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Departments of Preventive Medicine and Biometrics and of Physiology and Biophysics, School of Medicine, University of Colorado Health Sciences Center, Denver, Colorado 80262
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ABSTRACT |
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 TOP
 ABSTRACT
 INTRODUCTION
THE ISSUE EMBEDDED IN...
PHILOSOPHIES ABOUT MULTIPLE...
THE GENERAL STRATEGY
SIMULATED SAMPLE OBSERVATIONS
NEWMAN-KEULS PROCEDURE
BONFERRONI PROCEDURE
LEAST SIGNIFICANT DIFFERENCE...
FALSE DISCOVERY RATE PROCEDURE:...
SUMMARY
APPENDIX
REFERENCES
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Statistical procedures underpin the process of scientific discovery. As
researchers, one way we use these procedures is to test the validity of
a null hypothesis. Often, we test the validity of more than one null
hypothesis. If we fail to use an appropriate procedure to account for
this multiplicity, then we are more likely to reach a wrong scientific
conclusion
we are more likely to make a mistake. In physiology,
experiments that involve multiple comparisons are common: of the
original articles published in 1997 by the American Physiological
Society, ~40% cite a multiple comparison procedure. In this review,
I demonstrate the statistical issue embedded in multiple comparisons,
and I summarize the philosophies of handling this issue. I also
illustrate the three proceduresNewman-Keuls, Bonferroni, least
significant differencecited most often in my literature review; each
of these procedures is of limited practical value. Last, I demonstrate
the false discovery rate procedure, a promising development in multiple
comparisons. The false discovery rate procedure may be the best
practical solution to the problems of multiple comparisons that exist
within physiology and other scientific disciplines.
Bonferroni inequality, false discovery rate, least significant difference, Newman-Keuls, statistics
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INTRODUCTION |
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ABSTRACT
INTRODUCTION
THE ISSUE EMBEDDED IN...
PHILOSOPHIES ABOUT MULTIPLE...
THE GENERAL STRATEGY
SIMULATED SAMPLE OBSERVATIONS
NEWMAN-KEULS PROCEDURE
BONFERRONI PROCEDURE
LEAST SIGNIFICANT DIFFERENCE...
FALSE DISCOVERY RATE PROCEDURE:...
SUMMARY
APPENDIX
REFERENCES
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STATISTICAL PROCEDURES are inherent to
scientific discovery. As researchers, we use these procedures for two
main reasons: to obtain point and interval estimates about the value of
a population parameter, and to test the validity of a null hypothesis
(5). Point and interval estimates emphasize the magnitude
and uncertainty of the experimental results. The test of a null
hypothesis helps guard against an unwarranted scientific conclusion, or
it helps argue for a real experimental effect (18). When
more than one hypothesis is testedwhen multiple comparisons are
madethe validity of our scientific conclusions may be weakened if we
fail to use an appropriate multiple comparison procedure (6, 8,
11, 14, 19, 20).
In studies published recently by the American Physiological Society
(APS), the citation of a multiple comparison procedure is common (Table
1). This finding raises an important
question: do physiologists understand the philosophies and assumptions
behind competing multiple comparison procedures? This question is
relevant for three reasons: there are many procedures available,
textbooks of statistics (for example, Refs. 1, 13, and 18) provide little more than a cursory description of the procedures themselves, and there can be several solutions to the problem created by multiple comparisons.
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In this paper, I summarize the statistical issue embedded in
multiple comparisons, and I review the philosophies of handling this
issue. Then, I illustrate the three proceduresNewman-Keuls, Bonferroni, least significant differencecited most often in my literature review. Last, I review the false discovery rate, a promising
development in multiple comparisons.
Glossary
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Error rate for a single comparison |
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Error rate for a family of k comparisons |
| H0 | Null hypothesis |
| µ | Population mean |
| P | Achieved significance level |
| Pr{A} | Probability of event A |
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Sample mean |
 * |
Critical difference between two sample means |
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THE ISSUE EMBEDDED IN MULTIPLE COMPARISONS |
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TOP
ABSTRACT
INTRODUCTION
THE ISSUE EMBEDDED IN...
