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AIM 7: Define, calculate and interpret the coefficient of determination and coefficient of correlation.

1、Which of the following statements regarding the coefficient of determination is least accurate? The coefficient of determination:

A) may range from ?1 to +1.

B) is the percentage of the total variation in the dependent variable that is explained by the independent variable.

C) cannot decrease as independent variables are added to the model.

D) is the ratio of explained variation to total variation.

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The correct answer is A

In a simple regression, the coefficient of determination is calculated as the correlation coefficient squared and ranges from 0 to +1.

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2、A simple linear regression equation had a coefficient of determination (R2) of 0.8. What is the correlation coefficient between the dependent and independent variables and what is the covariance between the two variables if the variance of the independent variable is 4 and the variance of the dependent variable is 9?

       Correlation coefficient        Covariance

A) 0.89    5.34

B) 0.91    4.80

C) 0.89    4.80

D) 0.91    5.34

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The correct answer is A

The correlation coefficient is the square root of the R2, r = 0.89.

To calculate the covariance multiply the correlation coefficient by the product of the standard deviations of the two variables:

COV = 0.89 × √4 × √9 = 5.34

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3、Which term is least likely to apply to a regression model?

A) Goodness of fit.

B) R2.

C) Coefficient of determination. 

D) Coefficient of variation. 

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The correct answer is D

Goodness of fit, coefficient of determination and R2 are different names for the same concept. The coefficient of variation is not directly part of a regression model.

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4、Unlike the coefficient of determination, the coefficient of correlation:

A) measures the strength of association between the two variables more exactly.

B) can have an absolute value greater than 1.

C) indicates the percentage of variation explained by a regression model.

D) indicates whether the slope of the regression line is positive or negative.

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The correct answer is D

In a simple linear regression the coefficient of determination (R2) is the squared correlation coefficient, so it is positive even when the correlation is negative.

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5、An analyst performs two simple regressions. The first regression analysis has an R-squared of 0.40 and a beta coefficient of 1.2. The second regression analysis has an R-squared of 0.77 and a beta coefficient of 1.75. Which one of the following statements is most accurate?

A) The first regression equation has more explaining power than the second regression equation.

B) The second regression equation has more explaining power than the first regression equation.

C) The beta coefficient of the 2nd regression indicates that this regression has more explaining power than the first.

D) The R-squared of the first regression indicates that there is a 0.40 correlation between the independent and the dependent variables.

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The correct answer is B

The coefficient of determination (R-squared) is the percentage of variation in the dependent variable explained by the variation in the independent variable. The larger R-squared (0.77) of the second regression means that 77% of the variability in the dependent variable is explained by variability in the independent variable, while only 40% of that is explained in the first regression. This means that the second regression has more explaining power than the first regression. Note that the Beta is the slope of the regression line and doesn’t measure explaining power.