Is a model still usable if there are highly correlated indicators, low sample, negative eigenvalues?

[J V]

I'm trying to fit a model (previously described here) and I'm getting a bunch of warnings:

Warning messages:
1: In lav_data_full(data = data, group = group, cluster = cluster,  :
  lavaan WARNING: small number of observations (nobs < nvar)
  nobs = 93 nvar = 112
2: In lav_samplestats_step2(UNI = FIT, ov.names = ov.names, zero.add = zero.add,  :
  lavaan WARNING: correlation between variables q0069_0010par and q0051_0007par is (nearly) 1.0
3: 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
    (= 9.507439e-16) is close to zero. This may be a symptom that the
    model is not identified.

The model I'm trying to fit is fairly big, so I'm putting it in a separate post below.

My main concern is whether any of the above warnings imply that the resulting estimates are wrong in some big way. I can get error-free fits with certain subsets of factors and the results remain consistent all the way to the the full model, so it seems to me like the estimates should still be usable. And if not, I'd like to know what changes would be needed to be made to remedy this.

[J V]

The call to lavaan:

lavaan(lavmod, mydata, ordered = vars[-c(1,61)], int.ov.free = T, int.lv.free = F, std.lv = T, auto.var = T, auto.delta = F, parameterization = "delta")

The model:

