Boik_test | R Documentation |

This function calculates the LBI test statistic for testing the null hypothesis *H_0:* There is no interaction.
It returns an exact p-value when *p=2* where *p=min\{a-1,b-1\}*. It returns an exact Monte Carlo p-value when *p>2*. It also provides an asymptotic chi-squared p-value. Note that the p-value of the Boik.test is always one when *p=1*.

Boik_test(x, nsim = 10000, alpha = 0.05, report = TRUE)

`x` |
a numeric matrix, |

`nsim` |
a numeric value, the number of Monte Carlo samples for calculating an exact Monte Carlo p-value. The default value is 10000. |

`alpha` |
a numeric value, the level of the test. The default value is 0.05. |

`report` |
logical: if |

The LBI test statistic is *T_{B93}=(tr(R'R))^2/(p tr((R'R)^2))* where *p=min\{a-1,b-1\}* and *R* is the residual
matrix of the input data matrix, *x*, under the null hypothesis *H_0:* There is no interaction. This test rejects the null hypothesis of no interaction when *T_{B93}* is small.
Boik (1993) provided the exact distribution of *T_{B93}* when *p=2* under *H_0*. In addition, he provided an asymptotic distribution of *T_{B93}* under *H_0* when *q* tends to infinity where *q=max\{a-1,b-1\}*.
Note that the LBI test is powerful when the *a \times b* matrix of interaction terms has small rank and one singular value dominates the remaining singular values or
in practice, if the largest eigenvalue of *RR'* is expected to dominate the remaining eigenvalues.

An object of the class `ITtest`

, which is a list inducing following components:

`pvalue_exact` |
An exact Monte Carlo p-value when |

`pvalue_appro` |
An chi-squared asymptotic p-value. |

`statistic` |
The value of test statistic. |

`Nsim` |
The number of Monte Carlo samples that are used to estimate p-value. |

`data_name` |
The name of the input dataset. |

`test` |
The name of the test. |

`Level` |
The level of test. |

`Result` |
The result of the test at the alpha level with some descriptions on the type of significant interaction. |

Boik, R.J. (1993). Testing additivity in two-way classifications with no replications: the locally best invariant test. Journal of Applied Statistics 20(1): 41-55.

Shenavari, Z., Kharrati-Kopaei, M. (2018). A Method for Testing Additivity in Unreplicated Two-Way Layouts Based on Combining Multiple Interaction Tests. International Statistical Review 86(3): 469-487.

data(MVGH) Boik_test(MVGH, nsim = 1000)

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