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== 2024.04.19 ==
=== P-value ===
In [https://en.wikipedia.org/wiki/Statistical_hypothesis_testing null-hypothesis significance testing], the '''๐<img style="null" src=https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36>-value'''<sup id="cite_ref-2">[https://en.wikipedia.org/wiki/P-value#cite_note-2 [note 1]]</sup> is the probability of obtaining test results at least as extreme as the [https://en.wikipedia.org/wiki/Realization_(probability) result actually observed], under the assumption that the [https://en.wikipedia.org/wiki/Null_hypothesis null hypothesis] is correct.<sup id="cite_ref-3">[https://en.wikipedia.org/wiki/P-value#cite_note-3 [2]]</sup><sup id="cite_ref-ASA_4-0">[https://en.wikipedia.org/wiki/P-value#cite_note-ASA-4 [3]]</sup> A very small ''p''-value means that such an extreme observed [https://en.wikipedia.org/wiki/Outcome_(probability) outcome] would be very unlikely under the null hypothesis. Even though reporting ''p''-values of statistical tests is common practice in [https://en.wikipedia.org/wiki/Academic_publishing academic publications] of many quantitative fields, misinterpretation and [https://en.wikipedia.org/wiki/Misuse_of_p-values misuse of p-values] is widespread and has been a major topic in mathematics and [https://en.wikipedia.org/wiki/Metascience metascience].<sup id="cite_ref-5">[https://en.wikipedia.org/wiki/P-value#cite_note-5 [4]]</sup><sup id="cite_ref-6">[https://en.wikipedia.org/wiki/P-value#cite_note-6 [5]]</sup> 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".<sup id="cite_ref-7">[https://en.wikipedia.org/wiki/P-value#cite_note-7 [6]]</sup> That said, a 2019 task force by ASA has issued a statement on statistical significance and replicability, concluding with: "''p''-values and significance tests, when properly applied and interpreted, increase the rigor of the conclusions drawn from data".<sup id="cite_ref-ASA2019_8-0">[https://en.wikipedia.org/wiki/P-value#cite_note-ASA2019-8 [7]]</sup><br/> <br/>
In statistics, every conjecture concerning the unknown [https://en.wikipedia.org/wiki/Probability_distribution probability distribution] of a collection of random variables representing the observed data ๐<img style="null" src=https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab> in some study is called a ''statistical hypothesis''. If we state one hypothesis only and the aim of the statistical test is to see whether this hypothesis is tenable, but not to investigate other specific hypotheses, then such a test is called a [https://en.wikipedia.org/wiki/Statistical_hypothesis_testing null hypothesis test].
As our statistical hypothesis will, by definition, state some property of the distribution, the [https://en.wikipedia.org/wiki/Null_hypothesis null hypothesis] is the default hypothesis under which that property does not exist. The null hypothesis is typically that some parameter (such as a correlation or a difference between means) in the populations of interest is zero. Our hypothesis might specify the probability distribution of ๐<img style="null" src=https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab> precisely, or it might only specify that it belongs to some class of distributions. Often, we reduce the data to a single numerical statistic, e.g., ๐<img style="null" src=https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0>, whose marginal probability distribution is closely connected to a main question of interest in the study.
The ''p''-value is used in the context of null hypothesis testing in order to quantify the [https://en.wikipedia.org/wiki/Statistical_significance statistical significance] of a result, the result being the observed value of the chosen statistic ๐<img style="null" src=https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0>.<sup id="cite_ref-9">[https://en.wikipedia.org/wiki/P-value#cite_note-9 [note 2]]</sup> The lower the ''p''-value is, the lower the probability of getting that result if the null hypothesis were true. A result is said to be ''statistically significant'' if it allows us to reject the null hypothesis. All other things being equal, smaller ''p''-values are taken as stronger evidence against the null hypothesis.
Loosely speaking, rejection of the null hypothesis implies that there is sufficient evidence against it.
As a particular example, if a null hypothesis states that a certain summary statistic ๐<img style="null" src=https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0> follows the standard [https://en.wikipedia.org/wiki/Normal_distribution normal distribution] ๐(0,1),<img style="null" src=https://wikimedia.org/api/rest_v1/media/math/render/svg/56a1569ab3f65b7d30a222662aa537e7e4965344> then the rejection of this null hypothesis could mean that (i) the mean of ๐<img style="null" src=https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0> is not 0, or (ii) the [https://en.wikipedia.org/wiki/Variance variance] of ๐<img style="null" src=https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0> is not 1, or (iii) ๐<img style="null" src=https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0> is not normally distributed. Different tests of the same null hypothesis would be more or less sensitive to different alternatives. However, even if we do manage to reject the null hypothesis for all 3 alternatives, and even if we know that the distribution is normal and variance is 1, the null hypothesis test does not tell us which non-zero values of the mean are now most plausible. 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.<br/> <br/> full text link : [https://en.wikipedia.org/wiki/P-value https://en.wikipedia.org/wiki/P-value]
