VCOV does not appear to be positive definite

[Nicolas]

Dear all,

When I try to build a model including single-indicator latent variables (few of which are dummy coded albeit numeric), I keep on getting this message:

Warning message:
In lav_model_vcov(lavmodel = lavmodel, lavsamplestats = lavsamplestats,  :
  lavaan WARNING:
    The variance-covariance matrix of the estimated parameters (vcov)
    does not appear to be positive definite! The smallest eigenvalue
    (= -2.910431e-20) is smaller than zero. This may be a symptom that
    the model is not identified.

Should I ignore this or is this something I have to care about?

For the sake of clarity, I am copying-pasting below a very simple CFA model that illustrates my issue:

model<-'

#LV definition

NS =~ a*T1_NS1 + b*T1_NS2 + c*T1_NS3
DA =~ d*T1_DA1 + e*T1_DA2 + f*T1_DA4
Gender =~ 1*T1_Gender #my covariate, being dummy coded 0/1 and numeric

#Scaling the LV with effect coding

a==3-b-c
d==3-e-f

#Residuals

T1_NS1~~T1_NS1
T1_NS2~~T1_NS2
T1_NS3~~T1_NS3
T1_DA1~~T1_DA1
T1_DA2~~T1_DA2
T1_DA4~~T1_DA4
T1_Gender~~0*T1_Gender #none, measured without error

#Intercepts
T1_NS1~i1*1
T1_NS2~i2*1
T1_NS3~i3*1
T1_DA1~i4*1
T1_DA2~i5*1
T1_DA4~i6*1
T1_Gender~0*1 #none, will be estimated at the LV level
#Effect coding for the means
i1==0-i2-i3
i4==0-i5-i6

#Estimations for the LV means/Intercepts

NS~1
DA~1
Gender~1

#Variance-CoVariance

NS ~~ NS + DA + Gender
DA ~~ DA + Gender
Gender ~~ Gender

'
model.fit<-lavaan(model, data.df, estimator="MLR")

Warning message:
In lav_model_vcov(lavmodel = lavmodel, lavsamplestats = lavsamplestats,  :
  lavaan WARNING:
    The variance-covariance matrix of the estimated parameters (vcov)
    does not appear to be positive definite! The smallest eigenvalue
    (= 4.825014e-15) is close to zero. This may be a symptom that the
    model is not identified.
> print(summary(model.fit, fit.measures=TRUE, standardized=TRUE))
lavaan 0.6-2 ended normally after 73 iterations

  Optimization method                           NLMINB
  Number of free parameters                         27
  Number of equality constraints                     4

  Number of observations                           433

  Estimator                                         ML      Robust
  Model Fit Test Statistic                      59.380      44.499
  Degrees of freedom                                12          12
  P-value (Chi-square)                           0.000       0.000
  Scaling correction factor                                  1.334
    for the Yuan-Bentler correction (Mplus variant)

Model test baseline model:

  Minimum Function Test Statistic             1349.128     729.698
  Degrees of freedom                                21          21
  P-value                                        0.000       0.000

User model versus baseline model:

  Comparative Fit Index (CFI)                    0.964       0.954
  Tucker-Lewis Index (TLI)                       0.938       0.920

  Robust Comparative Fit Index (CFI)                         0.967
  Robust Tucker-Lewis Index (TLI)                            0.942

Loglikelihood and Information Criteria:

  Loglikelihood user model (H0)              -3225.504   -3225.504
  Scaling correction factor                                  1.501
    for the MLR correction
  Loglikelihood unrestricted model (H1)      -3195.814   -3195.814
  Scaling correction factor                                  1.616
    for the MLR correction

  Number of free parameters                         23          23
  Akaike (AIC)                                6497.009    6497.009
  Bayesian (BIC)                              6590.636    6590.636
  Sample-size adjusted Bayesian (BIC)         6517.646    6517.646

Root Mean Square Error of Approximation:

  RMSEA                                          0.095       0.079
  90 Percent Confidence Interval          0.072  0.120       0.058  0.101
  P-value RMSEA <= 0.05                          0.001       0.012

  Robust RMSEA                                               0.091
  90 Percent Confidence Interval                             0.064  0.121

Standardized Root Mean Square Residual:

  SRMR                                           0.045       0.045

Parameter Estimates:

  Information                                 Observed
  Observed information based on                Hessian
  Standard Errors                   Robust.huber.white

Latent Variables:
                   Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
  NS =~                                                                
    T1_NS1     (a)    0.911    0.033   28.037    0.000    0.854    0.828
    T1_NS2     (b)    1.055    0.026   40.167    0.000    0.988    0.809
    T1_NS3     (c)    1.033    0.029   36.027    0.000    0.968    0.920
  DA =~                                                                
    T1_DA1     (d)    1.112    0.086   13.003    0.000    0.706    0.801
    T1_DA2     (e)    0.796    0.080    9.933    0.000    0.505    0.682
    T1_DA4     (f)    1.092    0.060   18.133    0.000    0.693    0.804
  Gender =~                                                            
    T1_Gender         1.000                               0.495    1.000

