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In
statistics Statistics (from German: '' Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, indust ...
, the multiple comparisons, multiplicity or multiple testing problem occurs when one considers a set of
statistical inference Statistical inference is the process of using data analysis to infer properties of an underlying distribution of probability.Upton, G., Cook, I. (2008) ''Oxford Dictionary of Statistics'', OUP. . Inferential statistical analysis infers properti ...
s simultaneously or infers a subset of parameters selected based on the observed values. The more inferences are made, the more likely erroneous inferences become. Several statistical techniques have been developed to address that problem, typically by requiring a stricter significance threshold for individual comparisons, so as to compensate for the number of inferences being made.


History

The problem of multiple comparisons received increased attention in the 1950s with the work of statisticians such as Tukey and Scheffé. Over the ensuing decades, many procedures were developed to address the problem. In 1996, the first international conference on multiple comparison procedures took place in
Israel Israel (; he, יִשְׂרָאֵל, ; ar, إِسْرَائِيل, ), officially the State of Israel ( he, מְדִינַת יִשְׂרָאֵל, label=none, translit=Medīnat Yīsrāʾēl; ), is a country in Western Asia. It is situated ...
.


Definition

Multiple comparisons arise when a statistical analysis involves multiple simultaneous statistical tests, each of which has a potential to produce a "discovery." A stated confidence level generally applies only to each test considered individually, but often it is desirable to have a confidence level for the whole family of simultaneous tests. Failure to compensate for multiple comparisons can have important real-world consequences, as illustrated by the following examples: * Suppose the treatment is a new way of teaching writing to students, and the control is the standard way of teaching writing. Students in the two groups can be compared in terms of grammar, spelling, organization, content, and so on. As more attributes are compared, it becomes increasingly likely that the treatment and control groups will appear to differ on at least one attribute due to random
sampling error In statistics, sampling errors are incurred when the statistical characteristics of a population are estimated from a subset, or sample, of that population. Since the sample does not include all members of the population, statistics of the sample ( ...
alone. * Suppose we consider the efficacy of a
drug A drug is any chemical substance that causes a change in an organism's physiology or psychology when consumed. Drugs are typically distinguished from food and substances that provide nutritional support. Consumption of drugs can be via inhala ...
in terms of the reduction of any one of a number of disease symptoms. As more symptoms are considered, it becomes increasingly likely that the drug will appear to be an improvement over existing drugs in terms of at least one symptom. In both examples, as the number of comparisons increases, it becomes more likely that the groups being compared will appear to differ in terms of at least one attribute. Our confidence that a result will generalize to independent data should generally be weaker if it is observed as part of an analysis that involves multiple comparisons, rather than an analysis that involves only a single comparison. For example, if one test is performed at the 5% level and the corresponding null hypothesis is true, there is only a 5% risk of incorrectly rejecting the null hypothesis. However, if 100 tests are each conducted at the 5% level and all corresponding null hypotheses are true, the expected number of incorrect rejections (also known as
false positive A false positive is an error in binary classification in which a test result incorrectly indicates the presence of a condition (such as a disease when the disease is not present), while a false negative is the opposite error, where the test resul ...
s or
Type I error In statistical hypothesis testing, a type I error is the mistaken rejection of an actually true null hypothesis (also known as a "false positive" finding or conclusion; example: "an innocent person is convicted"), while a type II error is the fa ...
s) is 5. If the tests are statistically independent from each other, the probability of at least one incorrect rejection is approximately 99.4%. The multiple comparisons problem also applies to
confidence intervals In frequentist statistics, a confidence interval (CI) is a range of estimates for an unknown parameter. A confidence interval is computed at a designated ''confidence level''; the 95% confidence level is most common, but other levels, such as 9 ...
. A single confidence interval with a 95%
coverage probability In statistics, the coverage probability is a technique for calculating a confidence interval which is the proportion of the time that the interval contains the true value of interest. For example, suppose our interest is in the mean number of m ...
level will contain the true value of the parameter in 95% of samples. However, if one considers 100 confidence intervals simultaneously, each with 95% coverage probability, the expected number of non-covering intervals is 5. If the intervals are statistically independent from each other, the probability that at least one interval does not contain the population parameter is 99.4%. Techniques have been developed to prevent the inflation of false positive rates and non-coverage rates that occur with multiple statistical tests.