PHILOSOPHIES ABOUT MULTIPLE...
THE GENERAL STRATEGY
SIMULATED SAMPLE OBSERVATIONS
NEWMAN-KEULS PROCEDURE
BONFERRONI PROCEDURE
LEAST SIGNIFICANT DIFFERENCE...
FALSE DISCOVERY RATE PROCEDURE:...
SUMMARY
APPENDIX
REFERENCES
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To test a null hypothesis, we must formulate the hypothesis beforehand. Then, using data collected during the experiment, we must compute the observed value T of some test statistic. Last, we must compare the observed value T to a critical value T*, chosen from the distribution of the test statistic that is based on the null hypothesis. If T is more extreme than T*, then that is surprising if the null hypothesis is true, and we are entitled to become skeptical about the scientific validity of the null hypothesis.
Suppose we want to assess renal blood flow in two independent samples.
If our objective is to compare the underlying population means,
µ1 and µ2, then one pair of null and
alternative hypotheses, H0 and
H1, is
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. We can
use mathematical notation1 to
write this statement as
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(1) |
( / 2)]th percentile from the distribution of the test
statistic given that the null hypothesis is true. Equation 1
can be rewritten as
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(2) |
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Multiple comparisons.
Suppose we want to assess renal blood flow in three independent
samples.2 In this setting,
there are three alternative hypotheses,
H1-H3, that correspond to the
comparisons among population means:
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. If the three comparisons are considered to be a family,
then the family will have an error rate , where > . As a result, it is more likely that a
true null hypothesis will be rejected erroneously. This is the
statistical issue that lies at the heart of multiple comparison procedures.
To see why this issue warrants our attention, imagine that each of
k independent comparisons is tested at an error rate of .
Assume that the underlying populations are identical and that each of
the k null hypotheses is true. What is ,
the probability that at least one of the k comparisons will
reject a true null hypothesis? As in Eq. 2, the probability
of rejecting at least one H0 given that all
H0 are true can be written
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= . When the
number of comparisons increases, remains constant, but
increases. For example, if = 0.05, then for
k = 1, 2, 3, 4, 5, ... , 10,
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Misguided multiple comparisons. In many of the studies tallied in Table 1, a multiple comparison procedure was used to analyze several groups of observations made on the same subjects. In general, this use of a multiple comparison procedure is misguided: most procedures assume that the groups are independent, but repeated observations on a subject, for example, observations made during baseline and then during several periods after some intervention, create correlation among the groups (9). As a result, the true error variability is underestimated, and the observed values for the standard deviations of the group means underestimate the true variabilities (9). When most multiple comparison procedures are used to analyze groups of repeated observations, the outcome will be an inflated number of statistically significant differences among the group means (see APPENDIX).
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PHILOSOPHIES ABOUT MULTIPLE COMPARISONS |
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TOP
ABSTRACT
INTRODUCTION
THE ISSUE EMBEDDED IN...
PHILOSOPHIES ABOUT MULTIPLE...
THE GENERAL STRATEGY
SIMULATED SAMPLE OBSERVATIONS
NEWMAN-KEULS PROCEDURE
BONFERRONI PROCEDURE
LEAST SIGNIFICANT DIFFERENCE...
FALSE DISCOVERY RATE PROCEDURE:...
SUMMARY
APPENDIX
REFERENCES
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Would you tell me, please, which way I ought to go from here?
That depends a good deal on where you want to get
to.The Cat
L. Carroll in Alice's Adventures in Wonderland (1865)
When we decide the validity of a single comparison, we can make a
mistake: we can reject a true null hypothesis, or we can fail to reject
a false null hypothesis. When we decide the validity of k
comparisonsthis happens in most experimentswe are more likely to
reject a true null hypothesis. The challenge for any multiple
comparison procedure is to satisfy two conflicting requirements: reduce
the risk that we reject a true null hypothesis but maintain the
likelihood that we detect an experimental effect if it exists (7,
12, 17). The relative importance assigned to these requirements
has produced opposing philosophies about how to handle the issue of
multiple comparisons.
Focus on individual comparisons.