Covariances:
                   Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
  NS ~~                                                                
    DA                0.306    0.048    6.325    0.000    0.514    0.514
    Gender            0.051    0.023    2.195    0.028    0.110    0.110
  DA ~~                                                                
    Gender            0.066    0.017    3.870    0.000    0.209    0.209

Intercepts:
                   Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
   .T1_NS1    (i1)    0.452    0.126    3.573    0.000    0.452    0.438
   .T1_NS2    (i2)   -0.425    0.102   -4.161    0.000   -0.425   -0.348
   .T1_NS3    (i3)   -0.026    0.112   -0.236    0.814   -0.026   -0.025
   .T1_DA1    (i4)   -0.567    0.383   -1.482    0.138   -0.567   -0.643
   .T1_DA2    (i5)    0.989    0.364    2.714    0.007    0.989    1.333
   .T1_DA4    (i6)   -0.422    0.273   -1.545    0.122   -0.422   -0.489
   .T1_Gender         0.000                               0.000    0.000
    NS                3.734    0.048   78.068    0.000    3.987    3.987
    DA                4.399    0.034  129.853    0.000    6.926    6.926
    Gender            0.568    0.024   23.867    0.000    1.147    1.147

Variances:
                   Estimate  Std.Err  z-value  P(>|z|)   Std.lv  Std.all
   .T1_NS1            0.333    0.062    5.400    0.000    0.333    0.314
   .T1_NS2            0.515    0.058    8.883    0.000    0.515    0.345
   .T1_NS3            0.169    0.033    5.075    0.000    0.169    0.153
   .T1_DA1            0.279    0.066    4.247    0.000    0.279    0.359
   .T1_DA2            0.295    0.058    5.114    0.000    0.295    0.536
   .T1_DA4            0.263    0.069    3.827    0.000    0.263    0.354
   .T1_Gender         0.000                               0.000    0.000
    NS                0.877    0.075   11.683    0.000    1.000    1.000
    DA                0.403    0.041    9.767    0.000    1.000    1.000
    Gender            0.245    0.003   75.645    0.000    1.000    1.000

Constraints:
                                               |Slack|
    a - (3-b-c)                                  0.000
    d - (3-e-f)                                  0.000
    i1 - (0-i2-i3)                               0.000
    i4 - (0-i5-i6)                               0.000

NULL

> lavInspect(model.fit, "cov.all")
          T1_NS1 T1_NS2 T1_NS3 T1_DA1 T1_DA2 T1_DA4 T1_Gnd NS    DA    Gender
T1_NS1    1.062                                                             
T1_NS2    0.844  1.492                                                      
T1_NS3    0.826  0.957  1.106                                               
T1_DA1    0.310  0.359  0.351  0.778                                        
T1_DA2    0.222  0.257  0.251  0.357  0.550                                 
T1_DA4    0.304  0.352  0.345  0.490  0.351  0.744                          
T1_Gender 0.047  0.054  0.053  0.073  0.052  0.072  0.245                   
NS        0.799  0.926  0.906  0.340  0.243  0.334  0.051  0.877            
DA        0.278  0.322  0.316  0.449  0.321  0.440  0.066  0.306 0.403      
Gender    0.047  0.054  0.053  0.073  0.052  0.072  0.245  0.051 0.066 0.245

When I remove "Gender" from the model, everything is fine and gets back to normal....
Could you please share your thoughts with me?

Best,

Nicolas

[Terrence Jorgensen]

  lavaan WARNING:
    The variance-covariance matrix of the estimated parameters (vcov)
    does not appear to be positive definite! The smallest eigenvalue
    (= -2.910431e-20) is smaller than zero. This may be a symptom that
    the model is not identified.

Should I ignore this or is this something I have to care about?

This is barely below zero (in the 20th decimal place), which might just be a machine-precision issue.  I think you can ignore this.

When I remove "Gender" from the model, everything is fine and gets back to normal.... 

Could you please share your thoughts with me?

Remove all the single-indicator stuff, it is not necessary.  Just regress the factors on the observed variable "T1_Gender" and set fixed.x = TRUE, so you don't have to estimate any other parameters related to gender (nor do you have to assume it is normally distributed, if that is the reason for using MLR).

Terrence D. Jorgensen

Postdoctoral Researcher, Methods and Statistics

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

UvA web page: http://www.uva.nl/profile/t.d.jorgensen

[Nicolas]

Many thanks for your answer, Terrence!

[Rebekka Weidmann]

Here and somewhere else I found comments on not worrying about eigenvalues that are super close to zero. Is there a cut-off that could be used, any literature on that?