Design matrix
Encyclopedia
In statistics
, a design matrix is a matrix
of explanatory variables, often denoted by X, that 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.
The design matrix represents the independent variable
s in statistical models which describe observed data (often called dependent variables) in terms of other known variables (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 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.
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
, a design matrix is a matrix
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
of explanatory variables, often denoted by X, that is used in certain statistical models, e.g., the general linear model
General linear model
The general linear model is a statistical linear model.It may be written aswhere Y is a matrix with series of multivariate measurements, X is a matrix that might be a design matrix, B is a matrix containing parameters that are usually to be estimated and U is a matrix containing errors or...
.
It can contain indicator variables (ones and zeros) that indicate group membership in an ANOVA.
The design matrix represents the independent variable
Independent variable
The terms "dependent variable" and "independent variable" are used in similar but subtly different ways in mathematics and statistics as part of the standard terminology in those subjects...
s in statistical models which describe observed data (often called dependent variables) in terms of other known variables (explanatory variables). The theory relating to such models makes substantial use of matrix manipulations involving the design matrix: see for example linear regression
Linear regression
In statistics, linear regression is an approach to modeling the relationship between a scalar variable y and one or more explanatory variables denoted X. The case of one explanatory variable is called simple 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
ANCOVA
In statistics, analysis of covariance is a general linear model with a continuous outcome variable and two or more predictor variables where at least one is continuous and at least one is categorical . ANCOVA is a merger of ANOVA and regression for continuous variables...
, and linear regression
Linear regression
In statistics, linear regression is an approach to modeling the relationship between a scalar variable y and one or more explanatory variables denoted X. The case of one explanatory variable is called simple 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.