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[edit] Expected value of SSH

Consider one-way MANOVA with $G$ groups, each with $n g$ observations. Let $N = \sum_{g=1}^G n_g\!$ and let

$D = \begin{bmatrix} 1_{n_1} & \cdots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \cdots & 1_{n_g} \end{bmatrix}$

be the design matrix.

Let $Q$ be the $N \times N$ residual projection matrix defined by

$Q = I - 1_N(1_N'1_N)^{-1}1_N'=I-\frac{1}{N}U$

[edit] Analyzing SSH

We can find expressions for SSH in terms of the data and find expected values for SSH under a fixed effects or under a random effects model.

The following formula is used repeatedly to find the expected value of a quadratic form. If $Y$ is a random vector with $\operatorname{E}(Y)=\mu$ and $\operatorname{Var}(Y) = \Psi\!$ , and $Q\!$ is symmetric, then

$\operatorname{E}( Y'QY) = \mu'Q\mu + \operatorname{tr}(Q \Psi) \!$

We can model:

$\mathbf{Y} = D \bold{\mu} + \epsilon\!$

where

$\mu \sim N( 1\psi, \phi^2 I) \!$

and

$\epsilon \sim N(0, \sigma^2 I )\!$

and $μ$ is independent of $ε$ .

Thus

$\operatorname{E}(Y) = 1 \psi$ and $\operatorname{Var}(Y) = \phi^2 DD' + \sigma^2 I\!$

Consequently

$\operatorname{E}(SSTO)\!$	$=$	$\operatorname{E}( Y'QY ) = \psi 1' Q 1 \psi + \operatorname{tr}\left[ (\phi^2 DD' + \sigma^2 I )(I - \frac{1}{N}U) \right]$
	$=$	$0 + \operatorname{tr}\left[ \phi^2 DD' - \frac{\phi^2}{N}DD'U + \sigma^2 Q \right]$
	$=$	$\phi^2 N - \frac{\phi^2}{N}\operatorname{tr}(DD'U) + \sigma^2 \operatorname{tr}( Q )$
	$=$	$\phi^2 N - \frac{\phi^2}{N}\operatorname{tr}(1'DD'1) + \sigma^2 (N-1)$
	$=$	$\phi^2 N - \frac{\phi^2}{N}\sum_{g=1}^G n_g^2 + \sigma^2 (N-1)$
	$=$	$\phi^2 (N - \tilde{n}) + \sigma^2 (N-1)$

where $\tilde{n}= \sum_{g=1}^G \frac{n_g}{N} n_g$ is the group-size weighted mean of group sizes. With equal groups $\tilde{n} = N / G$ and

$E(SSTO) = \phi^2 N \frac{G-1}{G} + \sigma^2 (N-1) =\phi^2 (N-n) + \sigma^2 (N-1)$

Thus

$E(SSH)\!$	=	$E (SSTO) - E(SSE)\!$
	=	$\phi^2 (N - \tilde{n}) + \sigma^2 (N-1) - \sigma^2 (N-G)$
	=	$\phi^2 (N - \tilde{n}) + \sigma^2 (G-1)$

[edit] Multivariate response

If we are sampling from a p-variate distribution in which

$\mathbf{Y}_{ig} \sim \mbox{i.i.d.} N(\mathbf{\mu}_g , \Sigma)$

and

$\mathbf{\mu}_1,..., \mathbf{\mu}_G \sim \mbox{ i.i.d. } N(\mathbf{\psi}, \Phi), \mbox{ independently of } \mathbf{Y}_{ig}$

then the analogous results are:

E (S S E) = (N - G)Σ

and

$E(SSH) = (N - \overset{\sim}{n}) \Phi + (G-1) \Sigma$

Note that

$Var( \bar{\mathbf{Y}}_{\cdot g} ) = \Phi + \frac{1}{n_g} \Sigma$

and that the group-size weighted average of these variances is:

$\sum_{g=1}^G \frac{n_g}{N} Var( \bar{\mathbf{Y}}_{\cdot g} ) = \sum_{g=1}^G \frac{n_g}{N} \left[ \Phi + \frac{1}{n_g} \Sigma \right] = \Phi + \frac{G}{N} \Sigma$

The expectation of combinations of $S S H$ and $S S E$ of the form $k H S S H + k E S S E$ :

$k_H \!$	$k_E \!$	$E(k_H SSH + k_E SSE)\!$
1	0	$(N - \tilde{n}) \Phi + (G-1) \Sigma$
0	1	$(N - G)Σ$
$\frac{1}{N- \overset{\sim}{n}}$	0	$\Phi + \frac{G-1}{N - \overset{\sim}{n}} \Sigma$
$\frac{G}{N(G-1)}$	0	$\Phi + \frac{G}{N} \Sigma, \mbox{ with equal groups}$
$\frac{G}{N(G-1)}$	$- \frac{G}{N(N-G)}$	$Φ, with equal groups$