# Unbiased modified likelihood ratio test for simple separability of a variance-covariance matrix.

### Description

A likelihood ratio test (LRT) for simple separability of a variance-covariance matrix, modified to be unbiased in finite samples. The modification is a penalty-based homothetic transformation of the LRT statistic. The penalty value is optimized for a given mean model, which is left unstructured here. In the required function, the Id1 and Id2 variables correspond to the row and column subscripts, and are the second and third columns in the matrix (2d) data file, respectively; “value2d” refers to the observed variable, and is the fourth column in the matrix data file.

### Usage

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### Arguments

`formula` |
value2d~Id1+Id2 |

`rep` |
the replicate, also called the subject or individual, the first column in the matrix (2d) data file |

`data` |
the name of the matrix data |

`eps` |
the threshold in the stopping criterion for the iterative mle algorithm (estimation) |

`maxiter` |
the maximum number of iterations for the mle algorithm (estimation) |

`startmat` |
the value of the second factor variance-covariance matrix used for initialization, i.e., to start the mle algorithm (estimation) and obtain the initial estimate of the first factor variance-covariance matrix |

`sign.level` |
the significance level, or rejection rate in the testing of the null hypothesis of simple separability for a variance-covariance structure, when the unbiased modified LRT is used, i.e., the critical value in the chi-square test is derived by simulations from the sampling distribution of the LRT statistic |

`n.simul` |
the number of simulations used to build the sampling distribution of the LRT statistic under the null hypothesis, using the same characteristics as the i.i.d. random sample from a matrix normal distribution. At least 8000 simulations are recommended in applications with same characteristics as the example here. |

### Output

“Convergence”, TRUE or FALSE

“chi.df”, the theoretical number of degrees of freedom of the asymptotic chi-square distribution that would apply to the unmodified LRT statistic for simple separability of a variance-covariance structure

“Lambda”, the observed value of the unmodified LRT statistic

“critical.value”, the critical value at the specified significance level for the chi-square distribution with “chi.df” degrees of freedom

“Decision.lambda” will indicate whether or not the null hypothesis of separability was rejected, based on the theoretical LRT statistic

“Simulation.critical.value”, the critical value at the specified significance level that is derived from the sampling distribution of the unbiased modified LRT statistic

“Decision.lambda.simulation”, the decision (acceptance/rejection) regarding the null hypothesis of simple separability, made using the theoretical (biased unmodified) LRT

“Penalty”, the optimized penalty value used in the homothetic transformation between the biased unmodified and unbiased modified LRT statistics

“U1hat”, the estimated variance-covariance matrix for the rows

“Standardized_U1hat”, the standardized estimated variance-covariance matrix for the rows; the standardization is performed by dividing each entry of U1hat by entry(1, 1) of U1hat

“U2hat”, the estimated variance-covariance matrix for the columns

“Standardized_U2hat”, the standardized estimated variance-covariance matrix for the columns; the standardization is performed by multiplying each entry of U2hat by entry(1, 1) of U1hat

“Shat”, the sample variance-covariance matrix computed from the vectorized data matrices

### References

Manceur AM, Dutilleul P. 2013. Unbiased modified likelihood ratio tests for simple and double separability of a variance-covariance structure. Statistics and Probability Letters 83: 631-636.

### Examples

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