Proponents of this philosophy argue it is sufficient to control the
single comparison error rate , the probability that we reject a true
null hypothesis. They base this philosophy on the assumption that most
scientific comparisons are preplanned (2, 15, 16). This
assumption is naive and unrealistic: many experimental effects are
discovered only after an investigator exploresrummages throughthe data.
Control for multiple comparisons.
In general, physiologists examine the impact of an intervention on a
seta familyof related comparisons: for example, the impact of some
drug on renal blood flow and urinary excretion of hormones and
electrolytes, or a series of paired comparisons among several groups of
observations. In these situations, we base our scientific conclusions
on a family of comparisons: that is, multiple comparisons considered as
a single entity. As a result, it is not the single comparison error
rate that we must control but the family error rate
, the probability that we reject at least one true
null hypothesis in the family of comparisons (7, 8, 11-13,
17, 19-20). Multiple comparison procedures provide control
of the family error rate .
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THE GENERAL STRATEGY |
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TOP
ABSTRACT
INTRODUCTION
THE ISSUE EMBEDDED IN...
PHILOSOPHIES ABOUT MULTIPLE...
THE GENERAL STRATEGY
SIMULATED SAMPLE OBSERVATIONS
NEWMAN-KEULS PROCEDURE
BONFERRONI PROCEDURE
LEAST SIGNIFICANT DIFFERENCE...
FALSE DISCOVERY RATE PROCEDURE:...
SUMMARY
APPENDIX
REFERENCES
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Most multiple comparison procedures use the same basic strategy:
to make inferences about the population means for two groups, µ
and µ
, they compare the magnitude
of the difference between the sample means
and
to a critical difference *. If
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(3) |
µ. Procedures differ in
the statistics substituted for the coefficient c and the
quantity u. Table 2 lists the
statistics for the Newman-Keuls, Bonferroni, and least significant
difference tests.
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SIMULATED SAMPLE OBSERVATIONS |
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TOP
ABSTRACT
INTRODUCTION
THE ISSUE EMBEDDED IN...
PHILOSOPHIES ABOUT MULTIPLE...
THE GENERAL STRATEGY
SIMULATED SAMPLE OBSERVATIONS
NEWMAN-KEULS PROCEDURE
BONFERRONI PROCEDURE
LEAST SIGNIFICANT DIFFERENCE...
FALSE DISCOVERY RATE PROCEDURE:...
SUMMARY
APPENDIX
REFERENCES
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An article published recently in the Journal provides an ideal
framework with which to illustrate multiple comparison procedures. In
the experiment, Koch et al. (10) explored the heritability of running endurance, measured as distance run, in rats. I used the
observed sample statistics from 10 experimental groups (Fig. 1) as the empirical foundation for the
simulated sample
observations.3
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This is how I generated the simulated sample observationsthe data.
Let the random variable Yj represent the
distance run by a rat in group j, where
j = 1, 2, ... , 10. Assume that each Yj is distributed normally with mean
µj and variance
j2
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j using approximate values for the observed
group means and standard deviations (see Ref. 10, Tables 1 and 2). For
simplicity, I limited each sample to 10 observations. One set of 10 simulated samples is listed in Table 3.
For the rest of the review, I use the resulting sample means
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NEWMAN-KEULS PROCEDURE |
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TOP
ABSTRACT
INTRODUCTION
THE ISSUE EMBEDDED IN...
PHILOSOPHIES ABOUT MULTIPLE...
THE GENERAL STRATEGY
SIMULATED SAMPLE OBSERVATIONS
NEWMAN-KEULS PROCEDURE
BONFERRONI PROCEDURE
LEAST SIGNIFICANT DIFFERENCE...
FALSE DISCOVERY RATE PROCEDURE:...
SUMMARY
APPENDIX
REFERENCES
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The Newman-Keuls
procedure4 is a multiple
range test that compares the underlying population means of
r experimental groups. That is, it evaluates the null
hypothesis
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(4) |
at
, the single comparison error rate, by using studentized range
distributions to calculate critical differences (see Eq. 5).
