Multiple correlation
Encyclopedia
In statistics
, multiple correlation is a linear relationship among more than two variables. It is measured by the coefficient of multiple determination, denoted as R2, which is a measure of the fit of a linear regression
. A regression's R2 falls somewhere between zero and one (assuming a constant term has been included in the regression); a higher value indicates a stronger relationship among the variables, with a value of one indicating that all data points fall exactly on a line in multidimensional space and a value of zero indicating no relationship at all between the independent variables collectively and the dependent variable.
Unlike the coefficient of determination
in a regression involving just two variables, the coefficient of multiple determination is not computationally commutative: a regression of y on x and z will in general have a different R2 than will a regression of z on x and y. For example, suppose that in a particular sample the variable z is uncorrelated with both x and y, while x and y are linearly related to each other. Then a regression of z on y and x will yield an R2 of zero, while a regression of y on x and z will yield a positive R2.
), can be computed using the vector
c of cross-correlation
s between the predictor variables and the criterion variable, its transpose
c, and the matrix
Rxx of inter-correlations between predictor variables. The "fundamental equation of multiple regression analysis" is
The expression on the left side denotes the coefficient of multiple determination. The terms on the right side are the transposed vector c ' of cross-correlations, the inverse of the matrix Rxx of inter-correlations, and the vector c of cross-correlations. Note that if all the predictor variables are uncorrelated, the matrix Rxx is the identity matrix and R2 simply equals c c, the sum of the squared cross-correlations. Otherwise, the inverted matrix of the inter-correlations removes the redundant variance that results from the inter-correlations of the predictor variables.
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....
, multiple correlation is a linear relationship among more than two variables. It is measured by the coefficient of multiple determination, denoted as R2, which is a measure of the fit of a 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 regression's R2 falls somewhere between zero and one (assuming a constant term has been included in the regression); a higher value indicates a stronger relationship among the variables, with a value of one indicating that all data points fall exactly on a line in multidimensional space and a value of zero indicating no relationship at all between the independent variables collectively and the dependent variable.
Unlike the coefficient of determination
Coefficient of determination
In statistics, the coefficient of determination R2 is used in the context of statistical models whose main purpose is the prediction of future outcomes on the basis of other related information. It is the proportion of variability in a data set that is accounted for by the statistical model...
in a regression involving just two variables, the coefficient of multiple determination is not computationally commutative: a regression of y on x and z will in general have a different R2 than will a regression of z on x and y. For example, suppose that in a particular sample the variable z is uncorrelated with both x and y, while x and y are linearly related to each other. Then a regression of z on y and x will yield an R2 of zero, while a regression of y on x and z will yield a positive R2.
Fundamental equation of multiple regression analysis
The coefficient of multiple determination R2 (a scalarScalar (mathematics)
In linear algebra, real numbers are called scalars and relate to vectors in a vector space through the operation of scalar multiplication, in which a vector can be multiplied by a number to produce another vector....
), can be computed using the vector
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
c of cross-correlation
Correlation
In statistics, dependence refers to any statistical relationship between two random variables or two sets of data. Correlation refers to any of a broad class of statistical relationships involving dependence....
s between the predictor variables and the criterion variable, its transpose
Transpose
In linear algebra, the transpose of a matrix A is another matrix AT created by any one of the following equivalent actions:...
c, and the 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...
Rxx of inter-correlations between predictor variables. The "fundamental equation of multiple regression analysis" is
-
- R2 = c Rxx−1 c.
The expression on the left side denotes the coefficient of multiple determination. The terms on the right side are the transposed vector c ' of cross-correlations, the inverse of the matrix Rxx of inter-correlations, and the vector c of cross-correlations. Note that if all the predictor variables are uncorrelated, the matrix Rxx is the identity matrix and R2 simply equals c c, the sum of the squared cross-correlations. Otherwise, the inverted matrix of the inter-correlations removes the redundant variance that results from the inter-correlations of the predictor variables.