pvalue
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In statistical hypothesis testing, the pvalue or probability value or asymptotic significance is the probability for a given statistical model that, when the null hypothesis is true, the statistical summary (such as the sample mean difference between two compared groups) would be the same as or of greater magnitude than the actual observed results.^{[1]} The use of pvalues in statistical hypothesis testing is common in many fields of research^{[2]} such as physics, economics, finance, political science, psychology,^{[3]} biology, criminal justice, criminology, and sociology.^{[4]} Their misuse has been a matter of considerable controversy.
Italicisation, capitalisation and hyphenation of the term varies. For example, AMA style uses "P value," APA style uses "p value," and the American Statistical Association uses "pvalue."^{[5]}
Contents
Basic concepts
The pvalue is used in the context of null hypothesis testing in order to quantify the idea of statistical significance of evidence.^{[a]} Null hypothesis testing is a reductio ad absurdum argument adapted to statistics. In essence, a claim is assumed valid if its counterclaim is improbable.
As such, the only hypothesis that needs to be specified in this test and which embodies the counterclaim is referred to as the null hypothesis (that is, the hypothesis to be nullified). A result is said to be statistically significant if it allows us to reject the null hypothesis. That is, as per the reductio ad absurdum reasoning, the statistically significant result should be highly improbable if the null hypothesis is assumed to be true. The rejection of the null hypothesis implies that the correct hypothesis lies in the logical complement of the null hypothesis. However, unless there is a single alternative to the null hypothesis, the rejection of null hypothesis does not tell us which of the alternatives might be the correct one.
As a general example, if a null hypothesis is assumed to follow the standard normal distribution N(0,1), then the rejection of this null hypothesis can either mean (i) the mean is not zero, or (ii) the variance is not unity, or (iii) the distribution is not normal, depending on the type of test performed. However, supposing we manage to reject the zero mean hypothesis, even if we know the distribution is normal and variance is unity, the null hypothesis test does not tell us which nonzero value we should adopt as the new mean.
If is a random variable representing the observed data and is the statistical hypothesis under consideration, then the notion of statistical significance can be naively quantified by the conditional probability , which gives the likelihood of the observation if the hypothesis is assumed to be correct. However, if is a continuous random variable and an instance is observed, Thus, this naive definition is inadequate and needs to be changed so as to accommodate the continuous random variables.
Nonetheless, it helps to clarify that pvalues should not be confused with probability on hypothesis (as is done in Bayesian hypothesis testing) such as the probability of the hypothesis given the data, or the probability of the hypothesis being true, or the probability of observing the given data.
Definition and interpretation
The pvalue is defined as the probability, under the null hypothesis, here simply denoted by (but is often denoted , as opposed to , which is sometimes used to represent the alternative hypothesis), of obtaining a result equal to or more extreme than what was actually observed. Depending on how it is looked at, the "more extreme than what was actually observed" can mean (righttail event) or (lefttail event) or the "smaller" of and (doubletailed event). Thus, the pvalue is given by
 for right tail event,
 for left tail event,
 for double tail event.
The smaller the pvalue, the higher the significance because it tells the investigator that the hypothesis under consideration may not adequately explain the observation. The null hypothesis is rejected if any of these probabilities is less than or equal to a small, fixed but arbitrarily predefined threshold value , which is referred to as the level of significance. Unlike the pvalue, the level is not derived from any observational data and does not depend on the underlying hypothesis; the value of is instead set by the researcher before examining the data. The setting of is arbitrary. By convention, is commonly set to 0.05, 0.01, 0.005, or 0.001.
Since the value of that defines the left tail or right tail event is a random variable, this makes the pvalue a function of and a random variable in itself; under the null hypothesis, the pvalue is defined uniformly over interval, assuming is continuous. Thus, the pvalue is not fixed. This implies that pvalue cannot be given a frequency counting interpretation since the probability has to be fixed for the frequency counting interpretation to hold. In other words, if the same test is repeated independently bearing upon the same overall null hypothesis, it will yield different pvalues at every repetition. Nevertheless, these different pvalues can be combined using Fisher's combined probability test. It should further be noted that an instantiation of this random pvalue can still be given a frequency counting interpretation with respect to the number of observations taken during a given test, as per the definition, as the percentage of observations more extreme than the one observed under the assumption that the null hypothesis is true.
