Chapter 3 Association: Contingency, Correlation, and Regression

1. Chapter 3Association: Contingency, Correlation, and Regression • Learn …. How to examine links between two variables

2. Section 3.2 How Can We Explore the Association Between Two Quantitative Variables?

3. Scatterplot • Graphical display of two quantitative variables: • Horizontal Axis: Explanatory variable, x • Vertical Axis: Response variable, y

4. Example: Internet Usage and Gross National Product (GDP)

5. Positive Association • Two quantitative variables, x and y, are said to have a positive association when high values of x tend to occur with high values of y, and when low values of x tend to occur with low values of y

6. Negative Association • Two quantitative variables, x and y, are said to have a negative association when high values of x tend to occur with low values of y, and when low values of x tend to occur with high values of y

7. Example: Did the Butterfly Ballot Cost Al Gore the 2000 Presidential Election?

8. Linear Correlation: r • Measures the strength of the linear association between x and y • A positive r-value indicates a positive association • A negative r-value indicates a negative association • An r-value close to +1 or -1 indicates a strong linear association • An r-value close to 0 indicates a weak association

9. Calculating the correlation, r

10. Example: 100 cars on the lot of a used-car dealership Would you expect a positive association, a negative association or no association between the age of the car and the mileage on the odometer? • Positive association • Negative association • No association

11. Section 3.3 How Can We Predict the Outcome of a Variable?

12. Regression Line • Predicts the value for the response variable, y, as a straight-line function of the value of the explanatory variable, x

13. Example: How Can Anthropologists Predict Height Using Human Remains? • Regression Equation: • is the predicted height and is the length of a femur (thighbone), measured in centimeters

14. Example: How Can Anthropologists Predict Height Using Human Remains? • Use the regression equation to predict the height of a person whose femur length was 50 centimeters

15. Interpreting the y-Intercept • y-Intercept: • the predicted value for y when x = 0 • helps in plotting the line • May not have any interpretative value if no observations had x values near 0

16. Interpreting the Slope • Slope: measures the change in the predicted variable for every unit change in the explanatory variable • Example: A 1 cm increase in femur length results in a 2.4 cm increase in predicted height

17. Slope Values: Positive, Negative, Equal to 0

18. Residuals • Measure the size of the prediction errors • Each observation has a residual • Calculation for each residual:

19. Residuals • A large residual indicates an unusual observation • Large residuals can easily be found by constructing a histogram of the residuals

20. “Least Squares Method” Yields the Regression Line • Residual sum of squares: • The optimal line through the data is the line that minimizes the residual sum of squares

21. Regression Formulas for y-Intercept and Slope • Slope: • Y-Intercept:

22. The Slope and the Correlation • Correlation: • Describes the strength of the association between 2 variables • Does not change when the units of measurement change • It is not necessary to identify which variable is the response and which is the explanatory

23. The Slope and the Correlation • Slope: • Numerical value depends on the units used to measure the variables • Does not tell us whether the association is strong or weak • The two variables must be identified as response and explanatory variables • The regression equation can be used to predict the response variable

24. Section 3.4 What Are Some Cautions in Analyzing Associations?

25. Extrapolation • Extrapolation: Using a regression line to predict y-values for x-values outside the observed range of the data • Riskier the farther we move from the range of the given x-values • There is no guarantee that the relationship will have the same trend outside the range of x-values

26. Regression Outliers • Construct a scatterplot • Search for data points that are well removed from the trend that the rest of the data points follow

27. Influential Observation • An observation that has a large effect on the regression analysis • Two conditions must hold for an observation to be influential: • Its x-value is relatively low or high compared to the rest of the data • It is a regression outlier, falling quite far from the trend that the rest of the data follow

28. Which Regression Outlier is Influential?

29. Example: Does More Education Cause More Crime?

30. Correlation does not Imply Causation • A correlation between x and y means that there is a linear trend that exists between the two variables • A correlation between x and y, does not mean that x causes y

31. Lurking Variable • A lurking variable is a variable, usually unobserved, that influences the association between the variables of primary interest

32. Simpson’s Paradox • The direction of an association between two variables can change after we include a third variable and analyze the data at separate levels of that variable

33. Example: Is Smoking Actually Beneficial to Your Health?

34. Example: Is Smoking Actually Beneficial to Your Health?

35. Example: Is Smoking Actually Beneficial to Your Health?

36. Example: Is Smoking Actually Beneficial to Your Health?

37. Example: Is Smoking Actually Beneficial to Your Health? • An association can look quite different after adjusting for the effect of a third variable by grouping the data according to the values of the third variable

38. Data are available for all fires in Chicago last year on x = number of firefighters at the fires and y = cost of damages due to fire Would you expect the correlation to be negative, zero, or positive? • Negative • Zero • Positive

39. Data are available for all fires in Chicago last year on x = number of firefighters at the fires and y = cost of damages due to fire If the correlation is positive, does this mean that having more firefighters at a fire causes the damages to be worse? • Yes • No

40. Data are available for all fires in Chicago last year on x = number of firefighters at the fires and y = cost of damages due to fire Identify a third variable that could be considered a common cause of x and y: • Distance from the fire station • Intensity of the fire • Time of day that the fire was discovered