# Intimacy
Intim  =~ b_4601*q0046_0001+b_4602*q0046_0002+b_4603*q0046_0003+b_4604*q0046_0004+b_4605*q0046_0005+b_4606*q0046_0006+b_4607*q0046_0007+b_4608*q0046_0008+b_4609*q0046_0009+b_4610*q0046_0010+b_4611*q0046_0011
Intimp =~ b_4601*q0046_0001par+b_4602*q0046_0002par+b_4603*q0046_0003par+b_4604*q0046_0004par+b_4605*q0046_0005par+b_4606*q0046_0006par+b_4607*q0046_0007par+b_4608*q0046_0008par+b_4609*q0046_0009par+b_4610*q0046_0010par+b_4611*q0046_0011par
q0046_0001| q04601_1t*t1 + q04601_2t*t2 + q04601_3t*t3
q0046_0002| q04602_1t*t1 + q04602_2t*t2 + q04602_3t*t3
q0046_0003| q04603_1t*t1 + q04603_2t*t2 + q04603_3t*t3
q0046_0004| q04604_1t*t1 + q04604_2t*t2 + q04604_3t*t3
q0046_0005| q04605_1t*t1 + q04605_2t*t2 + q04605_3t*t3
q0046_0006| q04606_1t*t1 + q04606_2t*t2 + q04606_3t*t3
q0046_0007| q04607_1t*t1 + q04607_2t*t2 + q04607_3t*t3
q0046_0008| q04608_1t*t1 + q04608_2t*t2 + q04608_3t*t3
q0046_0009| q04609_1t*t1 + q04609_2t*t2 + q04609_3t*t3
q0046_0010| q04610_1t*t1 + q04610_2t*t2 + q04610_3t*t3
q0046_0011| q04611_1t*t1 + q04611_2t*t2 + q04611_3t*t3
q0046_0001par| q04601_1t*t1 + q04601_2t*t2 + q04601_3t*t3
q0046_0002par| q04602_1t*t1 + q04602_2t*t2 + q04602_3t*t3
q0046_0003par| q04603_1t*t1 + q04603_2t*t2 + q04603_3t*t3
q0046_0004par| q04604_1t*t1 + q04604_2t*t2 + q04604_3t*t3
q0046_0005par| q04605_1t*t1 + q04605_2t*t2 + q04605_3t*t3
q0046_0006par| q04606_1t*t1 + q04606_2t*t2 + q04606_3t*t3
q0046_0007par| q04607_1t*t1 + q04607_2t*t2 + q04607_3t*t3
q0046_0008par| q04608_1t*t1 + q04608_2t*t2 + q04608_3t*t3
q0046_0009par| q04609_1t*t1 + q04609_2t*t2 + q04609_3t*t3
q0046_0010par| q04610_1t*t1 + q04610_2t*t2 + q04610_3t*t3
q0046_0011par| q04611_1t*t1 + q04611_2t*t2 + q04611_3t*t3
Intim ~~ Intimp
# Satisfaction w/  Relationship
Satisf  =~ b_4701*q0047_0001+b_4702*q0047_0002+b_4703*q0047_0003+b_4704*q0047_0004+b_4705*q0047_0005+b_4706*q0047_0006+b_4707*q0047_0007
Satisfp =~ b_4701*q0047_0001par+b_4702*q0047_0002par+b_4703*q0047_0003par+b_4704*q0047_0004par+b_4705*q0047_0005par+b_4706*q0047_0006par+b_4707*q0047_0007par
q0047_0001| q04701_1t*t1 + q04701_2t*t2 + q04701_3t*t3 + q04701_4t*t4
q0047_0002| q04702_1t*t1 + q04702_2t*t2 + q04702_3t*t3 + q04702_4t*t4
q0047_0003| q04703_1t*t1 + q04703_2t*t2 + q04703_3t*t3 + q04703_4t*t4
q0047_0004| q04704_1t*t1 + q04704_2t*t2 + q04704_3t*t3 + q04704_4t*t4
q0047_0005| q04705_1t*t1 + q04705_2t*t2 + q04705_3t*t3 + q04705_4t*t4
q0047_0006| q04706_1t*t1 + q04706_2t*t2 + q04706_3t*t3 + q04706_4t*t4
q0047_0007| q04707_1t*t1 + q04707_2t*t2 + q04707_3t*t3 + q04707_4t*t4
q0047_0001par| q04701_1t*t1 + q04701_2t*t2 + q04701_3t*t3 + q04701_4t*t4
q0047_0002par| q04702_1t*t1 + q04702_2t*t2 + q04702_3t*t3 + q04702_4t*t4
q0047_0003par| q04703_1t*t1 + q04703_2t*t2 + q04703_3t*t3 + q04703_4t*t4
q0047_0004par| q04704_1t*t1 + q04704_2t*t2 + q04704_3t*t3 + q04704_4t*t4
q0047_0005par| q04705_1t*t1 + q04705_2t*t2 + q04705_3t*t3 + q04705_4t*t4
q0047_0006par| q04706_1t*t1 + q04706_2t*t2 + q04706_3t*t3 + q04706_4t*t4
q0047_0007par| q04707_1t*t1 + q04707_2t*t2 + q04707_3t*t3 + q04707_4t*t4
Satisf ~~ Satisfp
# Commitment
Commit  =~ b_5101*q0051_0001+b_5102*q0051_0002+b_5103*q0051_0003+b_5104*q0051_0004+b_5105*q0051_0005+b_5106*q0051_0006+b_5107*q0051_0007+b_5108*q0051_0008
