Coefficient of DeterminationSection 4.3

Alan Craig770-274-5242

acraig@gpc.edu

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Objectives 4.3

1. Compute and interpret the coefficient of determination.

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Coefficient of Determination

• The coefficient of determination, R2, measures the percentage of the total variation in the response variable that is explained by the least-squares regression.

• R2 is calculated by squaring the linear correlation coefficient, r.

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On the Calculator

r and R2 are part of the calculator output for a linear regression with DiagnosticsOn.

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Least-Squares Regression

• Recall that the least-squares regression line minimizes the sum of the squared

errors (residuals) = residuals2

Error or residual = actual - predicted

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Estimating y

If I have no information about values of the predictor variable x, then my best guess for y is the mean of y:

and the deviation is the actual value minus the mean.

y

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Actual

Deviation

yMean

yy

Total Deviation

y

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Estimating y

However, if I have additional data on the values of x and corresponding values of y, I can often do better by calculating the regression of y on x.

Part of the total deviation is now explained by the regression equation although some of the deviation is still unexplained (unless there is a perfect linear correlation).

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Actual

Deviation

yMean

Predictedyy ˆ

yy ˆ

Unexplained Deviation

Deviation explained

by the regression

y

y

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Deviation

Note that

That is,

y = mean of y

+ explained deviation

+ unexplained deviation

)ˆ()ˆ( yyyyyy

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Deviation

Or

That is,

Total deviation = explained deviation

+ unexplained deviation

)ˆ()ˆ( yyyyyy

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Variation

The total variation of y is

The explained variation is

The unexplained variation is

1

2

n

yy

1

ˆ 2

n

yy

1

ˆ 2

n

yy

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Variation

variationtotal

variationdunexplaine1

variationtotal

variationexplained

variationtotal

variationdunexplaine

variationtotal

variationexplained1

variationdunexplaine variationexplained variationtotal

R2

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Interpreting R2

Thus, R2 is the percentage of variation in the response variable, y, that is explained by the predictor variable x.

variationtotal

variationdunexplaine1

variationtotal

variationexplained2 R

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Interpreting R2

Using our example from Sections 4.1 and 4.2 (problem 10, p. 172), R2 =0.9835=98.35%, so the predictor variable, Carats, explains 98.35% of the variation in the response variable, Price.

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Questions

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