Another multiple range test is the Duncan
procedure.5 It is only the
specification of that differentiates the method of
Duncan from that of Newman-Keuls. The Duncan family error rate is
= 1 (1 )m
1, where m is the number of means
being compared. The Duncan multiple range test is a noted ancestor of
modern multiple comparison procedures, but because
grows with m, the test violates a basic tenet of multiple
comparisons: the control of despite a large number
of comparisons (see Ref. 12, p. 87-89).
The example.
To make inferences about the equality of two population means,
µ and µ, the Newman-Keuls procedure
uses the critical difference
*m, defined as
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(5) |
is the 100[1 ]th percentile from a studentized range
distribution with m means and 
degrees of freedom, and
SE{} is the standard error of the sample mean.
Using the pooled sample variance s2 = 6,883
(see Table 3), the standard error of the sample mean is estimated as
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= 0.05. In this
simulated experiment, there are = 90 degrees of freedom (see
Table 3). Because there can be groups of m = 2, 3, ... , 10 consecutive sample means, there are nine critical
differences to be calculated using Eq. 5 (Table
4).
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, to
its corresponding critical difference *m. If
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> *m, therefore draw no lines.
The next step is to evaluate groups of m = 3
consecutive means
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Practical considerations.
The Newman-Keuls procedure evaluates all r (r 1)/2
paired comparisons among r sample means from a balanced
design. The test assumes the r means are independent and are
based on identical numbers of observations (Ref. 12, p. 86). When it
compares more than three means, the Newman-Keuls procedure no longer
caps the family error rate at ; instead,
> (Ref. 8, p. 127). For this reason, the
Newman-Keuls procedure is of limited value for multiple comparisons.
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BONFERRONI PROCEDURE |
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TOP
ABSTRACT
INTRODUCTION
THE ISSUE EMBEDDED IN...
PHILOSOPHIES ABOUT MULTIPLE...
THE GENERAL STRATEGY
SIMULATED SAMPLE OBSERVATIONS
NEWMAN-KEULS PROCEDURE
BONFERRONI PROCEDURE
LEAST SIGNIFICANT DIFFERENCE...
FALSE DISCOVERY RATE PROCEDURE:...
SUMMARY
APPENDIX
REFERENCES
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The Bonferroni inequality is a probability inequality that does
control the family error rate . For a family of
k comparisons, the Bonferroni inequality defines the upper
bound of the family error rate to be
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is the error rate for each comparison. In other words,
the inequality assigns an error rate of
/ k to each comparison within the
family. Because can vary among comparisons, the general expression
for the family error rate is
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The example.
To make inferences about the equality of two population means,
µ and µ, the Bonferroni procedure
relies on the critical difference *, defined as
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(6) |
/ 2, is the 100[1 ( / 2)]th percentile from a t distribution with
degrees of freedom, and SE{ } is the standard error of the difference between the sample means.
If we define = 0.05, then for each of the 16 preplanned comparisons listed in Table 5
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= 90 degrees of freedom (see
Table 3), t/2, = 3.04. Using the
pooled sample variance s2 = 6,883, the standard
error of the difference between sample means is estimated as
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(7) |
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Practical considerations.
Although it is not a multiple comparison procedure per se, the
Bonferroni inequality can be used for multiple comparison problems. The
technique is valid regardless of whether the r sample means are independent or correlated (Ref. 12, p. 67). The Bonferroni inequality is appealing because it is versatile and simple.
Unfortunately, its appeal is diminished by the strict protection of the
single comparison error rate . As a consequence, the Bonferroni
inequality is conservative: it will be unable to detect some of the
actual differences among a family of k comparisons (see
Table 5).
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LEAST SIGNIFICANT DIFFERENCE PROCEDURE |
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TOP
ABSTRACT
INTRODUCTION
THE ISSUE EMBEDDED IN...
PHILOSOPHIES ABOUT MULTIPLE...
THE GENERAL STRATEGY
SIMULATED SAMPLE OBSERVATIONS
NEWMAN-KEULS PROCEDURE
BONFERRONI PROCEDURE
LEAST SIGNIFICANT DIFFERENCE...
FALSE DISCOVERY RATE PROCEDURE:...
