The following methods and formulas are used for testing the ratio between the test mean and the reference mean.

The ratio, *ρ*, is equal to the test mean, , divided by the reference mean, , as shown below:
where

The pooled variance for the reference periods, *S*^{2}_{RR}, is given by the following:
The pooled variance for the test periods, *S*^{2}_{TT}, is given by the following:
Let *S*_{TR} be defined as follows:

Term | Description |
---|---|

Y_{ijk} | Response for participant k during period j in sequence i (for more information, go to Methods and formulas for common concepts used in Equivalence Test for a 2x2 Crossover Design.) |

n_{i} | Number of participants in sequence i |

CI = [min(*C, ρ _{L}*), max(

CI = [*ρ _{L}*,

For a hypothesis of Test mean / reference mean > lower limit, the 100(1 – α)% lower bound is equal to* ρ _{L}*.

For a hypothesis of Test mean / reference mean < upper limit, the 100(1 – α)% upper bound is equal to *ρ _{U}*.

Term | Description |
---|---|

Mean of the reference periods (for more information, see the section on the Ratio) | |

Mean of the test periods (for more information, see the section on the Ratio) | |

n_{i} | Number of participants in sequence i |

v | Degrees of freedom |

α | Significance level for the test (alpha) |

t_{1-α,v} | Upper 1 – α critical value for a t-distribution with v degrees of freedom |

Let *t*_{1} be the t-value for the hypothesis, , and let *t*_{2} be the t-value for the hypothesis, , where is the ratio of the test mean to the reference mean for the populations. The t-values are calculated as follows:
where *S*^{2}_{RR} represents the pooled variance for the reference periods, *S*^{2}_{TT} represents the pooled variance for the test periods, and *S*_{TR} represents the overall standard deviation. For more information, see the section on the Pooled variance.

Term | Description |
---|---|

Mean of the reference periods (for more information, see the section on the Ratio) | |

Mean of the test periods (for more information, see the section on the Ratio) | |

δ_{1} | Lower equivalence limit |

δ_{2} | Upper equivalence limit |

The probability, *P*_{H0}, for each null hypothesis is given by the following:

If , then:

H_{0} |
P-Value |
---|---|

Term | Description |
---|---|

Λ | Unknown ratio of the mean of the test population to the mean of the reference population |

δ_{1} | Lower equivalence limit |

δ_{2} | Upper equivalence limit |

v | Degrees of freedom |

T | t-distribution with v degrees of freedom |

t_{1} | t-value for the hypothesis |

t_{2} | t-value for the hypothesis |

For information on how the t-values are calculated, see the section on t-values.