From: John Fox <jfox_at_mcmaster.ca>

Date: Sat 26 Aug 2006 - 00:30:32 EST

*>
*

> rownames(S.wh) <- colnames(S.wh) <-

+ c('Anomia67','Powerless67','Anomia71','Powerless71','Education','SEI')

Read 19 records

*>
*

*> sem.wh <- sem(model.wh, S.wh, 932)
*

*>
*

> summary(sem.wh)

Alienation67 -0.5498096

Alienation71 -0.2115088

Alienation67 -0.5498096

Alienation71 -0.5509147

Alienation67 0.0000000

Alienation71 -0.3394059

John Fox

Department of Sociology

McMaster University

Hamilton, Ontario

Canada L8S 4M4

905-525-9140x23604

http://socserv.mcmaster.ca/jfox

R-help@stat.math.ethz.ch 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. Received on Sat Aug 26 01:00:14 2006

Date: Sat 26 Aug 2006 - 00:30:32 EST

Dear Rense,

(This question was posted a few days ago when I wasn't reading my email.)

So-called effect decompositions are simple functions of the structural coefficients of the model, which in a model fit by sem() are contained in the $A component of the returned object. (See ?sem.) One approach, therefore, would be to put the coefficients in the appropriate locations of the estimated Beta, Gamma, Lamda-x, and Lambda-y matrices of the LISREL model, and then to compute the "effects" in the usual manner.

It should be possible to do this for the RAM formulation of the model as well, simply by distinguishing exogenous from endogenous variables. Here's an illustration using model C in the LISREL 7 Manual, pp. 169-177, for the Wheaton et al. "stability of alienation" data (a common example--I happen to have an old LISREL manual handy):

> S.wh <- matrix(c(

+ 11.834, 0, 0, 0, 0, 0, + 6.947, 9.364, 0, 0, 0, 0, + 6.819, 5.091, 12.532, 0, 0, 0, + 4.783, 5.028, 7.495, 9.986, 0, 0, + -3.839, -3.889, -3.841, -3.625, 9.610, 0, + -2.190, -1.883, -2.175, -1.878, 3.552, 4.503), + 6, 6)

> rownames(S.wh) <- colnames(S.wh) <-

+ c('Anomia67','Powerless67','Anomia71','Powerless71','Education','SEI')

*>
*

> model.wh <- specify.model()

1: Alienation67 -> Anomia67, NA, 1 2: Alienation67 -> Powerless67, lam1, NA 3: Alienation71 -> Anomia71, NA, 1 4: Alienation71 -> Powerless71, lam2, NA 5: SES -> Education, NA, 1 6: SES -> SEI, lam3, NA 7: SES -> Alienation67, gam1, NA 8: Alienation67 -> Alienation71, beta, NA 9: SES -> Alienation71, gam2, NA 10: Anomia67 <-> Anomia67, the1, NA 11: Anomia71 <-> Anomia71, the3, NA 12: Powerless67 <-> Powerless67, the2, NA 13: Powerless71 <-> Powerless71, the4, NA 14: Education <-> Education, thd1, NA 15: SEI <-> SEI, thd2, NA 16: Anomia67 <-> Anomia71, the13, NA 17: Alienation67 <-> Alienation67, psi1, NA 18: Alienation71 <-> Alienation71, psi2, NA 19: SES <-> SES, phi, NA 20:

Read 19 records

> summary(sem.wh)

Model Chisquare = 6.3349 Df = 5 Pr(>Chisq) = 0.27498
Chisquare (null model) = 17973 Df = 15
Goodness-of-fit index = 0.99773

Adjusted goodness-of-fit index = 0.99046
RMSEA index = 0.016934 90 % CI: (NA, 0.05092)
Bentler-Bonnett NFI = 0.99965

Tucker-Lewis NNFI = 0.99978

Bentler CFI = 0.99993

** BIC = -27.852
**
Normalized Residuals

Min. 1st Qu. Median Mean 3rd Qu. Max. -9.57e-01 -1.34e-01 -4.24e-02 -9.17e-02 6.43e-05 5.47e-01

Parameter Estimates

Estimate Std Error z value Pr(>|z|)

lam1 1.02656 0.053424 19.2152 0.0000e+00 Powerless67 <--- Alienation67 lam2 0.97089 0.049608 19.5712 0.0000e+00 Powerless71 <--- Alienation71 lam3 0.51632 0.042247 12.2214 0.0000e+00 SEI <--- SES gam1 -0.54981 0.054298 -10.1258 0.0000e+00 Alienation67 <--- SES beta 0.61732 0.049486 12.4746 0.0000e+00 Alienation71 <--- Alienation67 gam2 -0.21151 0.049862 -4.2419 2.2164e-05 Alienation71 <--- SES the1 5.06546 0.373464 13.5635 0.0000e+00 Anomia67 <--> Anomia67 the3 4.81176 0.397345 12.1098 0.0000e+00 Anomia71 <--> Anomia71 the2 2.21438 0.319740 6.9256 4.3423e-12 Powerless67 <--> Powerless67 the4 2.68322 0.331274 8.0997 4.4409e-16 Powerless71 <--> Powerless71 thd1 2.73051 0.517737 5.2739 1.3353e-07 Education <--> Education thd2 2.66905 0.182260 14.6442 0.0000e+00 SEI <--> SEI the13 1.88739 0.241627 7.8112 5.7732e-15 Anomia71 <--> Anomia67 psi1 4.70477 0.427511 11.0050 0.0000e+00 Alienation67 <--> Alienation67 psi2 3.86642 0.343971 11.2406 0.0000e+00 Alienation71 <--> Alienation71 phi 6.87948 0.659208 10.4360 0.0000e+00 SES <--> SES

