sem: General Structural Equation Models

Description

sem fits general structural equation models (with both observed and unobserved variables) by the method of maximum likelihood, assuming multinormal errors. Observed variables are also called indicators or manifest variables; unobserved variables are also called factors or latent variables.

Usage

sem(S, ram, N, param.names=paste("Param", 1:t, sep = ""), 
    var.names=paste("V", 1:m, sep=""), observed=1:n, 
    fixed.x=NULL, heywood=T, control=list())
    
start.values(S, ram, observed)

print.sem(x)

summary.sem(object, digits=5)

Arguments

covariance matrix among observed variables.

ram

RAM specification, which is a simple encoding of the path diagram for the model (see details below).

number of observations on which the covariance matrix is based.

param.names

names of the $t$ free parameters, given in their numerical order; default names are Param1, ..., Paramt.

var.names

names of the $m$ entries of the $v$ vector (typically the observed and latent variables --- see below), given in their numerical order; default names are Var1, ..., Varm.

observed

indices of the $n$ observed variables; default is 1 to $n$.

fixed.x

indices of fixed exogenous variables. Specifying these obviates the necessity of having to fix the variances and covariances among these variables (and produces correct degrees of freedom for the model chisquare).

heywood

if TRUE (the default) variances are constrained to be non-negative in the initial fitting process.

control

a list with up to two elements: optim.control, a list of control parameters to be passed to the optim function via its control argument; nlm.iterlim, the maximum number of iterati

object, x

an object of class sem returned by the sem function.

digits

number of digits for printed output.

Value

sem returns an object of class sem, with the following elements:
ramRAM matrix, including any rows generated for covariances among fixed exogenous variables; column 5 includes computed start values.
coefestimates of free parameters.
criterionfitting criterion --- negative log-liklihood divided by $N - 1$.
parvector of parameter estimates.
covestimated asymptotic covariance matrix of parameter estimates.
Sobserved covariance matrix.
Cmodel-reproduced covariance matrix.
ARAM $A$ matrix.
PRAM $P$ matrix.
n.fixnumber of fixed exogenous variables.
nnumber of observed variables.
Nnumber of observations.
mnumber of variables (observed plus unobserved).
tnumber of free parameters.
par.posnindices of free parameters.
convergenceconvergence code returned by nlm (a code > 2 indicates a problem).

Details

The model is set up using RAM ('reticular action model' -- don't ask!) notation -- a simple format for specifying general structural equation models by coding the `arrows' in the path diagram for the model (see, e.g., McArdle and McDonald, 1984). The variables in the $v$ vector in the model (typically, the observed and unobserved variables, but not error variables) are numbered from 1 to $m$. the RAM matrix contains one row for each (free or constrained) parameter of the model, and five columns, as follows: [object Object],[object Object],[object Object],[object Object],[object Object] sem fits the model by calling the optim and nlm optimizers sequentially to minimize the negative log-likelihood for the model. The L-BFGS-B method is employed by optim. The RAM formulation of the general structural equation model is given by the basic equation $$v = Av + u$$ where $v$ and $u$ are vectors of random variables (observed or unobserved), and the parameter matrix $A$ contains regression coefficients, symbolized by single-headed arrows in a path diagram. Another parameter matrix, $$P = E(uu')$$ contains covariances among the elements of $u$ (assuming that the elements of $u$ have zero means). Usually $v$ contains endogenous and exogenous observed and unobserved variables, but not error variables (see the examples below). The start.values function may be called directly, but is usually called by ram.

References

Bollen, K. A. (1989) Structural Equations With Latent Variables. Wiley. McArdle, J. J. and McDonald, R. P. (1984) Some algebraic properties of the reticular action model. British Journal of Mathematical and Statistical Psychology 37, 234--251. McDonald, R. P. and Hartmann, W. M. (1992) A procedure for obtaining initial values of parameters in the RAM model. Multivariate Behavioral Research 27, 57--76. Raftery, A. E. (1993) Bayesian model selection in structural equation models. In Bollen, K . A. and Long, J. S. (eds.) Testing Structural Equation Models, Sage.

