Those of you who aren't up on their statistics but still want to follow this argument really need to read this wikipedia article of p-value:
https://en.wikipedia.org/wiki/P-value
Relevant excerpts include:
In statistics, the p-value is a function of the observed sample results (a statistic) that is used for testing a statistical hypothesis. Before the test is performed, a threshold value is chosen, called the significance level of the test, traditionally 5% or 1% [1] and denoted as α.
If the p-value is equal to or smaller than the significance level (α), it suggests that the observed data are inconsistent with the assumption that the null hypothesis is true and thus that hypothesis must be rejected (but this does not automatically mean the alternative hypothesis can be accepted as true). When the p-value is calculated correctly, such a test is guaranteed to control the Type I error rate to be no greater than α.
An equivalent interpretation is that p-value is the probability of obtaining the observed sample results, or "more extreme" results, when the null hypothesis is actually true (here, "more extreme" is dependent on the way the hypothesis is tested).[2]
The smaller the p-value, the larger the significance because it tells the investigator that the hypothesis under consideration may not adequately explain the observation. The hypothesis H is rejected if any of these probabilities is less than or equal to a small, fixed
but arbitrarily pre-defined threshold value \alpha, which is referred to as the level of significance. Unlike the p-value, the \alpha level is not derived from any observational data and does not depend on the underlying hypothesis; the value of \alpha is instead determined by the consensus of the research community that the investigator is working in.
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 one-tailed. The researcher rolls the dice and observes that both dice show 6, yielding a test statistic of 12. The p-value 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 p-value without considering the experiment design.
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 two-tailed test. Suppose the researcher observes heads for each flip, yielding a test statistic of n and a p-value of 2/2n. If the coin was flipped only 5 times, the p-value would be 2/32 = 0.0625, which is not significant at the 0.05 level. But if the coin was flipped 10 times, the p-value 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 p-value and significance level. 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 p-value can never be below 0.05 for the coin example).
This demonstrates that in interpreting p-values, one must also know the sample size, which complicates the analysis.
History
While the modern use of p-values was popularized by Fisher in the 1920s, computations of p-values date back to the 1770s, where they were calculated by Pierre-Simon Laplace:[6]
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 p-value that the excess was a real, but unexplained, effect.
The p-value was first formally introduced by Karl Pearson, in his Pearson's chi-squared test,[7] using the chi-squared distribution and notated as capital P.[7] The p-values for the chi-squared distribution (for various values of χ2 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 p-value in statistics was popularized by Ronald Fisher,[8] and it plays a central role in Fisher's approach to statistics.[9]
In his influential book Statistical Methods for Research Workers (1925), Fisher proposes the level p = 0.05, or a 1 in 20 chance of being exceeded by chance, as a limit for statistical significance, and applies this to a normal distribution (as a two-tailed test), thus yielding the rule of two standard deviations (on a normal distribution) for statistical significance (see 68–95–99.7 rule).[10][d][11]
He then computes a table of values, similar to Elderton but, importantly, reverses the roles of χ2 and p. That is, rather than computing p for different values of χ2 (and degrees of freedom n), he computes values of χ2 that yield specified p-values, 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.[12] That allowed computed values of χ2 to be compared against cutoffs and encouraged the use of p-values (especially 0.05, 0.02, and 0.01) as cutoffs, instead of computing and reporting p-values themselves. The same type of tables were then compiled in (Fisher & Yates 1938), which cemented the approach.[11]
As an illustration of the application of p-values to the design and interpretation of experiments, in his following book The Design of Experiments (1935), Fisher presented the lady tasting tea experiment,[13] which is the archetypal example of the p-value.
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 p-value was 1/\binom{8}{4} = 1/70 \approx 0.014, 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:[14]
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 p-value of 1/\binom{6}{3} = 1/20 = 0.05, which would not have met this level of significance.[14] Fisher also underlined the frequentist interpretation of p, as the long-run 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 p-value for statistical inference in science with the Neyman–Pearson method, which he terms "
Acceptance Procedures".[15] Fisher emphasizes that while fixed levels such as 5%, 2%, and 1% are convenient, the exact p-value can be used, and the strength of evidence can and will be revised with further experimentation. In contrast, decision procedures require a clear-cut decision, yielding an irreversible action, and the procedure is based on costs of error, which, he argues,
are inapplicable to scientific research.
