Contrast in analysis of variance (ANOVA)

（someone asked me about contrast. So as a review, I tried to derive the equations for finding contrasts and their confidence intervals. Please leave a comment it there is anything wrong.)

Let's start by defining some notations:

1. Treatment:$\alpha_i$, where $i \in (1,...,p)$;
2. Contrast vector: c, which commonly is a column vector;
3. Response: $Y_{ij}$, where $j \in (1,...,n_i)$, where each level of treatment has $n_i} observations.
4. Random error term: $\epsilon \sim N(0,\Sigma)$, where $\Sigma = diag(\sigma^2)$.

As an example, our one way ANOVA model is:

$Y_i = \mu +\alpha_i+\epsilon_i$, 

in a matrix notation, it is:

$\bf {Y=X\beta}$

where $\bf{\beta} = (\mu \: \alpha_1\: ...\: \alpha_p)'$
Therefore, 

$SSE=(Y-X\hat\beta)'(Y-x\hat\beta)=[(I-P)Y]'[(I-P)Y]$
$=\Sigma_{i=1}^p \Sigma_{i=1}^n_i (y_{ij}-\bar{y}_{i.}) $

We know $c'\hat\beta$ follows a normal distribution with mean $c'\beta$, and variance

 $Var(c'\hat\beta)=c'(X'X)^-c\sigma^2$

where $(X'X)^-$ is the generalized inverse of $X'X$.

We can test $c'(X'X)^-(X'X)=c'$ for estimability.

The MLE of p estimable functions $\mu +\hat\beta = (\bar{Y_{1.}} \: ... .\: \bar{Y_{p.}})$

also, $\bar{Y} \sim N(\mu, \sigma^2/n)$  if  $Y_i \sim N(\mu, \sigma^2), i = 1,...,n$.

Therefore,

 $Var(c'\bar{Y})=c'Var( \bar{Y})c=SSE\sum\limits_{i=1}^p c_i^2/n_i$ (SSE is estimator of $\sigma^2$)

Finally, we can show:

 $\sum\limits_{i=1}^p c_i^2/n_i=c'(X'X)^-c$