Chapters 5-7 Correlation/Linear Regression

1. Chapters 5-7 Correlation/Linear Regression • Linear Relationships: If the explanatory and response variables show a straight-line pattern, then we say they follow a linear relationship. • Curved relationships and clusters are other forms to watch for.

2. Chapters 5-7 Correlation/Linear Regression • Linear Relationships: If the explanatory and response variables show a straight-line pattern, then we say they follow a linear relationship. • Curved relationships and clusters are other forms to watch for.

3. Chapters 5-7 Correlation/Linear Regression • Direction: If the relationship has a clear direction, we speak of either positive association or negative association. • Positive association: high values of the two variables tend to occur together • Negative association: high values of one variable tend to occur with low values of the other variable.

4. Chapters 5-7 Correlation/Linear Regression • Correlation is a number that determines the strength of a linear relationship between two quantitative variables. • Correlation is always between -1 and 1 inclusive • The sign of a correlation coefficient determines positive/negative association between the variables

5. Chapters 5-7 Correlation/Linear Regression • Strong correlation: If r is between 0.8 and 1 and -0.8 and -1 • Moderate correlation: If r is between 0.5 and 0.8 and -0.8 and -0.5 • Weak correlation: If r is between 0 and 0.5 and -0.5 and 0

6. Chapters 5-7 Correlation/Linear Regression • Correlation does not distinguish between X and Y • Correlation is unitless • Correlation measures the strength of linear relationship between two quantitative variables

7. Chapters 5-7 Correlation/Linear Regression

8. Choose the best description of the scatter plot • Moderate, negative, linear association • Strong, curved, association • Moderate, positive, linear association • Strong, negative, non-linear association • Weak, positive, linear association

9. Which of the following values is most likely to represent the correlation coefficient for the data shown in this scatterplot? • r = -0.67 • r = -0.10 • r = 0.71 • r = 0.96 • r = 1.00

10. Which of the following values is most likely to represent the correlation coefficient for the data shown in this scatterplot? • r = -0.67 • r = -0.10 • r = 0.71 • r = 0.96 • r = 1.00

11. Which of the following values is most likely to represent the correlation coefficient for the data shown in this scatterplot? • r = -0.67 • r = -0.10 • r = 0.71 • r = 0.96 • r = 1.00

12. Cautions about Correlation • It should only be used • To describe the relationship between 2 QUANTITATIVE variables • When the association is “linear enough” • When there are no outliers • Correlation does NOT imply causation

13. A teacher at an elementary school measures the heights of children on the playground and then makes a scatter plot of the children’s heights and reading test scores. The data meet the conditions for correlation so she calculates r = .79. Which conclusion is most accurate? Being taller causes students to read better Being shorter causes students to read better Taller students tend to have better reading scores Shorter students tend to have better reading scores

14. Chapters 5-7 Correlation/Linear Regression • Easiest to understand and analyze • Relationships are often linear • Variables with non-linear relationship can often be transformed into linear relationship through an appropriate transformation • Even when a relationship is non-linear, a linear model may provide an accurate approximation for a limited range of values. • Strength: The strength of a linear relationship is determined by how close the points in the scatterplot lie to a straight line

15. Least Square Regression Line - Calculations

16. Chapters 5-7 Correlation/Linear Regression • Not all data fall on a straight line! • Residual = Data – Model or • Residual = Observed Y – Predicted y

17. Chapters 5-7 Correlation/Linear Regression Example X= Fat Y= Calories 19 410 31 580 34 590 35 570 39 640 39 680 43 660

18. Chapters 5-7 Correlation/Linear Regression

19. Chapters 5-7 Correlation/Linear Regression

20. Chapters 5-7 Correlation/Linear Regression • S = 27.3340 R-Sq = 92.3% R-Sq(adj) = 90.7% Residual Plot

21. Chapters 5-7 Correlation/Linear Regression • Extrapolation: Reaching beyond the data • Outliers: Regression models are sensitive to outliers • Leverage: An unusual data point whose x value is far from the mean of the x values • A point with high leverage has the potential to change the regression line.

22. Chapters 5-7 Correlation/Linear Regression • Influential: A point is influential if omitting it from the analysis gives a very different model. • Influence depends on leverage and residual • Lurking variables: A variable that is not included in the construction of the linear model/study.

23. Chapters 5-7 Correlation/Linear Regression • Lurking variables may influence correlation and regression models. • Association is not causations!!

