Section 28 ANOVA Table
28.1 Analysis of Variance Table
\[ \large fm \leftarrow lm(SBP \sim Group, \space data=BP) \]
\[ \large anova(fm) \]
| Df | Sum Sq | Mean Sq | F value | Pr(>F) | |
|---|---|---|---|---|---|
| Group | 3 | 1521.638 | 507.2128 | 7.4983 | 5e-04 |
| Residuals | 36 | 2435.184 | 67.6440 | NA | NA |
| Total | 39 | 3956.822 | NA | NA | NA |
28.2 Explanation
Degrees of freedom (df)
\(\large n\) = observations per group
\(\large g\) = number of groups
Group df = \(\large (g - 1)\)
Residual df = \(\large g * (n - 1)\)
Total df = Treatment df + Residual df = \(\large (g * n) - 1\)
Sum of Squares due to Treatment (Group)
For each group calculate: n * (Group mean - overall mean)2
Add the values for the different groups together
\[ \large SST = n\sum\limits_{i=1}^{g} (\bar{y_i}-\bar{y})^2 \]
Sum of Squares due to Error
Residual Sum of Squares
For each observation calculate:
(Observed value - group mean)2
Add the values for the different observations together
\[ \large SSE = \sum\limits_{i=1}^{g} \sum\limits_{j=1}^{n} (y_{ij}-\bar{y_i})^2 \]
Total Sum of Squares = SS due to Treatment + SS due to Error
Mean Squares
Mean square = Sum of squares / degrees of freedom
\(\large MS = SS / df\)
F-value (Variance Ratio)
F value = Treatment MS / Residual MS
Pr(>F)
P-value: the probability of obtaining a variance ratio this large under the null hypothesis that the treatment means are all equal.
Under the null hypothesis the variance ratio has an F distribution.