lavaan warning message: some estimated variances are negative

Chan Jun Han

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Feb 22, 2016, 10:04:44 AM2/22/16

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Hi all,

I'm relatively new to lavaan and was running a sem model as follows:

optimism_model_1B = '#latent variables

social_well_being =~ 1*faces_total_sa + faces_total_sc
optimism =~ 1*opt_total_s
stress_in_life =~ 1*les_total_s
mental_health =~ 1*ghq_total_s
#Regression
mental_health ~ social_well_being + optimism + stress_in_life
optimism ~ social_well_being
# residual variances of manifest variables
faces_total_sa ~~ faces_total_sa
faces_total_sc ~~ faces_total_sc
opt_total_s ~~ opt_total_s
les_total_s ~~ les_total_s
ghq_total_s ~~ ghq_total_s
# factor variances
social_well_being ~~ 1*social_well_being
optimism ~~ 1*optimism
stress_in_life ~~ 1*stress_in_life
mental_health ~~ 1*mental_health
# factor covariances
faces_total_sa ~~ faces_total_sc
faces_total_sa ~~ les_total_s'

fit = lavaan(optimism_model_1B, data = student, estimator ="MLM", likelihood = "wishart", int.ov.free=TRUE, int.lv.free=FALSE)

summary(fit, fit.measures = TRUE, standardized=TRUE)

However I received the following warning messages in lavaan:

Warning messages:

1: In lav_object_post_check(lavobject) :

lavaan WARNING: some estimated variances are negative

2: In lav_object_post_check(lavobject) :

lavaan WARNING: observed variable error term matrix (theta) is not positive definite; use inspect(fit,"theta") to investigate.

The output of lavaan is given below:

lavaan (0.5-20) converged normally after 28 iterations

Number of observations 1918

Estimator ML Robust

Minimum Function Test Statistic 6.264 5.976

Degrees of freedom 3 3

P-value (Chi-square) 0.099 0.113

Scaling correction factor 1.048

for the Satorra-Bentler correction

Model test baseline model:

Minimum Function Test Statistic 2720.645 2216.119

Degrees of freedom 10 10

P-value 0.000 0.000

User model versus baseline model:

Comparative Fit Index (CFI) 0.999 0.999

Tucker-Lewis Index (TLI) 0.996 0.996

Loglikelihood and Information Criteria:

Loglikelihood user model (H0) -12249.722 -12249.722

Loglikelihood unrestricted model (H1) -12246.588 -12246.588

Number of free parameters 17 17

Akaike (AIC) 24533.444 24533.444

Bayesian (BIC) 24627.939 24627.939

Sample-size adjusted Bayesian (BIC) 24573.929 24573.929

Root Mean Square Error of Approximation:

RMSEA 0.024 0.023

90 Percent Confidence Interval 0.000 0.050 0.000 0.049

P-value RMSEA <= 0.05 0.947 0.958

Standardized Root Mean Square Residual:

SRMR 0.018 0.018

Parameter Estimates:

Information Expected

Standard Errors Robust.sem

Latent Variables:

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

social_well_being =~

faces_total_sa 1.000 1.000 0.998

faces_total_sc 1.175 0.049 23.987 0.000 1.175 1.175

optimism =~

opt_total_s 1.000 1.042 1.042

stress_in_life =~

les_total_s 1.000 1.000 1.000

mental_health =~

ghq_total_s 1.000 1.146 1.148

Regressions:

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

mental_health ~

social_wll_bng 0.224 0.022 10.259 0.000 0.195 0.195

optimism 0.435 0.024 18.361 0.000 0.395 0.395

stress_in_life 0.042 0.019 2.232 0.026 0.037 0.037

optimism ~

social_wll_bng 0.291 0.024 12.075 0.000 0.280 0.280

Covariances:

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

faces_total_sa ~~

faces_total_sc -0.418 0.063 -6.635 0.000 -0.418 -10.061

les_total_s -0.066 0.014 -4.583 0.000 -0.066 -97.005

Intercepts:

