Design matrix

In statistics, 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. In statistics, 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 X {displaystyle X} such that X i j {displaystyle X_{ij}} (the jth column of the ith row of X {displaystyle X} ) represents the value of the jth variable associated with the ith 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, β {displaystyle eta } is a vector of the model's coefficients (one for each variable), 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. The data matrix M would be a 10×4 matrix (meaning 10 rows and 4 columns). The datum in row i and column j of this matrix would be the answer of the i th person to the j th question. This section gives an example of simple linear regression—that is, regression with only a single explanatory variable—with seven observations.The seven data points are {yi, xi}, for i = 1, 2, …, 7. The simple linear regression model is where β 0 {displaystyle eta _{0}} is the y-intercept and β 1 {displaystyle eta _{1}} is the slope of the regression line. This model can be represented in matrix form as