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 hypotheses.
H_{0}: Δ ≤ δ_{1} | The difference (Δ) between the mean of the test population and the target is less than or equal to the lower equivalence limit (δ_{1}). |
H_{0}: Δ ≥ δ_{2} | The difference (Δ) between the mean of the test population and target is greater than or equal to the upper equivalence limit (δ_{2}). |
H_{1}: δ_{1}< Δ < δ_{2} | The difference (Δ) between the mean of the test population and the target value is greater than the lower equivalence limit (δ_{1}) and less than the upper equivalence limit (δ_{2}). |
By selecting a different alternative hypothesis when you perform the test, you can also evaluate additional sets of hypotheses. For more information, go to Hypotheses for 1-Sample Equivalence Test.
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.
Null hypothesis: | Difference ≤ -0.42 or Difference ≥ 0.42 |
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Alternative hypothesis: | -0.42 < Difference < 0.42 |
α level: | 0.05 |
Null Hypothesis | DF | T-Value | P-Value |
---|---|---|---|
Difference ≤ -0.42 | 27 | 5.0972 | 0.000 |
Difference ≥ 0.42 | 27 | -0.97605 | 0.169 |
In these results, Minitab tests two null hypotheses: 1) the difference between the population mean and the target is less than or equal to the lower equivalence limit of −0.42, and 2) the difference between the population mean and the target is greater than or equal to the upper equivalence limit of 0.42. The alternative hypothesis is that the difference between the population mean and the target is between the lower and upper equivalence limits (that is, the population mean is equivalent to the target).
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%.
Use the α-level to decide whether to reject or fail to reject the null hypothesis (H_{0}).
If the p-value is less than the α-level, then you reject H_{0} and claim that your results are statistically significant.
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.
For a 1-sample equivalence test, the total degrees of freedom are the number of observations in your sample minus 1 (n – 1).
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.
The t-value is the observed value of the t-test statistic that measures the difference between an observed sample statistic and its hypothesized population parameter, in units of standard error.
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 the difference relative to the random sampling variability, the greater the absolute value of the t-value for the test, and the stronger the evidence against the null hypothesis.
The t-value for the test is used to calculate the corresponding p-value. If the p-value is less than your significance level, you reject the null hypothesis and conclude that the results are statistically significant. For more information, see the section on 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.
Use the p-value to determine whether you have enough evidence to reject the following null hypotheses about the difference between the population mean and the target: 1) the difference is greater than the lower equivalence limit (noninferiority) and 2) the difference is less than the upper equivalence limit (nonsuperiority). By default, the equivalence test 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.
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.