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

faces_total_sa -0.000 0.023 -0.000 1.000 -0.000 -0.000

faces_total_sc -0.000 0.023 -0.000 1.000 -0.000 -0.000

opt_total_s 0.000 0.023 0.000 1.000 0.000 0.000

les_total_s -0.000 0.023 -0.000 1.000 -0.000 -0.000

ghq_total_s 0.000 0.023 0.000 1.000 0.000 0.000

social_wll_bng 0.000 0.000 0.000

optimism 0.000 0.000 0.000

stress_in_life 0.000 0.000 0.000

mental_health 0.000 0.000 0.000

Variances:

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

faces_total_sa 0.005 0.035 0.129 0.897 0.005 0.005

faces_total_sc -0.381 0.119 -3.208 0.001 -0.381 -0.381

opt_total_s -0.085 0.033 -2.580 0.010 -0.085 -0.085

les_total_s -0.000 0.051 -0.002 0.998 -0.000 -0.000

ghq_total_s -0.317 0.024 -13.013 0.000 -0.317 -0.318

social_wll_bng 1.000 1.000 1.000

optimism 1.000 0.922 0.922

stress_in_life 1.000 1.000 1.000

mental_health 1.000 0.761 0.761

Would appreciate if anyone can point out what's wrong with my analysis such that I have the error messages as prompted by lavaan.

Thank you.

Edward Rigdon

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Feb 22, 2016, 11:40:10 AM2/22/16

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Both of the warning messages are tied to the parameter estimates listed under "Variances" in the output. Note that the (residual) variance estimates for several of your observed variables are negative. Even one negative variance will typically make the matrix of residual covariances be not positive definite,so both messages are really telling you about the same thing.

There is a conflict in your syntax. You specify that the first loading for each factor must be fixed at 1, as in:

social_well_being =~ 1*faces_total_sa + faces_total_sc

. This constrains the variance of the associated factor to be equal to variance(y) - residual variance(y), where y is the observed variable whose loading is fixed to 1. But you also specify that the variance of each factor must be equal to 1, as in:

social_well_being ~~ 1*social_well_being

This is a conflict. .Where the factor is dependent on another factor, this specification affects the residual variance of the factor, ratehr than the total variance, but it is still likely to be inconfict with the loading-fixed-to-1 specification.

My advice is to delete the lines of code where you set factor variances (or residual variances) to 1:

# factor variances

social_well_being ~~ 1*social_well_being

optimism ~~ 1*optimism

stress_in_life ~~ 1*stress_in_life

mental_health ~~ 1*mental_health

Comment out those lines (or delete them), and I believe this particular issue will be removed.

--Ed Rigdon

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Chris

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Feb 22, 2016, 9:45:30 PM2/22/16

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Hi Edward,

Thank you for the reply. I tried to remove the factor variances like you suggested but end up with a different warning message instead,

optimism_model_1B = '#latent variables

social_well_being =~ 1*faces_total_sa + faces_total_sc

optimism =~ 1*opt_total_s

stress_in_life =~ 1*les_total_s

mental_health =~ 1*ghq_total_s

#Regression

mental_health ~ social_well_being + optimism + stress_in_life

optimism ~ social_well_being

# residual variances of manifest variables

faces_total_sa ~~ faces_total_sa

faces_total_sc ~~ faces_total_sc

opt_total_s ~~ opt_total_s

les_total_s ~~ les_total_s

ghq_total_s ~~ ghq_total_s

# factor covariances

faces_total_sa ~~ faces_total_sc

faces_total_sa ~~ les_total_s'

Warning messages:

1: In lav_model_vcov(lavmodel = lavmodel, lavsamplestats = lavsamplestats, :

lavaan WARNING: could not compute standard errors!

lavaan NOTE: this may be a symptom that the model is not identified.

