{ "id": "epdw4t1mjj", "slug": "statistical-hypothesis-testing", "title": "Statistical hypothesis testing", "updated": "2026-09-30", "topic_path": [ { "id": "physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/topics/physical" }, { "id": "physical.math", "label": "Mathematics and statistics", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math" }, { "id": "physical.math.statistics", "label": "Statistics and probability", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math.statistics" }, { "id": "physical.math.statistics.stats_statistical_inference", "label": "Statistical inference, estimation, sampling, and testing", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math.statistics.stats_statistical_inference" }, { "id": "physical.math.statistics.stats_statistical_inference.hypothesis_testing", "label": "Hypothesis testing", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math.statistics.stats_statistical_inference.hypothesis_testing" } ], "geo": [ { "id": "geo.nongeo.t1800.physical.math.statistics.stats_statistical_inference.hypothesis_testing", "label": "Non-geographic · 1800 to 1945: Hypothesis testing", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1800.physical.math.statistics.stats_statistical_inference.hypothesis_testing", "path": [ { "id": "geo.nongeo", "label": "Non-geographic", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo" }, { "id": "geo.nongeo.t1800", "label": "Non-geographic · 1800 to 1945", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1800" }, { "id": "geo.nongeo.t1800.physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1800.physical" }, { "id": "geo.nongeo.t1800.physical.math", "label": "Mathematics and statistics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1800.physical.math" }, { "id": "geo.nongeo.t1800.physical.math.statistics", "label": "Statistics and probability", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1800.physical.math.statistics" }, { "id": "geo.nongeo.t1800.physical.math.statistics.stats_statistical_inference", "label": "Statistical inference, estimation, sampling, and testing", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1800.physical.math.statistics.stats_statistical_inference" }, { "id": "geo.nongeo.t1800.physical.math.statistics.stats_statistical_inference.hypothesis_testing", "label": "Hypothesis testing", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1800.physical.math.statistics.stats_statistical_inference.hypothesis_testing" } ] }, { "id": "geo.nongeo.t1946.physical.math.statistics.stats_statistical_inference.hypothesis_testing", "label": "Non-geographic · 1946 to 2000: Hypothesis testing", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math.statistics.stats_statistical_inference.hypothesis_testing", "path": [ { "id": "geo.nongeo", "label": "Non-geographic", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo" }, { "id": "geo.nongeo.t1946", "label": "Non-geographic · 1946 to 2000", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946" }, { "id": "geo.nongeo.t1946.physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical" }, { "id": "geo.nongeo.t1946.physical.math", "label": "Mathematics and statistics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math" }, { "id": "geo.nongeo.t1946.physical.math.statistics", "label": "Statistics and probability", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math.statistics" }, { "id": "geo.nongeo.t1946.physical.math.statistics.stats_statistical_inference", "label": "Statistical inference, estimation, sampling, and testing", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math.statistics.stats_statistical_inference" }, { "id": "geo.nongeo.t1946.physical.math.statistics.stats_statistical_inference.hypothesis_testing", "label": "Hypothesis testing", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math.statistics.stats_statistical_inference.hypothesis_testing" } ] } ], "excerpt": "Statistical hypothesis testing is a method for deciding whether observed data are consistent with a null hypothesis, by computing a test statistic and reporting a p-value or a reject decision.", "snippet": "Statistical hypothesis testing is a method for deciding whether observed data are consistent with a null hypothesis, by computing a test statistic and reporting a p-value or a reject decision.", "node": "physical.math.statistics.stats_statistical_inference.hypothesis_testing", "markdown": "# Statistical hypothesis testing\n\nStatistical hypothesis testing is a method for deciding whether observed data are consistent with a null hypothesis, by computing a test statistic, comparing it with the distribution that statistic would follow if the null were true, and reporting a p-value and/or an accept–reject decision. It does not produce an effect estimate by itself, and the p-value is not the probability that the hypothesis is true.