Design matrix

In statistics and in particular in regression analysis, a design matrix, also known as model matrix or regressor matrix and often denoted by X, is a matrix of values of explanatory variables of a set of objects. 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 such that (the jth column of the ith row of ) represents the value of the jth variable associated with the ith object. A regression model may be represented via matrix multiplication as where X is the design matrix, is a vector of the model's coefficients (one for each variable), is a vector of random errors with mean zero, and y is the vector of predicted outputs for each object. The matrix of data has dimension n-by-p, where n is the number of samples observed, and p is the number of variables (features) measured in all samples. In this representation different rows typically represent different repetitions of an experiment, while columns represent different types of data (say, the results from particular probes). For example, suppose an experiment is run where 10 people are pulled off the street and asked four questions.