Second-order model with first-order std. loading of 1.0

[João Mota]

Dear lavaan community. I've tried to run a second-order CFA model in lavaan, in order to sequentially test different measurement models (unidimensional, correlated traits, second-order), trying to replicate the process used in a very similar test battery, using the syntax:

second_phys <- 'Physical =~ Global + cardio_shuttle + str_push + str_abs

Psychological =~ mot + conf + emo + phys

Social =~ ethics + collab + relation + cult

Cognitive =~ knowledge

PL =~  Physical + Psychological + Social + Cognitive

knowledge ~~ 92.8688*knowledge

emo ~~ phys'

second_run <- cfa(second_phys, full2, estimator = "MLR", missing = "FIML")

However, this results in negative variances:

"lavaan WARNING: some estimated lv variances are negative"

If I run the model by fixing the factor variances to 1 (std.lv =1) instead of using the default marker, the model runs, however, I get a standardized loading of 1.0 for one of the first-order factors (Psychological) - with a very large unstd. estimate.

second_run <- cfa(second_phys, full2, estimator = "MLR", missing = "FIML", std.lv =T)

summary(second_run, standardized = T, fit.measures = T, rs=T)

lavaan 0.6-9 ended normally after 1191 iterations

  Estimator                                         ML

  Optimization method                           NLMINB

  Number of model parameters                        43

                                                      

  Number of observations                           521

  Number of missing patterns                        11

                                                      

Model Test User Model:

                                               Standard      Robust

  Test Statistic                                382.556     359.177

  Degrees of freedom                                 61          61

  P-value (Chi-square)                            0.000       0.000

  Scaling correction factor                                   1.065

       Yuan-Bentler correction (Mplus variant)                     

Model Test Baseline Model:

  Test statistic                              2309.548    2027.494

  Degrees of freedom                                78          78

  P-value                                        0.000       0.000

  Scaling correction factor                                  1.139

User Model versus Baseline Model:

  Comparative Fit Index (CFI)                    0.856       0.847

  Tucker-Lewis Index (TLI)                       0.816       0.804

                                                                  

  Robust Comparative Fit Index (CFI)                         0.857

  Robust Tucker-Lewis Index (TLI)                            0.817

Loglikelihood and Information Criteria:

  Loglikelihood user model (H0)             -26502.470  -26502.470

  Scaling correction factor                                  1.207

      for the MLR correction                                      

  Loglikelihood unrestricted model (H1)     -26311.192  -26311.192

  Scaling correction factor                                  1.124

      for the MLR correction                                      

                                                                  

  Akaike (AIC)                               53090.941   53090.941

  Bayesian (BIC)                             53273.938   53273.938

  Sample-size adjusted Bayesian (BIC)        53137.446   53137.446

Root Mean Square Error of Approximation:

  RMSEA                                          0.101       0.097

  90 Percent confidence interval - lower         0.091       0.088

  90 Percent confidence interval - upper         0.110       0.106

  P-value RMSEA <= 0.05                          0.000       0.000

                                                                  

  Robust RMSEA                                               0.100

  90 Percent confidence interval - lower                     0.090

  90 Percent confidence interval - upper                     0.110

Standardized Root Mean Square Residual:

  SRMR                                           0.095       0.095

Parameter Estimates:

  Standard errors                             Sandwich

  Information bread                           Observed

  Observed information based on                Hessian

Latent Variables:

                   Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all

  Physical =~                                                           

    Global            5.982    0.588   10.174    0.000    7.896    0.463

    cardio_shuttle   13.196    0.848   15.562    0.000   17.419    0.792

    str_push         10.818    0.864   12.522    0.000   14.280    0.747

    str_abs           9.267    0.693   13.371    0.000   12.232    0.564

  Psychological =~                                                      

    mot               0.009    0.000   18.884    0.000   11.692    0.790

    conf              0.012    0.000   40.882    0.000   15.534    0.925

    emo               0.005    0.001    9.473    0.000    6.332    0.436

    phys              0.006    0.000   13.224    0.000    7.434    0.612

  Social =~                                                             

    ethics            7.354    0.749    9.822    0.000    7.977    0.670

    collab            8.926    0.869   10.274    0.000    9.682    0.851

    relation          9.154    0.583   15.710    0.000    9.928    0.738

    cult              6.231    0.952    6.545    0.000    6.758    0.346

  Cognitive =~                                                          

    knowledge        11.753    0.532   22.093    0.000   11.783    0.774

  PL =~                                                                 

    Physical          0.862    0.078   11.066    0.000    0.653    0.653

    Psychological  1247.039   26.392   47.251    0.000    1.000    1.000

    Social            0.420    0.099    4.254    0.000    0.387    0.387

    Cognitive         0.072    0.064    1.125    0.261    0.072    0.072

Covariances:

                   Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all

 .emo ~~                                                                

   .phys             30.538    6.576    4.644    0.000   30.538    0.243

Intercepts:

