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``Analysis of Variance.'' A Statistical Test for heterogeneity of Means by analysis of group Variances. 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 $j$th 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

$\displaystyle {\rm SST}$ $\textstyle \equiv$ $\displaystyle \sum_{i=1}^k \sum_{j=1}^n (y_{ij}-\bar{\bar y})^2$ (1)
  $\textstyle =$ $\displaystyle \sum_{i=1}^k \sum_{j=1}^n {y_{ij}}^2-{\left({\sum_{i=1}^k \sum_{j=1}^n y_{ij}}\right)^2\over Kn}$ (2)
$\displaystyle {\rm SSA}$ $\textstyle \equiv$ $\displaystyle {1\over n}\sum_{i=1}^k \left({\sum_{j=1}^n y_{ij}}\right)^2
-{1\over Kn}\left({\sum_{i=1}^k \sum_{j=1}^n y_{ij}}\right)^2$ (3)
$\displaystyle {\rm SSE}$ $\textstyle \equiv$ $\displaystyle \sum_{i=1}^k \sum_{j=1}^n (y_{ij}-\bar{y_i})^2$ (4)
  $\textstyle =$ $\displaystyle {\rm SST}-{\rm SSA},$ (5)

which are the total, treatment, and error sums of squares. Here, $\bar y_i$ is the mean of observations within Factor Level $i$, and $\bar{\bar y}$ 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={{\rm MSA}\over{\rm MSE}}.
\end{displaymath} (6)

Category SS °Freedom Mean Squared F-Ratio
Treatment SSA $K-1$ ${\rm MSA}\equiv{{\rm SSA}\over K-1}$ ${{\rm MSA}\over{\rm MSE}}$
Error SSE $K(n-1)$ ${\rm MSE}\equiv{{\rm SSE}\over K(n-1)}$  
Total SST $Kn-1$ ${\rm MST}\equiv{{\rm SST}\over Kn-1}$  

If the P-Value is small, reject the Null Hypothesis that all Means are the same for the different groups.

See also Factor Level, Replicate, Variance

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© 1996-9 Eric W. Weisstein