1. The geometry of the balanced ANOVA model (with fixed effects)


    Most usually, the mathematical treatment of Gaussian linear models starts with the matricial writing \(Y=X\beta+\sigma G\), where \(Y\) is a random vector modelling the \(n\) response values, \(X\) is a known matrix, \(\beta\) is a vector of unknown parameters, and \(G\) has the standard normal distribution on \(\mathbb{R}^n\).

    There are good reasons to use this matricial writing, however it is cleaner to treat the theory with the equivalent vector space notation \(Y = \mu + \sigma G\), where \(\mu\) is assumed to lie in a linear subspace \(W\) of \(\mathbb{R}^n\), corresponding to \(\text{Im}(X)\) in the matricial notation. For example, denoting by \(P_W\) the orthogonal projection on \(W\), the least-squares estimate \(\hat\mu\) of \(\mu\) is simply given by \(\hat\mu=P_Wy\) and the residuals are \(P_W^\perp y\), denoting by \(P^\perp_W\) the projection on the orthogonal complement of \(W\), and there is no need to consider \(W=\text{Im}(X)\) to derive the general principles of the theory. The balanced one-way ANOVA model, which is the topic of this article, illustrates this approach.

    Standard normal distribution on a vector space

    The main tool used to treat the theory of Gaussian linear models is the standard normal distribution on a linear space.

    Theorem and definition
    Let $X$ be a $\mathbb{R}^n$-valued random vector, and $W \subset \mathbb{R}^n$ be a linear space. Say that $X$ has the standard normal distribution on the vector space $W$, and then note $X \sim SN(W)$, if it takes its values in $W$ and its characteristic function is given by $$\mathbb{E} \textrm{e}^{i\langle w, X \rangle} = \textrm{e}^{-\frac12{\Vert w \Vert}^2} \quad \text{for all } w \in W.$$ The three following assertions are equivalent (and this is easy to prove):
    1. $X \sim SN(W)$;
    2. the coordinates of $X$ in some orthonormal basis of $W$ are i.i.d. standard normal random variables;
    3. the coordinates of $X$ in any orthonormal basis of $W$ are i.i.d. standard normal random variables.

    Of course we retrieve the standard normal distribution on \(\mathbb{R}^n\) when taking \(W=\mathbb{R}^n\).

    From this definition-theorem, the so-called Cochran's theorem is an obvious statement. More precisely, if \(U \subset W\) is a linear space, and \(Z=U^\perp \cap W\) is the orthogonal complement of \(U\) in \(W\), then the projection \(P_UX\) of \(X\) on \(U\) has the standard normal distribution on \(U\), similarly the projection \(P_ZX\) of \(X\) on \(Z\) has the standard normal distribution on \(Z\), and moreover \(P_UX\) and \(P_ZX\) are independent. This is straightforward to see from the definition-theorem of \(SN(W)\), and it is also easy to see that \({\Vert P_UX\Vert}^2 \sim \chi^2_{\dim(U)}\).

    The balanced ANOVA model

    The balanced ANOVA model is used to model a sample \(y=(y_{ij})\) with a tabular structure: \[y=\begin{pmatrix} y_{11} & \ldots & y_{1J} \\ \vdots & y_{ij} & \vdots \\ y_{I1} & \ldots & y_{IJ} \end{pmatrix}, \] \(y_{ij}\) denoting the \(j\)-th measurement in group \(i\). It is assumed that the \(y_{ij}\) are independent and the population mean depends on the group index \(i\). More precisely, the \(y_{ij}\) are modelled by random variables \(Y_{ij} \sim_{\text{iid}} {\cal N}(\mu_i, \sigma^2)\).

    So, how to write this model as \(Y=\mu + \sigma G\) where \(G \sim SN(\mathbb{R}^n)\) and \(\mu\) lies in a linear space \(W \subset \mathbb{R}^n\) ?

    Tensor product

    Here \(n=IJ\) and one should consider \(Y\) as the vector obtained by stacking the \(Y_{ij}\). For example if \(I=2\) and \(J=3\), we should write \[Y={(Y_{11}, Y_{12}, Y_{13}, Y_{21}, Y_{22}, Y_{23})}'.\]

    Actually this is not a good idea to loose the tabular structure. The appropriate approach for writing the balanced ANOVA model involves the tensor product. We keep the tabular structure of the data: \[Y = \begin{pmatrix} Y_{11} & Y_{12} & Y_{13} \\ Y_{21} & Y_{22} & Y_{23} \end{pmatrix}\] and we take \[G \sim SN(\mathbb{R}^I\otimes\mathbb{R}^J)\] where the tensor poduct \(\mathbb{R}^I\otimes\mathbb{R}^J\) of \(\mathbb{R}^I\) and \(\mathbb{R}^J\) is nothing but the space of matrices with \(I\) rows and \(J\) columns. Here \[\mu = \begin{pmatrix} \mu_1 & \mu_1 & \mu_1 \\ \mu_2 & \mu_2 & \mu_2 \end{pmatrix},\] lies in a linear space \(W \subset \mathbb{R}^I\otimes\mathbb{R}^J\) which is convenient to define with the help of the tensor product \(x \otimes y\) of two vectors \(x \in \mathbb{R}^I\) and \(y \in \mathbb{R}^J\), defined as the element of \(\mathbb{R}^I\otimes\mathbb{R}^J\) given by \[{(x \otimes y)}_{ij}=x_iy_j.\] Indeed, one has \[\mu = (\mu_1, \mu_2) \otimes (1,1,1),\] and then the linear space \(W\) in which \(\mu\) is assumed to lie is \[W = \mathbb{R}^I\otimes{\bf 1}_J.\]

    Moreover, there is a nice orthogonal decomposition of \(W\) corresponding to the usual other parameterization of the model: \[\boxed{\mu_i = m + \alpha_i} \quad \text{with } \sum_{i=1}^I\alpha_i=0.\] Indeed, writing \(\mathbb{R}^I=[{\bf 1}_I] \oplus {[{\bf 1}_I]}^\perp\) yields the following decomposition of \(\mu\): \[ \begin{align*} \mu = (\mu_1, \ldots, \mu_I) \otimes {\bf 1}_J & = \begin{pmatrix} m & m & m \\ m & m & m \end{pmatrix} + \begin{pmatrix} \alpha_1 & \alpha_1 & \alpha_1 \\ \alpha_2 & \alpha_2 & \alpha_2 \end{pmatrix} \\ & = \underset{\in \bigl([{\bf 1}_I]\otimes[{\bf 1}_J]\bigr)}{\underbrace{m({\bf 1}_I\otimes{\bf 1}_J)}} + \underset{\in \bigl([{\bf 1}_I]^{\perp}\otimes[{\bf 1}_J] \bigr)}{\underbrace{(\alpha_1,\ldots,\alpha_I)\otimes{\bf 1}_J}} \end{align*} \]

    Least-squares estimates

    With the theory introduced above, the least-squares estimates of \(m\) and the \(\alpha_i\) are given by \(\hat m({\bf 1}_I\otimes{\bf 1}_J) = P_U y\) and \(\hat\alpha\otimes{\bf 1}_J = P_Zy\) where \(U = [{\bf 1}_I]\otimes[{\bf 1}_J]\) and \(Z = {[{\bf 1}_I]}^{\perp}\otimes[{\bf 1}_J] = U^\perp \cap W\), and we also know that \(\hat m\) and the \(\hat\alpha_i\) are independent. The least-squares estimates of the \(\mu_i\) are given by \(\hat\mu_i=\hat m +\hat\alpha_i\). Deriving the expression of these estimates and their distribution is left as an exercise to the reader.