The fixed predefined level can be interpreted as the rate of falsely rejecting the null hypothesis (or type I error), since
 .
This also means that if we fix an instantiation of pvalue and allow to vary over , we can obtain an equivalent interpretation of pvalue in terms of level as the lowest value of that can be assumed for which the null hypothesis can be rejected for a given set of observations.
There is widespread agreement that pvalues are often misused and misinterpreted.^{[1]}^{[6]}^{[7]} One practice that has been particularly criticized is accepting the alternative hypothesis for any pvalue nominally less than .05 without other supporting evidence. Although pvalues are helpful in assessing how incompatible the data are with a specified statistical model, contextual factors must also be considered, such as "the design of a study, the quality of the measurements, the external evidence for the phenomenon under study, and the validity of assumptions that underlie the data analysis".^{[1]} Another concern is that the pvalue is often misunderstood as being the probability that the null hypothesis is true.^{[1]}^{[8]} Some statisticians have proposed replacing pvalues with alternative measures of evidence,^{[1]} such as confidence intervals,^{[9]}^{[10]} likelihood ratios,^{[11]}^{[12]} or Bayes factors,^{[13]}^{[14]}^{[15]} but there is heated debate on the feasibility of these alternatives.^{[16]}^{[17]} Others have suggested to remove fixed significance thresholds and to interpret pvalues as continuous indices of the strength of evidence against the null hypothesis.^{[18]}^{[19]}
Usage
The pvalue is widely used in statistical hypothesis testing, specifically in null hypothesis significance testing. In this method, as part of experimental design, before performing the experiment, one first chooses a model (the null hypothesis) and a threshold value for p, called the significance level of the test, traditionally 5% or 1% ^{[20]} and denoted as α. If the pvalue is less than the chosen significance level (α), that suggests that the observed data is sufficiently inconsistent with the null hypothesis that the null hypothesis may be rejected. However, that does not prove that the tested hypothesis is true. When the pvalue is calculated correctly, this test guarantees that the type I error rate is at most α^{[further explanation needed]}^{[citation needed]}. For typical analysis, using the standard α = 0.05 cutoff, the null hypothesis is rejected when p < .05 and not rejected when p > .05. The pvalue does not, in itself, support reasoning about the probabilities of hypotheses but is only a tool for deciding whether to reject the null hypothesis.
Calculation
Usually, is a test statistic, rather than any of the actual observations. A test statistic is the output of a scalar function of all the observations. This statistic provides a single number, such as the average or the correlation coefficient, that summarizes the characteristics of the data, in a way relevant to a particular inquiry. As such, the test statistic follows a distribution determined by the function used to define that test statistic and the distribution of the input observational data.
For the important case in which the data are hypothesized to follow the normal distribution, depending on the nature of the test statistic and thus the underlying hypothesis of the test statistic, different null hypothesis tests have been developed. Some such tests are ztest for normal distribution, ttest for Student's tdistribution, ftest for fdistribution. When the data do not follow a normal distribution, it can still be possible to approximate the distribution of these test statistics by a normal distribution by invoking the central limit theorem for large samples, as in the case of Pearson's chisquared test.
Thus computing a pvalue requires a null hypothesis, a test statistic (together with deciding whether the researcher is performing a onetailed test or a twotailed test), and data. Even though computing the test statistic on given data may be easy, computing the sampling distribution under the null hypothesis, and then computing its cumulative distribution function (CDF) is often a difficult problem. Today, this computation is done using statistical software, often via numeric methods (rather than exact formulae), but, in the early and mid 20th century, this was instead done via tables of values, and one interpolated or extrapolated pvalues from these discrete values^{[citation needed]}. Rather than using a table of pvalues, Fisher instead inverted the CDF, publishing a list of values of the test statistic for given fixed pvalues; this corresponds to computing the quantile function (inverse CDF).
Examples
Here a few simple examples follow, each illustrating a potential pitfall.
