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SEM model testing with identical goodness of fits

10 messages · William Revelle, John Fox, sun

sun
# 
HI,

   I am testing several models about three latent constructs that 
measure risk attitudes.
Two models with different structure obtained identical of fit measures 
from chisqure to BIC.
Model1 assumes three factors are correlated with  each other and model 
two assumes a higher order factor exist and three factors related to 
this higher factor instead of to each other.

Model1:
model.one <- specify.model()
	tr<->tp,e.trtp,NA
	tp<->weber,e.tpweber,NA
	weber<->tr,e.webertr,NA
	weber<->weber, e.weber,NA
	tp<->tp,e.tp,NA
	tr <->tr,e.trv,NA
	....

Model two
model.two <- specify.model()
	rsk->tp,e.rsktp,NA
	rsk->tr,e.rsktr,NA
	rsk->weber,e.rskweber,NA
	rsk<->rsk, NA,1
	weber<->weber, e.weber,NA
	tp<->tp,e.tp,NA
	tr <->tr,e.trv,NA
	 ....

the summary of both sem model gives identical fit indices, using same 
data set.

is there some thing wrong with this mode specification?

Thanks
John Fox
# 
Dear hyena,
On
more restrictive than the first, but that doesn't seem to be the case -- if
the two models have identical log-likelihoods and degrees of freedom, as you
seem to imply, then it's a good bet that the models are observationally
indistinguishable. On the other hand, you don't provide a whole lot of
information; it would have been much more informative had you shown the
input and output for both models.

John
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John Fox
# 
Dear hyena,

Actually, looking at this a bit more closely, the first models dedicate 6
parameters to the correlational and variational structure of the three
variables that you mention -- 3 variances and 3 covariances; the second
model also dedicates 6 parameters -- 3 factor loadings and 3 error variances
(with the variance of the factor fixed as a normalization). You don't show
the remaining structure of the models, but a good guess is that they are
observationally indistinguishable.

John
On
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sun
# 
Dear John,

    Thanks for the prompt reply! Sorry did not supply with more detailed 
information.

    The target model consists of three latent factors, general risk 
scale from Weber's domain risk scales, time perspective scale from 
Zimbardo(only future time oriented) and a travel risk attitude scale. 
Variables with "prob_" prefix are items of general risk scale, variables 
of "o1" to "o12" are items of future time perspective and "v5" to "v13" 
are items of travel risk scale.

  The purpose is to explore or find a best fit model that "correctly" 
represent the underlining relationship of three scales.  So far, the 
correlated model has the best fit indices, so I 'd like to check if 
there is a higher level factor that govern all three factors, thus the 
second model.

  The data are all 5 point Likert scale scores by respondents(N=397). 
The example listed bellow did not show "prob_" variables(their names are 
too long).

   Given the following model structure, if they are indeed 
observationally indistinguishable, is there some possible adjustments to 
test the higher level factor effects?

  Thanks,

###########################
#data example, partial
#########################
                     1                   1                     1        1
  id     o1 o2 o3 o4 o5 o6 o7 o8 o9 o10 o11 o12 o13 v5 v13 v14 v16 v17
14602  2  2  4  4  5  5  2  3  2   4   3   4   2  5   2   2   4   2
14601  2  4  5  4  5  5  2  5  3   4   5   4   5  5   3   4   4   2
14606  1  3  5  5  5  5  3  3  5   3   5   5   5  5   5   5   5   3
14610  2  1  4  5  4  5  3  4  4   2   4   2   1  5   3   5   5   5
14609  4  3  2  2  5  5  2  5  2   4   4   2   2  4   2   4   4   4

