Editing F-test (section)

==Common examples==

Common examples of the use of ''F''-tests include the study of the following cases

* [[File:One-way ANOVA Table generated using Matlab.jpg|thumb|One-way ANOVA table with 3 random groups that each has 30 observations. F value is being calculated in the second to last column]]The hypothesis that the [[Arithmetic mean|means]] of a given set of [[normal distribution|normally distributed]] populations, all having the same [[standard deviation]], are equal.  This is perhaps the best-known ''F''-test, and plays an important role in the [[analysis of variance]] (ANOVA).
** F test of [[analysis of variance]] (ANOVA) follows three assumptions 
**# [[Normality (statistics)]]
**# [[Homogeneity of variance]]
**# [[Independence (probability theory)|Independence of errors]] and [[Randomness|random sampling]] 

* The hypothesis that a proposed regression model fits the [[data]] well.  See [[Lack-of-fit sum of squares]].
* The hypothesis that a data set in a [[regression analysis]] follows the simpler of two proposed linear models that are [[Statistical model#Nested models|nested]] within each other.
* Multiple-comparison testing is conducted using needed data in already completed F-test, if F-test leads to rejection of null hypothesis and the factor under study has an impact on the dependent variable.<ref name=":0" />
** "''a priori'' comparisons"/ "planned comparisons"- a particular set of comparisons
** "pairwise comparisons"-all possible comparisons
*** i.e. Fisher's least significant difference (LSD) test, [[Tukey's Honestly Significant Difference|Tukey's honestly significant difference (HSD) test]],  [[Newman-Keuls test|Newman Keuls test]], Ducan's test
** "[[Post hoc analysis|''a posteriori'' comparisons]]"/ "[[Post hoc comparison|''post hoc'' comparisons]]"/ "[[Post hoc comparison|exploratory comparisons]]"- choose comparisons after examining the data
*** i.e.  [[Scheffé's method]] 
===''F''-test of the equality of two variances===
{{Main|F-test of equality of variances}}

The ''F''-test is [[robust statistics|sensitive]] to [[normal distribution|non-normality]].<ref>{{cite journal | last=Box | first=G. E. P. |author-link= George E. P. Box| journal=Biometrika | year=1953 | title=Non-Normality and Tests on Variances  | pages=318–335 | volume=40 | jstor=2333350 | issue=3/4 | doi=10.1093/biomet/40.3-4.318}}</ref><ref>{{cite journal | last=Markowski | first=Carol A |author2=Markowski, Edward P. | year = 1990 | title=Conditions for the Effectiveness of a Preliminary Test of Variance | journal=[[The American Statistician]] | pages=322–326 | volume=44 | jstor=2684360 | doi=10.2307/2684360 | issue=4}}</ref> In the [[analysis of variance]] (ANOVA), alternative tests include [[Levene's test]], [[Bartlett's test]], and the [[Brown–Forsythe test]]. However, when any of these tests are conducted to test the underlying assumption of [[homoscedasticity]] (''i.e.'' homogeneity of variance), as a preliminary step to testing for mean effects, there is an increase in the experiment-wise [[Type I error]] rate.<ref>{{cite journal |last=Sawilowsky |first=S. |year=2002 |title=Fermat, Schubert, Einstein, and Behrens–Fisher: The Probable Difference Between Two Means When σ<sub>1</sub><sup>2</sup> ≠ σ<sub>2</sub><sup>2</sup> |journal=Journal of Modern Applied Statistical Methods |volume=1 |issue=2 |pages=461–472 |doi=10.22237/jmasm/1036109940 |url=http://digitalcommons.wayne.edu/jmasm/vol1/iss2/55 |access-date=2015-03-30 |archive-url=https://web.archive.org/web/20150403095901/http://digitalcommons.wayne.edu/jmasm/vol1/iss2/55/ |archive-date=2015-04-03 |url-status=live |doi-access=free }}</ref>