Test for Equivalence Test with Paired Data

Find definitions and interpretation guidance for every result that is provided in the Test table of the equivalence test with paired data.

Null hypothesis and alternative hypothesis

The null and alternative hypotheses are mutually exclusive statements about a population. An equivalence test uses sample data to determine whether to reject the null hypothesis.
Null hypothesis
Minitab tests one or both of the following null hypotheses, depending on the alternative hypothesis you chose:
  • The difference (or ratio) between the mean of the test population and the mean of the reference population is greater than or equal to the upper equivalence limit.
  • The difference (or ratio) between the mean of the test population and the mean of the reference population is less than or equal to the lower equivalence limit.
Alternative Hypothesis
The alternative hypothesis states one or both of the following:
  • The difference (or ratio) between the mean of the test population and the mean of the reference population is less than the upper equivalence limit
  • The difference (or ratio) between the mean of the test population and the mean of the reference population is greater than the lower equivalence limit.

Interpretation

Use the null and alternative hypotheses to verify that the equivalence criteria are correct and that you have selected the appropriate alternative hypothesis to test.

Test Null hypothesis: Difference ≤ -0.5 or Difference ≥ 0.5 Alternative hypothesis: -0.5 < Difference < 0.5 α level: 0.05

In these results, Minitab tests two null hypotheses about the difference between the mean of the test population and the mean of the reference population: 1) the difference between the population means is less than or equal to the lower equivalence limit of −0.5, and 2) the difference between the population means is greater than or equal to the upper equivalence limit of 0.5. The alternative hypothesis is that the difference between the population means is between the lower and upper equivalence limits (that is, the mean of the test population is equivalent to the mean of the reference population).

α-level

The significance level (denoted by alpha or α) is the maximum acceptable level of risk for rejecting the null hypothesis when the null hypothesis is true (type I error). For example, if you perform an equivalence test using the default hypotheses, an α of 0.05 indicates a 5% risk of claiming equivalence when it is not actually true.

The α-level for an equivalence test also determines the confidence level for the confidence interval. By default, the confidence level is (1 – α) x 100%. If you use the alternative method of calculating the confidence interval, the confidence level is (1 – 2α) x 100%.

Interpretation

Use the α-level to decide whether to reject or fail to reject the null hypothesis (H0).

If the p-value is less than the α-level, then you reject H0 and claim that your results are statistically significant.

DF

The degrees of freedom (DF) indicate the amount of information that is available in your data to estimate the values of the unknown parameters, and to calculate the variability of these estimates.

The degrees of freedom for the equivalence test with paired data are determined by the sample size. For a sample with n pairs of observations, the total degrees of freedom is n - 1.

Interpretation

Minitab uses the degrees of freedom to calculate the test statistic.

Degrees of freedom are affected by the sample size. Increasing your sample size provides more information about the population, which increases the degrees of freedom.

T-value

The test statistic evaluates the size of the difference between two population means relative to the variation of the sample. If equivalence criteria are expressed in terms of a difference between the test mean and reference mean, or a ratio of test mean/reference mean using a lognormal transformation, the t-value measures the difference between the sample reference mean and the sample test mean in units of standard error. If equivalence criteria are expressed in terms of a ratio between the test mean and the reference mean, the t-value measures the difference between the sample test mean and a proportion of the reference mean, relative to the variability of both samples.

Interpretation

You can use the t-value to determine whether to reject the null hypothesis. However, most people use the p-value or the confidence interval because they are easier to interpret.

Generally, the greater the magnitude of difference or ratio relative to the sampling variability, the greater the absolute value of the t-value for the test, and the stronger the evidence against the null hypothesis.

For each test, Minitab uses the t-value to calculate the p-value.

P-value and decision

The p-value is a probability that measures the evidence against the null hypothesis. Lower probabilities provide stronger evidence against the null hypothesis.

Interpretation

Use the p-value to determine whether you have enough evidence to reject the following null hypotheses about the difference (or ratio) between the mean of the test population and the mean of the reference population: 1) the difference (or ratio) is greater than the lower equivalence limit (noninferiority) and 2) the difference (or ratio) is less than the upper equivalence limit (nonsuperiority). By default, the equivalence tests both of these null hypotheses and includes a p-value for each test.

For each null hypothesis, compare the p-value to the significance level for the test (denoted as alpha or α). An α of 0.05 is most common.

P-value ≤ α: The difference (or ratio) is within the equivalence limit
If the p-value is less than or equal to α, you reject the null hypothesis and conclude that the difference (or ratio) between the population means is within the equivalence limit.
P-value > α: The difference (or ratio) is not within the equivalence limit
If the p-value is greater than α, you fail to reject the null hypothesis. You do not have enough evidence to conclude that the difference (or ratio) between the population means is within the equivalence limit.
To demonstrate equivalence, p-values for both null hypotheses must be below the α-level. If the p-value for either test is greater than the α-level, you cannot claim equivalence.
Tip

To visually evaluate the results of an equivalence test, examine the results on the equivalence plot, which is easier to interpret than the p-values.

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