# Hypotheses for Equivalence Test for a 2x2 Crossover Design

The null and alternative hypotheses depend on the option that you select in Hypothesis about.

## Hypotheses for Test mean – reference mean

If you select a hypothesis about the difference between the test mean and the reference mean, Minitab tests two separate null hypotheses for the equivalence test.
Null hypotheses (default)
 H0: Δ ≤ δ1 The difference (Δ) between the mean of the test population and the mean of the reference population is less than or equal to the lower equivalence limit (δ1). H0: Δ ≥ δ2 The difference (Δ) between the mean of the test population and the mean of the reference population is greater than or equal to the upper equivalence limit (δ2).
Alternative hypothesis (default)
 H1: δ1< Δ < δ2 The difference (Δ) between the mean of the test population and the mean of the reference population is greater than the lower equivalence limit (δ1) and less than the upper equivalence limit (δ2).
You can also test the following hypotheses by selecting a different option for the alternative hypothesis.
Option Hypotheses
Test mean > reference mean H0: Test mean – reference mean (Δ) ≤ 0

H1: Test mean – reference mean (Δ) > 0

Test mean < reference mean H0: Test mean – reference mean (Δ) ≥ 0

H1: Test mean – reference mean (Δ) < 0

Test mean - reference mean > lower limit H0: Test mean – reference mean (Δ) ≤ δ1

H1: Test mean – reference mean (Δ) > δ1

Test mean - reference mean < upper limit H0: Test mean – reference mean (Δ) ≥ δ2

H1: Test mean – reference mean (Δ) < δ2

## Hypotheses for Test mean / reference mean

If you select a hypothesis about the ratio of the test mean to the reference mean, Minitab tests two separate null hypotheses for the equivalence test.

Null hypotheses (default)
 H0: ρ ≤ δ1 The ratio (ρ) of the mean of the test population to the mean of the reference population is less than or equal to the lower equivalence limit (δ1). H0: ρ ≥ δ2 The ratio (ρ) of the mean of the test population to the mean of the reference population is greater than or equal to the upper equivalence limit (δ2).
Alternative hypothesis (default)
 H1: δ1< ρ < δ2 The ratio (ρ) of the mean of the test population to the mean of the reference population is greater than the lower equivalence limit (δ1) and less than the upper equivalence limit (δ2).
If both null hypotheses are rejected, then the ratio falls within your equivalence interval and you can claim that the test mean and the reference mean are equivalent.
You can also test the following hypotheses by selecting a different option for the alternative hypothesis.
Option Hypotheses
Test mean / reference mean > lower limit H0: Test mean / reference mean (ρ) ≤ δ1

H1: Test mean / reference mean (ρ) > δ1

Test mean / reference mean < upper limit H0: Test mean / reference mean (ρ) ≥ δ2

H1: Test mean / reference mean (ρ) < δ2

## Hypotheses for Test mean / reference mean (by log transformation)

If you select a hypothesis about the ratio of the test mean to the reference mean using a log transformation, Minitab tests two separate null hypotheses for the equivalence test.

Null hypotheses (default)
 H0: ρ ≤ δ1 The ratio (ρ) of the mean of the test population to the mean of the reference population is less than or equal to the lower equivalence limit (δ1). H0: ρ ≥ δ2 The ratio (ρ) of the mean of the test population to the mean of the reference population is greater than or equal to the upper equivalence limit (δ2).
Alternative hypothesis (default)
 H1: δ1< ρ < δ2 The ratio (ρ) of the mean of the test population to the mean of the reference population is greater than the lower equivalence limit (δ1) and less than the upper equivalence limit (δ2).
If both null hypotheses are rejected, then the ratio falls within your equivalence interval and you can claim that the test mean and the reference mean are equivalent.
You can also test the following hypotheses by selecting a different option for the alternative hypothesis.
Option Hypotheses
Test mean / reference mean > lower limit H0: Test mean / reference mean (ρ) ≤ δ1

H1: Test mean / reference mean (ρ) > δ1

Test mean / reference mean < upper limit H0: Test mean / reference mean (ρ) ≥ δ2

H1: Test mean / reference mean (ρ) < δ2

## Hypotheses for carryover effect

To evaluate the carryover effects in an equivalence test for a 2x2 crossover design, the hypotheses are as follows:
• H0: The carryover effects are equal (λ1 = λ2)
• H1: The carryover effects are not equal (λ1 ≠ λ2)

## Hypotheses for treatment effect

To evaluate the treatment effects in an equivalence test for a 2x2 crossover design, the hypotheses are as follows:
• H0: The treatment effects are equal (φ1 = φ2)
• H1: The treatment effects are not equal (φ1 ≠ φ2)

## Hypotheses for period effect

To evaluate the period effects in an equivalence test for a 2x2 crossover design, the hypotheses are as follows:
• H0: The period effects are equal (π1 = π2)
• H1: The period effects are not equal (π1π2)
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