List of statistical tests

Statistical tests are used to test the fit between a hypothesis and the data. ^{[1] [2]} Choosing the right statistical test is not a trivial task. ^[1] The choice of the test depends on many properties of the research question. The vast majority of studies can be addressed by 30 of the 100 or so statistical tests in use. ^{[3] [4] [5]}

Explanation of properties

• Scaling of data: One of the properties of the tests is the scale of the data, which can be interval-based, ordinal or nominal. ^[3] Nominal scale is also known as categorical. ^[6] Interval scale is also known as numerical. ^[6] When categorical data has only two possibilities, it is called binary or dichotomous. ^[1]

• Assumptions, parametric and non-parametric: There are two groups of statistical tests, parametric and non-parametric. The choice between these two groups needs to be justified. Parametric tests assume that the data follow a particular distribution, typically a normal distribution, while non-parametric tests make no assumptions about the distribution. ^[7] Non-parametric tests have the advantage of being more resistant to misbehaviour of the data, such as outliers. ^[7] They also have the disadvantage of being less certain in the statistical estimate. ^[7]

• Type of data: Statistical tests use different types of data. ^[1] Some tests perform univariate analysis on a single sample with a single variable. Others compare two or more paired or unpaired samples. Unpaired samples are also called independent samples. Paired samples are also called dependent. Finally, there are some statistical tests that perform analysis of relationship between multiple variables like regression. ^[1]

• Number of samples: The number of samples of data.

• Exactness: A test can be exact or be asymptotic delivering approximate results.

List of statistical tests

Test name	Scaling	Assumptions	Data	Samples	Exact	Special case of	Application conditions
One sample t-test	interval	normal	univariate	1	No [8]	Location test
Paired difference test			paired	2		Location test
Unpaired t-test	interval	normal	unpaired	2	No [8]	Location test	Homoscedasticity [9]
Welch's t-test	interval	normal	unpaired	2	No [8]	Location test
Paired t-test	interval	normal	paired	2	No	Location test
F-test	interval	normal		2
Z-test	interval	normal		2	No		variance is known
Permutation test	interval	non-parametric	unpaired	≥2	Yes
Kruskal-Wallis ${\displaystyle H}$ test	ordinal	non-parametric	unpaired	≥2	Yes		small sample size [10]
Mann–Whitney ${\displaystyle U}$ test	ordinal	non-parametric	unpaired	2		Kruskal-Wallis test [11]
Wilcoxon signed-rank test	interval	non-parametric	paired	≥1		Location test
Sign test	ordinal	non-parametric	paired	2
Friedman test	ordinal	non-parametric	paired	>2		Location test
${\displaystyle \chi ^{2}}$ test	nominal [1]	non-parametric [12]			No		Contingency table, sample size > ca. 60, [1] any cell content ≥ 5, [13] marginal totals fixed [13]
Pearson's ${\displaystyle \chi ^{2}}$ test	nominal/ordinal	non-parametric			No	${\displaystyle \chi ^{2}}$ test
Median test	ordinal	non-parametric			No	Pearson's ${\displaystyle \chi ^{2}}$ test
Multinomial test	nominal	non-parametric	univariate	1	Yes	Location test
McNemar's test	binary	non-parametric [14]	paired	2	Yes	Cochran's ${\displaystyle Q}$ test [15]
Cochran's ${\displaystyle Q}$ test	binary	non-parametric	paired	≥2
Binomial test	binary	non-parametric	univariate	1	Yes	Multinomial test
Siegel–Tukey test	ordinal	non-parametric	unpaired	2
Chow test	interval	parametric	linear regression	2	No		Time series
Fisher's exact test	nominal	non-parametric	unpaired	≥2 [13]	Yes		Contingency table, marginal totals fixed [13]
Barnard's exact test	nominal	non-parametric	unpaired	2	Yes		Contingency table
Boschloo's test	nominal	non-parametric	unpaired	2	Yes		Contingency table
Shapiro–Wilk test	interval		univariate	1		Normality test	sample size between 3 and 5000 [16]
Kolmogorov–Smirnov test	interval			1		Normality test	distribution parameters known [16]
Shapiro-Francia test	interval		univariate	1		Normality test	Simplification of Shapiro–Wilk test
Lilliefors test	interval			1		Normality test