####################################
#correlated model, three scales corrlated to each other
model.correlated <- specify.model()
	weber<->tp,e.webertp,NA
	tp<->tr,e.tptr,NA
	tr<->weber,e.trweber,NA
	weber<->weber,NA,1
	tp<->tp,e.tp,NA
	tr <->tr,e.trv,NA
	weber -> prob_wild_camp,alpha2,NA
	weber -> prob_book_hotel_in_short_time,alpha3,NA
	weber -> prob_safari_Kenia, alpha4, NA
	weber -> prob_sail_wild_water,alpha5,NA
	weber -> prob_dangerous_sport,alpha7,NA
	weber -> prob_bungee_jumping,alpha8,NA
	weber -> prob_tornado_tracking,alpha9,NA
	weber -> prob_ski,alpha10,NA
	prob_wild_camp <-> prob_wild_camp, ep2,NA
	prob_book_hotel_in_short_time <-> prob_book_hotel_in_short_time,ep3,NA
	prob_safari_Kenia <-> prob_safari_Kenia, ep4, NA
	prob_sail_wild_water <-> prob_sail_wild_water,ep5,NA
	prob_dangerous_sport <-> prob_dangerous_sport,ep7,NA
	prob_bungee_jumping <-> prob_bungee_jumping,ep8,NA
	prob_tornado_tracking <-> prob_tornado_tracking,ep9,NA
	prob_ski <-> prob_ski,ep10,NA
	tp -> o1,NA,1
	tp -> o3,beta3,NA
	tp -> o4,beta4,NA
	tp -> o5,beta5,NA
	tp -> o6,beta6,NA
	tp -> o7,beta7,NA
	tp -> o9,beta9,NA
	tp -> o10,beta10,NA
	tp -> o11,beta11,NA
	tp -> o12,beta12,NA
	o1 <-> o1,eo1,NA
	o3 <-> o3,eo3,NA
	o4 <-> o4,eo4,NA
	o5 <-> o5,eo5,NA
	o6 <-> o6,eo6,NA
	o7 <-> o7,eo7,NA
	o9 <-> o9,eo9,NA
	o10 <-> o10,eo10,NA
	o11 <-> o11,eo11,NA
	o12 <-> o12,eo12,NA
	tr -> v5, NA,1
	tr -> v13, gamma2,NA
	tr -> v14, gamma3,NA
	tr -> v16,gamma4,NA
	tr -> v17,gamma5,NA
	v5 <-> v5,ev1,NA
	v13 <-> v13,ev2,NA
	v14 <-> v14,ev3,NA
	v16 <-> v16, ev4, NA
	v17 <-> v17,ev5,NA


sem.correlated <- sem(model.correlated, cov(riskninfo_s), 397)
summary(sem.correlated)
samelist = c('weber','tp','tr')
minlist=c(names(rk),names(tp))
maxlist = NULL
path.diagram(sem2,out.file = 
"e:/sem2.dot",same.rank=samelist,min.rank=minlist,max.rank = 
maxlist,edge.labels="values",rank.direction='LR')

