1. Statistics 200 Lecture #6 Thursday, September 8, 2016
Textbook: Sections 3.3 through 3.5
Objectives (for two quantitative variables and their relationship):
• Define and interpret residual
• Define and interpret correlation coefficient
• Interpret the square of the correlation coefficient (r-squared)
• Recognize various pitfalls in using regression
– Extrapolation is dangerous.
– Outliers can have a huge effect.
– Interpreting a linear relationship as causation is dangerous.

2. For which fitted line plot(s) does the
y-intercept have a logical interpretation?
line plot 1
line plot 2
line plot 3
line plots 1 & 3
line plots 1, 2, & 3

3. Residual: Deviation of Point from the Regression Line
= observed - predicted

4. Measuring strength and direction
We see a linear pattern in relationships so often that we use a statistic to characterize the strength and direction of the relationship.

5. Measuring strength and direction
The strength of correlation is determined by the ________of the points to a straight line.
The direction of correlation is determined by whether one variable generally increases or decreases when the other variable increases.
closeness

6. Measuring strength and direction
As a note, correlation can only be used when talking about linear (straight line) relationships.
Sometimes there definitely is a relationship, but the correlation may be zero because it isn’t a linear relationship.

7. Measuring strength and direction
Correlation is represented by the letter r
Correlation is sometimes called the Pearson product moment correlation, or the correlation coefficient.

8. Measuring strength and direction
For correlation: It doesn’t matter which variable you treat as the response and which variable you treat as the explanatory variable
For the regression equation, it DOES matter which variable you treat as the response and which you treat as the explanatory.

9. How is correlation (r) calculated?
The formula for calculating correlation looks quite complicated, but it is more easily explained in terms of standardized scores (z-scores)
Approximately, the correlation is the average product of standardized scores (z-scores) for variables x and y.

10. Interpreting correlation
Correlation values are always between __ and __
The further the correlation is from zero, the _________ the relationship
Whether the correlation is positive or negative indicates the ________ of the relationship. –1 1
stronger
direction

11. Interpreting correlation
If correlation is equal to 0, there is no linear relationship between the variables.
This also means that the best line to fit the relationship is exactly horizontal, such that y does not change with x.
If the correlation is –1 or 1, then all of the data points fall exactly on a line.

12. Clicker Question For the top scatterplot… A. r = .721 B. r = -0.193
C. r = -.927
For the bottom scatterplot…
A. r = 0.656
B. r =
C. R = 1.00

13. Related quantity: Squared correlation
The squared value for the correlation (r2) is often used to describe the strength of the linear relationship.
Since the r2 value is simply r squared, the possible values for r2 range from 0 to 1.

14. Squared Correlation (r2)
friendly quantity
Interpretation:
quantifies the amount of ________ in the ______
variable that can be __________ by the _________variable
possible values ____ to ____ (0% to 100%)
as it increases in value, the _________ the points are to the regression line.
variation
response
explained
explanatory 1
closer

15. Example: Squared Correlation
Squared Correlation Interpretation:
________ of the variation in ____can be explained by ____.
Pearson correlation of x and y = 0.844
Example: Squared Correlation
The regression equation is
y = x
S = 3.6
Pearson correlation of x and y = 0.844
r2 = ______________
(.844)*(.844) = .713
71.3% y x

16. Issues with regression
Several problems can arise when you are analyzing the relationship between two quantitative variables:
Extrapolation
Influential Outliers
Curvilinear Data
Combining Groups Inappropriately

17. Extrapolation
Extrapolation is when you use the regression equation to predict values _________the range of observed data.
For example, let’s look at height and weight data.
outside

18. Extrapolation
Here, we use height to predict weight, using a sample of adults.
Sample intercept is
– No logical interpretation
Weight = – Height

19. Extrapolation
Here, we use height to predict weight, using a sample of adults.
Sample intercept is
– No logical interpretation
Weight = – Height
Sample slope is
For every increase of 1”, predicted weight increases by lbs.

20. Extrapolation
Here, we use height to predict weight, using a sample of adults.
Sample intercept is
– No logical interpretation
Weight = – Height
Sample slope is
For every increase of 1”, predicted weight increases by lbs.
R-sq is 0.43 = 43%.
Height explains 43% of the variation observed in weight.

21. Extrapolation
This regression equation works fairly well for adults, but what happens for a child’s height?
If child is 40” tall,
use the equation to
predict their weight.
Weight = – Height
= – × 40
= –
= 11.1 pounds
Yikes! We can’t trust this value because we extrapolated outside the range of observed values to get it.

22. Influential Outliers - example
Consider this scatterplot and regression equation
R-sq is 0.
No linear relationship between variables

23. Influential Outliers - example
Consider this scatterplot and regression equation
R-sq is 0.
No relationship between variables
Slope of line is basically 0

24. Influential Outliers - example
Now we add a single influential outlier
R-sq is 8.5%. Possible
linear relationship between variables

25. Influential Outliers - example
Now we add a single influential outlier
R-sq is 8.5%. Possible
linear relationship between variables
Slope of line is
–.2745 : not zero!

26. Moral of the example
Influential outliers can have a huge effect on the relationship.
In some cases, there is no relationship at all unless you include one data point.
In cases like these, it may be best to remove the outlier before fitting a line to the data and making assumptions.

27. Curvilinear data
Be careful using linear regression on a curvilinear dataset.
Problem: If you use the equation, you will end up making incorrect estimates for the data.
Example: United States population is plotted by year.
Population = – Year
If we try to calculate the population for 2009, we get:
– × 2009 = mil
This was our population around 1999.

28. Combining groups inappropriately
Plot of “fastest speed ever reached in a car and height”.
What? Taller people speed more?!
Wait, maybe we combined some groups that we should have kept separate.
Terminology: Here, sex is a confounding variable!

29. Combining groups inappropriately
Separated by sex (M, F), we see that there is actually no apparent relationship between height and ‘fastest speed ever driven’.

30. Interpretations of observed association
There are four main ways for you to interpret an observed association between two quantitative variables:
There is causation
There may be causation
There is no causation
The response variable is causing a change in the explanatory variable (reverse causation)

31. Important note...
A strong correlation does not necessarily mean there is a causal relationship between two variables.
Most correlations come from observational studies and we can’t claim causation from observational studies!
All a strong correlation means is that there is an association between the variables.

32. Order these scatterplots in increasing order of r-squaredk j
Order these scatterplots in increasing order of r-squared m
A. j < k < m
B. m < j < k
C. j < m < k
D. k < m < j

33. Review: If you understood today’s lecture, you should be able to solve
3.27, 3.29, 3.33, 3.37, 3.39, 3.41, 3.45, 3.63, 3.75, 3.77.
Recall objectives (for two quantitative variables):
• Define and interpret residual
• Define and interpret correlation coefficient
• Interpret the square of the correlation coefficient (r-squared)
• Recognize various pitfalls in using regression
– Extrapolation is dangerous.
– Outliers can have a huge effect.
– Interpreting a linear relationship as causation is dangerous.