1. The Role of r2 in Regression Target Goal: I can use r2 to explain the variation of y that is explained by the LSRL.
D4: 3.2b
Hw: pg 191 – 43, 46, 48, 53, 63,

2. r : correlation coefficient r2 : the coefficient of determination
If the line ŷ is a poor model, the value of r2 turns out to be: too small, closer to 0.
If the line ŷ fit the data fairly well, the value of r2 turns out to be: larger, closer to 1.

3. What is the meaning of r2 in regression?
Squares of the deviations about ŷ

4. Least-Squares Regression
The Role of r2 in Regression
The standard deviation of the residuals gives us a numerical estimate of the average size of our prediction errors. There is another numerical quantity that tells us how well the least-squares regression line predicts values of the response y.
Least-Squares Regression
Definition:
The coefficient of determination r2 is the fraction of the variation in the values of y that is accounted for by the least-squares regression line of y on x. We can calculate r2 using the following formula:
where
and

5. Formula for r2 made up of these parts
SST: total sum of squares about the mean y bar.
SST = ∑(y – y bar)2
SSE: sum of the squares for error.
SSE = ∑(y – ŷ)2
r2: coefficient of determination.
r2 = SST – SSE
SST

6. Ex: Large r2
If , then the deviations and thus the ; in fact, if all of the points fell exactly on the regression line, SSE would be 0.
r2 = SST – SSE
SST
x is a good predictor of y
SSE would be small

7. For the data in this example, x: 0 5 10 y: 0 7 8
r2 = SST – SSE = 38 – 6 =
SST 38
Conclusion: We say that ____ of the ___________ is explained by the
__________________________.
0.842
84%
variation in y
least-squares regression of y on x.

8. r2 in Regression The coefficient of determination r2, is the
fraction of the variation in the values
that are explained
by least-squared regression of y on x.

9. Least-Squares Regression
The Role of r2 in Regression
r 2 tells us how much better the LSRL does at predicting values of y than simply guessing the mean y for each value in the dataset. Consider the example on page If we needed to predict a backpack weight for a new hiker, but didn’t know each hikers weight, we could use the average backpack weight as our prediction.
Least-Squares Regression
If we use the mean backpack weight as our prediction, the sum of the squared residuals is
SST = 83.87
If we use the LSRL to make our predictions, the sum of the squared residuals is
SSE = 30.90
SSE/SST = 30.97/83.87
SSE/SST = 0.368
Therefore, 36.8% of the variation in pack weight is unaccounted for by the least-squares regression line.
1 – SSE/SST = 1 – 30.97/83.87
r2 = 0.632
63.2 % of the variation in backpack weight is accounted for by the linear model relating pack weight to body weight.

10. Least-Squares Regression
Correlation and Regression Wisdom
Correlation and regression are powerful tools for describing the relationship between two variables. When you use these tools, be aware of their limitations
Least-Squares Regression
Fact 1. The distinction between explanatory and response variables is important in regression.

11. Facts about least-squared regression
Fact 2: There is a close connection between the slope of the least-squared regression line.
As the correlation grows less strong, the in response to changes in x.
correlation and
prediction ŷ moves less

12. Fact 3: Every LSRL passes through
Remember: When reporting a regression, give r2 as a measure of how successful the regression was in explaining the response.
When you see a correlation (r), square it to get a better feel for the
strength of the association.

13. Exercise: Predicting The Stock Market.
Some people think that the behavior of the stock market in January predicts its behavior for the rest of the year.
Take the explanatory variable x to be the percent change in a stock market index in January and the
response variable y to be the change in the index for the entire year.
We expect a positive correlation between x and y because the change during January contributes to the years full change.

14. Calculation from the data for the years 1960 to 1997 gives:
x bar = 1.75%, sx = 5.36
y-bar = 9.07%, sy = 15.35%
and r = 0.596

15. r = 0.596
a. What percent of the observed variation in yearly changes in the index with is explained by a straight-line relationship the changes during January?
The straight-line relationship is explained by r2 = or,
35.5% of the variations in yearly changes in the index is explained by the changes during January.

16. What is the equation of the least-squared regression line for predicting full-year change from January change?
Find b: ,
b = 1.707
Find a: ,
a = 6.083%

17. The regression equation is
ŷ = a + bx
ŷ = 6.083% x

18. Predictions The mean change in January is = 1.75%.
Use your regression line to predict the change in the index in a year in which the index rises 1.75% (x bar) in January.
Why could you have given this result w/out doing the calculation?
Every LSRL passes through (x bar, y bar). Recall y bar = 9.07%, so the predicted change is
ŷ = 9.07%.

19. Exercise: Class attendance and grades
A study of class attendance and grades among first year students at a state university showed that in general students who attended a higher percent of their classes earn higher grades.

20. Class attendance explained 16% of the variation in grade index among students. What is the numerical value of the correlation between percent of class attended and grade index? r2
r =
High attendance goes with high grades so the correlation must be positive.
0.40