#############################################
#high level latent scale, a high level factor exist
##############################################
model.rsk <- specify.model()
	rsk->tp,e.rsktp,NA
	rsk->tr,e.rsktr,NA
	rsk->weber,e.rskweber,NA
	rsk<->rsk, NA,1
	weber<->weber, e.weber,NA
	tp<->tp,e.tp,NA
	tr <->tr,e.trv,NA
	weber -> prob_wild_camp,NA,1
	weber -> prob_book_hotel_in_short_time,alpha3,NA
	weber -> prob_safari_Kenia, alpha4, NA
	weber -> prob_sail_wild_water,alpha5,NA
	weber -> prob_dangerous_sport,alpha7,NA
	weber -> prob_bungee_jumping,alpha8,NA
	weber -> prob_tornado_tracking,alpha9,NA
	weber -> prob_ski,alpha10,NA
	prob_wild_camp <-> prob_wild_camp, ep2,NA
	prob_book_hotel_in_short_time <-> prob_book_hotel_in_short_time,ep3,NA
	prob_safari_Kenia <-> prob_safari_Kenia, ep4, NA
	prob_sail_wild_water <-> prob_sail_wild_water,ep5,NA
	prob_dangerous_sport <-> prob_dangerous_sport,ep7,NA
	prob_bungee_jumping <-> prob_bungee_jumping,ep8,NA
	prob_tornado_tracking <-> prob_tornado_tracking,ep9,NA
	prob_ski <-> prob_ski,ep10,NA
	tp -> o1,NA,1
	tp -> o3,beta3,NA
	tp -> o4,beta4,NA
	tp -> o5,beta5,NA
	tp -> o6,beta6,NA
	tp -> o7,beta7,NA
	tp -> o9,beta9,NA
	tp -> o10,beta10,NA
	tp -> o11,beta11,NA
	tp -> o12,beta12,NA
	o1 <-> o1,eo1,NA
	o3 <-> o3,eo3,NA
	o4 <-> o4,eo4,NA
	o5 <-> o5,eo5,NA
	o6 <-> o6,eo6,NA
	o7 <-> o7,eo7,NA
	o9 <-> o9,eo9,NA
	o10 <-> o10,eo10,NA
	o11 <-> o11,eo11,NA
	o12 <-> o12,eo12,NA
	tr -> v5, NA,1
	tr -> v13, gamma2,NA
	tr -> v14, gamma3,NA
	tr -> v16,gamma4,NA
	tr -> v17,gamma5,NA
	v5 <-> v5,ev1,NA
	v13 <-> v13,ev2,NA
	v14 <-> v14,ev3,NA
	v16 <-> v16, ev4, NA
	v17 <-> v17,ev5,NA


sem.rsk <- sem(model.rsk, cov(riskninfo_s), 397)
summary(sem.rsk)


##############
#model one results
###############
  Model Chisquare =  680.79   Df =  227 Pr(>Chisq) = 0
  Chisquare (null model) =  2443.4   Df =  253
  Goodness-of-fit index =  0.86163
  Adjusted goodness-of-fit index =  0.83176
  RMSEA index =  0.07105   90% CI: (NA, NA)
  Bentler-Bonnett NFI =  0.72137
  Tucker-Lewis NNFI =  0.7691
  Bentler CFI =  0.79282
  SRMR =  0.069628
  BIC =  -677.56

  Normalized Residuals
    Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
-3.4800 -0.8490 -0.0959 -0.0186  0.6540  8.8500

