Dear Ruijie,
Your model is underidentified by virtue of two of the factors having only
one observed indicator each. No SEM software can magically estimate this
model as it stands. Beyond that, I won't comment on the wisdom of what
you're doing, such as computing covariances between ordinal variables -- but
see what I discovered below.
Removing these two variables and the associated factors produces the
following model:
--------- snip ------------
I01 I02 I03 I04 I05 I06 I07
0.09794939 0.09239769 0.08647698 0.40592964 0.14988296 0.34276336 0.09257290
I08 I09 I10 I11 I12 I13 I14
0.26288788 0.05673501 0.21562354 0.40159670 0.19999190 0.14969750 0.48787040
I15 I16 I17 I18 I19 I20 I21
0.03279129 0.11746460 0.20339207 0.03279129 0.22450179 0.49285671 0.11291786
I22 I23 I24 I25 I26 I27 I28
0.51844236 0.03279129 0.17390500 0.20982058 0.28746674 0.18587268 0.03279129
I29 I30 I31 I32 I33 I34 I35
0.11789736 0.19121352 0.44618622 0.24132578 0.50500808 0.12183229 0.08647698
I36 I37 I38 I39 I40 I41 I42
0.14183651 0.03279129 0.20705800 0.36721084 0.51768833 0.29210990 0.28739426
I43 I45 I46 I47 I48 I49 I50
0.11746460 0.13454976 0.43680464 0.09794939 0.52139099 0.05673501 0.09239769
I51 I54 I55 I56 I57 I58 I59
0.03279129 0.43984267 0.26013269 0.36354251 0.30622933 0.53958761 0.36898429
I60 I61 I62 I63 I64 I65 I66
0.16867489 0.22011795 0.08663745 0.27761032 0.07329198 0.52861343 0.24514452
I67 I68 I69 I70 I71
0.03279129 0.16616880 0.33665601 0.17020504 0.19965594
--------- snip ------------
Some of the standard deviations are very small, suggesting that the
corresponding variables must have been close to invariant in your data set.
If you haven't already done so, I think that you might back up and look more
closely at your data, and perhaps seek some competent local help.
I hope that this helps,
John
-----------------------------------------------
John Fox
Senator McMaster Professor of Social Statistics
Department of Sociology
McMaster University
Hamilton, Ontario, Canada
-----Original Message-----
From: r-help-bounces at r-project.org [mailto:r-help-bounces at r-project.org]
On Behalf Of Ruijie
Sent: Friday, February 08, 2013 9:56 PM
To: R-help at stat.math.ethz.ch
Subject: [R] Troubleshooting underidentification issues in structural
equation modelling (SEM)
Hi all, hope someone can help me out with this.
Background Introduction
I have a data set consisting of data collected from a questionnaire that
I
wish to validate. I have chosen to use confirmatory factor analysis to
analyse this data set.
Instrument
The instrument consists of 11 subscales. There is a total of 68 items in
the 11 subscales. Each item is scored on an integer scale between 1 to
4.
Confirmatory factor analysis (CFA) setup
I use the sem package to conduct the CFA. My code is as below:
cov.mat <-
as.matrix(read.table("http://dl.dropbox.com/u/1445171/cov.mat.csv",
sep = ",", header = TRUE))
rownames(cov.mat) <- colnames(cov.mat)
model <- cfa(file = "http://dl.dropbox.com/u/1445171/cfa.model.txt",
reference.indicators = FALSE)
cfa.output <- sem(model, cov.mat, N = 900, maxiter = 80000, optimizer
= optimizerOptim)
Warning message:In eval(expr, envir, enclos) : Negative parameter
variances.Model may be underidentified.
Straight off you might notice a few anomalies, let me explain.
- Why is the optimizer chosen to be optimizerOptim?
ANS: I originally stuck with the default optimizerSem but no matter how
many iterations I run, either I run out of memory first (8GB RAM setup)
or
it would report no convergence Things "seemed" a little better when I
switched to optimizerOptim where by it would conclude successfully but
throws up the error that the model is underidentified. Upon closer
inspection, I realise that the output shows convergence as TRUE but
iterations is NA so I am not sure what is exactly happening.
- The maxiter is too high.
ANS: If I set it to a lower value, it refuses to converge, although as
mentioned above, I doubt real convergence actually occurred.
Problem
So by now I guess that the model is really underidentified so I looked
for
resources to resolve this problem and found:
- http://davidakenny.net/cm/identify_formal.htm
- http://faculty.ucr.edu/~hanneman/soc203b/lectures/identify.html
I followed the 2nd link quite closely and applied the t-rule:
- I have 68 observed variables, providing me with 68 variances and
2278
covariances between variables = *2346 data points*.
- I also have 68 regression coefficients, 68 error variances of
variables, 11 factor variances and 55 factor covariances to estimate
making
it a total of 191 parameters.
- Since I will be fixing the variances of the 11 latent factors to 1
for
scaling, I would remove them from the parameters to estimate making
it a
total of *180 parameters to estimate*.
- My degrees of freedom is therefore 2346 - 180 = 2166, making it
an
over identified model by the t-rule.
Questions
1. Is the low variance of some of my items a possible cause for the
underidentification? I was advised previously to remove items with
zero
variance which led me to think about items which are very close to
zero.
Should they be removed too?
2. After reading much, I think but am not sure that it might be a
case
of empirical underidentification. Is there a systematic way of
diagnosing
what kind of underidentification it is? And what are my options to
proceed
with my analysis?
I have more questions but let's take it at these 2 for now. Thanks for
any
help!
Regards,
Ruijie (RJ)
--------
He who has a why can endure any how.
~ Friedrich Nietzsche
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