# Conditional independence test

### Description

Performs a conditional independence test between two variables given a conditioning set.

### Usage

1 2 3 4 5 6 7 8 9 10 11 12 13 14 | ```
## S4 method for signature 'ExpressionSet'
qpCItest(X, i=1, j=2, Q=c(), exact.test=TRUE, use=c("complete.obs", "em"),
tol=0.01, R.code.only=FALSE)
## S4 method for signature 'cross'
qpCItest(X, i=1, j=2, Q=c(), exact.test=TRUE, use=c("complete.obs", "em"),
tol=0.01, R.code.only=FALSE)
## S4 method for signature 'data.frame'
qpCItest(X, i=1, j=2, Q=c(), I=NULL, 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'
qpCItest(X, i=1, j=2, Q=c(), I=NULL, 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'
qpCItest(X, i=1, j=2, Q=c(), R.code.only=FALSE)
``` |

### Arguments

`X` |
data set where the test should be performed. 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` |
indexes or names of the variables in |

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

`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

When variables in `i, j`

and `Q`

are continuous and `I=NULL`

, this function
performs a conditional independence test using a t-test for zero partial regression coefficient
(Lauritzen, 1996, pg. 150). Note that the size of possible `Q`

sets 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
the number of variables in `Q`

.

When variables in `i, j`

and `Q`

are continuous and discrete (mixed data),
indicated with the `I`

argument when `X`

is a matrix, then mixed graphical
model theory (Lauritzen and Wermuth, 1989) is employed and, concretely, it is assumed
that data come from an homogeneous conditional Gaussian distribution. By default, with
`exact.test=TRUE`

, an exact likelihood ratio test for conditional independence is
performed (Lauritzen, 1996, pg. 192-194; Tur, Roverato and Castelo, 2014), otherwise an
asymptotic one is used.

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.

### Value

A list with class `"htest"`

containing the following components:

`statistic` |
in case of pure continuous data and |

`parameter` |
in case of pure continuous data and |

`p.value` |
the p-value for the test. |

`estimate` |
in case of pure continuous data ( |

`alternative` |
a character string describing the alternative hypothesis. |

`method` |
a character string indicating what type of conditional independence test was performed. |

`data.name` |
a character string giving the name(s) of the random variables involved in the conditional independence test. |

### 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.

Lauritzen, S.L. *Graphical models*. Oxford University Press, 1996.

Lauritzen, S.L and Wermuth, N. Graphical Models for associations between variables,
some of which are qualitative and some quantitative. *Ann. Stat.*, 17(1):31-57, 1989.

Tur, I., Roverato, A. and Castelo, R. Mapping eQTL networks with mixed graphical Markov models.
*Genetics*, 198:1377-1393, 2014.

### See Also

`qpCov`

`qpNrr`

`qpEdgeNrr`

### 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))
qpCItest(X, i=3, j=4, Q=1, long.dim.are.variables=FALSE)
qpCItest(X, i=3, j=4, Q=c(1,2), long.dim.are.variables=FALSE)
``` |