p-value
[4][5] In 2016, the American Statistical Association (ASA) made a formal statement that "p-values do not measure the probability that the studied hypothesis is true, or the probability that the data were produced by random chance alone" and that "a p-value, or statistical significance, does not measure the size of an effect or the importance of a result" or "evidence regarding a model or hypothesis".
[7] In statistics, every conjecture concerning the unknown probability distribution of a collection of random variables representing the observed data
The more independent observations from the same probability distribution one has, the more accurate the test will be, and the higher the precision with which one will be able to determine the mean value and show that it is not equal to zero; but this will also increase the importance of evaluating the real-world or scientific relevance of this deviation.
The 0.05 value (equivalent to 1/20 chances) was originally proposed by R. Fisher in 1925 in his famous book entitled "Statistical Methods for Research Workers".
In other words, it remains the case that very small values are relatively unlikely if the null-hypothesis is true, and that a significance test at level
In these circumstances the p-value is defined by taking the least favorable null-hypothesis case, which is typically on the border between null and alternative.
In this method, before conducting the study, one first chooses a model (the null hypothesis) and the alpha level α (most commonly 0.05).
[3][16] Some statisticians have proposed abandoning p-values and focusing more on other inferential statistics,[3] such as confidence intervals,[17][18] likelihood ratios,[19][20] or Bayes factors,[21][22][23] but there is heated debate on the feasibility of these alternatives.
[24][25] Others have suggested to remove fixed significance thresholds and to interpret p-values as continuous indices of the strength of evidence against the null hypothesis.
[28] That said, in 2019 a task force by ASA had convened to consider the use of statistical methods in scientific studies, specifically hypothesis tests and p-values, and their connection to replicability.
They also stress that p-values can provide valuable information when considering the specific value as well as when compared to some threshold.
In general, it stresses that "p-values and significance tests, when properly applied and interpreted, increase the rigor of the conclusions drawn from data".
For data of other nature, for instance, categorical (discrete) data, test statistics might be constructed whose null hypothesis distribution is based on normal approximations to appropriate statistics obtained by invoking the central limit theorem for large samples, as in the case of Pearson's chi-squared test.
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.
Here, the calculated p-value exceeds 0.05, meaning that the data falls within the range of what would happen 95% of the time, if the coin were fair.
However, had one more head been obtained, the resulting p-value (two-tailed) would have been 0.0414 (4.14%), in which case the null hypothesis would be rejected at the 0.05 level.
The difference between the two meanings of "extreme" appear when we consider a sequential hypothesis testing, or optional stopping, for the fairness of the coin.
If we consider every outcome that has equal or lower probability than "3 heads 3 tails" as "at least as extreme", then the p-value is exactly
Thus, the "at least as extreme" definition of p-value is deeply contextual and depends on what the experimenter planned to do even in situations that did not occur.
Considering more male or more female births as equally likely, the probability of the observed outcome is 1/282, or about 1 in 4,836,000,000,000,000,000,000,000; in modern terms, the p-value.
In modern terms, he rejected the null hypothesis of equally likely male and female births at the p = 1/282 significance level.
Ronald Fisher formalized and popularized the use of the p-value in statistics,[40][41] with it playing a central role in his approach to the subject.
so Fisher was willing to reject the null hypothesis (consider the outcome highly unlikely to be due to chance) if all were classified correctly.
Fisher reiterated the p = 0.05 threshold and explained its rationale, stating:[47] 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
[47] Fisher also underlined the 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".
[48] 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.
[50] It is used in multiple hypothesis testing to maintain statistical power while minimizing the false positive rate.
[52] It corresponds to the proportion of the posterior distribution that is of the median's sign, typically varying between 50% and 100%, and representing the certainty with which an effect is positive or negative.
