Description Usage Arguments Output References Examples

A likelihood ratio test (LRT) for double separability of a variance-covariance structure, 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 Id3, Id4 and Id5 variables correspond to the row, column and edge subscripts, and are the second, third and fourth columns in the tensor (3d) data file, respectively; “value3d” refers to the observed variable, and is the fifth column in the tensor data file.

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`formula` |
value3d~Id3+Id4+Id5 |

`rep` |
the replicate, also called subject or individual, the first column in the tensor (3d) data file |

`data` |
the name of the tensor 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) |

`startmatU2` |
the value of the second factor variance-covariance matrix used for initialization |

`startmatU3` |
the value of the third factor variance-covariance matrix used for initialization, i.e., startmatU3 together with startmatU2 are used to start the mle algorithm (estimation) and obtain the initial estimate of the first factor variance-covariance matrix U1 |

`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 tensor normal distribution. At least 8000 simulations are recommended in applications with same characteristics as the example here. |

“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 double 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”, the decision (acceptance/rejection) regarding the null hypothesis of double separability, made using the theoretical (biased unmodified) LRT

“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 double separability, made using the unbiased modified 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

“U3hat”, the estimated variance-covariance matrix for the edges

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

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.

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