In [[statistics]], [[regression analysis]] is a method for the explanation of phenomena and the prediction of future events. In the regression analysis, a [[correlation coefficient|coefficient of correlation]] ''r'' between [[random variable]]svariables ''X'' and ''Y'' is a quantitative index of associationco-movement between these two variables. In itsIts squared form, as athe [[coefficient of determination]] ''r''<sup> 2</sup>, indicates the amountfraction of [[variance]] in the criterion variable ''Y'' that is accounted for by the variation in the predictor variable ''X''. In the multiple regression analysis, the set of predictor variables (also called independent variables or explanatory variables) ''X''<sub>1</sub>, ''X''<sub>2</sub>, ... is used to explain variability of the criterion variable (also called the dependent variable) ''Y''. A multivariate counterpart of the coefficient of determination ''r''<sup> 2</sup> is the ''coefficient of multiple determination'', ''R''<sup> 2</sup> , which is frequently called simply the coefficient of determination. The [[square root]] of the coefficient of multiple determination is the '''coefficient of multiple correlation''', '''''R'''''. ▼{{Expert-subject|statistics|date=November 2008}}
▲In [[statistics]], [[regression analysis]] is a method for explanation of phenomena and prediction of future events. In the regression analysis, a [[correlation coefficient|coefficient of correlation]] ''r'' between [[random variable]]s ''X'' and ''Y'' is a quantitative index of association between these two variables. In its squared form, as a [[coefficient of determination]] ''r''<sup> 2</sup>, indicates the amount of [[variance]] in the criterion variable ''Y'' that is accounted for by the variation in the predictor variable ''X''. In the multiple regression analysis, the set of predictor variables ''X''<sub>1</sub>, ''X''<sub>2</sub>, ... is used to explain variability of the criterion variable ''Y''. A multivariate counterpart of the coefficient of determination ''r''<sup> 2</sup> is the ''coefficient of multiple determination'', ''R''<sup> 2</sup>. The [[square root]] of the coefficient of multiple determination is the '''coefficient of multiple correlation''', '''''R'''''.
==Conceptualization of multiple correlation==
An intuitive approach to the multiple regression analysis is to sum the squared correlations between the predictor variables and the criterion variable to obtain an index of the over-allstrength of the overall relationship between the predictor variables and the criterion variable. However, such a sum is often greater than one, suggesting that simple summation of the squared coefficients of correlations is not a correct procedure to employ. In fact, athe simple summation of squared coefficients of correlations between the predictor variables and the criterion variable ''is'' the correct procedure, butif and only inif one has the special case whenin which the predictor variables are not correlated. If the predictors are relatedcorrelated, their inter-correlations must be removed so that only the unique contributions of each predictor toward explanation of the criterion remain.
==Fundamental equation of multiple regression analysis==
Initially,The coefficient of multiple determination ''R''<sup>2</sup> (a [[matrixscalar (mathematics)|matrixscalar]] of correlations ''R''), is computed for all variables involved in the analysis. This matrix can be conceptualized as a supermatrix, consisting ofusing the [[Euclidean space|vector]] ''c'' of cross-correlations between the predictor variables and the criterion variable ''c'', its [[transpose]] ''c''’', and the matrix of intercorrelations between predictor variables ''R''<sub>''xx''</sub> of inter-correlations between predictor variables. The fundamental equation of the multiple regression analysis is
::''R''<sup>2</sup> = ''c''' ''R''<sub>''xx''</sub><sup>−1</sup> ''c''.
The expression on the left side signifiesdenotes the coefficient of multiple determination (the squared coefficient of multiple correlation). The expressionsterms on the right side are the transposed vector of cross-correlations ''c'' ' of cross-correlations, the matrix[[Matrix inversion|inverse]] of inter-correlationsthe matrix ''R''<sub>''xx''</sub> toof be inverted (cf., [[matrix inversion]])inter-correlations, and the vector of cross-correlations, ''c''. The premultiplication of the vector of cross-correlations by its transpose changes the coefficients of correlation into coefficients of determination. The inverted matrix of the inter-correlations removes the redundant variance that results from the of inter-correlations of the predictor set of variables. TheseThe not-redundantsquare cross-correlationsroot areof summedthe toresulting obtaincoefficient theof multiple determination is the coefficient of determinationmultiple correlation ''R''<sup>2</sup>. The squareNote rootthat ofif thisall coefficientthe ispredictor variables are uncorrelated, the matrix coefficient''R''<sub>''xx''</sub> ofis multiplethe correlationidentity matrix and ''R''<sup>2</sup> simply equals ''c''' ''c'', the sum of the squared cross-correlations.
==References==
| 