  Parameter Estimates
               Estimate  Std Error z value Pr(>|z|)
e.webertp     -0.058847 0.023473  -2.5070 1.2175e-02
e.tptrl     0.151913 0.031072   4.8890 1.0134e-06
e.trweber -0.255449 0.044469  -5.7444 9.2264e-09
e.tp           0.114260 0.038652   2.9562 3.1149e-03
e.trv          0.464741 0.068395   6.7950 1.0832e-11
alpha2         0.488106 0.051868   9.4105 0.0000e+00
alpha3         0.446255 0.052422   8.5127 0.0000e+00
alpha4         0.517707 0.050863  10.1784 0.0000e+00
alpha5         0.772128 0.045863  16.8356 0.0000e+00
alpha7         0.782098 0.045754  17.0934 0.0000e+00
alpha8         0.668936 0.048092  13.9095 0.0000e+00
alpha9         0.376798 0.052977   7.1124 1.1400e-12
alpha10        0.449507 0.051885   8.6635 0.0000e+00
ep2            0.761752 0.058103  13.1104 0.0000e+00
ep3            0.800857 0.060154  13.3134 0.0000e+00
ep4            0.731980 0.056002  13.0705 0.0000e+00
ep5            0.403819 0.040155  10.0565 0.0000e+00
ep7            0.388322 0.039930   9.7250 0.0000e+00
ep8            0.552524 0.046619  11.8519 0.0000e+00
ep9            0.858023 0.063098  13.5982 0.0000e+00
ep10           0.797945 0.059651  13.3770 0.0000e+00
beta3          1.670861 0.312656   5.3441 9.0871e-08
beta4          1.536421 0.292725   5.2487 1.5319e-07
beta5          1.530081 0.294266   5.1997 1.9966e-07
beta6          1.767803 0.329486   5.3653 8.0801e-08
beta7          0.870601 0.200366   4.3451 1.3924e-05
beta9          1.692284 0.312799   5.4101 6.2975e-08
beta10         1.009742 0.224155   4.5047 6.6480e-06
beta11         1.723416 0.324593   5.3095 1.0995e-07
beta12         1.452796 0.286857   5.0645 4.0940e-07
eo1            0.885742 0.065529  13.5168 0.0000e+00
eo3            0.681004 0.055626  12.2425 0.0000e+00
eo4            0.730277 0.057682  12.6603 0.0000e+00
eo5            0.732500 0.059305  12.3514 0.0000e+00
eo6            0.642921 0.055797  11.5226 0.0000e+00
eo7            0.913393 0.066903  13.6526 0.0000e+00
eo9            0.672777 0.054994  12.2336 0.0000e+00
eo10           0.883505 0.065198  13.5512 0.0000e+00
eo11           0.660627 0.055399  11.9249 0.0000e+00
eo12           0.758847 0.059582  12.7361 0.0000e+00
gamma2         0.689244 0.089575   7.6946 1.4211e-14
gamma3         0.880574 0.093002   9.4684 0.0000e+00
gamma4         1.083443 0.092856  11.6680 0.0000e+00
gamma5         0.589127 0.087252   6.7520 1.4584e-11
ev1            0.535257 0.050039  10.6968 0.0000e+00
ev2            0.779221 0.060274  12.9280 0.0000e+00
ev3            0.639632 0.054097  11.8239 0.0000e+00
ev4            0.454467 0.048438   9.3824 0.0000e+00
ev5            0.838702 0.062929  13.3277 0.0000e+00

#####################################
#model two results
##################################
Model Chisquare =  680.79   Df =  227 Pr(>Chisq) = 0
  Chisquare (null model) =  2443.4   Df =  253
  Goodness-of-fit index =  0.86163
  Adjusted goodness-of-fit index =  0.83176
  RMSEA index =  0.07105   90% CI: (NA, NA)
  Bentler-Bonnett NFI =  0.72137
  Tucker-Lewis NNFI =  0.7691
  Bentler CFI =  0.79282
  SRMR =  0.069627
  BIC =  -677.56

  Normalized Residuals
    Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
-3.4800 -0.8490 -0.0959 -0.0186  0.6540  8.8500