SUMMARY
APPENDIX
REFERENCES
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The least significant difference (LSD) procedure, developed by Sir
R. A. Fisher, preceded the Newman-Keuls multiple range test. Like
the Newman-Keuls test, the LSD procedure compares the underlying
population means of r experimental groups (see Eq. 4), and it sets the family error rate at the
single comparison error rate .
The example.
To make inferences about the equality of two population means,
µ and µ, the LSD procedure uses the
critical difference *, defined as
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(8) |
/ 2, is the
100[1 ( / 2)]th percentile from a
t distribution with degrees of freedom, and
SE{ } is
the standard error of the difference between the sample
means.6
If we define = 0.05, then because there are
= 90 degrees of freedom (see Table 3),
t / 2, = 1.99. As shown in Eq. 7,
SE{ } = 37.1. Therefore, by virtue of Eq. 8, the resulting
critical difference for the LSD procedure is
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Practical considerations.
The LSD procedure evaluates all r (r 1) /2 paired
comparisons among r sample means. In its protected form, the
procedure is done only if a preliminary analysis of variance is
statistically significant (18). When it compares more than
three means, the LSD procedure fails to maintain the family error rate
at (Ref. 8, p. 139). The solution to this
problem is to replace t / 2, in
Eq. 8 with a percentile from a studentized range
distribution: qr1,
(Ref. 8, p. 139) or qr,
(Ref. 12, p. 92).7
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FALSE DISCOVERY RATE PROCEDURE: A RECENT DEVELOPMENT |
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ABSTRACT
INTRODUCTION
THE ISSUE EMBEDDED IN...
PHILOSOPHIES ABOUT MULTIPLE...
THE GENERAL STRATEGY
SIMULATED SAMPLE OBSERVATIONS
NEWMAN-KEULS PROCEDURE
BONFERRONI PROCEDURE
LEAST SIGNIFICANT DIFFERENCE...
FALSE DISCOVERY RATE PROCEDURE:...
SUMMARY
APPENDIX
REFERENCES
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In most experiments, scientists strive to make a discovery: to
reject a null hypothesis. When an experiment involves a family of
k comparisons, a scientist is more likely to make a mistaken discovery. The false discovery rate
procedure8 is a promising
solution to the problem of multiple comparisons. This procedure
controls not the family error rate but the false
discovery rate f, the expected fraction of
null hypotheses rejected mistakenly
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= ; if at least one
null hypothesis is not true, then f 
(3). When we define the family
error rate , we also set an upper bound on the false
discovery rate f. But if we control
f rather than , we gain
statistical power, the ability to detect an experimental effect if it
exists (3, 4, 22).
The example. Unlike the preceding methods, the false discovery rate procedure operates on achieved significance levels (P values) to make inferences about a family of k comparisons. Let Pi represent the significance level associated with comparison i. To execute this procedure, we must complete three steps:
Step 1. Order the k comparisons by decreasing magnitude of Pi. Step 2. For i = k, k 1, ... , 1,
calculate the critical significance level
d*i as
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(9) |
d*i, then reject the null hypotheses
associated with the remaining i
comparisons.10
In the simulation, we selected k = 16
comparisons of interest. For each comparison, we evaluate the null
hypothesis H0: µ = µ by doing a t test. The P
values associated with the resulting t statistics vary from
0.723 
0.001 (Table 6).
If we define the false discovery rate f = 0.05, the magnitude of the family error rate we
have been using, then the critical significance level
d*i varies from 0.050 0.003. In
step 3, we declare comparisons 1-14 to be
statistically significant (see Table 6). Table 5 lists the inferences
for all 16 comparisons.
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Practical considerations. Because the false discovery rate procedure operates on actual P values, it is quite versatile. For example, the procedure can be employed when a family of k comparisons involves different test statistics such as Student t and Wilcoxon signed rank statistics (3, 4). The false discovery rate procedure is valid when the k comparisons are independent (a sample mean is part of only one comparison) or correlated (a sample mean is part of more than one comparison, as in the example) (3, 4, 22).
