Design matrix

In statistics, a design matrix (also known as regressor matrix or model matrix) is a matrix of values of explanatory variables of a set of objects, often denoted by X. Each row represents an individual object, with the successive columns corresponding to the variables and their specific values for that object. The design matrix is used in certain statistical models, e.g., the general linear model. It can contain indicator variables (ones and zeros) that indicate group membership in an ANOVA, or it can contain values of continuous variables.

The design matrix contains data on the independent variables (also called explanatory variables) in statistical models which attempt to explain observed data on a response variable (often called a dependent variable) in terms of the explanatory variables. The theory relating to such models makes substantial use of matrix manipulations involving the design matrix: see for example linear regression. A notable feature of the concept of 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.

The design matrix is defined to be a matrix X such that the j^th column of the i^th row of X represents the value of the j^th variable associated with the i^th object.

A regression model which is a linear combination of the explanatory variables may therefore be represented via matrix multiplication as

where X is the design matrix, ${\displaystyle \beta }$ is a vector of the model's coefficients (one for each variable), and y is the vector of predicted outputs for each object.

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