[1](https://www.amstat.org/asa/files/pdfs/p-valuestatement.pdf) A test produces two things: a measure of compatibility between the data and the null model, and, in the Neyman–Pearson tradition, a rule of behavior that controls how often the analyst will be wrong in the long run.[2](https://doi.org/10.1098/rsta.1933.0009)\n\n| Key fact | Detail |\n|---|---|\n| What a test produces | A p-value, the smallest significance level at which the null would be rejected, and optionally a reject/accept decision; never a probability that the hypothesis is true.[3](https://stats.libretexts.org/Bookshelves/Probability_Theory/Probability_Mathematical_Statistics_and_Stochastic_Processes_%28Siegrist%29/09%3A_Hypothesis_Testing/9.01%3A_Introduction_to_Hypothesis_Testing) |\n| p-value definition | The probability, under the null model, of a result at least as extreme as observed according to the test's specified measure of extremeness, if every model assumption, including the test hypothesis, were correct.[4](https://link.springer.com/content/pdf/10.1007/s10654-016-0149-3.pdf) |\n| Significance level | The maximum probability of a Type I error over distributions specified by the null; conventionally 0.1, 0.05, or 0.01.[3](https://stats.libretexts.org/Bookshelves/Probability_Theory/Probability_Mathematical_Statistics_and_Stochastic_Processes_%28Siegrist%29/09%3A_Hypothesis_Testing/9.01%3A_Introduction_to_Hypothesis_Testing) |\n| Power | \\( 1-\\beta \\), the probability of rejecting a false null; a planning floor of 80% is a common convention.[5](https://www.frontiersin.org/journals/psychology/articles/10.3389/fpsyg.2015.00223/full) |\n| Optimal test | For a simple null against a simple alternative, the likelihood ratio test is the most powerful at any given level (the Neyman–Pearson lemma).[6](https://www.stats.ox.ac.uk/~reinert/stattheory/chapter309.pdf) |\n| Test–interval equivalence | Rejecting the null hypothesis at level \\( \\alpha \\) is equivalent to \\( \\theta_0 \\) lying outside the \\( 1-\\alpha \\) confidence set.[3](https://stats.libretexts.org/Bookshelves/Probability_Theory/Probability_Mathematical_Statistics_and_Stochastic_Processes_%28Siegrist%29/09%3A_Hypothesis_Testing/9.01%3A_Introduction_to_Hypothesis_Testing) |\n\n## How it works\n\nThe mechanism rests on three objects. First, two hypotheses: the null \\( H_0 \\), the model under which the statistic's distribution is computed, and the alternative \\( H_1 \\), the class of models against which the null is compared. A hypothesis specifying a single distribution is simple; one specifying several is composite.[3](https://stats.libretexts.org/Bookshelves/Probability_Theory/Probability_Mathematical_Statistics_and_Stochastic_Processes_%28Siegrist%29/09%3A_Hypothesis_Testing/9.01%3A_Introduction_to_Hypothesis_Testing) Second, a test statistic \\( T \\), a summary of the data whose distribution under \\( H_0 \\) is known analytically or by simulation. The p-value is \\( p = P(T \\geq t(x) \\mid H_0) \\) for a simple null; a small p signals inconsistency with \\( H_0 \\).[6](https://www.stats.ox.ac.uk/~reinert/stattheory/chapter309.pdf)\n\nThird, an error structure. A Type I error rejects a true null; the size of the test is \\( \\sup_{P \\in H_0} P(\\text{reject } H_0) \\), which for a simple null reduces to the rejection probability \\( P(\\text{reject } H_0 \\mid H_0) \\), and a test of level \\( \\alpha \\) need only have size at most \\( \\alpha \\). A Type II error accepts a false null with probability \\( \\beta \\); power is \\( 1-\\beta \\).[6](https://www.stats.ox.ac.uk/~reinert/stattheory/chapter309.pdf) The Neyman–Pearson lemma states that for testing a simple hypothesis against a simple alternative at level \\( \\alpha \\), the most powerful test rejects when the likelihood ratio \\( L(\\theta_1;x)/L(\\theta_0;x) \\geq k_{\\alpha} \\), with \\( k_{\\alpha} \\) chosen so that \\( P(X \\in R \\mid H_0) = \\alpha \\).