fitDagLatent: Fitting Gaussian DAG models with one latent variable
In ggm: Graphical Markov Models with Mixed Graphs

Description Usage Arguments Details Value Author(s) References See Also Examples

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

1 2	fitDagLatent(amat, Syy, n, latent, norm = 1, seed, maxit = 9000, tol = 1e-06, pri = FALSE)

`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.

For the EM algorithm used see Kiiveri (1987).

`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.

Giovanni M. Marchetti

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.

fitDag, checkIdent

## 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)