ANOVA Table
📂Statistical TestANOVA Table
Definition
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
Source | df | SS | MS | F |
---|
Treatments | k−1 | SST | MST | MST/MSE |
Error | n−k | SSE | MSE | |
Total | n−1 | TSS | | |
Randomized Block Design
Source | df | SS | MS | F |
---|
Treatments | k−1 | SST | MST | MST/MSE |
Blocks | b−1 | SSB | MSB | |
Error | (k−1)(b−1) | SSE | MSE | |
Total | n−1 | TSS | | |
Explanation
The ANOVA table illustrates the process of calculating the F 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 k treatments, and from the j-th treatment, nj samples x1j,⋯,xnjj are obtained. For the total number of samples n=n1+⋯+nk, consider the sample mean as xˉ=∑ijxij/n; the total sum of squares TSS is expressed as follows.
TSS=j=1∑ki=1∑nj(xij−xˉ)2
Regarding the grand total G=∑ijxij, the correction for the mean CM is as follows.
CM=n1j=1∑ki=1∑nj(xij)2=n2G
The sum of squares for treatments SST is obtained as follows using the sample mean for each treatment j=1,⋯,k, expressed as xˉj.
SST=j=1∑knj(xˉj−xˉ)2
The sum of squares for error SSE is derived using sample variance for each treatment j=1,⋯,k and obtained as a pooled variance.
SSE=(n1−1)s12+⋯+(nk−1)sk2=TSS−SST
Mean squares MS are calculated by dividing each sum of squares SS by the degrees of freedom, resulting in the value MS=SS/df.
MST=MSE=k−1SSTn−kSSE
Finally, the F statistic is calculated as the ratio of MST to MSE as follows.
F=MSEMST=SSE/(n−k)SST/(k−1)
In a randomized block design, the number of blocks b is added, along with the sum of squares for blocks SSB and mean square MSB, and the degrees of freedom for MSE change to (b−1)(k−1).
SSB=MSB=MSE=F=i=1∑b(xi−xˉ)2b−1SSB(b−1)(k−1)SSEMSEMST=SSE/(b−1)(k−1)SST/(k−1)=SSE/(b−1)SST