Design matrix

In statistics, a design matrix is a matrix that is used in certain statistical models, e.g., the general linear model. It can contain indicator variables (ones and zeros) that indicates group membership in an ANOVA and it represents the independent variables. The advantage with a design matrix is that it is able to represent a number of different experimental designs and statistical models, e.g., ANOVA, ANCOVA, and linear regression.

One-way ANOVA
Example with a one-way analysis of variance (ANOVA) with 3 groups and 7 observations. The first column in the design matrix models the grand [global] mean of the ys, while the 3 remaining columns indicate the group membership of each observation. Here the first group consists of the 3 first observations and the next two groups each consist of two observations. $$ \begin{bmatrix}y_1 \\ y_2 \\ y_3 \\ y_4 \\ y_5 \\ y_6 \\ y_7 \end{bmatrix} = \begin{bmatrix}1 & 1 &0 &0 \\1 &1 &0 &0 \\ 1 & 1 & 0 & 0 \\ 1 & 0 & 1 & 0 \\ 1 & 0 & 1 & 0 \\ 1 & 0 & 0 & 1 \\ 1 & 0 & 0 & 1\end{bmatrix} \begin{bmatrix}\mu \\ \tau_1 \\ \tau_2 \\ \tau_3 \end{bmatrix} + \begin{bmatrix}u_1 \\ u_2 \\ u_3 \\ u_4 \\ u_5 \\ u_6 \\ u_7 \end{bmatrix} $$