[R] Structural Equation Model

[Chuck Cleland]

Marcos Sanches wrote:
>  I want to estimate parameters in a MIMIC model. I have one latent
> variable (ksi), four reflexive indicators (y1, y2, y3 and y4) and four
> formative indicators (x1, x2, x3, x4). Is there a way to do it in R? I
> know there is the SEM library, but it seems not to be possible to
> specify formative indicators, that is, observed exogenous variables
> which causes the latent variable. 

Marcos:
   A MIMIC model seems to work fine in sem().  Here is an example of a 
MIMIC model which also has 4 indicators of a single latent variable and 
4 covariates:

 > S.sch <- var(school)
 > S.sch[upper.tri(var(school))] <- 0

 > round(S.sch, 4)
         Y1      Y2      Y3      Y4      X1      X2      X3     X4
Y1  1.3586  0.0000  0.0000  0.0000  0.0000  0.0000  0.0000 0.0000
Y2  1.0586  1.3815  0.0000  0.0000  0.0000  0.0000  0.0000 0.0000
Y3  0.6709  0.6937  1.8192  0.0000  0.0000  0.0000  0.0000 0.0000
Y4  1.0452  1.1185  0.6584  3.6370  0.0000  0.0000  0.0000 0.0000
X1  0.4891  0.4929  0.3406  0.5244  1.1984  0.0000  0.0000 0.0000
X2  0.0011  0.0246  0.0236  0.0545  0.0177  0.2500  0.0000 0.0000
X3 -0.7325 -0.8166 -0.4524 -0.8481 -0.8759 -0.0017  4.5451 0.0000
X4  0.0614  0.0644  0.0110  0.0781  0.1070 -0.0009 -0.3961 0.1605

 > # n = 5198

 > model.sch <- matrix(c(
+         'Eta1 ->    Y1',         NA,  1,
+         'Eta1 ->    Y2', 'lambda21', NA,
+         'Eta1 ->    Y3', 'lambda31', NA,
+         'Eta1 ->    Y4', 'lambda41', NA,
+         'X1 ->    Eta1',  'gamma11', NA,
+         'X2 ->    Eta1',  'gamma12', NA,
+         'X3 ->    Eta1',  'gamma13', NA,
+         'X4 ->    Eta1',  'gamma14', NA,
+         'Eta1 <-> Eta1',     'psi1', NA,
+         'Y1 <->     Y1',   'theta1', NA,
+         'Y2 <->     Y2',   'theta2', NA,
+         'Y3 <->     Y3',   'theta3', NA,
+         'Y4 <->     Y4',   'theta4', NA,
+         'X1 <->     X1',    'phi11', NA,
+         'X2 <->     X2',    'phi22', NA,
+         'X3 <->     X3',    'phi33', NA,
+         'X4 <->     X4',    'phi44', NA,
+         'X1 <->     X2',    'phi12', NA,
+         'X1 <->     X3',    'phi13', NA,
+         'X1 <->     X4',    'phi14', NA,
+         'X2 <->     X3',    'phi23', NA,
+         'X2 <->     X4',    'phi24', NA,
+         'X3 <->     X4',    'phi34', NA), ncol=3, byrow=TRUE)

 > obs.vars.sch <- c('Y1', 'Y2', 'Y3', 'Y4', 'X1', 'X2', 'X3', 'X4')

 > sem.sch <- sem(model.sch, S.sch, 5198)

 > summary(sem.sch)

  Model Chisquare =  77.445   Df =  14 Pr(>Chisq) = 8.4002e-11
  Goodness-of-fit index =  0.99628
  Adjusted goodness-of-fit index =  0.99044
  RMSEA index =  0.029530   90 % CI: (0.0011724, 0.0011724)
  BIC =  -71.451

  Normalized Residuals
      Min.   1st Qu.    Median      Mean   3rd Qu.      Max.
-3.82e+00 -4.27e-02  1.44e-05  6.41e-02  5.52e-01  2.74e+00

  Parameter Estimates
             Estimate Std Error   z value   Pr(>|z|)
lambda21  1.05267671 0.0156238  67.37643 0.00000000   Y2 <--- Eta1
lambda31  0.65931621 0.0184649  35.70637 0.00000000   Y3 <--- Eta1
lambda41  1.04965996 0.0257337  40.78937 0.00000000   Y4 <--- Eta1
gamma11   0.32691518 0.0134334  24.33606 0.00000000   Eta1 <--- X1
gamma12   0.04487985 0.0263389   1.70394 0.08839245   Eta1 <--- X2
gamma13  -0.11473656 0.0073527 -15.60472 0.00000000   Eta1 <--- X3
gamma14  -0.12574905 0.0371760  -3.38253 0.00071822   Eta1 <--- X4
psi1      0.76836697 0.0218450  35.17364 0.00000000 Eta1 <--> Eta1
theta1    0.35328543 0.0131454  26.87518 0.00000000     Y1 <--> Y1
theta2    0.26742302 0.0133470  20.03617 0.00000000     Y2 <--> Y2
theta3    1.38220283 0.0282621  48.90661 0.00000000     Y3 <--> Y3
theta4    2.52933440 0.0525526  48.12959 0.00000000     Y4 <--> Y4
phi11     1.19841396 0.0235177  50.95794 0.00000000     X1 <--> X1
phi22     0.24998279 0.0049079  50.93426 0.00000000     X2 <--> X2
phi33     4.54509222 0.0891715  50.97021 0.00000000     X3 <--> X3
phi44     0.16053813 0.0031543  50.89539 0.00000000     X4 <--> X4
phi12     0.01769075 0.0075963   2.32886 0.01986650     X2 <--> X1
phi13    -0.87588544 0.0345801 -25.32916 0.00000000     X3 <--> X1
phi14     0.10701444 0.0062625  17.08800 0.00000000     X4 <--> X1
phi23    -0.00173226 0.0147860  -0.11716 0.90673658     X3 <--> X2
phi24    -0.00087825 0.0027789  -0.31604 0.75197217     X4 <--> X2
phi34    -0.39612152 0.0130628 -30.32432 0.00000000     X4 <--> X3

  Iterations =  24

   This example was taken from

http://statmodel.com/mplus/examples/continuous/cont2.html

and the results agree fairly closely.  However, there does seem to be a 
problem with the RMSEA 90% confidence interval above.  Thanks to John 
Fox for providing this package.

hope it helps,

Chuck Cleland

-- 
Chuck Cleland, Ph.D.
NDRI, Inc.
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