Definition: Treatment sum of squares
Index: The Book of Statistical Proofs ▷ Statistical Models ▷ Univariate normal data ▷ Analysis of variance ▷ Treatment sum of squares
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Metadata: ID: D183 | shortcut: trss | author: JoramSoch | date: 2022-12-14, 13:01.
Definition: Let there be an analysis of variance (ANOVA) model with one, two or multiple factors influencing the measured data $y$ (here, using the reparametrized version of one-way ANOVA):
\[\label{eq:anova} y_{ij} = \mu + \delta_i + \varepsilon_{ij}, \; \varepsilon_{ij} \overset{\mathrm{i.i.d.}}{\sim} \mathcal{N}(0, \sigma^2) \; .\]Then, the treatment sum of squares is defined as the explained sum of squares (ESS) for each main effect, i.e. as the sum of squared deviations of the average for each level of the factor, from the average across all observations:
\[\label{eq:trss} \mathrm{SS}_\mathrm{treat} = \sum_{i=1}^{k} \sum_{j=1}^{n_i} (\bar{y}_i - \bar{y})^2 \; .\]Here, $\bar{y}_i$ is the mean for the $i$-th level of the factor (out of $k$ levels), computed from $n_i$ values $y_{ij}$, and $\bar{y}$ is the mean across all values $y_{ij}$.
- Wikipedia (2022): "Analysis of variance"; in: Wikipedia, the free encyclopedia, retrieved on 2022-11-15; URL: https://en.wikipedia.org/wiki/Analysis_of_variance#Partitioning_of_the_sum_of_squares.
Metadata: ID: D183 | shortcut: trss | author: JoramSoch | date: 2022-12-14, 13:01.