1. Equivalence Testing
Dig it!

2. Outline Intro Two one-sided test approach
Alternative: regular CI approach
Tryon approach with “inferential” confidence intervals

3. 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
The observed p-value can only be used as 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 vs. standards

4. Conceptual approach
With our regular t-tests, to conclude there is a substantial difference you must observe a difference large enough to conclude it is not due to sampling error
The same approach applies with equivalence testing
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 from distributions centered on large effects
If the difference between means
falls in this range, we would
conclude the means belong to
equivalent groups.

5. Two one-sided tests (TOST)
One method is to test the joint null hypothesis that our mean difference is not as large as the upper value of a specified range and not below the lower bound of the specified range of equivalence
H0a: μ1 - μ2 > δ OR
H0b1: 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
1. Sometimes only this null is tested so as to claim ‘noninferiority’. Often used in clinical trials.

6. First we’d have to reject a null regarding a difference in which μ1 - μ2 < -δ
Then reject a difference of the opposite kind (same size though)

7. Two one-sided tests (TOST)
Having rejected both, we can safely conclude the small difference we see does not come from a distribution where the effect size is too big to ignore

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

9. Example
Scores from a life satisfaction scale given to groups from two different cultures of interest
First specify range of equivalence δ
Say, any score within 3 points of another
Group 1: M = 75, s = 3.2, N = 20
Group 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), but can be said to be unlikely to have fallen in that interval ‘by chance’.

12. Worked out
H01 is rejected if -t ≤ -tcv, and H02 is rejected if t ≥ tcv
df = = 38
Here we reject in both cases (.05 level)1 and conclude statistical equivalence
1.Note these are 2 one-sided tests, so the cv regards the one-tailed critical value tcv = 1.69

13. The CI Approach
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 between means falls entirely within the range of equivalency1
If either lower or upper end falls beyond do not claim equivalent
This is equivalent to the TOST outcome
1. In the previous example, the 95% CI for the difference between means is -.80 to 2.79.

14. Using Inferential Confidence Intervals
Decide on a ranged estimate that reflects your estimation of equivalence (δ)
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 (δ)

15. Equivalence Testing

16. Previous example Scores on the life satisfaction scale
First specify range of equivalence δ
Say, any score within 3 points of another
Group 1: M = 75, s = 3.2, N = 20
Group 2: M = 76, s = 2.4, N = 20
ICI95 Section 1 = to 76.06
ICI95 Section 2 = to 76.79
Rg = = 2.84

17. Example
The range observed by our ICIs is not larger than the equivalence range (δ)
Conclude the two classes scored similarly.1
1. At this point you might be confused how this parallels the other CI approach (slide 14). Note that the previous CI dealt with the difference score. Here we are dealing with the original scale units. However, if the ICIs overlap, you know the CI for the difference score would include zero.

18. 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

19. Results
Equivalent?

20. 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 simultaneously
Allows for easy communication of either outcome
Provides for a third outcome
Statistical indeterminancy
Say what??

21. Indeterminancy Neither statistically different or equivalent
Or perhaps both
With equivalence tests one may not be able to come to a solid conclusion
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

22. Figure from Jones et al (BMJ 1996) showing relationship
between equivalence and confidence intervals. This is from the first approach.

23. 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

24. 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
Furthermore, using these methods force you to think about what a meaningful difference is before you even start