One roll of a pair of dice
Suppose a researcher rolls a pair of dice once and assumes a null hypothesis that the dice are fair, not loaded or weighted toward any specific number/roll/result; uniform. The test statistic is "the sum of the rolled numbers" and is onetailed. The researcher rolls the dice and observes that both dice show 6, yielding a test statistic of 12. The pvalue of this outcome is 1/36 (because under the assumption of the null hypothesis, the test statistic is uniformly distributed) or about 0.028 (the highest test statistic out of 6×6 = 36 possible outcomes). If the researcher assumed a significance level of 0.05, this result would be deemed significant and the hypothesis that the dice are fair would be rejected.
In this case, a single roll provides a very weak basis (that is, insufficient data) to draw a meaningful conclusion about the dice. This illustrates the danger with blindly applying pvalue without considering the experiment design.
Five heads in a row
Suppose a researcher flips a coin five times in a row and assumes a null hypothesis that the coin is fair. The test statistic of "total number of heads" can be onetailed or twotailed: a onetailed test corresponds to seeing if the coin is biased towards heads, but a twotailed test corresponds to seeing if the coin is biased either way. The researcher flips the coin five times and observes heads each time (HHHHH), yielding a test statistic of 5. In a onetailed test, this is the upper extreme of all possible outcomes, and yields a pvalue of (1/2)^{5} = 1/32 ≈ 0.03. If the researcher assumed a significance level of 0.05, this result would be deemed significant and the hypothesis that the coin is fair would be rejected. In a twotailed test, a test statistic of zero heads (TTTTT) is just as extreme and thus the data of HHHHH would yield a pvalue of 2×(1/2)^{5} = 1/16 ≈ 0.06, which is not significant at the 0.05 level.
This demonstrates that specifying a direction (on a symmetric test statistic) halves the pvalue (increases the significance) and can mean the difference between data being considered significant or not.
Sample size dependence
Suppose a researcher flips a coin some arbitrary number of times (n) and assumes a null hypothesis that the coin is fair. The test statistic is the total number of heads and is a twotailed test. Suppose the researcher observes heads for each flip, yielding a test statistic of n and a pvalue of 2/2^{n}. If the coin was flipped only 5 times, the pvalue would be 2/32 = 0.0625, which is not significant at the 0.05 level. But if the coin was flipped 10 times, the pvalue would be 2/1024 ≈ 0.002, which is significant at the 0.05 level.
In both cases the data suggest that the null hypothesis is false (that is, the coin is not fair somehow), but changing the sample size changes the pvalue. In the first case, the sample size is not large enough to allow the null hypothesis to be rejected at the 0.05 level (in fact, the pvalue can never be below 0.05 for the coin example).
This demonstrates that in interpreting pvalues, one must also know the sample size, which complicates the analysis.
Alternating coin flips
Suppose a researcher flips a coin ten times and assumes a null hypothesis that the coin is fair. The test statistic is the total number of heads and is twotailed. Suppose the researcher observes alternating heads and tails with every flip (HTHTHTHTHT). This yields a test statistic of 5 and a pvalue of 1 (completely unexceptional), as that is the expected number of heads.
Suppose instead that the test statistic for this experiment was the "number of alternations" (that is, the number of times when H followed T or T followed H), which is onetailed. That would yield a test statistic of 9, which is extreme and has a pvalue of . That would be considered extremely significant, well beyond the 0.05 level. These data indicate that, in terms of one test statistic, the data set is extremely unlikely to have occurred by chance, but it does not suggest that the coin is biased towards heads or tails.
By the first test statistic, the data yield a high pvalue, suggesting that the number of heads observed is not unlikely. By the second test statistic, the data yield a low pvalue, suggesting that the pattern of flips observed is very, very unlikely. There is no "alternative hypothesis" (so only rejection of the null hypothesis is possible) and such data could have many causes. The data may instead be forged, or the coin may be flipped by a magician who intentionally alternated outcomes.
This example demonstrates that the pvalue depends completely on the test statistic used and illustrates that pvalues can only help researchers to reject a null hypothesis, not consider other hypotheses.
