11. Analysis of Variance
One-Way Analysis of Variance
[latexpage]$F=\dfrac{MS_{B}}{MS_{w}}$
$MS_{B}=\dfrac{SS_{B}}{df_{B}}=\dfrac{SS_{B}}{k-1}$
$MS_{W}=\dfrac{SS_{W}}{df_{W}}=\dfrac{SS_{W}}{N-k}$
$\begin{aligned}SS_{W}=\sum ^{k}_{i=1}\left( n_{i}-1\right) S_{i}^{2}\end{aligned}$
$\begin{aligned}SS_{B}=\sum ^{k}_{i=1}n_{i}\left( \overline{X}_{i}-\overline{X}_{T}\right) \\ =\left( \sum ^{k}_{i=1}n_{i}\left( \overline{X}_{i}^{2}\right) \right) -N\left( \overline{X}_{T}^{2}\right) \end{aligned}$
$\overline{X}_{T}=\dfrac{\displaystyle\sum ^{k}_{i=1}n_{i}\overline{X}_{i}}{N}$
Analysis of Variance (or ANOVA) uses the F statistic, which is calculated using the between-group mean squares ($MS_{B}$) and the within-group mean squares ($MS_{W}$). Essentially, this statistics examines if the variation between groups is significantly greater than the variation within the groups. For example, imagine you are working for a non-governmental organization seeking to increase economic development and you offer financial training and microfinancing opportunities to one group, cash grants to another group, and no incentives to a control group. How could you conclude which strategy was most effective one year later? If there was a wide variation within each group and relatively little variation between the groups, you could not definitively conclude that one strategy made a significant difference. However, if you found relatively little variation within each group but that the cash grant group had significantly increased incomes compared to the other groups, you could potentially conclude that there was a significant difference between these strategies.
