Coefficient of Determination Section 4.3 Alan Craig 770-274-5242 [email protected].
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Transcript of Coefficient of Determination Section 4.3 Alan Craig 770-274-5242 [email protected].
![Page 2: Coefficient of Determination Section 4.3 Alan Craig 770-274-5242 acraig@gpc.edu.](https://reader030.fdocuments.us/reader030/viewer/2022020117/56649cd75503460f9499e76b/html5/thumbnails/2.jpg)
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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.
![Page 4: Coefficient of Determination Section 4.3 Alan Craig 770-274-5242 acraig@gpc.edu.](https://reader030.fdocuments.us/reader030/viewer/2022020117/56649cd75503460f9499e76b/html5/thumbnails/4.jpg)
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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
![Page 7: Coefficient of Determination Section 4.3 Alan Craig 770-274-5242 acraig@gpc.edu.](https://reader030.fdocuments.us/reader030/viewer/2022020117/56649cd75503460f9499e76b/html5/thumbnails/7.jpg)
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Actual
Deviation
yMean
yy
Total Deviation
y
![Page 8: Coefficient of Determination Section 4.3 Alan Craig 770-274-5242 acraig@gpc.edu.](https://reader030.fdocuments.us/reader030/viewer/2022020117/56649cd75503460f9499e76b/html5/thumbnails/8.jpg)
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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).
![Page 9: Coefficient of Determination Section 4.3 Alan Craig 770-274-5242 acraig@gpc.edu.](https://reader030.fdocuments.us/reader030/viewer/2022020117/56649cd75503460f9499e76b/html5/thumbnails/9.jpg)
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Actual
Deviation
yMean
Predictedyy ˆ
yy ˆ
Unexplained Deviation
Deviation explained
by the regression
y
y
![Page 10: Coefficient of Determination Section 4.3 Alan Craig 770-274-5242 acraig@gpc.edu.](https://reader030.fdocuments.us/reader030/viewer/2022020117/56649cd75503460f9499e76b/html5/thumbnails/10.jpg)
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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
• ???????????????