Coin flipping
As an example of a statistical test, an experiment is performed to determine whether a coin flip is fair (equal chance of landing heads or tails) or unfairly biased (one outcome being more likely than the other).
Suppose that the experimental results show the coin turning up heads 14 times out of 20 total flips. The null hypothesis is that the coin is fair, and the test statistic is the number of heads. If a righttailed test is considered, the pvalue of this result is the chance of a fair coin landing on heads at least 14 times out of 20 flips. That probability can be computed from binomial coefficients as
This probability is the pvalue, considering only extreme results that favor heads. This is called a onetailed test. However, the deviation can be in either direction, favoring either heads or tails. The twotailed pvalue, which considers deviations favoring either heads or tails, may instead be calculated. As the binomial distribution is symmetrical for a fair coin, the twosided pvalue is simply twice the above calculated singlesided pvalue: the twosided pvalue is 0.115.
In the above example:
 Null hypothesis (H_{0}): The coin is fair, with Prob(heads) = 0.5
 Test statistic: Number of heads
 Level of significance: 0.05
 Observation O: 14 heads out of 20 flips; and
 Twotailed pvalue of observation O given H_{0} = 2*min(Prob(no. of heads ≥ 14 heads), Prob(no. of heads ≤ 14 heads))= 2*min(0.058, 0.978) = 2*0.058 = 0.115.
Note that the Prob(no. of heads ≤ 14 heads) = 1  Prob(no. of heads ≥ 14 heads) + Prob(no. of head = 14) = 1  0.058 + 0.036 = 0.978; however, symmetry of the binomial distribution makes it an unnecessary computation to find the smaller of the two probabilities. Here, the calculated pvalue exceeds 0.05, so the observation is consistent with the null hypothesis, as it falls within the range of what would happen 95% of the time were the coin in fact fair. Hence, the null hypothesis at the 5% level is not rejected. Although the coin did not fall evenly, the deviation from the expected outcome is small enough to be consistent with chance.
However, had one more head been obtained, the resulting pvalue (twotailed) would have been 0.0414 (4.14%). The null hypothesis is rejected when a 5% cutoff is used.
Distribution
When the null hypothesis is true, if it takes the form , and the underlying random variable is continuous, then the probability distribution of the pvalue is uniform on the interval [0,1]. By contrast, if the alternative hypothesis is true, the distribution is dependent on sample size and the true value of the parameter being studied.^{[2]}^{[21]}
The distribution of pvalues for a group of studies is called a pcurve.^{[22]} The curve is affected by four factors: the proportion of studies that examined false null hypotheses, the power of the studies that investigated false null hypotheses, the alpha levels, and publication bias.^{[23]} A pcurve can be used to assess the reliability of scientific literature, such as by detecting publication bias or phacking.^{[22]}^{[24]}
History
Computations of pvalues date back to the 1700s, where they were computed for the human sex ratio at birth, and used to compute statistical significance compared to the null hypothesis of equal probability of male and female births.^{[25]} John Arbuthnot studied this question in 1710,^{[26]}^{[27]}^{[28]}^{[29]} and examined birth records in London for each of the 82 years from 1629 to 1710. In every year, the number of males born in London exceeded the number of females. Considering more male or more female births as equally likely, the probability of the observed outcome is 0.5^{82}, or about 1 in 4,836,000,000,000,000,000,000,000; in modern terms, the pvalue. This is vanishingly small, leading Arbuthnot that this was not due to chance, but to divine providence: "From whence it follows, that it is Art, not Chance, that governs." In modern terms, he rejected the null hypothesis of equally likely male and female births at the p = 1/2^{82} significance level. This is and other work by Arbuthnot is credited as "… the first use of significance tests …"^{[30]} the first example of reasoning about statistical significance,^{[31]} and "… perhaps the first published report of a nonparametric test …",^{[27]} specifically the sign test; see details at Sign test § History.
The same question was later addressed by PierreSimon Laplace, who instead used a parametric test, modeling the number of male births with a binomial distribution:^{[32]}
In the 1770s Laplace considered the statistics of almost half a million births. The statistics showed an excess of boys compared to girls. He concluded by calculation of a pvalue that the excess was a real, but unexplained, effect.