Examples

Run this code

# ------------- Duncan, Haller and Portes peer-influences model ----------------------
# A nonrecursive SEM with unobserved endogenous variables and fixed exogenous variables

R.DHP <- matrix(c(      # lower triangle of correlation matrix
            1,      0,      0,      0,      0,      0,      0,      0,      0,      0,            
            .6247,   1,     0,      0,      0,      0,      0,      0,      0,      0,            
            .3269,  .3669,  1,      0,      0,      0,      0,      0,      0,      0,            
            .4216,  .3275,  .6404,  1,      0,      0,      0,      0,      0,      0,
            .2137,  .2742,  .1124,  .0839,  1,      0,      0,      0,      0,      0,
            .4105,  .4043,  .2903,  .2598,  .1839,  1,      0,      0,      0,      0,
            .3240,  .4047,  .3054,  .2786,  .0489,  .2220,  1,      0,      0,      0,
            .2930,  .2407,  .4105,  .3607,  .0186,  .1861,  .2707,  1,      0,      0,
            .2995,  .2863,  .5191,  .5007,  .0782,  .3355,  .2302,  .2950,  1,      0,
            .0760,  .0702,  .2784,  .1988,  .1147,  .1021,  .0931, -.0438,  .2087,  1
            ), ncol=10, byrow=T)

R.DHP <- R.DHP + t(R.DHP) - diag(diag(R.DHP))   # form symmetric correlation matrix
            
ram.dhp <- matrix(c(
#               heads   to      from    param  start
                1,       1,     11,      0,     1,
                1,       2,     11,      1,     NA, # lam21
                1,       3,     12,      0,     1,
                1,       4,     12,      2,     NA, # lam42
                1,      11,      5,      3,     NA, # gam11
                1,      11,      6,      4,     NA, # gam12
                1,      11,      7,      5,     NA, # gam13
                1,      11,      8,      6,     NA, # gam14
                1,      12,      7,      7,     NA, # gam23
                1,      12,      8,      8,     NA, # gam24
                1,      12,      9,      9,     NA, # gam25
                1,      12,     10,     10,     NA, # gam26
                1,      11,     12,     11,     NA, # beta12
                1,      12,     11,     12,     NA, # beta21
                2,       1,      1,     13,     NA, # theta1
                2,       2,      2,     14,     NA, # theta2
                2,       3,      3,     15,     NA, # theta3
                2,       4,      4,     16,     NA, # theta4
                2,      11,     11,     17,     NA, # psi11
                2,      12,     12,     18,     NA, # psi22
                2,      11,     12,     19,     NA  # psi12
                ), ncol=5, byrow=T)

params.dhp <- c('lam21', 'lam42', 'gam11', 'gam12', 'gam13', 'gam14',
                 'gam23',  'gam24',  'gam25',  'gam26',
                 'beta12', 'beta21', 'theta1', 'theta2', 'theta3', 'theta4',
                 'psi11', 'psi22', 'psi12')
                 
vars.dhp <- c('ROccAsp', 'REdAsp', 'FOccAsp', 'FEdAsp', 'RParAsp', 'RIQ',
                'RSES', 'FSES', 'FIQ', 'FParAsp', 'RGenAsp', 'FGenAsp')
                
sem.dhp <- sem(R.DHP, ram.dhp, 329, params.dhp, vars.dhp, fixed.x=5:10)

summary(sem.dhp)
    
