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[edit] Expected value of SSH

Consider one-way MANOVA with G groups, each with ng 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(SSE) = (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 SSH and SSE of the form kHSSH + kESSE:

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