The pvalue was first formally introduced by Karl Pearson, in his Pearson's chisquared test,^{[33]} using the chisquared distribution and notated as capital P.^{[33]} The pvalues for the chisquared distribution (for various values of χ^{2} and degrees of freedom), now notated as P, was calculated in (Elderton 1902), collected in (Pearson 1914, pp. xxxi–xxxiii, 26–28, Table XII).
The use of the pvalue in statistics was popularized by Ronald Fisher,^{[34]} and it plays a central role in his approach to the subject.^{[35]} In his influential book Statistical Methods for Research Workers (1925), Fisher proposed the level p = 0.05, or a 1 in 20 chance of being exceeded by chance, as a limit for statistical significance, and applied this to a normal distribution (as a twotailed test), thus yielding the rule of two standard deviations (on a normal distribution) for statistical significance (see 68–95–99.7 rule).^{[36]}^{[b]}^{[37]}
He then computed a table of values, similar to Elderton but, importantly, reversed the roles of χ^{2} and p. That is, rather than computing p for different values of χ^{2} (and degrees of freedom n), he computed values of χ^{2} that yield specified pvalues, specifically 0.99, 0.98, 0.95, 0,90, 0.80, 0.70, 0.50, 0.30, 0.20, 0.10, 0.05, 0.02, and 0.01.^{[38]} That allowed computed values of χ^{2} to be compared against cutoffs and encouraged the use of pvalues (especially 0.05, 0.02, and 0.01) as cutoffs, instead of computing and reporting pvalues themselves. The same type of tables were then compiled in (Fisher & Yates 1938), which cemented the approach.^{[37]}
As an illustration of the application of pvalues to the design and interpretation of experiments, in his following book The Design of Experiments (1935), Fisher presented the lady tasting tea experiment,^{[39]} which is the archetypal example of the pvalue.
To evaluate a lady's claim that she (Muriel Bristol) could distinguish by taste how tea is prepared (first adding the milk to the cup, then the tea, or first tea, then milk), she was sequentially presented with 8 cups: 4 prepared one way, 4 prepared the other, and asked to determine the preparation of each cup (knowing that there were 4 of each). In that case, the null hypothesis was that she had no special ability, the test was Fisher's exact test, and the pvalue was so Fisher was willing to reject the null hypothesis (consider the outcome highly unlikely to be due to chance) if all were classified correctly. (In the actual experiment, Bristol correctly classified all 8 cups.)
Fisher reiterated the p = 0.05 threshold and explained its rationale, stating:^{[40]}
It is usual and convenient for experimenters to take 5 per cent as a standard level of significance, in the sense that they are prepared to ignore all results which fail to reach this standard, and, by this means, to eliminate from further discussion the greater part of the fluctuations which chance causes have introduced into their experimental results.
He also applies this threshold to the design of experiments, noting that had only 6 cups been presented (3 of each), a perfect classification would have only yielded a pvalue of which would not have met this level of significance.^{[40]} Fisher also underlined the interpretation of p, as the longrun proportion of values at least as extreme as the data, assuming the null hypothesis is true.
In later editions, Fisher explicitly contrasted the use of the pvalue for statistical inference in science with the Neyman–Pearson method, which he terms "Acceptance Procedures".^{[41]} Fisher emphasizes that while fixed levels such as 5%, 2%, and 1% are convenient, the exact pvalue can be used, and the strength of evidence can and will be revised with further experimentation. In contrast, decision procedures require a clearcut decision, yielding an irreversible action, and the procedure is based on costs of error, which, he argues, are inapplicable to scientific research.
Related quantities
A closely related concept is the Evalue,^{[42]} which is the expected number of times in multiple testing that one expects to obtain a test statistic at least as extreme as the one that was actually observed if one assumes that the null hypothesis is true. The Evalue is the product of the number of tests and the pvalue.
