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Chapter 5: Bivariate Data
Definition: Coefficient of Determination
The coefficient of determination measures the proportion of the variability in
the dependent variable (y variable) that is explained by the regression model
through the independent variable (x variable).
Next we will present the properties of this statistics.
Properties of the Coefficient of Determination
The coefficient of determination is obtained by squaring the value of the
linear correlation coefficient and is sometimes expressed as a percentage.
The symbol used to denote the coefficient of determination is
.
Note that 0
1 or equivalently 0
100%.
values close to 1 or 100% would imply that the model is explaining
most of the variation in the dependent variable and may be a very useful
model.
values close to 0 or 0% would imply that the model is explaining very
little of the variation in the dependent variable and may not be a useful
model.
Example 5-8:
What is the value of the coefficient of determination for the
model in
Example 5-3
.
Solution:
Using the information in
Example 5–3
, we can compute the
correlation coefficient using the formula. Thus,
( ) ( )( )
√{ ( ) ( )
} { ( ) ( )
}
Thus, the coefficient of determination is
= (- 0.939)
2
= 0.882 or 88.2
percent. That is, the regression model can explain about 88.2 percent of the
variation in the
-values. This would be a reasonable model to use for
prediction because of the large
value.