Two-way analysis of variance: Difference between revisions

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{{short description|Statistical test examining influence of two categorical variables on one continuous variable}}

In [[statistics]], the '''two-way [[analysis of variance]]''' ('''ANOVA''') is an extension of the [[One-way analysis of variance|one-way ANOVA]] that examines the influence of two different [[Categorical variable|categorical]] [[independent variables]] on one [[Continuous function|continuous]] [[dependent variable]]. The two-way ANOVA not only aims at assessing the [[main effect]] of each independent variable but also if there is any [[Interaction (statistics)|interaction]] between them.

==History==

In 1925, [[Ronald Fisher]] mentions the two-way ANOVA in his celebrated book, ''[[Statistical Methods for Research Workers]]'' (chapters 7 and 8). In 1934, [[Frank Yates]] published procedures for the unbalanced case.<ref>{{cite journal |last=Yates |first=Frank |date=March 1934 |title=The analysis of multiple classifications with unequal numbers in the different classes |jstor=2278459 |journal=Journal of the American Statistical Association |volume=29 |issue=185 |pages=51–66 |doi=10.1080/01621459.1934.10502686}}</ref> Since then, an extensive literature has been produced. The topic was reviewed in 1993 by [[Yasunori Fujikoshi]].<ref>{{cite journal |last=Fujikoshi |first=Yasunori |date=1993 |title=Two-way ANOVA models with unbalanced data |journal=Discrete Mathematics |volume=116 |issue=1 |pages=315–334 |doi=10.1016/0012-365X(93)90410-U |doi-access=free }}</ref> In 2005, [[Andrew Gelman]] proposed a different approach of ANOVA, viewed as a [[multilevel model]].<ref>{{cite journal |last=Gelman |first=Andrew |date=February 2005 |title=Analysis of variance? why it is more important than ever |journal=The Annals of Statistics |volume=33 |issue=1 |pages=1–53 | arxiv=math/0504499|doi=10.1214/009053604000001048 |s2cid=125025956 }}</ref>

==Data set==

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==Assumptions==

Following [[Andrew Gelman|Gelman]] and [[Jennifer Hill|Hill]], the assumptions of the ANOVA, and more generally the [[general linear model]], are, in decreasing order of importance:<ref>{{cite book |lastlast1=Gelman |firstfirst1=Andrew |last2=Hill |first2=Jennifer|author2-link=Jennifer Hill |date=18 December 2006 |title= Data Analysis Using Regression and Multilevel/Hierarchical Models |url=http://www.cambridge.org/us/academic/subjects/statistics-probability/statistical-theory-and-methods/data-analysis-using-regression-and-multilevelhierarchical-models |publisher=[[Cambridge University Press]] |pages=45–46 |isbn=978-0521867061 }}</ref>

# the data points are relevant with respect to the scientific question under investigation;

# the mean of the response variable is influenced additively (if not interaction term) and linearly by the factors;

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! Calculation

! Sum

! ''N''

! <math>\sigma^2</math>

|-

| Individual

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Finally, the sums of squared deviations required for the [[analysis of variance]] can be calculated.<ref>{{cite book|last=Mecklin|first=Christopher|title=STA 265 Notes (Methods of Statistics and Data Science)|date=20 October 2020|access-date=6 December 2024|chapter-url=https://bookdown.org/cmecklin/sta265notes/anova-with-interaction.html|chapter=Chapter 7: ANOVA with Interaction|via=bookdown.org}}</ref>

{| class="wikitable"

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! Factor

! Sum

! ''N''

! <math>\sigma^2</math>

! Total

! Environment

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|-

| Composite (correction factor<ref>{{cite book|chapter-url=https://iastate.pressbooks.pub/quantitativeplantbreeding/chapter/the-analysis-of-variance-anova/|title=Quantitative Methods for Plant Breeding|chapter=Chapter 8: The Analysis of Variance (ANOVA)|last1=Moore|first1=Ken|last2=Mowers|first2=Ron|last3=Harbur|first3=M.L.|last4=Merrick|first4=Laura|last5=Mahama|first5=Anthony Assibi|publisher=Iowa State University Digital Press|editor-last1=Suza|editor-first1=W.P.|editor-last2=Lamkey|editor-first2=K.R.|year=2023|access-date=6 December 2024}}</ref>)

| Composite

| 504.6

| 1

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|-

| Squared deviations (<math>\sigma^2</math>)

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|

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| 2

| 9

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| Mean square variance

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|

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| 14.668

| 10.4

| 8.0495

| 9.426

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==See also==

* [[Analysis of variance]]

* [[F -test]] (''Includes a one-way ANOVA example'')

* [[Mixed model]]

* [[Multivariate analysis of variance|Multivariate analysis of variance (MANOVA)]]