Deviance Statistics

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$\begin{aligned} \text{Deviance} &= -2 \sum_{i=1}^K \left[ y_i \log \hat{p}_i + (1 - y_i) \log \left(1 - \hat{p}_i \right) - y_i \log y_i - (1 - y_i) \log \left(1 - y_i \right) \right] \\ &= -2 \sum_{i=1}^K \left[ y_i \log \left( \frac{\hat{p}_i}{y_i}\right) + (1-y_i) \log \left( \frac{1-\hat{p}_i}{1-y_i} \right) \right] \\ &= 2 \sum_{i=1}^K \left[ y_i \log \left( \frac{y_i}{\hat{p}_i}\right) + (1-y_i) \log \left( \frac{1-y_i}{1-\hat{p}_i} \right) \right]. \end{aligned}$

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$\begin{aligned} D_{\mathcal{M}} &= -2 (\log L_{\mathcal{M}} - \log L_{\mathcal{S}}) \\ &= -2 \left[ - \min_{\beta} \sum_{i=1}^n \frac{\left( y_i - \sum_{j=1}^p \beta_j X_{ij} \right)^2}{2 \sigma^2} \right] \\ &= \frac{1}{\sigma^2} \min_{\beta} \sum_{i=1}^n \left( y_i - \sum_{j=1}^p \beta_j X_{ij} \right)^2. \end{aligned}$