Equivalence testing

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					Equivalence Testing


      Dig it!
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
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
Tests of Equivalence
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
TOST
Tests of Equivalence

   Specify a range? Isn’t that subjective?

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

   See if the difference between means is
    significantly different from the specified
    allowable difference
   Must reject two null hypotheses
   H01: 1  2  
   H02: 1  2  
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
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       Top is traditional null search for sig diff
    interval (-3,3)           Bottom the two null approach for equiv
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)* and
    conclude statistical equivalence
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.
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
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 ()
Equivalence Testing
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
Example

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

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


                                                               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
   Provides for a third outcome
       Statistical indeterminancy
       Say what??
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
Figure from Jones et al (BMJ 1996) showing relationship
between equivalence and confidence intervals
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

				
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posted:9/14/2012
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