Equivalence Testing

Dig it!

Outline

Intro Two one-sided test approach

Alternative: regular CI approach Tryon approach with “inferential”

confidence intervals

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

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 meansfalls in this range, we would conclude the means belong to equivalent groups.

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

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)

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

Tests of Equivalence

Specify a range? Isn’t that subjective?

Base it on: Previous research Practical considerations Your knowledge of the scale of

measurement

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

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

1 2 3 1 2 3

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’.

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)1 and conclude statistical equivalence

2 2

2 2

(76 75) 3 22.25

.893.2 2.420 20

(76 75) ( 3) 44.47

.893.2 2.420 20

t

t

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

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)

Equivalence Testing

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 = 73.95 to 76.06 ICI95 Section 2 = 75.21 to 76.79 Rg = 76.79 - 73.95 = 2.84

Example

The range observed by our ICIs is not larger than the equivalence range (δ)

Conclude the two classes scored similarly.1

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

40 9.29 12

47 11.03 12A

B

X s n

X s n

1 2

1 2

1 2

1 2

2 2

95

2 2

95

Y Y

Y Y

x

Y Y

x

Y Y

s sE

s s

t t E

s st t

s s

Results

Equivalent?

2 2 2 2

95

2.68

3.18

2.68 3.18 2.68 3.18 4.16.710

2.68 3.18 2.68 3.18 5.86(11) 2.20 for both groups

A

B

s

s

E

t

40 2.20(.71)(2.68) 40 4.19 35.81 44.119

47 2.20(.71)(3.18) 47 4.97 42.03 51.97

35.81 51.97 16.16

ICI A to

ICI B to

Range to

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??

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

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

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

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