1. 3.2 Day 3

2. The Standard Deviation of the Residuals
What does this number tell us?

3. 𝑅 2
“R-squared” has many different definitions that are all technically correct
For example, from your book:

4. 𝑅 2 : More useful definitions
How much better the regression (“our model”) does at predicting the values of the outcome variable than simply guessing the mean
THE MOST COMMON (and intuitive) definition: the proportion of the variance in our outcome variable (y) that can be explained by our predictor variable (x)
The square of the correlation
More useful for calculations than for interpretations

5. Interpretation of 𝑅 2 Will always be between 0 and 1
0 would mean that our regression is explaining 0% of the variance in the outcome
1 would mean that our regression is explaining 100% of the variance in the outcome
If 𝑅 2 =.64, that means that the regression is explaining 64% of the variance in y (the response variable)

6. Calculating 𝑅 2
Most of the time, you will be given 𝑅 2 in some way, and asked to interpret it
If you know (or are given) the correlation coefficient, how do you get 𝑅 2 ?

7. Calculating 𝑅 2
Most of the time, you will be given 𝑅 2 in some way, and asked to interpret it
If you know (or are given) the correlation coefficient, how do you get 𝑅 2 ?
Square it ( 𝑅 2 is literally r…..squared)
Not sure why one is capitalized while the other isn’t
If not, you can calculate it on your calculator
YOU DO NOT NEED TO CALCULATE 𝑅 2 by hand

8. Calculator We need to turn diagnostics on 2ND----0 (catalog)
Use down arrow to scroll down to DiagnosticsOn
Hit enter twice

9. Let’s do a regression now
Income
Housing Price
Arvada
$69,581
$249,200
Parker
$105,041
$342,583
Westminster
$70,212
$275,300
Gunbarrel
$81,554
$458,286
Lafayette
$71,758
$318,135
Denver
$58,003
$316,700
Elizabeth
$56,439
$203,745
We’ll use average household income to predict average house price in Colorado cities
Use your calculator to perform a linear regression
What proportion of the variation in housing prices can be explained by differences in income?

10. Let’s do a regression now
Household Income
Housing Price
Arvada
$69,581
$249,200
Parker
$105,041
$342,583
Westminster
$70,212
$275,300
Gunbarrel
$81,554
$458,286
Lafayette
$71,758
$318,135
Denver
$58,003
$316,700
Elizabeth
$56,439
$203,745
We’ll use average household income to predict average house price in Colorado cities
Use your calculator to perform a linear regression
𝐻𝑜𝑢𝑠𝑒 =116, (𝐼𝑛𝑐𝑜𝑚𝑒)
What proportion of the variation in housing prices can be explained by differences in income?
About 29%

11. Let’s do a regression now
Household Income
Housing Price
Arvada
$69,581
$249,200
Parker
$105,041
$342,583
Westminster
$70,212
$275,300
Gunbarrel
$81,554
$458,286
Lafayette
$71,758
$318,135
Denver
$58,003
$316,700
Elizabeth
$56,439
$203,745
Use your calculator to perform a linear regression
𝐻𝑜𝑢𝑠𝑒 =116, (𝐼𝑛𝑐𝑜𝑚𝑒)
What proportion of the variation in housing prices can be explained by differences in income?
About 29% ( 𝑅 2 )
Notice that r=.535
=.29

13. Regression Wisdom (and Limitations)
Be careful with which variable is x (explanatory variable) and which is y (response variable)
You will get different regression equations depending on which way you do it
Though r and R-squared will be the same
You can only use linear regression effectively when the relationship is, in fact, linear
Good news: many relationships are linear
There are more advanced techniques for non-linear relationships

14. Regression Wisdom (and Limitations)

15. Regression Wisdom (and Limitations)

16. Regression Wisdom (and Limitations)