[6](https://www.stats.ox.ac.uk/~reinert/stattheory/chapter309.pdf) When no uniformly most powerful test exists for a composite alternative, the generalized likelihood ratio test gives a tractable solution with good large-sample properties; its statistic \\( -2\\log\\Lambda \\) is approximately \\( \\chi^2_p \\) under the null, with \\( p \\) the number of independent restrictions, provided the regularity conditions of [Wilks' theorem](https://www.edgepedia.org/wilks-theorem) hold.[7](https://www.stat.uchicago.edu/~stigler/Stat244/ch6r2-opt-withfigs.pdf)\n\n## How it is done\n\nThe canonical steps are: state the null and alternative hypotheses; choose the significance level and the test's directionality; compute the test statistic and its null distribution, analytically or by simulation; compute the p-value; and compare it with the threshold to reach a decision.[8](https://data102.org/ds-102-book/content/chapters/01/hypothesis-testing/) Good practice pairs the p-value with effect sizes, uncertainty measures, and intervals, and predefines the analyses before looking at the data.[9](https://pmc.ncbi.nlm.nih.gov/articles/PMC5187603/)\n\n## Origin\n\nThe modern framework distinguishes \"significance testing\" from \"hypothesis testing\".[10](https://link.springer.com/content/pdf/10.1007/978-1-4614-1412-4_19.pdf) Neyman and Pearson set out preliminary test criteria in their 1928 [Biometrika](https://www.edgepedia.org/biometrika) paper, where they applied Fisher's likelihood, crediting Fisher with the term, as a comparative measure of two hypotheses.[11](https://doi.org/10.1093/biomet/20a.3-4.263) Their 1933 paper in Philosophical Transactions of the Royal Society A framed a test as a rule of behavior: calculate a character \\( x \\) of the facts, reject \\( H \\) if \\( x > x_0 \\), accept otherwise, so that in the long run \\( H \\) is rejected when true not more than once in a hundred times.[2](https://doi.org/10.1098/rsta.1933.0009) A companion 1933 paper treated testing in relation to probabilities a priori.[12](https://doi.org/10.1017/s030500410001152x) The tradition of the 0.05 threshold traces to the practice of statistical significance testing.[13](https://link.springer.com/article/10.1007/s12630-023-02557-5) Fisher later reversed position on fixed levels, writing that no scientific worker has a fixed level of significance at which from year to year, and in all circumstances, he rejects hypotheses (Fisher 1956, p. 42).[14](https://faculty.fiu.edu/~blissl/PearsonFisher.pdf) The procedure most widely taught today, null hypothesis significance testing, is an ill-defined hybrid of the two theories rather than either one.[5](https://www.frontiersin.org/journals/psychology/articles/10.3389/fpsyg.2015.00223/full)\n\n## Variants\n\nTests differ along several axes. One-tailed tests reject only in a prespecified direction; a two-sided p-value can be computed by doubling the smaller of the two one-sided p-values.[15](https://methods.egap.org/guides/analysis-procedures/hypothesis-testing_en.html) Exact tests compute the null distribution exactly, while asymptotic tests rely on large-sample approximations such as the \\( \\chi^2 \\) limit of likelihood ratio statistics.[16](https://eml.berkeley.edu/~mcfadden/e240a_sp00/ch7.pdf)\n\nChoice of test follows the data situation. [Student's t-test](https://www.edgepedia.org/students-t-test), in one-sample, two-sample, and paired forms, suits continuous, approximately normal data.[17](https://www.ncbi.nlm.nih.gov/sites/books/NBK553048/) For binary data in unpaired samples, [Fisher's exact test](https://www.edgepedia.org/fishers-exact-test) uses the 2×2 table, while the chi-square test is a less precise alternative whose accuracy depends on the setting and deteriorates when expected cell counts are small.[18](https://pmc.ncbi.nlm.nih.gov/articles/PMC2881615/) • [27](https://www.biostathandbook.com/small.html) ANOVA generalizes the unpaired t-test to more than two groups but does not identify which groups differ, so multiple-testing methods are needed.[18](https://pmc.ncbi.nlm.nih.gov/articles/PMC2881615/) For ordinal or non-normal data, the [Mann–Whitney U test](https://www.edgepedia.org/mann-whitney-u-test) (Wilcoxon rank sum test) compares two unpaired samples; the Wilcoxon signed rank test handles paired non-normal samples; and the Kruskal–Wallis test extends Mann–Whitney to more than two unpaired samples.