fitDagLatent {ggm}

Fitting Gaussian DAG models with one latent variable

Description

Fits by maximum likelihood a Gaussian DAG model where one of the nodes of the graph is latent and it is marginalised over.

Usage

fitDagLatent(amat, Syy, n, latent, norm = 1, seed,
             maxit = 9000, tol = 1e-06, pri = FALSE)

Arguments

amat

a square matrix with dimnames representing the adjacency matrix of the DAG.

Syy

a symmetric positive definite matrix, with dimnames, the sample covariance matrix of the observed variables. The set of the observed nodes of the graph must be a subset of the set of the names of the variables in Syy.

n

a positive integer, the sample size.

latent

the name of the latent variable.

norm

an integer, the kind of normalization of the latent variable. If norm=1, the latent is scaled to have unit variance. If norm=2, the latent is scaled to have unit partial variance given its parents.

seed

an integer, used by set.seed to specify a random starting point of the EM algorithm.

maxit

an integer denoting the maximum number of iterations allowed for the EM algorithm. If the convergence criterion is not satisfied within maxit iterations the algorithms stops and a warning message is returned.

tol

a small real value, denoting the tolerance used in testing convergence.

pri

logical, if pri=TRUE then the value of the deviance at each iteration is printed.

Details

For the EM algorithm used see Kiiveri (1987).

Value

Shat

the fitted covariance matrix of all the variables including the latent one. The latent variable is the last. If norm=1 then the variance of the latent variable is constrained to 1.

Ahat

a square matrix of the fitted regression coefficients. The entry Ahat[i,j] is minus the regression coefficient of variable i in the regression equation j. Thus there is a non zero partial regression coefficient Ahat[i,j] corresponding to each non zero value amat[j,i] in the adjacency matrix.

Dhat

a vector containing the partial variances of each variable given the parents. If norm=2 then the partial variance of the latent variable is constrained to 1.

dev

the ‘deviance’ of the model.

df

the degrees of freedom of the model.

it

the number of EM algorithm iterations at convergence.

Author(s)

Giovanni M. Marchetti

References

Kiiveri,H. T. (1987). An incomplete data approach to the analysis of covariance structures. Psychometrika, 52, 4, 539–554.

Joreskog, K.G. and Goldberger, A.S. (1975). Estimation of a model with multiple indicators and multiple causes of a single latent variable. Journal of the American Statistical Association, 10, 631–639.

See Also

fitDag, checkIdent

Examples

## data from Joreskog and Goldberger (1975)
V <- matrix(c(1,     0.36,   0.21,  0.10,  0.156, 0.158,
              0.36,  1,      0.265, 0.284, 0.192, 0.324,
              0.210, 0.265,  1,     0.176, 0.136, 0.226,
              0.1,   0.284,  0.176, 1,     0.304, 0.305, 
              0.156, 0.192,  0.136, 0.304, 1,     0.344,
              0.158, 0.324,  0.226, 0.305, 0.344, 1),     6,6)
nod <- c("y1", "y2", "y3", "x1", "x2", "x3")
dimnames(V) <- list(nod,nod)
dag <- DAG(y1 ~ z, y2 ~ z, y3 ~ z, z ~ x1 + x2 + x3, x1~x2+x3, x2~x3) 
fitDagLatent(dag, V, n=530, latent="z", seed=4564)
fitDagLatent(dag, V, n=530, latent="z", norm=2, seed=145)