The quantity r2 thus defined is known as the (sample) coefficient of determination and is the most commonly used measure of the goodness of fit of a regression line. Verbally, r2 measures the proportion or percentage of the total variation in Y explained by the regression model.

Two properties of r2 may be noted:

1. It is a nonnegative quantity. (Why?)

2. Its limits are 0 < r2 < 1. An r2 of 1 means a perfect fit, that is, Yi _ Yi for each i. On the other hand, an r2 of zero means that there is no relationship between the regressand and the regressor whatsoever (i.e., j2 _ 0). In this case, as (3.1.9) shows, Yri _ j _ Y, that is, the best prediction of any Y value is simply its mean value. In this situation therefore the regression line will be horizontal to the X axis.

Although r2 can be computed directly from its definition given in (3.5.5), it can be obtained more quickly from the following formula:


If we divide the numerator and the denominator of (3.5.6) by the sample size n (or n — 1 if the sample size is small), we obtain

where Sy2 and Sx2 are the sample variances of Y and X, respectively. Since f2 = x^/xx2, Eq. (3.5.6) can also be expressed as r 2 = (E xyQ2 X x2X yi

an expression that may be computationally easy to obtain.

Given the definition of r2, we can express ESS and RSS discussed earlier as follows:


Was this article helpful?

0 0
Rules Of The Rich And Wealthy

Rules Of The Rich And Wealthy

Learning About The Rules Of The Rich And Wealthy Can Have Amazing Benefits For Your Life And Success. Discover the hidden rules and beat the rich at their own game. The general population has a love / hate kinship with riches. They resent those who have it, but spend their total lives attempting to get it for themselves. The reason an immense majority of individuals never accumulate a substantial savings is because they don't comprehend the nature of money or how it works.

Get My Free Ebook

Post a comment