# Restricted Likelihood Ratio Test and Generalized F-test for Zero Variance Components

### Description

`rlr.test`

tests whether certain variance components are zeros
using restricted likelihood ratio test and generalized F-test.

### Usage

1 | ```
rlr.test(Y, X, Z, Sigma, m0, nsim = 5000L, seed = 130623L)
``` |

### Arguments

`Y` |
response vector of length |

`X` |
fixed effects design matrix of dimension |

`Z` |
a list of random effects design matrices.
Each matrix should have |

`Sigma` |
a list of random effects correlation structures. Each matrix should be symmetric and positive definite, and match the dimension of the corresponding random effects design matrix. |

`m0` |
an integer indicating the number of nuisance variance components.
Should be between |

`nsim` |
number of simulations from the null distribution. If zero, REML estimates are computed but tests are not performed. |

`seed` |
a seed to be set before simulating from the null distribution. |

### Value

A list containing the following components:

`RLRT` |
a vector of the test statistic and the p-value of restricted likelihood ratio test. |

`GFT` |
a vector of the test statistic and the p-value of generalized F-test. |

`H0.estimate` |
REML estimate of variance components (including the error term) under the null hypothesis. |

`H1.estimate` |
REML estimate of variance components (including the error term) under the alternative hypothesis. |

### Author(s)

Yichi Zhang

### References

Zhang, Y., Staicu, A.-M., and Maity, A. (2014). Testing for a Subset of Variance Components in Linear Mixed Models with Application to Testing for Additivity. Submitted.

### See Also

`pseudo.rlr.test`

, `score.test`

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 | ```
# two-way random effects ANOVA
n1 <- 5L
n2 <- 6L
n0 <- 4L
n <- n1 * n2 * n0
X <- cbind(rep(1, n))
A <- gl(n1, n2 * n0)
Z1 <- model.matrix(~ -1 + A, contrasts.arg = contr.treatment)
B <- rep(gl(n2, n0), n1)
Z2 <- model.matrix(~ -1 + B, contrasts.arg = contr.treatment)
Z3 <- model.matrix(~ -1 + B : A, contrasts.arg = contr.treatment)
set.seed(1L)
Y <- (X %*% 1
+ Z1 %*% rnorm(ncol(Z1), 0, 0.7)
+ Z2 %*% rnorm(ncol(Z2), 0, 0.3)
+ Z3 %*% rnorm(ncol(Z3), 0, 0.5)
+ rnorm(n, 0, 1))
Z <- list(Z1, Z2, Z3)
Sigma <- lapply(Z, function(z) diag(ncol(z)))
# tests interaction effects
rlr.test(Y, X, Z, Sigma, 2L, 2000L, 2L)
# tests overall effects
rlr.test(Y, X, Z, Sigma, 1L, 2000L, 3L)
``` |

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