Two-way analysis of variance: Difference between revisions

Revision as of 23:29, 29 December 2018

Revision as of 06:00, 23 January 2019

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Upon observing variation among all <math>n</math> data points, for instance via a [[histogram]], "[[Probability theory|probability]] may be used to describe such variation".<ref>{{cite journal |last=Kass |first=Robert E |date=1 February 2011 |title=Statistical inference: The big picture |url=http://projecteuclid.org/euclid.ss/1307626554 |journal=[[Statistical Science]] |volume=26 |issue=1 |pages=1–9 |doi=10.1214/10-sts337|pmid=21841892 |pmc=3153074 |arxiv=1106.2895 }}</ref> Let us hence denote by <math>Y_{ijk}</math> the [[random variable]] which observed value <math>y_{ijk}</math> is the <math>k</math>-th measure for treatment <math>(i,j)</math>. The '''two-way ANOVA''' models all these variables as varying [[Independence (probability theory)|independently]] and [[Normal distribution|normally]] around a mean, <math>\mu_{ij}</math>, with a constant variance, <math>\sigma^2</math> ([[homoscedasticity]]):

<math>Y_{ijk} \, | \, \mu_{ij}, \sigma^2 \; \overset{\mathrm{i.i.d.}}{\sim} \; \mathcal{N}(\mu_{ij}, \sigma^2)</math>.

Specifically, the mean of the response variable is modeled as a [[linear combination]] of the explanatory variables:

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Another equivalent way of describing the two-way ANOVA is by mentioning that, besides the variation explained by the factors, there remains some [[statistical noise]]. This amount of unexplained variation is handled via the introduction of one random variable per data point, <math>\epsilon_{ijk}</math>, called [[Errors and residuals in statistics|error]]. These <math>n</math> random variables are seen as deviations from the means, and are assumed to be independent and normally distributed:

<math>Y_{ijk} = \mu_{ij} + \epsilon_{ijk} \text{ with } \epsilon_{ijk} \overset{\mathrm{i.i.d.}}{\sim} \mathcal{N}(0, \sigma^2)</math>.

==Assumptions==