# Equivalence Testing by lL4X730

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```									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       If the difference between means
equivalence testing                  falls in this range, we would
conclude the means belong to
   To conclude there is not a           equivalent groups.

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
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: 1  2  3
   H02: 1  2  3

   By rejecting H01 we conclude the
difference is less than 3
   By rejecting H02 we conclude the
difference is greater than -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
(76  75)  3      2
t                         2.25
2
3.2 2.4    2     .89

20      20
(76  75)  (3)       4
t                           4.47
2
3.2 2.4     2     .89

20      20

   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
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
sY1  sY2
2     2

E
sY1  sY2
X A  40 s  9.29 n  12
t x  t95 E
X B  47 s  11.03 n  12
sY1  sY2
2     2

t x  t95
sY1  sY2
Results
s A  2.68
sB  3.18
2.682  3.182   2.682  3.182 4.16
E                                         .710
2.68  3.18     2.68  3.18   5.86
t95 (11)  2.20 for both groups

ICI A  40  2.20(.71)(2.68)  40  4.19  35.81to 44.119
ICI B  47  2.20(.71)(3.18)  47  4.97  42.03 to 51.97

Range  35.81 to 51.97  16.16

   Equivalent?
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 relationship
between 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

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