                   Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all

   .Global           99.952    0.751  133.149    0.000   99.952    5.866

   .cardio_shuttle   49.124    0.968   50.765    0.000   49.124    2.234

   .str_push         35.776    0.845   42.356    0.000   35.776    1.871

   .str_abs          48.381    0.967   50.028    0.000   48.381    2.232

   .mot              74.844    0.656  114.167    0.000   74.844    5.059

   .conf             68.280    0.744   91.792    0.000   68.280    4.064

   .emo              69.880    0.646  108.224    0.000   69.880    4.807

   .phys             75.022    0.539  139.313    0.000   75.022    6.174

   .ethics           81.511    0.529  154.116    0.000   81.511    6.846

   .collab           85.005    0.505  168.165    0.000   85.005    7.476

   .relation         77.688    0.597  130.065    0.000   77.688    5.778

   .cult             58.663    0.864   67.866    0.000   58.663    3.008

   .knowledge        68.802    0.676  101.804    0.000   68.802    4.520

   .Physical          0.000                               0.000    0.000

   .Psychological     0.000                               0.000    0.000

   .Social            0.000                               0.000    0.000

   .Cognitive         0.000                               0.000    0.000

    PL                0.000                               0.000    0.000

Variances:

                   Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all

   .knowledge        92.869                              92.869    0.401

   .Global          228.040   15.618   14.601    0.000  228.040    0.785

   .cardio_shuttle  179.945   20.754    8.670    0.000  179.945    0.372

   .str_push        161.792   22.619    7.153    0.000  161.792    0.442

   .str_abs         320.239   20.174   15.873    0.000  320.239    0.682

   .mot              82.129    8.444    9.726    0.000   82.129    0.375

   .conf             40.935   16.046    2.551    0.011   40.935    0.145

   .emo             171.193   11.941   14.337    0.000  171.193    0.810

   .phys             92.378    8.266   11.176    0.000   92.378    0.626

   .ethics           78.153   11.707    6.676    0.000   78.153    0.551

   .collab           35.552    9.543    3.726    0.000   35.552    0.275

   .relation         82.205   11.817    6.957    0.000   82.205    0.455

   .cult            334.758   19.017   17.603    0.000  334.758    0.880

   .Physical          1.000                               0.574    0.574

   .Psychological     1.000                               0.000    0.000

   .Social            1.000                               0.850    0.850

   .Cognitive         1.000                               0.995    0.995

    PL                1.000                               1.000    1.000

I've also tried running the model with the option bounds=T, which provides an equivalent solution. 

Can this results be derived from having very low correlations among some of the factors (as per the correlated traits solution, abbreviated below), and thus making this an implausable factor structure (other that not having an acceptable fit)? Or is there any plausible technical solution that I'm overlooking?

Covariances:

                   Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all

 .emo ~~                                                                

   .phys             29.731    6.528    4.555    0.000   29.731    0.239

  Physical ~~                                                           

    Psychological    59.167    8.081    7.322    0.000    0.661    0.661

    Social            0.555    3.857    0.144    0.886    0.009    0.009

    Cognitive        18.036    5.478    3.292    0.001    0.201    0.201

  Psychological ~~                                                      

    Social           39.445    6.982    5.650    0.000    0.419    0.419

    Cognitive         8.385    8.701    0.964    0.335    0.060    0.060

  Social ~~                                                             

    Cognitive         8.557    6.315    1.355    0.175    0.091    0.091

Thanks in advance for any help you can provide me.

Best regards.

[Terrence Jorgensen]

Can this results be derived from having very low correlations among some of the factors

That seems like the most likely problem.  Some factors are basically uncorrelated with others, especially Social, so the higher-order model could have estimation trouble and probably fit terribly.  FYI, output is easier to read if you change its font to fixed-width (e.g., Courier New) after pasting.

Terrence D. Jorgensen

Assistant Professor, Methods and Statistics

Research Institute for Child Development and Education, the University of Amsterdam

http://www.uva.nl/profile/t.d.jorgensen

[Edward Rigdon]

     The loading for Psychological on PL is 1247. That and the residual variances makes me wonder whether there is a wide range of variances across these observed variables. If that is true, then lavaan might behave better if variances across observed variables were made more similar through dividing some variables by constants, like 10.

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[João Mota]

Thank you Terrence and Edward for such a quick response.

These are the variances of the observed variables - by calling varTable().

name	mean	var
Global	99.98153	290.8596
cardio_shuttle	49.49499	483.5477
str_push	36.10505	366.0456
str_abs	48.57028	471.5494
mot	74.91001	219.2770
conf	68.36833	282.8268
emo	69.91564	211.7064
phys	75.06398	147.9372
ethics	81.52887	142.0586
collab	85.02625	129.5413
relation	77.70997	181.1360
cult	58.67829	381.1825
knowledge	68.80692	232.1721

Should I try to equalize all these variances into the same order of magnitude (say, 200 ish?), or is that an overkill?

Thanks again.

Regards.

João Mota

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[Edward Rigdon]

I was mistaken--that range of variances should not be causing a problem. But loadings, for either first order or second order factors, should not range from 0 to 1247.

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[João Mota]

Hi again Edward, thanks again for taking time to help folks around here.

Hmm, okay; I figured it should indeed be a modelling inadequacy (i.e., model a second-order factor with such low covariances between first-order factors). And I should have added to my first query that I ran a bifactor model using the same definition of group factors (first-order factors) with no issue: the Psychological observed variables have a higher saturation (loadings) on the general factor than all others. Could that be a plausible argument for why the second-order model is misbehaving?

Best regards.   

João Mota

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