Commitp =~ b_5101*q0051_0001par+b_5102*q0051_0002par+b_5103*q0051_0003par+b_5104*q0051_0004par+b_5105*q0051_0005par+b_5106*q0051_0006par+b_5107*q0051_0007par+b_5108*q0051_0008par
q0051_0001| q05101_1t*t1 + q05101_2t*t2
q0051_0002| q05102_1t*t1 + q05102_2t*t2
q0051_0003| q05103_1t*t1 + q05103_2t*t2
q0051_0004| q05104_1t*t1 + q05104_2t*t2
q0051_0005| q05105_1t*t1 + q05105_2t*t2
q0051_0006| q05106_1t*t1 + q05106_2t*t2
q0051_0007| q05107_1t*t1 + q05107_2t*t2
q0051_0008| q05108_1t*t1 + q05108_2t*t2
q0051_0001par| q05101_1t*t1 + q05101_2t*t2
q0051_0002par| q05102_1t*t1 + q05102_2t*t2
q0051_0003par| q05103_1t*t1 + q05103_2t*t2
q0051_0004par| q05104_1t*t1 + q05104_2t*t2
q0051_0005par| q05105_1t*t1 + q05105_2t*t2
q0051_0006par| q05106_1t*t1 + q05106_2t*t2
q0051_0007par| q05107_1t*t1 + q05107_2t*t2
q0051_0008par| q05108_1t*t1 + q05108_2t*t2
Commit ~~ Commitp
# Trust
Trust  =~ b_5801*q0058_0001+b_5802*q0058_0002+b_5803*q0058_0003+b_5804*q0058_0004+b_5805*q0058_0005+b_5806*q0058_0006+b_5807*q0058_0007+b_5808*q0058_0008+b_5809*q0058_0009+b_5810*q0058_0010+b_5811*q0058_0011+b_5812*q0058_0012+b_5813*q0058_0013+b_5814*q0058_0014+b_5815*q0058_0015+b_5816*q0058_0016+b_5817*q0058_0017
Trustp =~ b_5801*q0058_0001par+b_5802*q0058_0002par+b_5803*q0058_0003par+b_5804*q0058_0004par+b_5805*q0058_0005par+b_5806*q0058_0006par+b_5807*q0058_0007par+b_5808*q0058_0008par+b_5809*q0058_0009par+b_5810*q0058_0010par+b_5811*q0058_0011par+b_5812*q0058_0012par+b_5813*q0058_0013par+b_5814*q0058_0014par+b_5815*q0058_0015par+b_5816*q0058_0016par+b_5817*q0058_0017par
q0058_0001| q05801_1t*t1 + q05801_2t*t2 + q05801_3t*t3 + q05801_4t*t4 + q05801_5t*t5
q0058_0002| q05802_1t*t1 + q05802_2t*t2 + q05802_3t*t3 + q05802_4t*t4 + q05802_5t*t5
q0058_0003| q05803_1t*t1 + q05803_2t*t2 + q05803_3t*t3 + q05803_4t*t4 + q05803_5t*t5
q0058_0004| q05804_1t*t1 + q05804_2t*t2 + q05804_3t*t3 + q05804_4t*t4 + q05804_5t*t5
q0058_0005| q05805_1t*t1 + q05805_2t*t2 + q05805_3t*t3 + q05805_4t*t4 + q05805_5t*t5
q0058_0006| q05806_1t*t1 + q05806_2t*t2 + q05806_3t*t3 + q05806_4t*t4 + q05806_5t*t5
q0058_0007| q05807_1t*t1 + q05807_2t*t2 + q05807_3t*t3 + q05807_4t*t4 + q05807_5t*t5
q0058_0008| q05808_1t*t1 + q05808_2t*t2 + q05808_3t*t3 + q05808_4t*t4 + q05808_5t*t5
q0058_0009| q05809_1t*t1 + q05809_2t*t2 + q05809_3t*t3 + q05809_4t*t4 + q05809_5t*t5
q0058_0010| q05810_1t*t1 + q05810_2t*t2 + q05810_3t*t3 + q05810_4t*t4 + q05810_5t*t5
q0058_0011| q05811_1t*t1 + q05811_2t*t2 + q05811_3t*t3 + q05811_4t*t4 + q05811_5t*t5
q0058_0012| q05812_1t*t1 + q05812_2t*t2 + q05812_3t*t3 + q05812_4t*t4 + q05812_5t*t5
q0058_0013| q05813_1t*t1 + q05813_2t*t2 + q05813_3t*t3 + q05813_4t*t4 + q05813_5t*t5
q0058_0014| q05814_1t*t1 + q05814_2t*t2 + q05814_3t*t3 + q05814_4t*t4 + q05814_5t*t5
q0058_0015| q05815_1t*t1 + q05815_2t*t2 + q05815_3t*t3 + q05815_4t*t4 + q05815_5t*t5
q0058_0016| q05816_1t*t1 + q05816_2t*t2 + q05816_3t*t3 + q05816_4t*t4 + q05816_5t*t5
q0058_0017| q05817_1t*t1 + q05817_2t*t2 + q05817_3t*t3 + q05817_4t*t4 + q05817_5t*t5
q0058_0001par| q05801_1t*t1 + q05801_2t*t2 + q05801_3t*t3 + q05801_4t*t4 + q05801_5t*t5
q0058_0002par| q05802_1t*t1 + q05802_2t*t2 + q05802_3t*t3 + q05802_4t*t4 + q05802_5t*t5
q0058_0003par| q05803_1t*t1 + q05803_2t*t2 + q05803_3t*t3 + q05803_4t*t4 + q05803_5t*t5
q0058_0004par| q05804_1t*t1 + q05804_2t*t2 + q05804_3t*t3 + q05804_4t*t4 + q05804_5t*t5