##    Model Chisquare =  26.697   Df =  15 Pr(>Chisq) = 0.031302
##    Goodness-of-fit index =  0.98439
##    Adjusted goodness-of-fit index =  0.94275 
##    BIC =  -94.782 
##    
##    Normalized Residuals
##       Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##    -0.8010 -0.1180  0.0000 -0.0120  0.0398  1.5700 
##    
##    Parameter Estimates
##            Estimate Std Error  z value   Pr(>|z|)                     
##    lam21   1.062673  0.091968 11.55488 0.0000e+00  REdAsp <--- RGenAsp
##    lam42   0.929732  0.071152 13.06682 0.0000e+00  FEdAsp <--- FGenAsp
##    gam11   0.161220  0.038487  4.18892 1.4015e-05 RGenAsp <--- RParAsp
##    gam12   0.249647  0.044581  5.59989 1.0724e-08     RGenAsp <--- RIQ
##    gam13   0.218402  0.043476  5.02350 2.5369e-07    RGenAsp <--- RSES
##    gam14   0.071836  0.050335  1.42714 7.6769e-02    RGenAsp <--- FSES
##    gam23   0.061879  0.051738  1.19602 1.1584e-01    FGenAsp <--- RSES
##    gam24   0.228863  0.044495  5.14361 1.3476e-07    FGenAsp <--- FSES
##    gam25   0.349030  0.044551  7.83443 2.3315e-15     FGenAsp <--- FIQ
##    gam26   0.159529  0.040129  3.97540 3.5131e-05 FGenAsp <--- FParAsp
##    beta12  0.184245  0.096209  1.91505 2.7743e-02 RGenAsp <--- FGenAsp
##    beta21  0.235502  0.119744  1.96671 2.4608e-02 FGenAsp <--- RGenAsp
##    theta1  0.412143  0.052211  7.89385 1.4433e-15 ROccAsp <--> ROccAsp
##    theta2  0.336146  0.053323  6.30399 1.4504e-10   REdAsp <--> REdAsp
##    theta3  0.311197  0.046665  6.66880 1.2895e-11 FOccAsp <--> FOccAsp
##    theta4  0.404601  0.046733  8.65773 0.0000e+00   FEdAsp <--> FEdAsp
##    psi11   0.280987  0.046311  6.06743 6.4985e-10 RGenAsp <--> RGenAsp
##    psi22   0.263832  0.044901  5.87586 2.1033e-09 FGenAsp <--> FGenAsp
##    psi12  -0.022620  0.051650 -0.43795 3.3071e-01 RGenAsp <--> FGenAsp

# -------------------- Wheaton et al. alienation data ----------------------
    

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,
  -21.899,  -18.831,  -21.748,  -18.775,  35.522,  450.288)
  , 6, 6)
  
S.wh <- S.wh + t(S.wh) - diag(diag(S.wh))

# This is the model in the SAS manual for PROC CALIS: A Recursive SEM with
# latent endogenous and exogenous variables.
# Curiously, both factor loadings for two of the latent variables are fixed.
 
ram.wh <- matrix(c(             
                1, 1, 7, 0, 1,
                1, 2, 7, 0, .833,
                1, 3, 8, 0, 1,
                1, 4, 8, 0, .833,
                1, 5, 9, 0, 1,
                1, 6, 9, 1, NA,
                1, 7, 9, 2, NA,
                1, 8, 7, 3, NA,
                1, 8, 9, 4, NA,
                2, 1, 1, 5, NA,
                2, 2, 2, 6, NA,
                2, 3, 3, 5, NA,
                2, 4, 4, 6, NA,
                2, 5, 5, 7, NA,
                2, 6, 6, 8, NA,
                2, 1, 3, 9, NA,
                2, 2, 4, 9, NA,
                2, 7, 7, 10, NA,
                2, 8, 8, 11, NA,
                2, 9, 9, 12, NA
                ), ncol=5, byrow=T)

par.names.wh <- c('lamb', 'gam1', 'beta', 'gam2', 
            'the1', 'the2', 'the3', 'the4', 'the5', 'psi1', 'psi2', 'phi')
            
var.names.wh <- c('Anomia67','Powerless67','Anomia71','Powerless71','Education',
    'SEI', 'Alienation67','Alienation71','SES')

sem.wh <- sem(S.wh, ram.wh, 932, par.names.wh, var.names.wh)

summary(sem.wh)