2: In lav_model_test(lavmodel = lavmodel, lavpartable = lavpartable, :

lavaan WARNING: could not compute scaled test statistic

I also tried to not restrict the factor variances to 1 but rather letting the model freely estimate them. In that scenario, I obtain just 1 warning message.

optimism_model_1B = '#latent variables

social_well_being =~ 1*faces_total_sa + faces_total_sc

optimism =~ 1*opt_total_s

stress_in_life =~ 1*les_total_s

mental_health =~ 1*ghq_total_s

#Regression

mental_health ~ social_well_being + optimism + stress_in_life

optimism ~ social_well_being

# residual variances of manifest variables

faces_total_sa ~~ faces_total_sa

faces_total_sc ~~ faces_total_sc

opt_total_s ~~ opt_total_s

les_total_s ~~ les_total_s

ghq_total_s ~~ ghq_total_s

# factor variances

social_well_being ~~ social_well_being

optimism ~~ optimism

stress_in_life ~~ stress_in_life

mental_health ~~ mental_health

# factor covariances

faces_total_sa ~~ faces_total_sc

faces_total_sa ~~ les_total_s'

Warning message:

In lav_model_vcov(lavmodel = lavmodel, lavsamplestats = lavsamplestats, :

lavaan WARNING: could not compute standard errors!

lavaan NOTE: this may be a symptom that the model is not identified.

I am just thinking whether it could be because I have a low number of indicator variables for each latent variable because I read somewhere that there is a recommended number of indicators per latent variable for proper solutions and model identification?

Edward Rigdon

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Feb 22, 2016, 10:28:12 PM2/22/16

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Chris--

What function are you using? If you are using the lavaan function, then you would nee to fix residual variances for the observed variables that are single indicators for their factors (the indicators for optimism, stress in life and mental health). If you were using the sem function, then by default these residual variances would be fixed to 0--though you could set them to different values. With the lavaan function, I can see how both fixing first loadings and fixing factor variances could conceal the identification problem. But if you just switch to the sem function, I believe that will eliminate the warning about an identification problem.

--Ed Rigdon

Chris

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Feb 23, 2016, 5:28:32 AM2/23/16

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Hi Edward,

I was initially using the lavaan function, but tried to switch to the sem function as you suggested and left the residual variances specifications to the automated default by sem.

optimism_model_1B = '#latent variables

social_well_being =~ faces_total_sa + faces_total_sc

optimism =~ opt_total_s

stress_in_life =~ les_total_s

mental_health =~ ghq_total_s

#Regression

mental_health ~ social_well_being + optimism + stress_in_life

optimism ~ social_well_being'

fit = sem(optimism_model_1B, data = student, estimator ="MLM", likelihood = "wishart")

Initially the warning messages disappear, but as I tried to introduce some factor covariances through modification indices, one warning message struck back

optimism_model_1B = '#latent variables

social_well_being =~ faces_total_sa + faces_total_sc

optimism =~ opt_total_s

stress_in_life =~ les_total_s

mental_health =~ ghq_total_s

#Regression

mental_health ~ social_well_being + optimism + stress_in_life

optimism ~ social_well_being

#factor covariances

faces_total_sa ~~ les_total_s'

Warning message:

In lav_object_post_check(lavobject) :

lavaan WARNING: observed variable error term matrix (theta) is not positive definite; use inspect(fit,"theta") to investigate.

Am I still doing something wrong in my model specification? Thanks so much for your help thus far.

Edward Rigdon

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Feb 23, 2016, 9:28:39 AM2/23/16

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Chris--

In your new model syntax, the lines:

# factor covariances

faces_total_sa ~~ faces_total_sc

faces_total_sa ~~ les_total_s'

refer to observed variables, not factors. In fact, since these variables are dependent on factors, these lines refer to the residual variances of these observed variables. Your ability to allow free residual covariances is limited by the factor model that you are attempting to estimate. The only common factors in your model are social well being, optimism, stress in life and mental health. Some of the covariances or residual covariances among these common factors are free by default. Some cannot be freed without wrecking the regression relationships that you are trying to estimate.

--Ed Rigdon

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