q0058_0005par| q05805_1t*t1 + q05805_2t*t2 + q05805_3t*t3 + q05805_4t*t4 + q05805_5t*t5
q0058_0006par| q05806_1t*t1 + q05806_2t*t2 + q05806_3t*t3 + q05806_4t*t4 + q05806_5t*t5
q0058_0007par| q05807_1t*t1 + q05807_2t*t2 + q05807_3t*t3 + q05807_4t*t4 + q05807_5t*t5
q0058_0008par| q05808_1t*t1 + q05808_2t*t2 + q05808_3t*t3 + q05808_4t*t4 + q05808_5t*t5
q0058_0009par| q05809_1t*t1 + q05809_2t*t2 + q05809_3t*t3 + q05809_4t*t4 + q05809_5t*t5
q0058_0010par| q05810_1t*t1 + q05810_2t*t2 + q05810_3t*t3 + q05810_4t*t4 + q05810_5t*t5
q0058_0011par| q05811_1t*t1 + q05811_2t*t2 + q05811_3t*t3 + q05811_4t*t4 + q05811_5t*t5
q0058_0012par| q05812_1t*t1 + q05812_2t*t2 + q05812_3t*t3 + q05812_4t*t4 + q05812_5t*t5
q0058_0013par| q05813_1t*t1 + q05813_2t*t2 + q05813_3t*t3 + q05813_4t*t4 + q05813_5t*t5
q0058_0014par| q05814_1t*t1 + q05814_2t*t2 + q05814_3t*t3 + q05814_4t*t4 + q05814_5t*t5
q0058_0015par| q05815_1t*t1 + q05815_2t*t2 + q05815_3t*t3 + q05815_4t*t4 + q05815_5t*t5
q0058_0016par| q05816_1t*t1 + q05816_2t*t2 + q05816_3t*t3 + q05816_4t*t4 + q05816_5t*t5
q0058_0017par| q05817_1t*t1 + q05817_2t*t2 + q05817_3t*t3 + q05817_4t*t4 + q05817_5t*t5
Trust ~~ Trustp
# Risky Sexual Behavior
SexRisk  =~ b_6901*q0069_0001+b_6902*q0069_0002+b_6903*q0069_0003+b_6904*q0069_0004+b_6905*q0069_0005+b_6906*q0069_0006+b_6907*q0069_0007+b_6908*q0069_0008+b_6909*q0069_0009+b_6910*q0069_0010+b_6911*q0069_0011+b_6912*q0069_0012+b_6913*q0069_0013
SexRiskp =~ b_6901*q0069_0001par+b_6902*q0069_0002par+b_6903*q0069_0003par+b_6904*q0069_0004par+b_6905*q0069_0005par+b_6906*q0069_0006par+b_6907*q0069_0007par+b_6908*q0069_0008par+b_6909*q0069_0009par+b_6910*q0069_0010par+b_6911*q0069_0011par+b_6912*q0069_0012par+b_6913*q0069_0013par
q0069_0001| q06901_1t*t1 + q06901_2t*t2 + q06901_3t*t3
q0069_0002| q06902_1t*t1 + q06902_2t*t2 + q06902_3t*t3
q0069_0003| q06903_1t*t1 + q06903_2t*t2 + q06903_3t*t3
q0069_0004| q06904_1t*t1 + q06904_2t*t2 + q06904_3t*t3
q0069_0006| q06906_1t*t1 + q06906_2t*t2 + q06906_3t*t3
q0069_0007| q06907_1t*t1 + q06907_2t*t2 + q06907_3t*t3
q0069_0009| q06909_1t*t1 + q06909_2t*t2 + q06909_3t*t3
q0069_0010| q06910_1t*t1 + q06910_2t*t2 + q06910_3t*t3
q0069_0011| q06911_1t*t1 + q06911_2t*t2 + q06911_3t*t3
q0069_0012| q06912_1t*t1 + q06912_2t*t2 + q06912_3t*t3
q0069_0013| q06913_1t*t1 + q06913_2t*t2 + q06913_3t*t3
q0069_0001par| q06901_1t*t1 + q06901_2t*t2 + q06901_3t*t3
q0069_0002par| q06902_1t*t1 + q06902_2t*t2 + q06902_3t*t3
q0069_0003par| q06903_1t*t1 + q06903_2t*t2 + q06903_3t*t3
q0069_0004par| q06904_1t*t1 + q06904_2t*t2 + q06904_3t*t3
q0069_0006par| q06906_1t*t1 + q06906_2t*t2 + q06906_3t*t3
q0069_0007par| q06907_1t*t1 + q06907_2t*t2 + q06907_3t*t3
q0069_0009par| q06909_1t*t1 + q06909_2t*t2 + q06909_3t*t3
q0069_0010par| q06910_1t*t1 + q06910_2t*t2 + q06910_3t*t3
q0069_0011par| q06911_1t*t1 + q06911_2t*t2 + q06911_3t*t3
q0069_0012par| q06912_1t*t1 + q06912_2t*t2 + q06912_3t*t3
q0069_0013par| q06913_1t*t1 + q06913_2t*t2 + q06913_3t*t3
q0069_0005    | q06905_1t*t1 + q06905_2t*t2
q0069_0008    | q06908_1t*t1 + q06908_2t*t2
q0069_0005par | q06905_1t*t1 + q06905_2t*t2
q0069_0008par | q06908_1t*t1 + q06908_2t*t2
SexRisk ~~ SexRiskp
SexRisk  ~ ageB*age    + intiB*Intim  + satiB*Satisf  + commB*Commit  + trusB*Trust
SexRiskp ~ ageB*agepar + intiB*Intimp + satiB*Satisfp + commB*Commitp + trusB*Trustp