##  Model Chisquare =  13.485   Df =  9 Pr(>Chisq) = 0.14186
##  Goodness-of-fit index =  0.99527
##  Adjusted goodness-of-fit index =  0.98896 
##  BIC =  -64.177
##  
##  Normalized Residuals
##      Min.   1st Qu.    Median      Mean   3rd Qu.      Max. 
##  -1.26e+00 -1.31e-01  2.09e-05 -2.87e-02  1.14e-01  8.75e-01 
##  
##  Parameter Estimates
##      Estimate Std Error  z value   Pr(>|z|)                               
##  lamb   5.36882  0.433985  12.3710 0.0000e+00                   SEI <--- SES
##  gam1  -0.62994  0.056128 -11.2233 0.0000e+00          Alienation67 <--- SES
##  beta   0.59311  0.046820  12.6679 0.0000e+00 Alienation71 <--- Alienation67
##  gam2  -0.24086  0.055202  -4.3633 6.4058e-06          Alienation71 <--- SES
##  the1   3.60786  0.200588  17.9864 0.0000e+00         Anomia67 <--> Anomia67
##  the2   3.59494  0.165233  21.7567 0.0000e+00   Powerless67 <--> Powerless67
##  the3   2.99366  0.498975   5.9996 9.8888e-10       Education <--> Education
##  the4 259.57813 18.321380  14.1680 0.0000e+00                   SEI <--> SEI
##  the5   0.90580  0.121710   7.4423 4.9516e-14         Anomia67 <--> Anomia71
##  psi1   5.67049  0.422905  13.4084 0.0000e+00 Alienation67 <--> Alienation67
##  psi2   4.51480  0.334991  13.4774 0.0000e+00 Alienation71 <--> Alienation71
##  phi    6.61633  0.639515  10.3459 0.0000e+00                   SES <--> SES


# The same model, but treating one loading for each latent variable as free.

ram.wh.1 <- matrix(c(           
                1, 1, 7, 0, 1,
                1, 2, 7,13, NA,
                1, 3, 8, 0, 1,
                1, 4, 8,13, NA,
                1, 5, 9, 0, 1,
                1, 6, 9, 1, NA,
                1, 7, 9, 2, NA,
                1, 8, 7, 3, NA,
                1, 8, 9, 4, NA,
                2, 1, 1, 5, NA,
                2, 2, 2, 6, NA,
                2, 3, 3, 5, NA,
                2, 4, 4, 6, NA,
                2, 5, 5, 7, NA,
                2, 6, 6, 8, NA,
                2, 1, 3, 9, NA,
                2, 2, 4, 9, NA,
                2, 7, 7, 10,NA,
                2, 8, 8, 11,NA,
                2, 9, 9, 12,NA
                ), ncol=5, byrow=T)

par.names.wh.1 <- c('lambx', 'gam1', 'beta', 'gam2', 
            'the1', 'the2', 'the3', 'the4', 'the5', 'psi1', 'psi2', 'phi',
            'lamby')

sem.wh.1 <- sem(S.wh, ram.wh.1, 932,  par.names.wh.1, var.names.wh)

summary(sem.wh.1)