Classification of multiple hypothesis tests


Controlling procedures

If ''m'' independent comparisons are performed, the ''
family-wise error rate In statistics, family-wise error rate (FWER) is the probability of making one or more false discoveries, or type I errors when performing multiple hypotheses tests. Familywise and Experimentwise Error Rates Tukey (1953) developed the concept of a ...
'' (FWER), is given by : \bar = 1-\left( 1-\alpha_ \right)^m. Hence, unless the tests are perfectly positively dependent (i.e., identical), \bar increases as the number of comparisons increases. If we do not assume that the comparisons are independent, then we can still say: : \bar \le m \cdot \alpha_, which follows from
Boole's inequality In probability theory, Boole's inequality, also known as the union bound, says that for any finite or countable set of events, the probability that at least one of the events happens is no greater than the sum of the probabilities of the individu ...
. Example: 0.2649=1-(1-.05)^6 \le .05 \times 6 = 0.3 There are different ways to assure that the family-wise error rate is at most \bar. The most conservative method, which is free of dependence and distributional assumptions, is the
Bonferroni correction In statistics, the Bonferroni correction is a method to counteract the multiple comparisons problem. Background The method is named for its use of the Bonferroni inequalities. An extension of the method to confidence intervals was proposed by Ol ...
\alpha_\mathrm=/m. A marginally less conservative correction can be obtained by solving the equation for the family-wise error rate of m independent comparisons for \alpha_\mathrm. This yields \alpha_ = 1-^, which is known as the
Šidák correction In statistics, the Šidák correction, or Dunn–Šidák correction, is a method used to counteract the problem of multiple comparisons. It is a simple method to control the familywise error rate. When all null hypotheses are true, the method pr ...
. Another procedure is the Holm–Bonferroni method, which uniformly delivers more power than the simple Bonferroni correction, by testing only the lowest p-value (i=1) against the strictest criterion, and the higher p-values (i>1) against progressively less strict criteria. \alpha_\mathrm=/(m-i+1). For continuous problems, one can employ
Bayesian Thomas Bayes (/beɪz/; c. 1701 – 1761) was an English statistician, philosopher, and Presbyterian minister. Bayesian () refers either to a range of concepts and approaches that relate to statistical methods based on Bayes' theorem, or a followe ...
logic to compute m from the prior-to-posterior volume ratio. Continuous generalizations of the
Bonferroni Carlo Emilio Bonferroni (28 January 1892 – 18 August 1960) was an Italian mathematician who worked on probability theory. Biography Bonferroni studied piano and conducting in Turin Conservatory and at University of Turin under Giuseppe Pean ...
and
Šidák correction In statistics, the Šidák correction, or Dunn–Šidák correction, is a method used to counteract the problem of multiple comparisons. It is a simple method to control the familywise error rate. When all null hypotheses are true, the method pr ...
are presented in.


Multiple testing correction

Multiple testing correction refers to making statistical tests more stringent in order to counteract the problem of multiple testing. The best known such adjustment is the
Bonferroni correction In statistics, the Bonferroni correction is a method to counteract the multiple comparisons problem. Background The method is named for its use of the Bonferroni inequalities. An extension of the method to confidence intervals was proposed by Ol ...
, but other methods have been developed. Such methods are typically designed to control the
family-wise error rate In statistics, family-wise error rate (FWER) is the probability of making one or more false discoveries, or type I errors when performing multiple hypotheses tests. Familywise and Experimentwise Error Rates Tukey (1953) developed the concept of a ...
or the
false discovery rate In statistics, the false discovery rate (FDR) is a method of conceptualizing the rate of type I errors in null hypothesis testing when conducting multiple comparisons. FDR-controlling procedures are designed to control the FDR, which is the expec ...
.