[Terrence Jorgensen]

The warning messages are pretty clear.  You cannot trust results when you have more variables than observations.  I'm surprised the model would even run.  Results are already questionable when you estimate more parameters than you have observations, and there are certainly more model parameters than variables (at a minimum, there is one (residual) variance per variable, plus parameters representing relationships between variables).  Furthermore, you have categorical data.  Simulations seem to indicate that even when you have more observations than parameters (and certainly more than variables), estimates still do not stabilize until you have moderately large samples (e.g., N >= 500 or 1000).

http://dx.doi.org/10.1080/10705511.2014.859510

http://dx.doi.org/10.1080/10705511.2014.882658

Sorry to say it, but if I were reviewing your paper, I would not even be able to recommend a revision unless you could increase your sample size by an order of magnitude to be able to fit a model this large.

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

[J V]

Oh dear. Thanks for the references.

I realize I don't have enough sample size to reliably test even a fraction of the factors of interest, but I see they all review ML-based estimators. Do you know if the same hard limitations would apply using Bayesian estimation? This is more of a pilot study so even if parameter posteriors are very wide, some approximation of their location would be useful.

[Terrence Jorgensen]

Do you know if the same hard limitations would apply using Bayesian estimation? 

Unlike these frequentist methods, Bayes is not built on asymptotic theory, although more data always helps to draw more confident conclusions.  If there are convergence problems with MCMC estimation, you can help it out by specifying more informative priors, but make sure to perform a sensitivity analysis with different priors to see how much your results depend on them.  Hopefully you will find the blavaan package helpful.