qpPAC {qpgraph} | R Documentation

## Estimation of partial correlation coefficients

### Description

Estimates partial correlation coefficients (PACs) for a Gaussian graphical model with undirected graph G and their corresponding p-values for the null hypothesis of zero-partial correlation.

### Usage

    ## S4 method for signature 'ExpressionSet'
    qpPAC(X, g, return.K=FALSE, tol=0.001,
                                    matrix.completion=c("HTF", "IPF"), verbose=TRUE,
                                    R.code.only=FALSE)
    ## S4 method for signature 'data.frame'
    qpPAC(X, g, return.K=FALSE, long.dim.are.variables=TRUE,
                                 tol=0.001, matrix.completion=c("HTF", "IPF"),
                                 verbose=TRUE, R.code.only=FALSE)
    ## S4 method for signature 'matrix'
    qpPAC(X, g, return.K=FALSE, long.dim.are.variables=TRUE,
                             tol=0.001, matrix.completion=c("HTF", "IPF"),
                             verbose=TRUE, R.code.only=FALSE)

### Arguments

`X` | data set from where to estimate the partial correlation coefficients. It can be an ExpressionSet object, a data frame or a matrix.
`g` | either a `qpGraph` object, or a `graphNEL`, `graphAM` or `graphBAM` object, or an adjacency matrix of an undirected graph.
`return.K` | logical; if TRUE this function also returns the concentration matrix `K`; if FALSE it does not return it (default).
`long.dim.are.variables` | logical; if TRUE it is assumed that when `X` is a data frame or a matrix, the longer dimension is the one defining the random variables (default); if FALSE, then random variables are assumed to be at the columns of the data frame or matrix.
`tol` | maximum tolerance in the application of the IPF algorithm.
`matrix.completion` | algorithm to employ in the matrix completion operations employed to construct a positive definite matrix with the zero pattern specified in `g`
`verbose` | show progress on the calculations.
`R.code.only` | logical; if FALSE then the faster C implementation is used (default); if TRUE then only R code is executed.

### Details

In the context of maximum likelihood estimation (MLE) of PACs it is a necessary condition for the existence of MLEs that the sample size `n` is larger than the clique number `w(G)` of the graph `G`. If the sample size `n` is larger than the maximum boundary of the input graph `bd(G)`, then the default matrix completion algorithm HTF by Hastie, Tibshirani and Friedman (2009) can be used (see the function `qpHTF()` for details), which has the avantage that is faster than IPF (see the function `qpIPF()` for details).

The PAC estimation is done by first obtaining a MLE of the covariance matrix using the `qpIPF` function and the p-values are calculated based on the estimation of the standard errors (see Roverato and Whittaker, 1996) and performing Wald tests based on the asymptotic chi-squared distribution.

### Value

A list with two matrices, one with the estimates of the PACs and the other with their p-values. If `return.K=TRUE` then the MLE of the inverse covariance is also returned as part of the list.

### Author(s)

R. Castelo and A. Roverato

### References

Castelo, R. and Roverato, A. A robust procedure for Gaussian graphical model search from microarray data with p larger than n. J. Mach. Learn. Res., 7:2621-2650, 2006.

Castelo, R. and Roverato, A. Reverse engineering molecular regulatory networks from microarray data with qp-graphs. J. Comp. Biol., 16(2):213-227, 2009.

Hastie, T., Tibshirani, R. and Friedman, J.H. The Elements of Statistical Learning, Springer, 2009.

Roverato, A. and Whittaker, J. Standard errors for the parameters of graphical Gaussian models. Stat. Comput., 6:297-302, 1996.

### See Also

`qpGraph` `qpCliqueNumber` `qpClique` `qpGetCliques` `qpIPF`

### Examples

    require(mvtnorm)

    nVar <- 50  ## number of variables
    maxCon <- 5 ## maximum connectivity per variable
    nObs <- 30  ## number of observations to simulate

    set.seed(123)

    A <- qpRndGraph(p=nVar, d=maxCon)
    Sigma <- qpG2Sigma(A, rho=0.5)
    X <- rmvnorm(nObs, sigma=as.matrix(Sigma))

    nrr.estimates <- qpNrr(X, verbose=FALSE)

    qpg <- qpGraph(nrr.estimates, epsilon=0.5)
    qpg$g

    pac.estimates <- qpPAC(X, g=qpg, verbose=FALSE)

    ## distribution absolute values of the estimated
    ## partial correlation coefficients of the present edges
    summary(abs(pac.estimates$R[upper.tri(pac.estimates$R) & A]))

    ## distribution absolute values of the estimated
    ## partial correlation coefficients of the missing edges
    summary(abs(pac.estimates$R[upper.tri(pac.estimates$R) & !A]))

[Package qpgraph version 2.16.0 Index]