  Parameter Estimates
            Estimate  Std Error z value  Pr(>|z|)
e.rsktp      0.187069 0.045642   4.09859 4.1567e-05
e.rsktrl  0.812070 0.131731   6.16462 7.0652e-10
e.rskweber  -0.153542 0.038132  -4.02660 5.6589e-05
e.weber     0.214671 0.046260   4.64056 3.4746e-06
e.tp        0.079263 0.028484   2.78270 5.3909e-03
e.trv      -0.194712 0.197101  -0.98788 3.2321e-01
alpha3      0.914263 0.131132   6.97206 3.1233e-12
alpha4      1.060649 0.143622   7.38499 1.5254e-13
alpha5      1.581889 0.177961   8.88898 0.0000e+00
alpha7      1.602316 0.182893   8.76095 0.0000e+00
alpha8      1.370476 0.164966   8.30764 0.0000e+00
alpha9      0.771961 0.128670   5.99955 1.9787e-09
alpha10     0.920922 0.136148   6.76413 1.3411e-11
ep2         0.761752 0.058109  13.10909 0.0000e+00
ep3         0.800856 0.060155  13.31314 0.0000e+00
ep4         0.731979 0.056003  13.07044 0.0000e+00
ep5         0.403818 0.040155  10.05643 0.0000e+00
ep7         0.388322 0.039932   9.72459 0.0000e+00
ep8         0.552523 0.046620  11.85175 0.0000e+00
ep9         0.858024 0.063099  13.59811 0.0000e+00
ep10        0.797943 0.059651  13.37694 0.0000e+00
beta3       1.670904 0.310681   5.37820 7.5234e-08
beta4       1.536444 0.290968   5.28045 1.2887e-07
beta5       1.530096 0.292603   5.22926 1.7019e-07
beta6       1.767838 0.327427   5.39918 6.6945e-08
beta7       0.870626 0.199814   4.35718 1.3175e-05
beta9       1.692309 0.310816   5.44473 5.1885e-08
beta10      1.009760 0.223270   4.52259 6.1088e-06
beta11      1.723432 0.322488   5.34417 9.0830e-08
beta12      1.452761 0.285172   5.09434 3.4997e-07
eo1         0.885741 0.065519  13.51880 0.0000e+00
eo3         0.681003 0.055625  12.24265 0.0000e+00
eo4         0.730278 0.057683  12.66029 0.0000e+00
eo5         0.732501 0.059307  12.35108 0.0000e+00
eo6         0.642919 0.055799  11.52215 0.0000e+00
eo7         0.913394 0.066900  13.65310 0.0000e+00
eo9         0.672778 0.054994  12.23360 0.0000e+00
eo10        0.883503 0.065197  13.55124 0.0000e+00
eo11        0.660630 0.055397  11.92534 0.0000e+00
eo12        0.758852 0.059582  12.73619 0.0000e+00
gamma2      0.689244 0.089545   7.69720 1.3989e-14
gamma3      0.880580 0.092955   9.47317 0.0000e+00
gamma4      1.083430 0.092789  11.67631 0.0000e+00
gamma5      0.589119 0.087233   6.75338 1.4444e-11
ev1         0.535258 0.050034  10.69783 0.0000e+00
ev2         0.779219 0.060273  12.92808 0.0000e+00
ev3         0.639627 0.054096  11.82402 0.0000e+00
ev4         0.454472 0.048437   9.38269 0.0000e+00
ev5         0.838705 0.062929  13.32769 0.0000e+00
John Fox wrote:
John Fox
# 
Dear hyena,
On
Both models are very odd. In the first, each of tr, weber, and tp has direct
effects on different subsets of the endogenous variables. The implicit claim
of these models is that, e.g., prob_* are conditionally independent of tr
and tp given weber, and that the correlations among prob_* are entirely
accounted for by their dependence on weber. The structural coefficients are
just the simple regressions of each prob_* on weber. The second model is the
same except that the variances and covariances among weber, tr, and tp are
parametrized differently. I'm not sure why you set the models up in this
manner, and why your research requires a structural-equation model. I would
have expected that each of the prob_*, v*, and o* variables would have
comprised indicators of a latent variable (risk-taking, etc.). The models
that you specified seem so strange that I think that you'd do well to try to
find competent local help to sort out what you're doing in relationship to
the goals of the research. Of course, maybe I'm just having a failure of
imagination.
It's problematic to treat ordinal variables if they were metric (and to fit
SEMs of this complexity to a small sample).
No. Because the models necessarily fit the same, you'd have to decide
between them on grounds of plausibility. Moreover both models fit very
badly.

Regards,
 John
prob_book_hotel_in_short_time,ep3,NA
prob_book_hotel_in_short_time,ep3,NA
6
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sun
# 
Dear John,

   Thanks for the reply.

Maybe I had used  wrong terminology, as you pointed out, in fact, 
variables "prob*", "o*" and "v*" are indicators of three latent 
variables(scales): weber, tp, and  tr respectively. So variables 
"prob*", "o*" and "v*" are exogenous variables. e.g., variable 
"prob_dangerous_sport" is the answers of question "how likely do you 
think you will engage  a dangerous sport? (1-very unlikely to 5- very 
likely). Variables weber, tr, tp are latent variables representing risk 
attitudes in different domains(recreation, planned behaviour, travel 
choice ).   Hope this make sense of the models.

