[R] SEM model testing with identical goodness of fits (2)
William Revelle
lists at revelle.net
Sun Mar 15 17:11:54 CET 2009
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:
>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:
>>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
>>
>>>-----Original Message-----
>>>From: r-help-bounces at r-project.org [mailto:r-help-bounces at r-project.org]
>>On
>>>Behalf Of hyena
>>>Sent: March-14-09 5:07 PM
>>>To: r-help at stat.math.ethz.ch
>>>Subject: [R] SEM model testing with identical goodness of fits
>>>
>>>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
>>>
>>>______________________________________________
>>>R-help at r-project.org mailing list
>>>https://stat.ethz.ch/mailman/listinfo/r-help
>>>PLEASE do read the posting guide
>>http://www.R-project.org/posting-guide.html
>>>and provide commented, minimal, self-contained, reproducible code.
>>
>>______________________________________________
>>R-help at r-project.org mailing list
>>https://stat.ethz.ch/mailman/listinfo/r-help
>>PLEASE do read the posting guide http://www.R-project.org/posting-guide.html
>>and provide commented, minimal, self-contained, reproducible code.
>>
>
>______________________________________________
>R-help at r-project.org mailing list
>https://stat.ethz.ch/mailman/listinfo/r-help
>PLEASE do read the posting guide http://www.R-project.org/posting-guide.html
>and provide commented, minimal, self-contained, reproducible code.
--
William Revelle http://personality-project.org/revelle.html
Professor http://personality-project.org/personality.html
Department of Psychology http://www.wcas.northwestern.edu/psych/
Northwestern University http://www.northwestern.edu/
Attend ISSID/ARP:2009 http://issid.org/issid.2009/
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