That shows that the false match probability depends on δ and σ only through the ratio. (The argument is a little more complicated when v is small, because the ratio is a random quantity, but the conclusion will be the same.) Also, when v is large, the quantity which is distributed as a standard normal distribution. So the probability can be written

where Φ(·) denotes the standard cumulative normal distribution function (for example, Φ(1.645) = 0.95). So, for large values of v, the nonlinear equation can be solved for Kα, so that the probability of interest does not exceed α. For small values of v, Kα is the 100(1 − α)% point of the non-central t distribution with v degrees of freedom and noncentrality parameter (Ref. 14).

Values of Kα are given in Table F.1 below, for various values of α (0.30, 0.25, 0.20, 0.10, 0.05, 0.01, and 0.0004), degrees of freedom (4, 40, 100, and 200), and δ / σ (0.25, 0.33, 0.50, 1, 1.5, 2, and 3). The theory for Hotelling’s T2

TABLE F.1 Values of Kα(n,v) Used in Equivalence t Test (Need to Multiply by

α = 0.30, n = 3

(δ / σ)

 

0.25

0.33

0.50

1

1.5

2

3

v = 4

0.43397

0.44918

0.49809

0.81095

1.35161

1.94726

3.12279

40

0.40683

0.42113

0.46725

0.77043

1.31802

1.92530

3.13875

100

0.40495

0.41919

0.46511

0.76783

1.31622

1.92511

3.14500

200

0.40435

0.41857

0.46443

0.76697

1.31563

1.92510

3.14734

α = 0.30, n = 5

(δ / σ)

 

0.25

0.33

0.50

1

2

3

v = 4

0.44761

0.47385

0.56076

1.11014

2.63496

4.12933

40

0.41965

0.44436

0.52681

1.07231

2.63226

4.19067

100

0.41771

0.44232

0.52445

1.06984

2.63546

4.20685

200

0.41710

0.44167

0.52370

1.06906

2.63664

4.21278



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