See also
 Bonferroni correction
 Confidence interval
 Counternull
 False discovery rate
 Fisher's method of combining pvalues
 Generalized pvalue
 Holm–Bonferroni method
 Multiple comparisons
 Null hypothesis
 prep
 pvalue fallacy
 Statistical hypothesis testing
Notes
 ^ Note that the statistical significance of a result does not imply that the result is scientifically significant as well.
 ^ To be precise the p = 0.05 corresponds to about 1.96 standard deviations for a normal distribution (twotailed test), and 2 standard deviations corresponds to about a 1 in 22 chance of being exceeded by chance, or p ≈ 0.045; Fisher notes these approximations.
References
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 ^ Babbie, E. (2007). The practice of social research 11th ed. Thomson Wadsworth: Belmont, California.
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 ^ "Scientists Perturbed by Loss of Stat Tool to Sift Research Fudge from Fact". Scientific American. April 16, 2015.
 ^ Goodman SN (1999). "Toward evidencebased medical statistics. 1: The P value fallacy". Annals of Internal Medicine. 130 (12): 995–1004. doi:10.7326/000348191301219990615000008. PMID 10383371.
 ^ Colquhoun, David (2014). "An investigation of the false discovery rate and the misinterpretation of pvalues". Royal Society Open Science. 1: 140216. doi:10.1098/rsos.140216.
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Further reading
 Pearson, Karl (1900). "On the criterion that a given system of deviations from the probable in the case of a correlated system of variables is such that it can be reasonably supposed to have arisen from random sampling" (PDF). Philosophical Magazine. Series 5. 50 (302): 157–175. doi:10.1080/14786440009463897.
 Elderton, William Palin (1902). "Tables for Testing the Goodness of Fit of Theory to Observation". Biometrika. 1 (2): 155–163. doi:10.1093/biomet/1.2.155.
 Fisher, Ronald (1925). Statistical Methods for Research Workers. Edinburgh: Oliver & Boyd. ISBN 0050021702.
 Fisher, Ronald A. (1971) [1935]. The Design of Experiments (9th ed.). Macmillan. ISBN 0028446909.
 Fisher, R. A.; Yates, F. (1938). Statistical tables for biological, agricultural and medical research. London.
 Stigler, Stephen M. (1986). The history of statistics : the measurement of uncertainty before 1900. Cambridge, Mass: Belknap Press of Harvard University Press. ISBN 0674403401.
 Hubbard, Raymond; Bayarri, M. J. (November 2003), P Values are not Error Probabilities (PDF), archived from the original (PDF) on 20130904, a working paper that explains the difference between Fisher's evidential pvalue and the Neyman–Pearson Type I error rate α.
 Hubbard, Raymond; Armstrong, J. Scott (2006). "Why We Don't Really Know What Statistical Significance Means: Implications for Educators" (PDF). Journal of Marketing Education. 28 (2): 114–120. doi:10.1177/0273475306288399. Archived from the original on May 18, 2006.
 Hubbard, Raymond; Lindsay, R. Murray (2008). "Why P Values Are Not a Useful Measure of Evidence in Statistical Significance Testing" (PDF). Theory & Psychology. 18 (1): 69–88. doi:10.1177/0959354307086923.
 Stigler, S. (December 2008). "Fisher and the 5% level". Chance. 21 (4): 12. doi:10.1007/s0014400800333.
 Dallal, Gerard E. (2012). The Little Handbook of Statistical Practice.
 Biau, D.J.; Jolles, B.M.; Porcher, R. (March 2010). "P value and the theory of hypothesis testing: an explanation for new researchers". Clin Orthop Relat Res. 463 (3): 885–892. doi:10.1007/s1199900911644. PMC 2816758 .
 Reinhart, Alex. Statistics Done Wrong: The Woefully Complete Guide. No Starch Press. p. 176. ISBN 9781593276201.
External links
Wikimedia Commons has media related to Pvalue. 
 Free online pvalues calculators for various specific tests (chisquare, Fisher's Ftest, etc.).
 Understanding pvalues, including a Java applet that illustrates how the numerical values of pvalues can give quite misleading impressions about the truth or falsity of the hypothesis under test.
 StatQuest: P Values, clearly explained on YouTube
 StatQuest: Pvalue pitfalls and power calculations on YouTube