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shaikh22ammar
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Theorem 11.2.8 in Casella & Berger defines the ANOVA statistic as a maxima of T^2 statistic as:
<br /> \sup_{\sum a_i = 0} T_a^2 = \sup_{\sum a_i = 0} \left( <br /> \left( S^2_p \sum a_i^2 / n_i \right)^{-1/2} \left( \sum a_i \bar Y_{i \cdot} - \sum a_i \theta_i\right)<br /> \right)^2 = \left( S^2_p \right)^{-1} \sum n_i \left( \bar Y_{i \cdot} - \bar{\bar Y} - \theta_i + \bar{\theta} \right)^2<br />
where all the summations are from 1 to k the no. of treatments and S^2_p, n_i, \theta_i, \bar Y_{i \cdot} are the pooled sample variance, no. of observations of treatment i, its mean, and sample mean respectively. The term inside the square between equals signs follows t distribution but for whatever reason the supremum of the square follows (k-1) F(k-1, n-k), as opposed to t^2.
<br /> \sup_{\sum a_i = 0} T_a^2 = \sup_{\sum a_i = 0} \left( <br /> \left( S^2_p \sum a_i^2 / n_i \right)^{-1/2} \left( \sum a_i \bar Y_{i \cdot} - \sum a_i \theta_i\right)<br /> \right)^2 = \left( S^2_p \right)^{-1} \sum n_i \left( \bar Y_{i \cdot} - \bar{\bar Y} - \theta_i + \bar{\theta} \right)^2<br />
where all the summations are from 1 to k the no. of treatments and S^2_p, n_i, \theta_i, \bar Y_{i \cdot} are the pooled sample variance, no. of observations of treatment i, its mean, and sample mean respectively. The term inside the square between equals signs follows t distribution but for whatever reason the supremum of the square follows (k-1) F(k-1, n-k), as opposed to t^2.
