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ANOVA


"Analysis of Variance." A statistical test for heterogeneity of means by analysis of group variances. ANOVA is implemented as ANOVA[data] in the Wolfram Language package ANOVA` .

To apply the test, assume random sampling of a variate Y with equal variances, independent errors, and a normal distribution. Let n be the number of replicates (sets of identical observations) within each of K factor levels (treatment groups), and y_(ij) be the jth observation within factor level i. Also assume that the ANOVA is "balanced" by restricting n to be the same for each factor level.

Now define the sum of square terms

SST=sum_(i=1)^(k)sum_(j=1)^(n)(y_(ij)-y^_^_)^2
(1)
=sum_(i=1)^(k)sum_(j=1)^(n)y_(ij)^2-((sum_(i=1)^(k)sum_(j=1)^(n)y_(ij))^2)/(Kn)
(2)
SSA=1/nsum_(i=1)^(k)(sum_(j=1)^(n)y_(ij))^2-1/(Kn)(sum_(i=1)^(k)sum_(j=1)^(n)y_(ij))^2
(3)
SSE=sum_(i=1)^(k)sum_(j=1)^(n)(y_(ij)-y^__i)^2
(4)
=SST-SSA,
(5)

which are the total, treatment, and error sums of squares. Here, y^__i is the mean of observations within factor level i, and y^_^_ is the "group" mean (i.e., mean of means). Compute the entries in the following table, obtaining the P-value corresponding to the calculated F-ratio of the mean squared values

 F=(MSA)/(MSE).
(6)
category degrees freedomSSmean squaredF-ratio
modelK-1SSAMSA=(SSA)/(K-1)(MSA)/(MSE)
errorK(n-1)SSEMSE=(SSE)/(K(n-1))
totalKn-1SSTMST=(SST)/(Kn-1)

If the P-value is small, reject the null hypothesis that all means are the same for the different groups.


See also

Factor Level, Least Squares Fitting, MANOVA, Replicate, Variance

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References

Miller, R. G. Beyond ANOVA: Basics of Applied Statistics. Boca Raton, FL: Chapman & Hall, 1997.

Referenced on Wolfram|Alpha

ANOVA

Cite this as:

Weisstein, Eric W. "ANOVA." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ANOVA.html

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