# Thread: Variance of an Addative Model

1. ## Variance of an Addative Model

Let $\displaystyle X_{ijk}$ be normal distributed with mean $\displaystyle \mu_{ijk}$ and common variance $\displaystyle \sigma^2$. i = 1,...,a ; j = 1,...,b;k = 1,...,c

Find the variance of $\displaystyle \hat{\alpha_i} = \overline{X}_{i..} - \overline{X}_{...}$

$\displaystyle Var(\hat{\alpha_1}) = Var(\overline{X}_{1..} - \frac{\overline{X}_{1..} + ... + \overline{X}_{a..}}{a})=Var(\frac{(a-1)\overline{X}_{1..} - ... - \overline{X}_{a..}}{a}) =$ $\displaystyle a^{-2}[(a-1)^2 \frac{\sigma^2}{n} + (a-1) \frac{\sigma^2}{n}]$

where n = abc. Then I would multiply it by 'a' to find $\displaystyle Var(\hat{\alpha_i})$ Have I made a mistake somewhere?

2. I n=bc instead, but I don't follow your comment on multiplying by a, which may then agree with my computation.

Originally Posted by statmajor
Let $\displaystyle X_{ijk}$ be normal distributed with mean $\displaystyle \mu_{ijk}$ and common variance $\displaystyle \sigma^2$. i = 1,...,a ; j = 1,...,b;k = 1,...,c

Find the variance of $\displaystyle \hat{\alpha_i} = \overline{X}_{i..} - \overline{X}_{...}$

$\displaystyle Var(\hat{\alpha_1}) = Var(\overline{X}_{1..} - \frac{\overline{X}_{1..} + ... + \overline{X}_{a..}}{a})=Var(\frac{(a-1)\overline{X}_{1..} - ... - \overline{X}_{a..}}{a}) =$ $\displaystyle a^{-2}[(a-1)^2 \frac{\sigma^2}{n} + (a-1) \frac{\sigma^2}{n}]$

where n = abc. Then I would multiply it by 'a' to find $\displaystyle Var(\hat{\alpha_i})$ Have I made a mistake somewhere?

3. I calculated $\displaystyle Var(\hat{\alpha_1})$, and not $\displaystyle Var(\hat{\alpha_i})$, so I'll need to sum up $\displaystyle Var(\hat{\alpha_1})$ from i= 1,...,a to get $\displaystyle Var(\hat{\alpha_i})$

4. Are the observations independent?

If they are, then $\displaystyle V(\hat{\alpha}_{i}) = \sigma^{2}\Big(\frac{1}{bc} + \frac{1}{abc} \Big)$