Statistical Methodologies with Medical Applications by Poduri S. R. S. Rao;

Author:Poduri, S. R. S. Rao;
Language: eng
Format: epub
Publisher: John Wiley & Sons, Incorporated
Published: 2016-11-18T00:00:00+00:00

10.4.3 Multiple correlation, adjusted R2 and partial correlation

The fraction of the SST that is attributed to the regression is , which is the square of the multiple correlation coefficient. It is the square of the correlation of the observed Yi and the estimator Ŷi. For the regression of Y on the p independent variables, , which shows that the F‐ratio increases as R2 increases.

As the number of independent variables in a regression model is increased, the SSE does not increase, SSR does not decrease and R2 does not decrease. With the averages of the SSR and SST, the Adjusted R2, denoted by , is obtained from

(10.39)

As a result, becomes smaller than R2. For the regression of LDL on Age, Weight and Fitness in Example 10.5, . Since , .

For the partial correlation, consider three variables (X1, X2, X3). From a sample of size n observed on these variables, the residuals of the regression of X1 on X3 and of X2 on X3 respectively are

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