Confidence intervals in equivalence testing

When you perform an equivalence test in Minitab, the options you select for the analysis determine the type of confidence interval that Minitab displays.

(1 – alpha) x 100% confidence interval for equivalence
If use the default settings to perform an equivalence test, Minitab displays a confidence interval for equivalence. The confidence level for this interval is set at (1 – alpha) x 100%. For example, using the default alpha-level of 0.05, Minitab displays a 95% confidence interval for equivalence.
Difference: Mean(C2) - Mean(C1) 95% CI for Difference SE Equivalence Equivalence Interval 1.2773 0.52438 (0, 2.19282) (-2, 2) CI is not within the equivalence interval. Cannot claim equivalence.

In these results, using the default (1 – alpha) x 100% method and an alpha of 0.05 produces a 95% CI for equivalence of (0, 2.1928).

Like a standard confidence interval, the confidence interval for equivalence is calculated using information about the point estimate of the difference (or ratio), the sample size, and the variability in the data. However, the (1 – alpha) x 100% confidence interval for equivalence is specifically derived to closely correspond with the results of an alpha-level equivalence test. For this reason, the confidence interval for equivalence also considers the additional information of the lower and upper limits of the equivalence interval. Because the confidence interval incorporates this additional information, a (1 – alpha) x 100% confidence interval for equivalence is in most cases tighter than a standard (1 – alpha) x 100% confidence interval that is calculated for a t-test.
(1 – 2 alpha) x 100% confidence interval
If you change the Options settings to use a (1–2 alpha) x 100% confidence interval for an equivalence test, Minitab displays a (1–2 alpha) x 100% confidence interval. The confidence level for this interval is set at (1 – 2 alpha) x 100%. For example, using the default alpha-level of 0.05, Minitab displays a 90% confidence interval. This standard confidence interval corresponds with the confidence interval you would obtain using a standard t-test at the same confidence level.
The (1 – 2 alpha) x 100% confidence interval is sometimes requested by regulatory agencies. This method often produces the same upper confidence limit or lower confidence limit as the default (1 – alpha) x 100% confidence interval for equivalence. In many cases, both limits may be the same or be very close. However, the standard confidence interval is generally less powerful in the equivalence testing context and may not always correspond as closely with the results of the alpha-level equivalence test—sometimes being more conservative and sometimes more liberal.
Difference: Mean(C2) - Mean(C1) Difference SE 90% CI Equivalence Interval 1.2773 0.52438 (0.361822, 2.19282) (-2, 2) CI is not within the equivalence interval. Cannot claim equivalence.

In these results, which are calculated for the same data, using the alternative (1 – 2 alpha) x 100% method and an alpha of 0.05 produces a 90% CI of (0.36182, 2.1928). The upper confidence limit is identical to that of the (1 – alpha) x 100% default method, (2.1928). However, the lower confidence limit for the alternative method is slightly higher (0.36182) than the lower confidence limit of the default method (0). Note that both overall conclusions are the same: "CI is not within the equivalence interval. Cannot claim equivalence."

Note

To see the exact calculations used for each confidence interval, go to Methods and Formulas for equivalence tests.

Hsu, J.C., Hwang, J.T.G., Liu, H. K., and Ruberg, S. J. (1994). Confidence Intervals Associated with Tests for Bioequivalence. Biometrika 81, 103-114.

Berger, R.L. and Hsu, J.C. (1996). Bioequivalence Trials, Intersection-Union Tests and Equivalence Confidence Sets. Statistical Science. Vol 11, 283-319.

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