Large-scale multiple testing

Traditional methods for multiple comparisons adjustments focus on correcting for modest numbers of comparisons, often in an
analysis of variance Analysis of variance (ANOVA) is a collection of statistical models and their associated estimation procedures (such as the "variation" among and between groups) used to analyze the differences among means. ANOVA was developed by the statistician ...
. A different set of techniques have been developed for "large-scale multiple testing", in which thousands or even greater numbers of tests are performed. For example, in
genomics Genomics is an interdisciplinary field of biology focusing on the structure, function, evolution, mapping, and editing of genomes. A genome is an organism's complete set of DNA, including all of its genes as well as its hierarchical, three-dim ...
, when using technologies such as
microarray A microarray is a multiplex lab-on-a-chip. Its purpose is to simultaneously detect the expression of thousands of genes from a sample (e.g. from a tissue). It is a two-dimensional array on a solid substrate—usually a glass slide or silicon ...
s, expression levels of tens of thousands of genes can be measured, and genotypes for millions of genetic markers can be measured. Particularly in the field of
genetic association Genetic association is when one or more genotypes within a population co-occur with a phenotypic trait more often than would be expected by chance occurrence. Studies of genetic association aim to test whether single-locus alleles or genotype fre ...
studies, there has been a serious problem with non-replication — a result being strongly statistically significant in one study but failing to be replicated in a follow-up study. Such non-replication can have many causes, but it is widely considered that failure to fully account for the consequences of making multiple comparisons is one of the causes. It has been argued that advances in
measurement Measurement is the quantification of attributes of an object or event, which can be used to compare with other objects or events. In other words, measurement is a process of determining how large or small a physical quantity is as compared ...
and
information technology Information technology (IT) is the use of computers to create, process, store, retrieve, and exchange all kinds of data . and information. IT forms part of information and communications technology (ICT). An information technology syste ...
have made it far easier to generate large datasets for exploratory analysis, often leading to the testing of large numbers of hypotheses with no prior basis for expecting many of the hypotheses to be true. In this situation, very high
false positive rate In statistics, when performing multiple comparisons, a false positive ratio (also known as fall-out or false alarm ratio) is the probability of falsely rejecting the null hypothesis for a particular test. The false positive rate is calculated as th ...
s are expected unless multiple comparisons adjustments are made. For large-scale testing problems where the goal is to provide definitive results, the
family-wise error rate In statistics, family-wise error rate (FWER) is the probability of making one or more false discoveries, or type I errors when performing multiple hypotheses tests. Familywise and Experimentwise Error Rates Tukey (1953) developed the concept of a ...
remains the most accepted parameter for ascribing significance levels to statistical tests. Alternatively, if a study is viewed as exploratory, or if significant results can be easily re-tested in an independent study, control of the
false discovery rate In statistics, the false discovery rate (FDR) is a method of conceptualizing the rate of type I errors in null hypothesis testing when conducting multiple comparisons. FDR-controlling procedures are designed to control the FDR, which is the expec ...
(FDR) is often preferred. The FDR, loosely defined as the expected proportion of false positives among all significant tests, allows researchers to identify a set of "candidate positives" that can be more rigorously evaluated in a follow-up study. The practice of trying many unadjusted comparisons in the hope of finding a significant one is a known problem, whether applied unintentionally or deliberately, is sometimes called "p-hacking."


Assessing whether any alternative hypotheses are true

A basic question faced at the outset of analyzing a large set of testing results is whether there is evidence that any of the alternative hypotheses are true. One simple meta-test that can be applied when it is assumed that the tests are independent of each other is to use the
Poisson distribution In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known co ...
as a model for the number of significant results at a given level α that would be found when all null hypotheses are true. If the observed number of positives is substantially greater than what should be expected, this suggests that there are likely to be some true positives among the significant results. For example, if 1000 independent tests are performed, each at level α = 0.05, we expect 0.05 × 1000 = 50 significant tests to occur when all null hypotheses are true. Based on the Poisson distribution with mean 50, the probability of observing more than 61 significant tests is less than 0.05, so if more than 61 significant results are observed, it is very likely that some of them correspond to situations where the alternative hypothesis holds. A drawback of this approach is that it overstates the evidence that some of the alternative hypotheses are true when the
test statistic A test statistic is a statistic (a quantity derived from the sample) used in statistical hypothesis testing.Berger, R. L.; Casella, G. (2001). ''Statistical Inference'', Duxbury Press, Second Edition (p.374) A hypothesis test is typically specifi ...
s are positively correlated, which commonly occurs in practice. . On the other hand, the approach remains valid even in the presence of correlation among the test statistics, as long as the Poisson distribution can be shown to provide a good approximation for the number of significant results. This scenario arises, for instance, when mining significant frequent itemsets from transactional datasets. Furthermore, a careful two stage analysis can bound the FDR at a pre-specified level. Another common approach that can be used in situations where the
test statistic A test statistic is a statistic (a quantity derived from the sample) used in statistical hypothesis testing.Berger, R. L.; Casella, G. (2001). ''Statistical Inference'', Duxbury Press, Second Edition (p.374) A hypothesis test is typically specifi ...
s can be standardized to Z-scores is to make a normal quantile plot of the test statistics. If the observed quantiles are markedly more dispersed than the normal quantiles, this suggests that some of the significant results may be true positives.