By exploratory analysis, it had shown consistencies(Cronbach alpha) in 
each scale(latent variable tr, tp, weber), and significant correlations 
among  these three scales. The two models mentioned in previous posts 
are the efforts to find out if there is a more general factor that can 
account for the correlations and make the three scales its sub scales. 
In this sense, SEM is used more of a CFA (sem is the only packages I 
know to do so, i did not search very hard of course).

  And Indeed the model fit is quite bad.

regards,
John Fox wrote:
William Revelle
# 
Dear Hyena,

Your model  is of three correlated factors accounting for the 
observed variables.
Those three correlations may be accounted for equally well by 
correlations (loadings) of the lower order factors with a general 
factor.
Those two models are  indeed equivalent models and will, as a 
consequence have exactly equal fits and dfs.

Call the three correlations rab, rac, rbc.  Then a higher order 
factor model will have loadings of
fa, fb and fc, where fa*fb = rab, fa*bc = rac, and fb*fc = rbc.
You can solve for fa, fb and fc in terms of  factor inter-correlations.

You can not compare the one to the other, for they are equivalent models.

You can examine how much of the underlying variance of the original 
items is due to the general factor by considering a bi-factor 
solution where the general factor loads on each of the observed 
variables and a set of residual group factors account for the 
covariances within your three domains.  This can be done in an 
Exploratory Factor Analysis (EFA) context using the omega function in 
the psych package. It is possible to then take that model and test it 
using John Fox's sem package to evaluate the size of each of the 
general and group factor loadings.   (A discussion of how to do that 
is at http://www.personality-project.org/r/book/psych_for_sem.pdf ).

Bill
At 4:25 PM +0800 3/15/09, hyena wrote:

  
    
  


  


      

    

    

      
      

        


  

        

      John Fox
    

    

      
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Dear Hyena,

OK -- I see that what you're trying to do is simply to fit a confirmatory
factor-analysis model. 

The two models that you're considering aren't really different -- they are,
as I said, observationally equivalent, and fit the data poorly. You can
*assume* a common higher-level factor and estimate the three loadings on it
for the lower-level factors, but you can't test this model against the first
model. 

I'm not sure what you gain from the CFA beyond what you learned from an
exploratory factor analysis. Using the same data first in an EFA and then
for a CFA essentially invalidates the CFA, which is no longer confirmatory.
One would, then, expect a CFA following an EFA to fit the data well, since
the CFA was presumably specified to do so, but I suspect that a closer
examination of the EFA will show that the items don't divide so neatly into
the three sets.

Regards,
 John

            

      
      
      
    

        
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      sun
    

    

      
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Thanks for the clear clarification. The suggested bi-factor solution 
sounds attractive. I am going to check it in details.

regards,

        
William Revelle wrote:

            

      
      
      
    

        

      
      
    

        

      
      
    

            

      
      
      
    

        

      
      
    

            

      
      
      
    

  
  


  


      

    

    

      
      

        


  

        

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The purpose of carrying this CFA is to test the validity of a new 
developed scale "tr" with "v*" items, other two scales "weber" and "tp" 
are existing scales that measures specific risk attitudes. I am not sure 
if a simple correlation analysis is adequate to this purpose or not, 
thus the CFA test.

Further, although a PCA has tested the dimensionality of all items, they 
are not divided as PCA result suggested, rather, their original grouping 
remains. The indicators are indeed not very well divided in PCA, mainly, 
"o*" items are located in two components.

Originally, the EFA has been carried out on the first half of the sample 
and CFA on the second half. Due to the low fit indices from CFA of the 
partial sample, the full sample is tested in CFA to see  if sample size 
affects much, and the results is as poor as before.

It seems the time to read more about scale developing. And thanks for 
all these inputs.

regards,

        
John Fox wrote: