1. Chapter 9 Regression Wisdom
*Subsets
*Extrapolation
*Outliers, Leverage, and Influence Points
*Lurking Variables

2. Subsets
The data should be homogeneous (of the same or a similar kind or nature)
If the data is made up of two or more groups that have been thrown together, it is usually best to fit different linear models to each group
Residual plots can help find subsets in the data

3. Cereal – without subgroups

4. Cereal – with subgroups

5. Extrapolation
Although linear models provide an easy way to predict values of y for a given value of x, it is unsafe to predict for values of x far from the ones used to find the linear model
Such extrapolation may pretend to see into the future, but the predictions should not be trusted
Example: data was collected from 1945 – 2000 in Massachusetts of the number of women in elected positions. We should NOT use the model to predict how many women will hold office in 2015

6. Homework
a: 1900 – 1940 there is a linear pattern; 1940 – 1970 the data is curved up; 1970 – 2000 there is a strong linear pattern
b: relatively strong from 1970 – 2000
c: no, on the whole graph. If we look at 1970 – 2000 then yes there would be a high correlation
d: no. its not straight enough

7. Homework
a: plot the data from 1955 – The scatterplot has a slight curve. Check the residual plot!! Residual plot has a pattern to it, so it is not a good place to use a linear model. If you did find an equation the predicted value would be 25.3 years.
b: not too much. The data is not straight enough to use a linear regression.
c: 50 years is too far from the data to make a prediction

8. Homework
a: knowing only the R2 value is not enough to use a linear regression. We need to check a residual plot and the 3 conditions (straight enough, quantitative variables, and outliers)
b: no, a linear model might not even fit

9. Homework
a: for every degree the temp rises the cost will go down $2.13
b: The cost when the temp is 00F
c: Too high, the residual is negative showing the model overestimates the cost.
d: cost = $
e: actual = $106.70
f: No, the residual plot has a curve to it. The data are probably not linear
g: no, there would be no change. The relationship does not depend on the units

10. Outlier Any data point that stands away from the others
In regression, outliers can be extraordinary in two ways
having a large residual
having a high leverage

11. Remember Linear models do not fit values with large residuals well
Large residuals always need a second look

12. Leverage
Data points whose x-values are far from the mean of x are said to exert leverage of a linear model.
High leverage points pull the line close to them
large effect of the line
can completely determine the slope and the y-intercept
with a high enough leverage their residuals can be deceptively small

13. Leverage Points A linear regression goes through the point
Think of this point as the fulcrum of a lever
The father away a point is from the fulcrum the more leverage it has
High Leverage has the potential to change the regression line

14. High Leverage Points
How to decide if the point will change the regression model
Find the regression model with and without the leverage point
The point is influential if there is a big change in the model

15. Influence Depends on both leverage and residual
high leverage point whose y value is on the line the point is NOT influential
moderate leverage point with a very high residual the point is influential
YOU HAVE TO CHECK THE MODELS!!

16. Unusual Points
Unusual points can sometimes tell us more about a model or data than any other point
A model based on 1 point is unlikely to be helpful to understand the rest of the data
Looking at 1 point against the rest of the data is the best way to understand the point

17. Warning! Do NOT throw away points!!!!
Take out unusual points to look at the model without them
Throwing them away can give us a false sense of how accurate the model is
Look for the unusual points in the scatterplot
they can hide in the residual plots

18. Checking In
Each of these scatterplots shows an unusual point. For each, tell whether the point is a high leverage point, would have a rage residual, or is influential.

19. Causation No matter how strong the association…
No matter how large the R2 value…
No matter how straight the line is…
you can NOT conclude from the regression alone that one variable CAUSES another

20. Lurking Variable Only for observational data
opposed to data from a designed experiment
We can not be sure that a lurking variable is not the cause of a strong or weak association

21. Life Expectancy
The relationship between life expectance (years) and availability of doctors (measured as √(doctors/person)) for the countries of the world

22. Life Expectancy
The relationship between life expectancy (years) and the availability of TVs ((measured as √(TVs/person)) for the countries of the world

23. Means vary less than individual values
Warning!!
Summarized Data: can give a false sense of how good an association is
Means vary less than individual values
Weight (lb) against height (in) for a sample of men. R2 = 41.5%
Mean weight (lb) against height (in). R2 = 80.1%

24. Homework # 10
slope = -.1 for every mph you increase your mpg decreases by .1
y-int = 32 the y-int would be your mpg at 0 mph.
the residuals are negative, so the model is overestimating mpg
27 mpg
predicted = 27.5 mpg + 1 (residual) = 28.5 mpg
strong but not linear
no. the residual plot shows the data is not linear

25. Homework # 11 a high leverage, low residual
no, not influential to the slope
correlation would decrease
the slope would stay about the same because the point is on the line

26. Homework # 11 b
1.high leverage, small residual (remember the point is pulling the line towards it) 2. yes, influential 3. correlation would weekend and become less negative 4. the slope would increase toward 0

27. Homework # 11 c some leverage, high residual slightly influential
correlation would increase because scatter would decrease
slope would increase

28. Homework # 11 d low/no leverage, high residual not influential
correlation would become stronger
slope would increase

29. Homework # 15
stronger, the point has high leverage and is influential so its pulling the line toward it. slope and correlation would both increase
you could take the humans out. Now your data is for non-human mammals.
moderately strong
for every year an animal is expected to line it has to live 15.5 days in its mother before being born
270.4 days

30. Homework # 16
hippos would make the association stronger because it is farther from the pattern
increase
no, there must be a good reason to take out points
yes, the slope changed from 15.5 to that is a big difference

31. Homework # 19
No! There is a high leverage point with point: without point: There is a large change in R2 and the slope

32. Homework #20
only 7% of the variation in time is accounted for by the regression on year
we can’t say with such a bad regression
probably not, the point doesn’t have much leverage
15.9% is better, it appears that swimmers are taking 14 minutes off there time each year

33. Homework # 22
2 subgroups: 1965 – 1985; linear and positive 1994 – 1998; linear and flat (horizontal)

34. Homework # 23
a) the graph is clearly nonlinear, however from about 1972 and on appears to be a positive linear relationship b) In 2010 CPI = $218.60

35. Homework # 24
not including Costa Rica the data has a strong negative linear association
Costa Rica has 25 babies/woman. It has to be a mistake, because it is impossible
r = .814 and R2 = 66.4% without Costa Rica
w/Costa Rica
w/out C.R.
e) the model with C.R. is not appropriate, the residual plot has some pattern. Without C.R. the residual plot has an even amount of scatter with no pattern
f) slope: the life expectancy goes down 4.36 years for every baby a woman has. the y-intercept says a woman with no children should live to be 86.8 years old.
g) there could be a lurking variable also effecting life expectancy