The false discovery rate procedure has two important benefits. First, it allows us to make an inference, with 100[1 (f / 2)]% confidence, about the direction
of a statistical difference (4, 22). For example, because
f = 0.05, we can declare, with 97.5%
confidence, that µ2 < µ8 (see Table
6). This is a stronger inference than the simple declaration
µ2 µ8 (Ref. 8, p. 27-39).
Second, the statistical results for a set of primary comparisons are
largely consistent despite substantial changes in the number of
secondary comparisons included within the family (22).
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SUMMARY |
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TOP
ABSTRACT
INTRODUCTION
THE ISSUE EMBEDDED IN...
PHILOSOPHIES ABOUT MULTIPLE...
THE GENERAL STRATEGY
SIMULATED SAMPLE OBSERVATIONS
NEWMAN-KEULS PROCEDURE
BONFERRONI PROCEDURE
LEAST SIGNIFICANT DIFFERENCE...
FALSE DISCOVERY RATE PROCEDURE:...
SUMMARY
APPENDIX
REFERENCES
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We dare not seek a single multiple comparison procedure for all experiments.
Adapted from John W. Tukey (1994)
This remark, written by a pioneer in the area of multiple comparisons, reflects the range of multiple comparison problems that manifest themselves in scientific research. Over the last 50-60 years, statisticians have explored numerous approaches in an effort to address these problems (8, 12). In physiology, as in other disciplines, experiments that involve problems of multiple comparisons are common.
In this review, I have shown that, as researchers, we are more likely to reject a true null hypothesis if we fail to use a multiple comparison procedure when we analyze a family of comparisons. I have also illustrated the three procedures cited most often in APS journals: Newman-Keuls, Bonferroni, and LSD. Unfortunately, each of these is of limited value. In many experimental situations, the Newman-Keuls and LSD procedures fail to control the family error rate, the probability that we reject at least one true null hypothesis. In contrast, the Bonferroni inequality is overly conservative: it fails to detect some of the actual differences that exist within the family.
Finally, I have reviewed the false discovery rate: a versatile, simple, and powerful approach to multiple comparisons. As Tukey suggests, it is perhaps unrealistic to expect that a single multiple comparison procedure will suffice for all situations: a statistical procedure designed specifically for a particular experimental situation will perform better than a general procedure. Nevertheless, there is growing evidence (4, 22) that the false discovery rate procedure may be the best practical solution to the problems of multiple comparisons that exist within science.
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APPENDIX |
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ABSTRACT
INTRODUCTION
THE ISSUE EMBEDDED IN...
PHILOSOPHIES ABOUT MULTIPLE...
THE GENERAL STRATEGY
SIMULATED SAMPLE OBSERVATIONS
NEWMAN-KEULS PROCEDURE
BONFERRONI PROCEDURE
LEAST SIGNIFICANT DIFFERENCE...
FALSE DISCOVERY RATE PROCEDURE:...
SUMMARY
APPENDIX
REFERENCES
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For all but one of the multiple comparison procedures listed in Table 1, an important assumption is that the r experimental groups are independent (12).11 In many studies that use these multiple comparison procedures, however, the r groups are not independent. This happens because investigators make repeated observations on each subject: these observations are correlated by virtue of individual biological makeup (9). Therefore, the true error variability is underestimated, and the observed values for the standard deviations of the group means underestimate the true variabilities (9).
To appreciate the impact of correlation on variability, imagine an
investigation in a sample of n subjects. In each subject, some random variable X is measured during two experimental
conditions: a control period and a subsequent intervention period. Let
the random variable measured during the control period be designated X1 and that during the intervention period be
designated X2. Assume that
X1 and X2 are distributed
normally
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2|1, the standard deviation of the
conditional distribution of X2 given that
X1 equals a specific value, depends on the
correlation 
between X1 and
X2
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0, the standard deviation of the variable measured during a second condition, given the value of the first measurement, is reduced by a factor of
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ACKNOWLEDGEMENTS |
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I thank Dr. Steven L. Britton (Department of Physiology and Molecular Medicine, Medical College of Ohio) and colleagues for permission to cite their study.
