checkIdent: Identifiability of a model with one latent variable

Description

Checks four sufficient conditions for identifiability of a Gaussian DAG model with one latent variable.

Usage

checkIdent(amat, latent)

Value

a vector of length four, indicating if the model is identified according to the conditions of theorems 1 and 2 in Stanghellini \& Wermuth (2005). The answer is TRUE if the condition holds and thus the model is globally identified or FALSE if the condition fails, and thus we do not know if the model is identifiable.

Arguments

amat: a square matrix with dimnames, representing the adjacency matrix of a DAG.
latent: an integer representing the latent variables among the nodes, or the name of the node.

Author

Giovanni M. Marchetti

Details

Stanghellini and Wermuth (2005) give some sufficient conditions for checking if a Gaussian model that factorizes according to a DAG is identified when there is one hidden node over which we marginalize. Specifically, the function checks the conditions of Theorem 1, (i) and (ii) and of Theorem 2 (i) and (ii).

References

Stanghellini, E. \& Wermuth, N. (2005). On the identification of path-analysis models with one hidden variable. Biometrika, 92(2), 337-350.

Examples

Run this code

## See DAG in Figure 4 (a) in Stanghellini & Wermuth (2005)
d <- DAG(y1 ~ y3, y2 ~ y3 + y5, y3 ~ y4 + y5, y4 ~ y6)
checkIdent(d, "y3")  # Identifiable
checkIdent(d, "y4")  # Not identifiable?

## See DAG in Figure 5 (a) in Stanghellini & Wermuth (2005)
d <- DAG(y1 ~ y5+y4, y2 ~ y5+y4, y3 ~ y5+y4)
checkIdent(d, "y4")  # Identifiable
checkIdent(d, "y5")  # Identifiable

## A simple function to check identifiability for each node

is.ident <- function(amat){
### Check suff. conditions on each node of a DAG.
   p <- nrow(amat)
   ## Degrees of freedom
     df <- p*(p+1)/2 - p  - sum(amat==1) - p + 1
   if(df <= 0)
       warning(paste("The degrees of freedom are ", df))
    a <- rownames(amat)
    for(i in a) {
      b <- checkIdent(amat, latent=i)
      if(TRUE %in% b)
        cat("Node", i, names(b)[!is.na(b)], "\n")
      else
        cat("Unknown.\n")
    }
  }

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