See also

• List of probability distributions

References

1. Parab, Shraddha; Bhalerao, Supriya (2010). "Choosing statistical test". International Journal of Ayurveda Research. 1 (3): 187–191. doi: 10.4103/0974-7788.72494 . ISSN   0974-7788. PMC   2996580 . PMID   21170214.
2. "Entscheidbaum" (in German). Retrieved 8 February 2024.
3. Nayak, Barun K; Hazra, Avijit (2011). "How to choose the right statistical test?". Indian Journal of Ophthalmology. 59 (2): 85–86. doi: 10.4103/0301-4738.77005 . ISSN   0301-4738. PMC   3116565 . PMID   21350275.
4. Lewis, Nancy D.; Lewis, Nigel Da Costa; Lewis, N. D. (2013). 100 Statistical Tests in R: What to Choose, how to Easily Calculate, with Over 300 Illustrations and Examples. Heather Hills Press. ISBN   978-1-4840-5299-0.
5. Kanji, Gopal K. (18 July 2006). 100 Statistical Tests. SAGE. ISBN   978-1-4462-2250-8.
6. "What is the difference between categorical, ordinal and interval variables?". stats.oarc.ucla.edu. Retrieved 10 February 2024.
7. Huth, R.; Pokorná, L. (1 March 2004). "Parametric versus non-parametric estimates of climatic trends" . Theoretical and Applied Climatology. 77 (1): 107–112. Bibcode:2004ThApC..77..107H. doi:10.1007/s00704-003-0026-3. ISSN   1434-4483. S2CID   121539673.
8. de Winter, J.C.F. (2019). "Using the Student's t-test with extremely small sample sizes". Practical Assessment, Research, and Evaluation. 18. doi:10.7275/e4r6-dj05.
9. "t-Test für unabhängige Stichproben". Hochschule Luzern (in German). Retrieved 10 February 2024.
10. Choi, Won; Lee, Jae Won; Huh, Myung-Hoe; Kang, Seung-Ho (11 January 2003). "An Algorithm for Computing the Exact Distribution of the Kruskal–Wallis Test" . Communications in Statistics - Simulation and Computation. 32 (4): 1029–1040. doi:10.1081/SAC-120023876. ISSN   0361-0918. S2CID   123037097.
11. McKight, Patrick E.; Najab, Julius (30 January 2010). "Kruskal-Wallis Test". The Corsini Encyclopedia of Psychology. Wiley. p. 1. doi:10.1002/9780470479216.corpsy0491. ISBN   978-0-470-17024-3.
12. McHugh, Mary L. (15 June 2013). "The Chi-square test of independence". Biochemia Medica. 23 (2): 143–149. doi:10.11613/BM.2013.018. PMC   3900058 . PMID   23894860.
13. Warner, Pamela (1 October 2013). "Testing association with Fisher's Exact test". Journal of Family Planning and Reproductive Health Care. 39 (4): 281–284. doi:10.1136/jfprhc-2013-100747. ISSN   1471-1893. PMID   24062499.
14. Károly, Héberger; Róbert, Rajkó (1999). Pair-Correlation Method with parametric and non-parametric test-statistics for variable selection. Description of computer program and application for environmental data case studies. szef. pp. 82–91.
15. Carpi, Angelo; Rossi, Giuseppe; Coscio, Giancarlo Di; Iervasi, Giorgio; Nicolini, Andrea; Carpi, Federico; Mechanick, Jeffrey I.; Bartolazzi, Armando (2010). "Galectin-3 detection on large-needle aspiration biopsy improves preoperative selection of thyroid nodules: a prospective cohort study". Annals of Medicine. 42 (1): 70–78. doi:10.3109/07853890903439778. ISSN   1365-2060.
16. Ahmad, Fiaz; Khan, Rehan Ahmad (8 September 2015). "A power comparison of various normality tests". Pakistan Journal of Statistics and Operation Research. 11 (3): 331–345. doi: 10.18187/pjsor.v11i3.845 . ISSN   1816-2711.