .0179944                       .284747    .3554285
 var(e.indg_control_corruption)|   .0516799   .0037934                       .044755    .0596762
        var(e.indg_govt_effect)|   .0418468   .0028414                      .0366324    .0478035
var(e.indg_political_stability)|   .2844561   .0160206                      .2547273    .3176544
 var(e.indg_regulatory_quality)|   .0568253      .0035                      .0503633    .0641166
           var(e.indg_rule_law)|   .0147087   .0020161                      .0112434    .0192419
        var(GOVERNANCE_QUALITY)|   .5061483   .0423535                      .4295868    .5963547
------------------------------------------------------------------------------------------------
LR test of model vs. saturated: chi2(9)   =    123.50, Prob > chi2 = 0.0000

. 
end of do-file

. do "C:\Users\Utente\AppData\Local\Temp\STD281c_000000.tmp"

. estat gof, stats(all)

----------------------------------------------------------------------------
Fit statistic        |      Value   Description
---------------------+------------------------------------------------------
Likelihood ratio     |
          chi2_ms(9) |    123.497   model vs. saturated
            p > chi2 |      0.000
         chi2_bs(15) |   6140.712   baseline vs. saturated
            p > chi2 |      0.000
---------------------+------------------------------------------------------
Population error     |
               RMSEA |      0.140   Root mean squared error of approximation
 90% CI, lower bound |      0.119
         upper bound |      0.163
              pclose |      0.000   Probability RMSEA <= 0.05
---------------------+------------------------------------------------------
Information criteria |
                 AIC |   3941.369   Akaike's information criterion
                 BIC |   4021.927   Bayesian information criterion
---------------------+------------------------------------------------------
Baseline comparison  |
                 CFI |      0.981   Comparative fit index
                 TLI |      0.969   Tucker-Lewis index
---------------------+------------------------------------------------------
Size of residuals    |
                SRMR |      0.013   Standardized root mean squared residual
                  CD |      0.990   Coefficient of determination
----------------------------------------------------------------------------

. 
end of do-file

According to Comparison of Model Fit Indices Used in Structural Equation Modeling Under Multivariate Normality by S. Cangur and I. Ercan. The RMSEA is above the threshold of 0.10 (not satisfactory fit of the data); TLI is above the acceptable value of 0.95 (and exactly equals to 0.97, the cut-off used mostly in researchers); the CFI shows a good fit of the data and finally, the SMRS is below the acceptable value of 0.05. Therefore, I would ask if you might suggest how to interpret the overall fit of my model and if it is finally safe to perform an EFA to create the index.
Thank you again to all of you for showing interest and for providing useful insights on the issue.

[Nick Cox]

On #14 most useful map projections of showing continental or larger areas preserve area. Map projections are not condemned by alluding to the Mercator projection, which does have one rationale, but is otherwise the worst projection for most purposes.

[Christopher Bratt]

I don't use Stata for SEM/CFA, so I didn't quite understand your model. Instead, I will give more general comments.

I agree, the model has too low fit. The common cutoff for the RMSEA is .05 (or .06), or at least not above .08. The common cutoff for the SRMR is 0.08, I believe.

But more important: SEM and CFA have the advantage of handling measurement error and allowing us to test the model (can the model explain the data?).

another advantage is that SEM/CFA invites us to have a clear theoretical model before we turn to the data. For instance: what factor model do we expect? Drawing on paper might help.

Three genereral comments:

- Are you sure there is theoretical reason to expect a factor model in your case? I'm not sure there is. If not, then drop CFA. EFA (rather than PCA) can be an alternative when you don't have any theory and really want a factor model. If a factor model makes no theoretical sense (the idea of a latent variable is not applicable), then use a different approach.

- If you use CFA, spend time thinking through which model you expect. Then, not earlier, test your model.

- When running a CFA model and finding too low fit, think through which additional paths make sense. Use modification indices (and residuals) to guide your model development, but any such exploratory adaptation of the model to the data has to be theoretically reasonable.

[Erik Ruzek]

Erik Ruzek Sorry for being difficult. But...



would not really let us test the model with CFA. The model would be as complex as the data (df = 0). We would need to use four or more indicators, or fix parameters prior to the analysis, or for instance impose invariant parameters in longitudinal data to get df > 0,

Without a doubt. The original poster had 6 variables I believe. I was just trying to say that taking the mean of a set of items is an equivalent to an implicit factor model in which the loadings are specified to be equal with no residual covariances among the items. You can specify such a model in a CFA model and test it against a model where you relax some or all of those assumptions.