Following the principles of " standardized partial regression ", of which "'path analysis "'is a pictorially oriented version, Sewall Wright analyzed the paths of gene-flow during sexual reproduction, and established the " strengths of contribution " ( " coefficients of determination " ) of various components to the overall result.
22.
An " R " 2 ( coefficient of determination ) close to one does not guarantee that the model fits the data well, because as Anscombe's quartet shows, a high " R " 2 can occur in the presence of misspecification of the functional form of a relationship or in the presence of outliers that distort the true relationship.
23.
The coefficient of multiple correlation is computed as the square root of the coefficient of determination, but under the particular assumptions that an intercept is included and that the best possible linear predictors are used, whereas the coefficient of determination is defined for more general cases, including those of nonlinear prediction and those in which the predicted values have not been derived from a model-fitting procedure.
24.
The coefficient of multiple correlation is computed as the square root of the coefficient of determination, but under the particular assumptions that an intercept is included and that the best possible linear predictors are used, whereas the coefficient of determination is defined for more general cases, including those of nonlinear prediction and those in which the predicted values have not been derived from a model-fitting procedure.
25.
However, the fit of the data beyond the threshold is not as well as in the blue figure that has been made using the principle of minimization of the sum of squares of devations of the observed values from the regression lines over the whole domain of explanatory variable X ( i . e . maximization of the coefficient of determination ), while the partial regression is designed only to find the point where the horizontal trend changes into a sloping trend.
26.
"Similarity of EEG spatial configuration of each prototype map with each of the 10 maps is computed using the coefficient of determination to omit the maps'polarities . . . . Separately for each class the prototype maps are updated combining all assigned maps by computing the first spatial principal component ( Principal Component Analysis ) of the maps and thereby maximizing the common variance while disregarding the map polarity . " This process is repeated several times using different randomly selected prototype maps from among the collected data to use for statistical comparison and variance determination.
27.
In fact, many other functional forms appear approximately linear on the log log scale, and simply evaluating the goodness of fit of a linear regression on logged data using the coefficient of determination ( " R " 2 ) may be invalid, as the assumptions of the linear regression model, such as Gaussian error, may not be satisfied; in addition, tests of fit of the log log form may exhibit low statistical power, as these tests may have low likelihood of rejecting power laws in the presence of other true functional forms.
28.
Where " R " 2 is the coefficient of determination and " VAR " err and " VAR " tot are the variance of the estimated residuals and the sample variance of the dependent variable . " SS " " err " ( the sum of squared predictions errors, equivalently the residual sum of squares ), " SS " " tot " ( the total sum of squares ), and " SS " " reg " ( the sum of squares of the regression, equivalently the explained sum of squares ) are given by
