I know that model sum of squares is the ratio of the between-group sum of squares to the model degrees of freedom, and that the between-group sum of squares is a variation between cluster means (and the smaller the variation the closer they are) and that the model has k-1 degrees of freedom where...

I have two questions about contrasts in ANOVA for testing the hypothesis that $\sum_i c_i y_i = 0$. How is the sum of squares for contrasts $\{c_i\}$, $SS_C = \frac{(\sum_i {c_i \bar{y_i}})^2}{\sum_i c_i^2 /n_i}$, derived? I understand the denominator is the variance of the numerator, but I do n...

I'm studying regression analysis but I'm struggling with really understanding how degrees of freedom are calculated. For example, if we have the simple scenario where $Y_i=\beta_0+\beta_1 X_i + \epsilon_i$ (and all the standard assumptions hold) then I read $\frac{1}{\sigma^2} \sum_{i=1}^n (\ha...

In the context of a linear regression, say \begin{align} y_{i} & = \beta_{0} + \beta_{1}x_{1i} + \beta_{2}x_{2i} + \ldots + \beta_{k}x_{ki} + \epsilon_{i} \end{align} the F-test is \begin{align} F & = \frac{\sum_{i=1}^{n}(\hat{y}_{i} - \bar{y})^{2}/k}{\sum_{i=1}^{n}(y_{i} - \hat{y})^{2}/(n - k -...