[R] SEM model testing with identical goodness of fits (2)

John Fox jfox at mcmaster.ca
Sun Mar 15 17:46:17 CET 2009


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

> -----Original Message-----
> From: r-help-bounces at r-project.org [mailto:r-help-bounces at r-project.org]
On
> Behalf Of hyena
> Sent: March-15-09 12:00 PM
> To: r-help at stat.math.ethz.ch
> Subject: Re: [R] SEM model testing with identical goodness of fits (2)
> 
> 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:
> > Dear hyena,
> >
> >> -----Original Message-----
> >> From: r-help-bounces at r-project.org
[mailto:r-help-bounces at r-project.org]
> > On
> >> Behalf Of hyena
> >> Sent: March-15-09 4:25 AM
> >> To: r-help at stat.math.ethz.ch
> >> Subject: Re: [R] SEM model testing with identical goodness of fits (2)
> >>
> >> 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.
> >
> > 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.
> >
> >>   The data are all 5 point Likert scale scores by respondents(N=397).
> >
> > It's problematic to treat ordinal variables if they were metric (and to
fit
> > SEMs of this complexity to a small sample).
> >
> >> 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?
> >
> > 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
> >
> >>   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.
> >
> > ______________________________________________
> > 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.




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