Non Positive Definite Matrix
Message/Author
 Erik Thoonen posted on Thursday, October 20, 2005 - 2:18 pm
At the moment i'm doing a CFA (first- and second-order) for 23 items which should reflect three factors:

------

ANALYSIS: TYPE = GENERAL MISSING H1;
ESTIMATOR = ML;
MODEL: gedvis BY gv1* gv2 gv3 gv4 gv5 gv6 gv7 gv8;
partic BY par1* par5 par6 par7 par8;
samenw BY sam1* sam2 sam3 sam4 sam5 sam6 sam7 sam8 sam9;
gedvis@1 partic@1 samenw@1;
gedvis WITH partic samenw;
partic WITH samenw;
OUTPUT: MODINDICES RESIDUAL stand;
----

Altough this syntax did work well for my data of 2003, running this syntax for 2004 gives me the next error:

NO CONVERGENCE IN THE MISSING DATA ESTIMATION OF THE SAMPLE STATISTICS. THIS MAY BE DUE TO SPARSE DATA LEADING TO A SINGULAR COVARIANCE MATRIX ESTIMATE.

After listwise deletion, I did run it again, got the same error and so runned my syntax just for "cooperation" (=SAM). Too i used the BASIC-command and did get the next error:

RESULTS FOR BASIC ANALYSIS

ESTIMATED SAMPLE STATISTICS
THE MISSING DATA EM ALGORITHM FOR THE H1 MODEL HAS NOT CONVERGED DUE TO ESTIMATED COVARIANCE MATRIX BEING NON POSITIVE DEFINITE.
PROBLEM INVOLVING VARIABLE: 5

I understand SAM5 is here the problem, but leaving out this variable brings me to SAM8 as problem involving variable. Leaving this one out too, the problems are solved (as i can estimate now).

What is/could be the main problem of this error (especially the last one)? What is wrong with my data? And how can i solve it?

Thanks!
 Linda K. Muthen posted on Thursday, October 20, 2005 - 6:30 pm
 ClaudiaBergomi posted on Friday, February 08, 2013 - 4:04 am
Hello,

I am running the following model:
MODEL: AKZ BY Akz1 Akz2;
GEi BY GewI1 GewI2;
GEA BY GewA1 GewA2;
BEW BY Bewu1 Bewu2;
NRE BY Nrea1 Nrea2;
EXP BY Expo1 Expo2;
REL BY Rela1 Rela2;
EIN BY eins1 eins2;
gew by gei gea;
atti by EXP AKZ NRE REL EIN;
atte by BEW gew;
atte with atti;

in three independent samples (n=300, n=200 and n=160). In two of them the model works (i.e. I do not get any error message and all coefficients look fine) but in the n=300 sample I get the following error message.

WARNING: THE LATENT VARIABLE COVARIANCE MATRIX (PSI) IS NOT POSITIVE
DEFINITE. THIS COULD INDICATE A NEGATIVE VARIANCE/RESIDUAL VARIANCE FOR A
LATENT VARIABLE, A CORRELATION GREATER OR EQUAL TO ONE BETWEEN TWO LATENT
VARIABLES, OR A LINEAR DEPENDENCY AMONG MORE THAN TWO LATENT VARIABLES.
PROBLEM INVOLVING VARIABLE ATTENT.

Since the model works in two samples I guess the problem does not have to do with the model itself. How could I fix the problem?

Thanks.
 Linda K. Muthen posted on Friday, February 08, 2013 - 5:55 am
It sounds like the model is not correct for the sample of 300. Look at TECH4 to see what the problem is. It is likely that attent has a negative residual variance.
 ClaudiaB posted on Monday, February 11, 2013 - 2:50 am
I think the problem is that ATTENT and ATTITU have a correlation greater than 1.

According to the fit indices, it is the model that suits best to the data (in all three samples) when compared to other hierarchical models. Even in the sample of 300, despite the error message, the Chi square is significantly better than in the comparison models. Thus the results seem to suggest that this is the model that I should retain.

Is there some other way I can calculate this model in this sample?
 Linda K. Muthen posted on Monday, February 11, 2013 - 6:44 am
When two variables correlate greater than one, they cannot both be kept in the model. Any fit statistics and results should not be interpreted in this case. It appears the same model is not appropriate for this sample.
 ClaudiaB posted on Tuesday, February 12, 2013 - 3:55 am
Thank you Linda.