[18](https://pmc.ncbi.nlm.nih.gov/articles/PMC2881615/)\n\n## Applications\n\nHypothesis testing is embedded in clinical trial regulation: agencies often require significance levels of \\( \\alpha = 0.05 \\) with pre-specified hypotheses for trial approval.[19](https://www.mdpi.com/2227-7390/14/2/300) Regulatory practice is not exclusively frequentist. A 2026 review describes FDA guidance adopting Bayesian posterior probabilities to justify whether clinical efficacy trials are needed at all, recommending pre-specified decision criteria based on posterior probabilities, documented prior sources, and results presented under multiple prior specifications including a reference prior.[20](https://www.frontiersin.org/journals/medicine/articles/10.3389/fmed.2026.1790396/full)\n\n## Limitations and alternatives\n\nThe known failure modes are well documented. A p-value is conditional on the entire modeling framework and set of analytic decisions that produced it, so post-hoc adjustments can invalidate it as a single well-defined test.[21](https://www.nature.com/articles/s41599-026-08318-1) [Conducting](https://www.edgepedia.org/conducting) ten tests and reporting only those with \\( P < 0.05 \\) amounts to cherry-picking, data dredging, or p-hacking, a main cause of spurious significance.[9](https://pmc.ncbi.nlm.nih.gov/articles/PMC5187603/) The 0.05 threshold is conventional and arbitrary: a shift in incidence from 46% to 46.4% can move a P-value from 0.047 to 0.054, which is why exact p-values should be reported.[9](https://pmc.ncbi.nlm.nih.gov/articles/PMC5187603/) Power is often low in practice: van Zwet and colleagues estimated the median achieved power among more than 20,000 randomized trials in the Cochrane Database at only 0.13.[13](https://link.springer.com/article/10.1007/s12630-023-02557-5) Multiple comparisons inflate error: the Bonferroni inequality bounds the family-wise error of \\( m \\) tests by \\( m \\cdot \\alpha \\), so significance is declared only if \\( \\min_i p_i \\leq \\alpha/m \\), where \\( \\alpha \\) is the prespecified family-wise level.[6](https://www.stats.ox.ac.uk/~reinert/stattheory/chapter309.pdf)\n\nThe [American Statistical Association](https://www.edgepedia.org/american-statistical-association)'s 2016 statement codified six principles, including that p-values do not measure the probability that a hypothesis is true and do not measure effect size or importance.[1](https://www.amstat.org/asa/files/pdfs/p-valuestatement.pdf) Alternatives include estimation with confidence intervals, which correspond to the range of effect sizes whose test yields \\( P \\geq 0.05 \\).[4](https://link.springer.com/content/pdf/10.1007/s10654-016-0149-3.pdf) Bayesian methods compare hypotheses through Bayes factors, an approach Jeffreys developed in 1935, converting prior odds to posterior odds; Bayes factors give a more continuous measure of evidence than p-values but are sensitive to the choice of prior distribution, which can decisively affect results.[22](https://www.ssph-journal.org/journals/international-journal-of-public-health/articles/10.3389/ijph.2025.1608258/full) Bayes factors can also be built from test statistics themselves.[23](https://doi.org/10.1111/j.1467-9868.2005.00521.x) Tukey's three-decision formulation, developed with Jones, asserts the sign of a parameter above or below a null value or says nothing, replacing the accept/reject dichotomy.[24](https://doi.org/10.1037/1082-989x.5.4.411) McShane, Gal, Gelman, Robert, and Tackett argue p-values and similar measures should not be thresholded and should not take priority over prior evidence, mechanism, design quality, and costs.[25](https://doi.org/10.1080/00031305.2018.1527253) A shift from testing to estimation with effect sizes, confidence intervals, and meta-analysis has been promoted as \"the New Statistics.