Equivalence Testing

1. Equivalence Testing Dig it!

2. Tests of Equivalence • As has been mentioned, the typical method of NHST applied to looking for differences between groups does not technically allow us to conclude equivalence just because we do not reject null • p is a measure of evidence against the null, not for it • Having a small sample would allow us to the retain the null • Often this conclusion is reached anyway • Stated differently, absence of evidence does not imply evidence of absence • Altman & Bland,1995 • Examples of usage: • generic drug vs established drug • efficacy of counselling therapies

3. Tests of Equivalence • To conclude there is a substantial difference you must observe a difference large enough to conclude it is not due to sampling error • To conclude there is not a substantial difference you must observe a difference small enough to reject that closeness is not due to sampling error

4. Tests of Equivalence

5. Two one-sided tests (TOST) • One method is to test the joint null hypothesis that our mean difference score is not as large as the upper value of the specified range and not below the lower bound of the specified range of equivalence • H01: μ1 - μ2> δ OR • H02: μ1 - μ2< -δ* • By rejecting both of these hypotheses, we can conclude that | μ1 - μ2|< δ, or that our difference falls within the range specified

6. TOST

7. Tests of Equivalence • Specify a range? Isn’t that subjective? • Base it on: • Previous research • Practical considerations • Your knowledge of the scale of measurement

8. TOST • See if the difference between means is significantly different from the specified allowable difference • Must reject two null hypotheses • H01: • H02:

9. Example • Scores from the midterms of two sections of a stat class • First specify range of equivalence  • Say, any score within 3 points of another • Section 1: M = 75, s = 3.2, N = 20 • Section 2: M = 76, s = 2.4, N = 20

10. Example • H01: • H02: • By rejecting H01 we conclude the difference is less than 3 • By rejecting H02 we conclude the difference is greater than -3

11. Fuzzy yet? • Recall that the size difference we are looking for is one that is 3 units. • This would hold whether the first mean was 3 above the second mean or vice versa • Hence we are looking for a difference that lies in the μ1 – μ2 interval (-3,3) Top is traditional null search for sig diff Bottom the two null approach for equiv

12. Worked out • H01 is rejected if -t ≤ -tcv, and H02 is rejected if t ≥ tcv • Df = 20+20-2 = 38 • Here we reject in both cases (.05 level)* and conclude statistical equivalence

13. Another way to look at it • H0: -3 ≤ μ1-μ2 ≤ 3 • In this formulation we reject if either the lower bound of a CI on the mean difference exceeds the upper value in the null hypothesis, or our upper bound of the CI for the mean difference is lower than the lower value of the null hypothesis • In other words, we reject the notion of equivalence if our CI for the difference between means falls outside the H0 range.

14. The CI Approach • So another (and perhaps easier) method is to specify a range of values that would constitute equivalency among groups • -δ to δ • Determine the appropriate confidence interval for the mean difference between the groups • See if the CI for the difference score falls entirely within the range of equivalency • If either lower or upper end falls beyond do not claim equivalent • This is equivalent to the TOST outcome

15. Using Inferential Confidence Intervals • Decide on a ranged estimate that reflects your estimation of equivalence () • In other words, if my ranged estimate is smaller than this, I will conclude equivalence • Establish inferential CIs for each variable’s mean • Create a new range that includes the lower bound from the smaller mean, and the upper bound from the larger mean • Represents the maximum probable difference • See if this CI range (Rg) is smaller than the specified maximum amount of difference allowed to still claim equivalence ()

16. Equivalence Testing

17. Previous example • Scores from the midterms of two sections of a stat class • First specify range of equivalence  • Say, any score within 3 points of another • Section 1: M = 75, s = 3.2, N = 20 • Section 2: M = 76, s = 2.4, N = 20 • ICI95 Section 1 = 73.95 to 76.06 • ICI95 Section 2 = 75.21 to 76.79 • Rg = 76.79 - 73.95 = 2.84

18. Example • The range observed by our ICIs is not larger than the equivalence range () • Conclude the two classes scored similarly.

19. Another Example • Anxiety measures are taken from two groups of clients who’d been exposed to different types of therapies (A & B) • We’ll say the scale goes from 0 to 100 • First establish your range of equivalence

20. Results • Equivalent?

21. Which method? • Tryon’s proposal using ICIs is perhaps preferable in that: • NHST is implicit rather than explicit • Retains respective group information • Covers both tests of difference and equivalence • Provides for a third outcome • Statistical indeterminancy • Say what??

22. Indeterminancy • Neither statistically different or equivalent • Or perhaps both • Judgment must be suspended as there is no evidence for or against any hypothesis • May help in warding off interpretation of ‘marginally significant’ findings as trends

23. Figure from Jones et al (BMJ 1996) showing relationship between equivalence and confidence intervals

24. Note on sample size • It was mentioned how we couldn’t conclude equivalence from a difference test because small samples could easily be used to show nonsignificance • Power is not necessarily the same for tests of equivalence and difference • However the idea is the same, in that with larger samples we will be more likely to conclude equivalence

25. Summary • Confidence intervals are an important component statistical analysis and should always be reported • Non-significance on a test of difference does not allow us to assume equivalence • Methods exist to test the group equivalency, and should be implemented whenever that is the true goal of the research question