Path analysis: CFI = 1, RMSEA = 0

Question

I was running a path analysis model but it seems that model fit indexes were perfect: CFI = 1.00, RMSEA = 0.00. However, perfect model fits usually indicate saturated model. But it seems that my model is NOT the case since I have extra degrees of freedom. Then, how to interpret CFI and RMSEA? Thanks so much for your help!

lavaan (0.5-21) converged normally after  39 iterations

  Number of observations                           109

  Number of missing patterns                         6

  Estimator                                         ML
  Minimum Function Test Statistic                6.199
  Degrees of freedom                                11
  P-value (Chi-square)                           0.860

Model test baseline model:

  Minimum Function Test Statistic              150.084
  Degrees of freedom                                20
  P-value                                        0.000

User model versus baseline model:

  Comparative Fit Index (CFI)                    1.000
  Tucker-Lewis Index (TLI)                       1.067

Loglikelihood and Information Criteria:

  Loglikelihood user model (H0)              -1000.419
  Loglikelihood unrestricted model (H1)       -997.320

  Number of free parameters                         19
  Akaike (AIC)                                2038.838
  Bayesian (BIC)                              2089.974
  Sample-size adjusted Bayesian (BIC)         2029.936

Root Mean Square Error of Approximation:

  RMSEA                                          0.000
  90 Percent Confidence Interval          0.000  0.054
  P-value RMSEA <= 0.05                          0.941

Standardized Root Mean Square Residual:

  SRMR                                           0.052

Parameter Estimates:

  Information                                 Observed
  Standard Errors                             Standard

Regressions:
                          Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
  SelfEsteem ~                                                                 
    EnglishNam (a)          -0.382    0.184   -2.073    0.038   -0.382   -0.200
  Well_Being ~                                                                 
    SelfEsteem (b)           0.668    0.095    6.998    0.000    0.668    0.558
  EnglishName ~                                                                
    RmmbrChnsN              -0.057    0.035   -1.623    0.105   -0.057   -0.204
    PrnncChnsN              -0.064    0.032   -1.981    0.048   -0.064   -0.249
  MentalHealth ~                                                               
    SelfEsteem (c)           0.779    0.088    8.846    0.000    0.779    0.656
  GeneralPhysicalHealth ~                                                      
    SelfEsteem (d)           0.335    0.099    3.368    0.001    0.335    0.314

Covariances:
                   Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
 .Well_Being ~~                                                         
   .MentalHealth      0.085    0.079    1.076    0.282    0.085    0.105
   .GnrlPhysclHlth    0.196    0.091    2.153    0.031    0.196    0.214
 .MentalHealth ~~                                                       
   .GnrlPhysclHlth    0.191    0.083    2.308    0.021    0.191    0.233

    Intercepts:
                   Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
   .SelfEsteem        5.605    0.126   44.424    0.000    5.605    5.880
   .Well_Being        0.860    0.525    1.638    0.101    0.860    0.754
   .EnglishName       1.014    0.132    7.701    0.000    1.014    2.031
   .MentalHealth      0.708    0.485    1.460    0.144    0.708    0.626
   .GnrlPhysclHlth    3.756    0.548    6.854    0.000    3.756    3.700

Variances:
                   Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
   .SelfEsteem        0.872    0.119    7.356    0.000    0.872    0.960
   .Well_Being        0.896    0.122    7.329    0.000    0.896    0.689
   .EnglishName       0.206    0.029    7.127    0.000    0.206    0.826
   .MentalHealth      0.728    0.101    7.201    0.000    0.728    0.569
   .GnrlPhysclHlth    0.929    0.129    7.211    0.000    0.929    0.901

Answer 1

I read somewhere online that modeling problems exist when the chi-square contribution is less than the degrees of freedom for any given step of a model (i.e., baseline fit for testing configural invariance or step comparing metric model to configural model, etc.) I have never run into this problem before and I do not really understand it. However, across the board, this seems to be the case for all of the models that have corresponding "perfect fit."