\"[26](https://doi.org/10.1177/0956797613504966)\n\n## References\n\n1. [American Statistical Association Releases Statement on Statistical Significance and P-Values](https://www.amstat.org/asa/files/pdfs/p-valuestatement.pdf)\n2. [Jerzy Neyman, Egon Sharpe Pearson (1933). IX. On the problem of the most efficient tests of statistical hypotheses. Philosophical Transactions of the Royal Society of London Series A Containing Papers of a Mathematical or Physical Character.](https://doi.org/10.1098/rsta.1933.0009)\n3. [9.01: Introduction to Hypothesis Testing (stats.libretexts.org)](https://stats.libretexts.org/Bookshelves/Probability_Theory/Probability_Mathematical_Statistics_and_Stochastic_Processes_%28Siegrist%29/09%3A_Hypothesis_Testing/9.01%3A_Introduction_to_Hypothesis_Testing)\n4. [Statistical tests, P values, confidence intervals, and power: a guide to misinterpretations](https://link.springer.com/content/pdf/10.1007/s10654-016-0149-3.pdf)\n5. [Fisher, Neyman-Pearson or NHST? A tutorial for teaching data testing](https://www.frontiersin.org/journals/psychology/articles/10.3389/fpsyg.2015.00223/full)\n6. [3. Hypothesis Testing (Oxford statistical theory lecture notes, G. Reinert)](https://www.stats.ox.ac.uk/~reinert/stattheory/chapter309.pdf)\n7. [Chapter 6. Testing Hypotheses (optimal hypothesis testing, Neyman–Pearson lemma with proof)](https://www.stat.uchicago.edu/~stigler/Stat244/ch6r2-opt-withfigs.pdf)\n8. [Hypothesis Testing - Data, Inference, and Decisions (Berkeley Data 102 textbook)](https://data102.org/ds-102-book/content/chapters/01/hypothesis-testing/)\n9. [The American Statistical Association statement on P-values explained](https://pmc.ncbi.nlm.nih.gov/articles/PMC5187603/)\n10. [The Fisher, Neyman-Pearson Theories of Testing (Lehmann, 1993)](https://link.springer.com/content/pdf/10.1007/978-1-4614-1412-4_19.pdf)\n11. [J. NEYMAN, E. S. PEARSON (1928). ON THE USE AND INTERPRETATION OF CERTAIN TEST CRITERIA FOR PURPOSES OF STATISTICAL INFERENCE. Biometrika.](https://doi.org/10.1093/biomet/20a.3-4.263)\n12. [J. Neyman, E. S. Pearson (1933). The testing of statistical hypotheses in relation to probabilities a priori. Mathematical Proceedings of the Cambridge Philosophical Society.](https://doi.org/10.1017/s030500410001152x)\n13. [Interpreting frequentist hypothesis tests: insights from Bayesian inference (Canadian Journal of Anesthesia, 2023)](https://link.springer.com/article/10.1007/s12630-023-02557-5)\n14. [Karl Pearson and R. A. Fisher on Statistical Tests: A 1935 Exchange from Nature](https://faculty.fiu.edu/~blissl/PearsonFisher.pdf)\n15. [10 Things to Know About Hypothesis Testing (EGAP Methods Guide)](https://methods.egap.org/guides/analysis-procedures/hypothesis-testing_en.html)\n16. [Chapter 7. Hypothesis Testing (McFadden, econometrics lecture notes)](https://eml.berkeley.edu/~mcfadden/e240a_sp00/ch7.pdf)\n17. [T Test - StatPearls](https://www.ncbi.nlm.nih.gov/sites/books/NBK553048/)\n18. [Choosing Statistical Tests](https://pmc.ncbi.nlm.nih.gov/articles/PMC2881615/)\n19. [Statistical Hypothesis Testing: A Comprehensive Review of Theory, Methods, and Applications](https://www.mdpi.com/2227-7390/14/2/300)\n20. [The FDA's adoption of Bayesian methodology: transforming clinical trial justification from biosimilars to broader drug development (Frontiers in Medicine, 2026)](https://www.frontiersin.org/journals/medicine/articles/10.3389/fmed.2026.1790396/full)\n21. [The tyranny of the p: when significance misleads (Humanities and Social Sciences Communications, 2026)](https://www.nature.com/articles/s41599-026-08318-1)\n22. [Decision Rules in Frequentist and Bayesian Hypothesis Testing: P-Value and Bayes Factor (IJPH, 2025)](https://www.ssph-journal.org/journals/international-journal-of-public-health/articles/10.3389/ijph.2025.1608258/full)\n23. [Valen E. Johnson (2005). Bayes Factors Based on Test Statistics. Journal of the Royal Statistical Society Series B (Statistical Methodology).](https://doi.org/10.1111/j.1467-9868.2005.00521.x)\n24. [Lyle V. Jones, John W. Tukey (2000). A sensible formulation of the significance test.. Psychological Methods.](https://doi.org/10.1037/1082-989x.5.4.411)\n25. [Blakeley B. McShane and colleagues (2019). Abandon Statistical Significance. The American Statistician.](https://doi.org/10.1080/00031305.2018.1527253)\n26. [Geoff Cumming (2013). The New Statistics. Psychological Science.](https://doi.org/10.1177/0956797613504966)\n27. 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Edgepedia Community License 1.0.", "speakable": "Statistical hypothesis testing is a method for deciding whether observed data are consistent with a null hypothesis, by computing a test statistic and reporting a p-value or a reject decision." }