##  Model Chisquare =  12.673   Df =  8 Pr(>Chisq) = 0.12360
##  Goodness-of-fit index =  0.99553
##  Adjusted goodness-of-fit index =  0.98828 
##  BIC =  -56.36 
##  
##  Normalized Residuals
##      Min.   1st Qu.    Median      Mean   3rd Qu.      Max. 
##  -9.97e-01 -1.40e-01  3.22e-05 -2.87e-02  1.00e-01  7.59e-01 
##  
##  Parameter Estimates
##      Estimate Std Error  z value   Pr(>|z|)                               
##  lambx   5.35309  0.432585  12.3746 0.0000e+00                   SEI <--- SES
##  gam1   -0.62130  0.056143 -11.0665 0.0000e+00          Alienation67 <--- SES
##  beta    0.59427  0.047041  12.6331 0.0000e+00 Alienation71 <--- Alienation67
##  gam2   -0.23581  0.054685  -4.3122 8.0802e-06          Alienation71 <--- SES
##  the1    3.74499  0.249822  14.9907 0.0000e+00         Anomia67 <--> Anomia67
##  the2    3.49380  0.200754  17.4034 0.0000e+00   Powerless67 <--> Powerless67
##  the3    2.97422  0.499650   5.9526 1.3196e-09       Education <--> Education
##  the4  260.12995 18.297861  14.2164 0.0000e+00                   SEI <--> SEI
##  the5    0.90380  0.121819   7.4192 5.8842e-14         Anomia67 <--> Anomia71
##  psi1    5.47364  0.464062  11.7951 0.0000e+00 Alienation67 <--> Alienation67
##  psi2    4.36409  0.362721  12.0315 0.0000e+00 Alienation71 <--> Alienation71
##  phi     6.63578  0.640430  10.3614 0.0000e+00                   SES <--> SES
##  lamby   0.86262  0.033383  25.8400 0.0000e+00  Powerless67 <--- Alienation67 


# ----------------------- Thurstone data ---------------------------------------
#  Second-order confirmatory factor analysis, from the SAS manual for PROC CALIS

R.thur <- matrix(c(
        1.,       0,      0,      0,      0,      0,      0,      0,      0, 
         .828,   1.,      0,      0,      0,      0,      0,      0,      0, 
         .776,   .779,   1.,      0,      0,      0,      0,      0,      0, 
         .439,   .493,    .460,   1.,     0,      0,      0,      0,      0, 
         .432,   .464,    .425,   .674,   1.,     0,      0,      0,      0, 
         .447,   .489,    .443,   .590,    .541,  1.,     0,      0,      0, 
         .447,   .432,    .401,   .381,    .402,   .288,  1.,     0,      0, 
         .541,   .537,    .534,   .350,    .367,   .320,   .555,  1.,     0, 
         .380,   .358,    .359,   .424,    .446,   .325,   .598,   .452,  1.
            ), ncol=9, byrow=T)
R.thur <- R.thur + t(R.thur) - diag(diag(R.thur))

ram.thur <- matrix(c(
#       heads   to      from    param   start
        1,       1,     10,      1,     NA,     # lam11
        1,       2,     10,      2,     NA,     # lam21
        1,       3,     10,      3,     NA,     # lam31
        1,       4,     11,      4,     NA,     # lam42
        1,       5,     11,      5,     NA,     # lam52
        1,       6,     11,      6,     NA,     # lam62
        1,       7,     12,      7,     NA,     # lam73
        1,       8,     12,      8,     NA,     # lam83
        1,       9,     12,      9,     NA,     # lam93
        1,      10,     13,     10,     NA,     # gam1
        1,      11,     13,     11,     NA,     # gam2
        1,      12,     13,     12,     NA,     # gam3
        2,       1,      1,     13,     NA,     # th1
        2,       2,      2,     14,     NA,     # th2
        2,       3,      3,     15,     NA,     # th3
        2,       4,      4,     16,     NA,     # th4
        2,       5,      5,     17,     NA,     # th5
        2,       6,      6,     18,     NA,     # th6
        2,       7,      7,     19,     NA,     # th7
        2,       8,      8,     20,     NA,     # th8
        2,       9,      9,     21,     NA,     # th9
        2,      10,     10,      0,     1,
        2,      11,     11,      0,     1,
        2,      12,     12,      0,     1,
        2,      13,     13,      0,     1
        ), ncol=5, byrow=T)
        
par.names.thur <- c('lam11', 'lam21', 'lam31', 'lam42', 'lam52', 'lam62', 
                'lam73', 'lam83', 'lam93', 'gam1', 'gam2', 'gam3',
                'th1', 'th2', 'th3', 'th4', 'th5', 'th6', 'th7', 'th8', 'th9')
                
var.names.thur <- c('Sentences','Vocabulary','Sent.Completion','First.Letters',
                '4.Letter.Words','Sufixes','Letter.Series','Pedigrees',
                'Letter.Group', 'F1', 'F2','F3','F4')

        
sem.thur <- sem(R.thur, ram.thur, 213, par.names.thur, var.names.thur)

summary(sem.thur)