See also

* ''q''-value ;Key concepts *
Family-wise error rate In statistics, family-wise error rate (FWER) is the probability of making one or more false discoveries, or type I errors when performing multiple hypotheses tests. Familywise and Experimentwise Error Rates Tukey (1953) developed the concept of a ...
*
False positive rate In statistics, when performing multiple comparisons, a false positive ratio (also known as fall-out or false alarm ratio) is the probability of falsely rejecting the null hypothesis for a particular test. The false positive rate is calculated as th ...
*
False discovery rate In statistics, the false discovery rate (FDR) is a method of conceptualizing the rate of type I errors in null hypothesis testing when conducting multiple comparisons. FDR-controlling procedures are designed to control the FDR, which is the expec ...
(FDR) * False coverage rate (FCR) *
Interval estimation In statistics, interval estimation is the use of sample data to estimate an '' interval'' of plausible values of a parameter of interest. This is in contrast to point estimation, which gives a single value. The most prevalent forms of interva ...
*
Post-hoc analysis In a scientific study, post hoc analysis (from Latin '' post hoc'', "after this") consists of statistical analyses that were specified after the data were seen. They are usually used to uncover specific differences between three or more group mean ...
*
Experimentwise error rate In statistics, family-wise error rate (FWER) is the probability of making one or more false discoveries, or type I errors when performing multiple hypotheses tests. Familywise and Experimentwise Error Rates Tukey (1953) developed the concept of a ...
*
Statistical hypothesis testing A statistical hypothesis test is a method of statistical inference used to decide whether the data at hand sufficiently support a particular hypothesis. Hypothesis testing allows us to make probabilistic statements about population parameters. ...
;General methods of alpha adjustment for multiple comparisons *
Closed testing procedure In statistics, the closed testing procedure is a general method for performing more than one hypothesis test simultaneously. The closed testing principle Suppose there are ''k'' hypotheses ''H''1,..., ''H'k'' to be tested and the overall type I ...
*
Bonferroni correction In statistics, the Bonferroni correction is a method to counteract the multiple comparisons problem. Background The method is named for its use of the Bonferroni inequalities. An extension of the method to confidence intervals was proposed by Ol ...
*Boole– Bonferroni bound * Duncan's new multiple range test * Holm–Bonferroni method * Harmonic mean p-value procedure * Benjamini–Hochberg procedure ;Related concepts *
Testing hypotheses suggested by the data In statistics, hypotheses suggested by a given dataset, when tested with the same dataset that suggested them, are likely to be accepted even when they are not true. This is because circular reasoning (double dipping) would be involved: somethin ...
*
Texas sharpshooter fallacy The Texas sharpshooter fallacy is an informal fallacy which is committed when differences in data are ignored, but similarities are overemphasized. From this reasoning, a false conclusion is inferred. This fallacy is the philosophical or rhetorical ...
*
Model selection Model selection is the task of selecting a statistical model from a set of candidate models, given data. In the simplest cases, a pre-existing set of data is considered. However, the task can also involve the design of experiments such that the ...
*
Look-elsewhere effect The look-elsewhere effect is a phenomenon in the statistical analysis of scientific experiments where an apparently statistically significant observation may have actually arisen by chance because of the sheer size of the parameter space to be sear ...
*
Data dredging Data dredging (also known as data snooping or ''p''-hacking) is the misuse of data analysis to find patterns in data that can be presented as statistically significant, thus dramatically increasing and understating the risk of false positives. T ...


References


Further reading

* F. Betz, T. Hothorn, P. Westfall (2010), ''Multiple Comparisons Using R'', CRC Press * S. Dudoit and M. J. van der Laan (2008), ''Multiple Testing Procedures with Application to Genomics'', Springer * * * P. H. Westfall and S. S. Young (1993), ''Resampling-based Multiple Testing: Examples and Methods for p-Value Adjustment'', Wiley * P. Westfall, R. Tobias, R. Wolfinger (2011) ''Multiple comparisons and multiple testing using SAS'', 2nd edn, SAS Institute
A gallery of examples of implausible correlations sourced by data dredging
{{Statistics Statistical hypothesis testing