The model is actually giving the same error even in the sample of 200, I overlooked that before. It thus works only in the sample of 160. The correlation which is greater than 1 in both other samples (200 and 300), is very near to 1 (.97) even in the sample of 160. Which makes me suspicious with reference to the model.

I checked that the error does not have to do with the small number of observed variables and it does not: I get the same error when I calculate the model from the items (4 to 5 per factor instead of 2 parcels).

I am puzzled because the model is theoretically meaningful and would be consistent with previous results. I am not sure if the model just does not fit for my samples or if I did some mistake in the model specification. I would be glad for any tip about possible causes I could check. Thank you.
 Linda K. Muthen posted on Tuesday, February 12, 2013 - 6:08 am
It sounds like you should try an EFA to see if the CFA specified is reasonable for the data. It sounds like it is not. You sample sizes are small for the size of your model.
 Raffaele Zanoli posted on Sunday, March 15, 2015 - 5:36 pm
I have been running a CFA on data from different countries. In one country, the model (which is a standard TPB model) continues to give problems of positive definite matrix). Dropping some variable does not help neither adding WITH statemnts.
Inspecting TECH4 one factor (PBC) has correlation above 1 with other two factors. Which means it probably they are not distinguishable or dependent.

In my acse CFA is just preliminary to SEM, where this dependencies are indeed modelled. In fact, when I run the same model as SEM adding ON statements the problem resolves. My question is:
1) is it legitimate to run a full SEM model if the CFA (measurement) exhibits problems of non positive definite PSI? Or, put in other words, when you have a full SEM model in mind, should one start directly with that or should before run a separate CFA model only?
2) when I test for the country measurement invariance using MODEL: CONFIGURAL METRIC, I obviously get the same error message for that specific country. However, since I cannot model ON variables using MODEL, it means that I CANNOT test equivalence of a causal structure in that way, as is quite clear reading the manual ("only BY statements are allowed", "no partial measurement invariance is allowed"). To do that I need to run two separate models for the configural and metric model and do the testing, or are there any shortcuts?
 Bengt O. Muthen posted on Sunday, March 15, 2015 - 7:54 pm
1) I think it is a good idea to do CFA on the measurement part first. I would investigate why SEM doesn't show the high correlation when CFA does.

2) You can yourself set up the restrictons for a configural, metric, or scalar model for the measurement part of a SEM. And do the testing by comparing two runs.
 Raffaele Zanoli posted on Monday, March 16, 2015 - 3:51 am
This is indeed what I knew from my training!
However, here is the full SEM model:
ANALYSIS:
ESTIMATOR IS MLM;
MODEL:
BENEFITS BY v4-v6;
PBC BY v7-v9;
MN BY v10-v12;
SN BY v12-v15;
ATU BY v16-v18;
ATU ON BENEFITS;
v19 ON ATU PBC MN SN;
When I run CFA (without the ON statements) I get:

ESTIMATED CORRELATION MATRIX FOR THE LATENT VARIABLES
BENEFITS PBC MN SN ATU
________ ________ ________ ________ ________
BENEFITS 1.000
PBC 1.001 1.000
MN 0.868 0.937 1.000
SN 0.670 1.028 0.681 1.000
ATU 0.917 0.956 0.880 0.632 1.000

When I run the SEM model I get:

ESTIMATED CORRELATION MATRIX FOR THE LATENT VARIABLES
BENEFITS PBC MN SN ATU
________ ________ ________ ________ ________
BENEFITS 1.000
PBC 0.827 1.000
MN 0.890 0.784 1.000
SN 0.674 0.853 0.681 1.000
ATU 0.934 0.772 0.831 0.630 1.000
V19 0.754 0.826 0.714 0.623 0.719

Any explanation?
 Bengt O. Muthen posted on Monday, March 16, 2015 - 9:14 am
The explanation is that the SEM model imposes strong further restrictions on the relationships among the variables. It says that ATU is influenced only by Benefits and not by the other factors. That could be a misspecification that changes the correlations. If you just-identify your structural part of the model, the problem will re-occur.
 Alexander Tokarev posted on Friday, March 31, 2017 - 8:18 am
Dear Professors Muthen,

When one runs CFA, and the output highlights non-positive definite covariance matrix problem,

1) Does the fit get affected by this?

2) Is it even meaningful to interpret the fit indices?

3) Can I have trust in the correctness of item loadings, and their significance?

Kind regards

Alex
 Linda K. Muthen posted on Friday, March 31, 2017 - 1:48 pm
The results are not interpretable. You should send the output and your license number to support@statmodel.com.