# Non-rejection rate estimation for a pair of variables

### Description

Estimates the non-rejection rate for one pair of variables.

### Usage

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 | ```
## S4 method for signature 'ExpressionSet'
qpEdgeNrr(X, i=1, j=2, q=1, restrict.Q=NULL, fix.Q=NULL,
nTests=100, alpha=0.05, exact.test=TRUE,
use=c("complete.obs", "em"), tol=0.01,
R.code.only=FALSE)
## S4 method for signature 'data.frame'
qpEdgeNrr(X, i=1, j=2, q=1, I=NULL, restrict.Q=NULL, fix.Q=NULL,
nTests=100, alpha=0.05, long.dim.are.variables=TRUE,
exact.test=TRUE, use=c("complete.obs", "em"), tol=0.01,
R.code.only=FALSE)
## S4 method for signature 'matrix'
qpEdgeNrr(X, i=1, j=2, q=1, I=NULL, restrict.Q=NULL, fix.Q=NULL,
nTests=100, alpha=0.05, long.dim.are.variables=TRUE,
exact.test=TRUE, use=c("complete.obs", "em"), tol=0.01,
R.code.only=FALSE)
## S4 method for signature 'SsdMatrix'
qpEdgeNrr(X, i=1, j=2, q=1, restrict.Q=NULL, fix.Q=NULL,
nTests=100, alpha=0.05, R.code.only=FALSE)
``` |

### Arguments

`X` |
data set from where the non-rejection rate should be estimated. It
can be either an |

`i` |
index or name of one of the two variables in |

`j` |
index or name of the other variable in |

`q` |
order of the conditioning subsets employed in the calculation. |

`I` |
indexes or names of the variables in |

`restrict.Q` |
indexes or names of the variables in |

`fix.Q` |
indexes or names of the variables in |

`nTests` |
number of tests to perform for each pair for variables. |

`alpha` |
significance level of each test. |

`long.dim.are.variables` |
logical; if TRUE it is assumed that when data are in a data frame or in 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. |

`exact.test` |
logical; if |

`use` |
a character string defining the way in which calculations are done in the
presence of missing values. It can be either |

`tol` |
maximum tolerance controlling the convergence of the EM algorithm employed
when the argument |

`R.code.only` |
logical; if FALSE then the faster C implementation is used (default); if TRUE then only R code is executed. |

### Details

The estimation of the non-rejection rate for a pair of variables is calculated as the fraction of tests that accept the null hypothesis of conditional independence given a set of randomly sampled q-order conditionals.

Note that the possible values of `q`

should be in the range 1 to
`min(p,n-3)`

, where `p`

is the number of variables and `n`

the number of observations. The computational cost increases linearly with
`q`

.

When `I`

is set different to `NULL`

then mixed graphical model theory
is employed and, concretely, it is assumed that the data comes from an homogeneous
conditional Gaussian distribution. In this setting further restrictions to the
maximum value of `q`

apply, concretely, it cannot be smaller than
`p`

plus the number of levels of the discrete variables involved in the
marginal distributions employed by the algorithm. By default, with
`exact.test=TRUE`

, an exact test for conditional independence is employed,
otherwise an asymptotic one will be used. Full details on these features can
be found in Tur, Roverato and Castelo (2014).

The argument `I`

specifying what variables are discrete actually applies only
when `X`

is a matrix object since in the other cases data types are specified
for each data columns or slot.

### Value

An estimate of the non-rejection rate for the particular given pair of variables.

### 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.

Tur, I., Roverato, A. and Castelo, R. Mapping eQTL networks with mixed graphical models.
*Submitted*, http://arxiv.org/abs/1402.4547, 2014.

### See Also

`qpNrr`

`qpAvgNrr`

`qpHist`

`qpGraphDensity`

`qpClique`

`qpCov`

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 | ```
require(mvtnorm)
nObs <- 100 ## number of observations to simulate
## the following adjacency matrix describes an undirected graph
## where vertex 3 is conditionally independent of 4 given 1 AND 2
A <- matrix(c(FALSE, TRUE, TRUE, TRUE,
TRUE, FALSE, TRUE, TRUE,
TRUE, TRUE, FALSE, FALSE,
TRUE, TRUE, FALSE, FALSE), nrow=4, ncol=4, byrow=TRUE)
Sigma <- qpG2Sigma(A, rho=0.5)
X <- rmvnorm(nObs, sigma=as.matrix(Sigma))
qpEdgeNrr(X, i=3, j=4, q=1, long.dim.are.variables=FALSE)
qpEdgeNrr(X, i=3, j=4, q=2, long.dim.are.variables=FALSE)
``` |

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