Iterations = 58

*>
**> A <- sem.wh$A # structural coefficients
*

> exog <- apply(A, 1, function(x) all(x == 0))

> endog <- !exog

> (B <- A[endog, endog, drop=FALSE]) # direct effects, endogenous ->

endogenous

Anomia67 Powerless67 Anomia71 Powerless71 Education SEI Anomia67 0 0 0 0 0 0 Powerless67 0 0 0 0 0 0 Anomia71 0 0 0 0 0 0 Powerless71 0 0 0 0 0 0 Education 0 0 0 0 0 0 SEI 0 0 0 0 0 0 Alienation67 0 0 0 0 0 0 Alienation71 0 0 0 0 0 0 Alienation67 Alienation71 Anomia67 1.0000000 0.000000 Powerless67 1.0265597 0.000000 Anomia71 0.0000000 1.000000 Powerless71 0.0000000 0.970892 Education 0.0000000 0.000000 SEI 0.0000000 0.000000 Alienation67 0.0000000 0.000000 Alienation71 0.6173153 0.000000

> (C <- A[endog, exog, drop=FALSE]) # direct effects, exogenous ->

endogenous

SES Anomia67 0.0000000 Powerless67 0.0000000 Anomia71 0.0000000 Powerless71 0.0000000 Education 1.0000000 SEI 0.5163168

Alienation67 -0.5498096

Alienation71 -0.2115088

*> I <- diag(nrow(B))
*

> IBinv <- solve(I - B)

> (Ty <- IBinv - I) # total effects, endogenous -> endogenous

Anomia67 Powerless67 Anomia71 Powerless71 Education SEI Anomia67 0 0 0 0 0 0 Powerless67 0 0 0 0 0 0 Anomia71 0 0 0 0 0 0 Powerless71 0 0 0 0 0 0 Education 0 0 0 0 0 0 SEI 0 0 0 0 0 0 Alienation67 0 0 0 0 0 0 Alienation71 0 0 0 0 0 0 Alienation67 Alienation71 Anomia67 1.0000000 0.000000 Powerless67 1.0265597 0.000000 Anomia71 0.6173153 1.000000 Powerless71 0.5993465 0.970892 Education 0.0000000 0.000000 SEI 0.0000000 0.000000 Alienation67 0.0000000 0.000000 Alienation71 0.6173153 0.000000

> (Tx <- IBinv %*% C) # total effects, exogenous -> endogenous

SES Anomia67 -0.5498096 Powerless67 -0.5644124 Anomia71 -0.5509147 Powerless71 -0.5348786 Education 1.0000000 SEI 0.5163168

Alienation67 -0.5498096

Alienation71 -0.5509147

> Ty - B # indirect effects, endogenous -> endogenous

Anomia67 Powerless67 Anomia71 Powerless71 Education SEI Anomia67 0 0 0 0 0 0 Powerless67 0 0 0 0 0 0 Anomia71 0 0 0 0 0 0 Powerless71 0 0 0 0 0 0 Education 0 0 0 0 0 0 SEI 0 0 0 0 0 0 Alienation67 0 0 0 0 0 0 Alienation71 0 0 0 0 0 0 Alienation67 Alienation71 Anomia67 0.0000000 0 Powerless67 0.0000000 0 Anomia71 0.6173153 0 Powerless71 0.5993465 0 Education 0.0000000 0 SEI 0.0000000 0 Alienation67 0.0000000 0 Alienation71 0.0000000 0

> Tx - C # indirect effects, exogenous -> endogenous

SES Anomia67 -0.5498096 Powerless67 -0.5644124 Anomia71 -0.5509147 Powerless71 -0.5348786 Education 0.0000000 SEI 0.0000000

Alienation67 0.0000000

Alienation71 -0.3394059

These results agree with those in the LISREL manual (and for another example there as well), but I haven't checked the method carefully.

It would, of course, be simple to encapsulate the steps above in a function, but here's a caveat: The idea of indirect and total effects makes sense to me for a recursive model, and for the exogenous variables in a nonrecursive model, where they are the reduced-form coefficients (supposing, of course, that the model makes sense in the first place, which is often problematic), but not for the endogenous variables in a nonrecursive model. That is why I haven't put such a function in the sem package; perhaps I should reconsider.

Having said that, I'm ashamed to add that I believe that I was the person who suggested the definition of total and indirect effects currently used for these models.

Finally, you can get standardized effects similarly by using standardized structural coefficients. In the sem package, these are computed and printed by standardized.eoefficients(). This function doesn't return the standardized A matrix in a usable form, but could be made to do so.

Regards,

John

John Fox

Department of Sociology

McMaster University

Hamilton, Ontario

Canada L8S 4M4

905-525-9140x23604

http://socserv.mcmaster.ca/jfox

R-help@stat.math.ethz.ch 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. Received on Sat Aug 26 01:00:14 2006

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