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ANOVA Table 📂Statistical Test

ANOVA Table

Definition 1

A table summarizing the results from analysis of variance (ANOVA) is called an ANOVA table. The format of the ANOVA table may vary slightly depending on the experimental design.

Completely Randomized Design

SourcedfSSMSF
Treatmentsk1k-1SSTMSTMST/MSE
Errornkn-kSSEMSE
Totaln1n-1TSS

Randomized Block Design

SourcedfSSMSF
Treatmentsk1k-1SSTMSTMST/MSE
Blocksb1b-1SSBMSB
Error(k1)(b1)(k-1)(b-1)SSEMSE
Totaln1n-1TSS

Explanation

The ANOVA table illustrates the process of calculating the FF statistic in analysis of variance. For undergraduates, it might initially seem like a mere memorization or computational task, but after some study, one will realize it ultimately involves deriving two values that follow a chi-squared distribution to obtain a value that follows an F-distribution. These fill-in-the-blank problems might seem challenging due to exams, but the genuinely important detail is the F statistic located at the top right of the table.

Calculation Method

Let’s calculate the figures in the ANOVA table in detail. Consider a completely randomized experiment with kk treatments, and from the jj-th treatment, njn_{j} samples x1j,,xnjjx_{1 j} , \cdots , x_{n_{j} j} are obtained. For the total number of samples n=n1++nkn = n_{1} + \cdots + n_{k}, consider the sample mean as xˉ=ijxij/n\bar{x} = \sum_{ij} x_{ij} / n; the total sum of squares TSS\text{TSS} is expressed as follows. TSS=j=1ki=1nj(xijxˉ)2 \text{TSS} = \sum_{j=1}^{k} \sum_{i=1}^{n_{j}} \left( x_{ij} - \bar{x} \right)^{2} Regarding the grand total G=ijxijG = \sum_{ij} x_{ij}, the correction for the mean CM\text{CM} is as follows. CM=1nj=1ki=1nj(xij)2=Gn2 \text{CM} = {\frac{ 1 }{ n }} \sum_{j=1}^{k} \sum_{i=1}^{n_{j}} \left( x_{ij} \right)^{2} = {\frac{ G }{ n^{2} }} The sum of squares for treatments SST\text{SST} is obtained as follows using the sample mean for each treatment j=1,,kj = 1 , \cdots , k, expressed as xˉj\bar{x}_{j}. SST=j=1knj(xˉjxˉ)2 \text{SST} = \sum_{j=1}^{k} n_{j} \left( \bar{x}_{j} - \bar{x} \right)^{2} The sum of squares for error SSE\text{SSE} is derived using sample variance for each treatment j=1,,kj = 1 , \cdots , k and obtained as a pooled variance. SSE=(n11)s12++(nk1)sk2=TSSSST \text{SSE} = \left( n_{1} - 1 \right) s_{1}^{2} + \cdots + \left( n_{k} - 1 \right) s_{k}^{2} = \text{TSS} - \text{SST} Mean squares MSMS are calculated by dividing each sum of squares SSSS by the degrees of freedom, resulting in the value MS=SS/dfMS = SS / \text{df}. MST=SSTk1MSE=SSEnk \begin{align*} \text{MST} =& {\frac{ \text{SST} }{ k - 1 }} \\ \text{MSE} =& {\frac{ \text{SSE} }{ n - k }} \end{align*} Finally, the FF statistic is calculated as the ratio of MSTMST to MSEMSE as follows. F=MSTMSE=SST/(k1)SSE/(nk) F = {\frac{ \text{MST} }{ \text{MSE} }} = {\frac{ \text{SST} / (k - 1) }{ \text{SSE} / (n - k) }}

In a randomized block design, the number of blocks bb is added, along with the sum of squares for blocks SSB\text{SSB} and mean square MSB\text{MSB}, and the degrees of freedom for MSEMSE change to (b1)(k1)(b-1)(k-1). SSB=i=1b(xixˉ)2MSB=SSBb1MSE=SSE(b1)(k1)F=MSTMSE=SST/(k1)SSE/(b1)(k1)=SSTSSE/(b1) \begin{align*} \text{SSB} =& \sum_{i=1}^{b} \left( x_{i} - \bar{x} \right)^{2} \\ \text{MSB} =& {\frac{ \text{SSB} }{ b - 1 }} \\ \text{MSE} =& {\frac{ \text{SSE} }{ (b-1)(k-1) }} \\ F =& {\frac{ \text{MST} }{ \text{MSE} }} = {\frac{ \text{SST}/ (k - 1) }{ \text{SSE} / (b-1)(k-1) }} = {\frac{ \text{SST}}{ \text{SSE} / (b-1) }} \end{align*}


  1. Mendenhall. (2012). Introduction to Probability and Statistics (13th Edition): p452. ↩︎