24. Summary • r is a number between -1 and 1 • r = 1 or r = -1 indicates a perfect correlation case where all data points lie on a straight line • r > 0 indicates positive association • r < 0 indicates negative association • r value does not change when units of measurement are changed (correlation has no units!) • Correlation treats X and Y symmetrically. The correlation of X with Y is the same as the correlation of Y with X

25. Summary • Quantitative variable condition: Do not apply correlation to categorical variables • Correlation can be misleading if the relationship is not linear • Outliers distort correlation dramatically. Report correlation with/without outliers.

26. More Examples for Checking Linear Enough ConditionAll four data sets have r = .82

27. In which case is a linear model appropriate? B. A. C. D.

28. A. Linear model appropriate; residual plot shows no pattern B. Linear model not appropriate; clear pattern of residuals

29. C. Graph has an outlier; outlier is clear on the residual plot D. Linear model not appropriate; clear pattern of residuals

30. Calculating r with the TI-83/84 • The first time you do this: • Press 2nd, CATALOG (above 0) • Scroll down to DiagnosticOn • Press ENTER, ENTER • Read “Done” • Your calculator will remember this setting even when turned off

31. Calculating r with the TI-83/84 • Press STAT, ENTER • If there are old values in L1: • Highlight L1, press CLEAR, then ENTER • If there are old values in L2: • Highlight L2, press CLEAR, then ENTER • Enter predictor (x) values in L1 • Enter response (y) values in L2 • Pairs must line up • There must be the same number of predictor and response values

32. Calculating r with the TI-83/84 • Press STAT, > (to CALC) • Scroll down to LinReg(ax+b), press ENTER, ENTER • Read r at bottom of screen

33. Re-Expression with the TI-83/84 • Most common re-expressions are built in. • To see what’s available, try • STAT • CALC • Scroll down to see • 5:QuadReg • 6:CubicReg • 9:LnReg • 0:ExpReg • A:PwrReg

34. Example • X: Age in months • Y: Height in inches • X: 18 19 20 21 22 23 24 • Y: 29.9 30.3 30.7 31 31.38 31.45 31.9

35. Chapters 5-7 Correlation/Linear Regression • Linear Model: Height = 24.212 +.321 * Age Correlation: r = .992 Examples • Age = 24 months, Observed Height = 31.9 • Predicted Height = 31.916 • Residual = 31.9 – 31.916 = .016

36. Chapters 5-7 Correlation/Linear Regression • Age = 20 years (20*12 = 240) • Predicted Height ~ 8.5 ft!! • Residual = BIG! • Be aware of Extrapolation!

37. Example 4.Relationship between calories and sugar content: A researcher tracked the sugar content and calorie of 15 baked goods and found the following information: Average sugar content: 7.0 grams Standard deviation of sugar content: 4.4 grams Average calories: 107.0 grams Standard deviation of calories: 19.5 grams Correlation between sugar content and calories: 0.564

38. Solution to Example a) Find a linear model that describes this example: b_{1}=r S_{y}/S_{x} = 0.564*19.5/4.4 = 2.5 calories per gram of sugar b_{0}= mean of (Y) –b{1}mean of (X) = 107 -2.50*7 = 89.5 Linear Model: y = b_{0}+b_{1}x y= 89.5 + 2.5x or better calories = 89.5 +2.50* sugar b) How many calories are there in a muffin with 6.5 grams of sugar? calories = 89.5 +2.50* 6.5 = 105.75

39. Chapters 5-7 Correlation/Linear Regression: Re-expressing Data • Example: The data shows the number of academic journals published on the Internet and during the last decade.

40. Chapters 5-7 Correlation/Linear Regression: Re-expressing Data

41. Chapters 5-7 Correlation/Linear Regression: Re-expressing Data • Re-express data to linearize:

42. Chapters 5-7 Correlation/Linear Regression: Re-expressing Data

44. Chapter 10 Re-expressing Data • Least Square Regression Line has the following equation: Log(journals) = 1.22 + 0.346 * Year Problem: How many journals will be published online in year 2000?

45. Chapter 10 Re-expressing Data Answer Log(journals) = 1.22+ 0.346*9 =4.334 Answer: 21577.44 (10^(4.334))

46. Chapter 10 Re-expressing Data Why Re-expressing data? • Make a distribution of a variable more symmetric • Make the spread of several groups more alike, even if their centers differ • Make the form of a scatterplot more nearly linear • Make the scatter in a scatterplot spreadout more evenly rather than thickening at one end.

47. Chapter 10 Re-expressing Data The Ladder of Powers: Power 2: the square of the data values y^2 • Try this for unimodal distributions that are skewed to the left. Power 1: No change at all Power ½: the square root of the data values Y^(1/2) • Try this for counted data Power 0: the logarithm of the data values y • Try this for measurements that cannot be negative • Especially those that grow by percentage increases • Salries and populations are good examples.