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FOOTNOTES |
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1 In comments about my review of statistical concepts (Ref. 5), one referee wrote that my exposition was mathematical and therefore unfriendly. I use mathematics for two reasons: mathematics is one dialect of the language of science, and the precision of mathematical notation simplifies communication and clarifies reasoning. Nevertheless, because I appreciate that readers will have different levels of comfort with mathematics, I integrate the mathematics with text summaries.
2
For r experimental groups, there are
r (r 1)/2 paired comparisons possible.
3 Statistical calculations and exercises were executed using SAS Release 6.12 (SAS Institute, Cary, NC, 1996).
4 This procedure is known also by the name Student-Newman-Keuls.
5 Nearly 6% (18 / 321) of the reviewed manuscripts that report a multiple comparison procedure used the Duncan procedure.
6
Because = , this
critical difference is simply the allowance used to obtain a 100(1 )% confidence interval for the difference
(see
Ref. 5, Eq. A2).
7 When the latter coefficient is used in Eq. 8, the method is called the wholly (or honestly) significant difference procedure.
8 This procedure is available within SAS Release 6.12 by using the fdr option in Proc MultTest.
9 Because of the artificial nature of null hypotheses (5), this is a rare occurrence.
10
If i < k when
Pi d*i, then there
will be k i null hypotheses that cannot be rejected.
11 The lone exception is the Bonferroni inequality, which allows the r experimental groups to be correlated.
Address for reprint requests and other correspondence: D. Curran-Everett, Department of Preventive Medicine and Biometrics, B-195, University of Colorado Health Sciences Center, 4200 East 9th Ave., Denver, CO 80262 (E-mail: dcurran-{at}carbon.cudenver.edu).
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TOP
ABSTRACT
INTRODUCTION
THE ISSUE EMBEDDED IN...
PHILOSOPHIES ABOUT MULTIPLE...
THE GENERAL STRATEGY
SIMULATED SAMPLE OBSERVATIONS
NEWMAN-KEULS PROCEDURE
BONFERRONI PROCEDURE
LEAST SIGNIFICANT DIFFERENCE...
FALSE DISCOVERY RATE PROCEDURE:...
SUMMARY
APPENDIX
REFERENCES
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1.
Altman, DG.
Practical Statistics for Medical Research. New York: Chapman & Hall, 1991, p. 210-212.
2.
Armitage, P,
and
Berry G.
Statistical Methods in Medical Research (3rd ed.). Cambridge, MA: Blackwell Scientific Publications, 1994, p. 224-228.
3.
Benjamini, Y,
and
Hochberg Y.
Controlling the false discovery rate: a practical and powerful approach to multiple testing.
J R Stat Soc Ser B
57:
289-300,
1995.
4.
Benjamini, Y,
Hochberg Y,
and
Kling Y.
False discovery rate controlling procedures for pairwise comparisons. Working Paper 93-02 Tel Aviv, Israel: Department of Statistics and Operation Research, Tel Aviv University, 1993.
5.
Curran-Everett, D,
Taylor S,
and
Kafadar K.
Fundamental concepts in statistics: elucidation and illustration.
J Appl Physiol
85:
775-786,
1998
6.
Godfrey, K.
Comparing the means of several groups.
N Engl J Med
313:
1450-1456,
1985[Abstract].
7.
Hochberg, Y,
and
Benjamini Y.
More powerful procedures for multiple significance testing.
Stat Med
9:
811-818,
1990[Web of Science][Medline].
8.
Hsu, JC.
Multiple Comparisons: Theory and Methods. New York: Chapman & Hall, 1996.
9.
Jones, RH.
Longitudinal Data with Serial Correlation: A State-Space Approach. New York: Chapman & Hall, 1993.
10.
Koch, LG,
Meredith TA,
Fraker TD,
Metting PJ,
and
Britton SL.
Heritability of treadmill running endurance in rats.
Am J Physiol Regulatory Integrative Comp Physiol
275:
R1455-R1460,
1998
11.
Ludbrook, J.
On making multiple comparisons in clinical and experimental pharmacology and physiology.
Clin Exp Pharmacol Physiol
18:
379-392,
1991[Web of Science][Medline].
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) and 2 male (
) rats at the extremes
of observed running endurance were paired and bred to produce the
subsequent generation.