##  Model Chisquare =  38.196   Df =  24 Pr(>Chisq) = 0.033101
##  Goodness-of-fit index =  0.95957
##  Adjusted goodness-of-fit index =  0.9242 
##  BIC =  -143.21
##  
##  Normalized Residuals
##      Min.   1st Qu.    Median      Mean   3rd Qu.      Max. 
##  -9.75e-01 -4.17e-01  2.51e-05  4.02e-02  9.41e-02  1.63e+00 
##  
##  Parameter Estimates
##      Estimate Std Error z value   Pr(>|z|)                                     
##  lam11  0.51511  0.064966  7.9289 1.1102e-15                    Sentences <--- F1
##  lam21  0.52030  0.065163  7.9845 6.6613e-16                   Vocabulary <--- F1
##  lam31  0.48742  0.062423  7.8083 2.8866e-15              Sent.Completion <--- F1
##  lam42  0.52112  0.063136  8.2540 1.1102e-16                First.Letters <--- F2
##  lam52  0.49707  0.059673  8.3299 0.0000e+00               4.Letter.Words <--- F2
##  lam62  0.43807  0.056479  7.7563 4.3299e-15                      Sufixes <--- F2
##  lam73  0.45243  0.071371  6.3391 1.1552e-10                Letter.Series <--- F3
##  lam83  0.41729  0.061037  6.8367 4.0519e-12                    Pedigrees <--- F3
##  lam93  0.40763  0.064524  6.3175 1.3294e-10                 Letter.Group <--- F3
##  gam1   1.44386  0.264190  5.4652 2.3117e-08                           F1 <--- F4
##  gam2   1.25382  0.216593  5.7888 3.5439e-09                           F2 <--- F4
##  gam3   1.40656  0.279333  5.0354 2.3842e-07                           F3 <--- F4
##  th1    0.18150  0.028400  6.3907 8.2566e-11             Sentences <--> Sentences
##  th2    0.16493  0.027797  5.9334 1.4837e-09           Vocabulary <--> Vocabulary
##  th3    0.26713  0.033468  7.9817 7.7716e-16 Sent.Completion <--> Sent.Completion
##  th4    0.30150  0.050686  5.9484 1.3536e-09     First.Letters <--> First.Letters
##  th5    0.36450  0.052358  6.9617 1.6808e-12   4.Letter.Words <--> 4.Letter.Words
##  th6    0.50641  0.059962  8.4455 0.0000e+00                 Sufixes <--> Sufixes
##  th7    0.39033  0.061599  6.3367 1.1735e-10     Letter.Series <--> Letter.Series
##  th8    0.48137  0.065388  7.3617 9.0816e-14             Pedigrees <--> Pedigrees
##  th9    0.50510  0.065227  7.7437 4.7740e-15       Letter.Group <--> Letter.Group


#------------------------- Kerchoff/Kenney path analysis ---------------------
# An observed-variable recursive SEM from the LISREL manual

R.kerch <- matrix(c(
    1,      0,      0,      0,      0,      0,      0,
    -.100,  1,      0,      0,      0,      0,      0,
     .277,  -.152,  1,      0,      0,      0,      0,
     .250,  -.108,  .611,   1,      0,      0,      0,
     .572,  -.105,  .294,   .248,   1,      0,      0,
     .489,  -.213,  .446,   .410,   .597,   1,      0,
     .335,  -.153,  .303,   .331,   .478,   .651,   1),
     ncol=7, byrow=T)