Misunderstandings
Despite the ubiquity of p-value tests, this particular test for statistical significance has been criticized for its inherent shortcomings and the potential for misinterpretation.
The data obtained by comparing the p-value to a significance level will yield one of two results: either the null hypothesis is rejected,
or the null hypothesis cannot be rejected at that significance level (which however does not imply that the null hypothesis is true). In Fisher's formulation, there is a disjunction: a low p-value means either that the null hypothesis is true and a highly improbable event has occurred or that the null hypothesis is false.
However, people interpret the p-value in many incorrect ways and try to draw other conclusions from p-values, which do not follow.
The p-value does not in itself allow reasoning about the probabilities of hypotheses, which requires multiple hypotheses or a range of hypotheses, with a prior distribution of likelihoods between them, as in Bayesian statistics. There, one uses a likelihood function for all possible values of the prior instead of the p-value for a single null hypothesis.
The p-value refers only to a single hypothesis, called the null hypothesis and does not make reference to or allow conclusions about any other hypotheses, such as the alternative hypothesis in Neyman–Pearson statistical hypothesis testing. In that approach,one instead has a decision function between two alternatives, often based on a test statistic, and computes the rate of Type I and type II errors as α and β. However, the p-value of a test statistic cannot be directly compared to these error rates α and β. Instead, it is fed into a decision function.
There are several common misunderstandings about p-values.[16][17]
The p-value is not the probability that the null hypothesis is true or the probability that the alternative hypothesis is false. It is not connected to either. In fact, frequentist statistics does not and cannot attach probabilities to hypotheses. Comparison of Bayesian and classical approaches shows that a p-value can be very close to zero and the posterior probability of the null is very close to unity (if there is no alternative hypothesis with a large enough a priori probability that would explain the results more easily), Lindley's paradox. There are also a priori probability distributions bin which the posterior probability and the p-value have similar or equal values.[18]
The p-value is not the probability that a finding is "merely a fluke." Calculating the p-value is based on the assumption that every finding is a fluke, the product of chance alone. Thus, the probability that the result is due to chance is in fact unity. The phrase "the results are due to chance" is used to mean that the null hypothesis is probably correct. However, that is merely a restatement of the inverse probability fallacy since the p-value cannot be used to figure out the probability of a hypothesis being true.
The p-value is not the probability of falsely rejecting the null hypothesis. That error is a version of the so-called prosecutor's fallacy.
The p-value is not the probability that replicating the experiment would yield the same conclusion. Quantifying the replicability of an experiment was attempted through the concept of p-rep.
The significance level, such as 0.05, is not determined by the p-value. Rather, the significance level is decided by the person conducting the experiment (with the value 0.05 widely used by the scientific community) before the data are viewed, and it is compared against the calculated p-value after the test has been performed. (
However, reporting a p-value is more useful than simply saying that the results were or were not significant at a given level and allows readers to decide for themselves whether to consider the results significant.)
The p-value does not indicate the size or importance of the observed effect. The two vary together,
however, and the larger the effect, the smaller the sample size that will be required to get a significant p-value (see effect size).
Criticisms
Critics of p-values point out that the criterion used to decide "statistical significance" is based on an arbitrary choice of level (often set at 0.05).[19] If significance testing is applied to hypotheses that are known to be false in advance, a non-significant result will simply reflect an insufficient sample size;
a p-value depends only on the information obtained from a given experiment.
The p-value is incompatible with the likelihood principle and depends on the experiment design, the test statistic in question. That is, the definition of "more extreme" data depends on the sampling methodology adopted by the investigator;[20] for example, the situation in which the investigator flips the coin 100 times, yielding 50 heads, has a set of extreme data that is different from the situation in which the investigator continues to flip the coin until 50 heads are achieved yielding 100 flips.[21] That is to be expected, as the experiments are different experiments, and the sample spaces and the probability distributions for the outcomes are different even though the observed data (50 heads out of 100 flips) are the same for the two experiments.
Fisher proposed p as an informal measure of evidence against the null hypothesis.
He called on researchers to combine p in the mind with other types of evidence for and against that hypothesis such as the a priori plausibility of the hypothesis and the relative strengths of results from previous studies.[22]
In very rare cases, the use of p-values has been banned by certain journals.[23]
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