R.kerch <- R.kerch + t(R.kerch) - diag(diag(R.kerch))
    
ram.kerch<-matrix(c(
#   heads   to      from    param   start
    1,      5,      1,       1,     NA,     # gam51
    1,      5,      2,       2,     NA,     # gam52
    1,      5,      3,       3,     NA,     # gam53
    1,      5,      4,       4,     NA,     # gam54
    1,      6,      1,       5,     NA,     # gam61
    1,      6,      2,       6,     NA,     # gam62
    1,      6,      3,       7,     NA,     # gam63
    1,      6,      4,       8,     NA,     # gam64
    1,      6,      5,       9,     NA,     # beta65
    1,      7,      1,      10,     NA,     # gam71
    1,      7,      2,      11,     NA,     # gam72
    1,      7,      3,      12,     NA,     # gam73
    1,      7,      4,      13,     NA,     # gam74
    1,      7,      5,      14,     NA,     # beta75
    1,      7,      6,      15,     NA,     # beta76
    2,      5,      5,      16,     NA,     # psi5
    2,      6,      6,      17,     NA,     # psi6
    2,      7,      7,      18,     NA),    # psi7
    ncol=5, byrow=T)
    
par.names.kerch <- c('gam51','gam52','gam53','gam54','gam61','gam62',
    'gam63','gam64','beta65','gam71','gam72','gam73','gam74',
    'beta75','beta76', 'psi5','psi6','psi7')
    
var.names.kerch <- c('Intelligence','Siblings','FatherEd','FatherOcc',
    'Grades','EducExp','Occup.Asp')

sem.kerch <- sem(R.kerch, ram.kerch, 737, par.names.kerch, 
    var.names.kerch,fixed.x=1:4)
    
summary(sem.kerch)

##  Model Chisquare =  6.537e-13   Df =  0 Pr(>Chisq) = NA
##  Goodness-of-fit index =  1
##  Adjusted goodness-of-fit index =  NA 
##  BIC =  NA 
##  
##  Normalized Residuals
##      Min.   1st Qu.    Median      Mean   3rd Qu.      Max. 
##  -3.61e-07 -9.77e-14 -7.45e-15 -3.27e-08  1.12e-15  1.27e-13 
##  
##  Parameter Estimates
##          Estimate Std Error  z value   Pr(>|z|)                            
##  gam51   0.525902  0.031182 16.86530 0.0000e+00    Grades <--- Intelligence
##  gam52  -0.029942  0.030149 -0.99314 1.6032e-01        Grades <--- Siblings
##  gam53   0.118966  0.038259  3.10951 9.3699e-04        Grades <--- FatherEd
##  gam54   0.040603  0.037785  1.07456 1.4129e-01       Grades <--- FatherOcc
##  gam61   0.160270  0.032710  4.89979 4.7970e-07   EducExp <--- Intelligence
##  gam62  -0.111779  0.026876 -4.15899 1.5983e-05       EducExp <--- Siblings
##  gam63   0.172719  0.034306  5.03461 2.3941e-07       EducExp <--- FatherEd
##  gam64   0.151852  0.033688  4.50758 3.2785e-06      EducExp <--- FatherOcc
##  beta65  0.405150  0.032838 12.33799 0.0000e+00         EducExp <--- Grades
##  gam71  -0.039405  0.034500 -1.14215 1.2670e-01 Occup.Asp <--- Intelligence
##  gam72  -0.018825  0.028222 -0.66700 2.5238e-01     Occup.Asp <--- Siblings
##  gam73  -0.041333  0.036216 -1.14126 1.2688e-01     Occup.Asp <--- FatherEd
##  gam74   0.099577  0.035446  2.80924 2.4829e-03    Occup.Asp <--- FatherOcc
##  beta75  0.157912  0.037443  4.21738 1.2358e-05       Occup.Asp <--- Grades
##  beta76  0.549593  0.038260 14.36486 0.0000e+00      Occup.Asp <--- EducExp
##  psi5    0.650995  0.033946 19.17743 0.0000e+00          Grades <--> Grades
##  psi6    0.516652  0.026943 19.17590 0.0000e+00        EducExp <--> EducExp
##  psi7    0.556617  0.029026 19.17644 0.0000e+00    Occup.Asp <--> Occup.Asp

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