Chapter 4: More on Two-Variable (Bivariate) Data

1. Chapter 4:More on Two-Variable (Bivariate) Data

2. 4.1 Transforming Relationships • Animal’s Brain Weight vs. Weight of Body Outliers r=.86

3. Drop Outliers Logarithm r=.50

4. Plot Logarithm vs. Logarithm r=.96 The vertical spread about the LSRL is similar everywhere, so the predictions of brain weight from body weight will be pretty precise (high r2) – in LOG SCALE

5. Working with a function of our original measurements can greatly simplify statistical analysis. • Transforming- How?

6. Recall… • Chapter 1 we did Linear Transformations • Took a set of data and transformed it linearly • Called: SHIFTING C to F Meters to Miles

7. A Linear Transformation CANNOT make a curved relationship between 2 variables “straight” • Resort to common non-linear functions like the logarithm, positive & negative powers • We can transform either one of the explanatory/response variables OR BOTH • when we do we will call the variable “t”

8. Real World Example: We measure fuel consumption of a car in miles per gallon Engineers measure it in gallons per mile (how many gallons of fuel the car needs to travel 1 mile) Reciprocal Transformation: 1/f(t) My Car- 25 miles per gallon 1/25=.04 gallons per mile

9. Monotonic Function • A monotonic function f(t) moves in one direction as its argument “t” increases • Monotonic Increasing • Monotonic Decreasing

10. Monotonic Increasing: a + bt slope b>0 Positive “t”

11. Monotonic Decreasing: a + bt slope b<0 Positive “t”

12. Nonlinear monotonic transformations change data enough to change form or relations between 2 variables, yet preserve order and allow recovery of original data.

13. Strategy: • If the variable that you want to transform has values that are 0 or negative apply linear transformation (add a constant) to get all positive. • Choose power or logarithmic transformation that approximately straightens the scatterplot.

14. Ladder of Power Transformations: Power Function: tP

15. Power Functions: • Monotonic Power Function For t > 0…. 1. Positive p – are monotonic increasing 2. Negative p – are monotonic decreasing

16. Monotonic Decreasing- Hard to interpret because reversed order of original data point • We want to make all tP therefore monotonic increasing. We can apply a LINEAR TRANSFORMATION

17. Linear Transformation:

19. This is a line This is log t

20. Concavity of Power Functions: P is greater than 1 = - Push out right tail & pull in left tail - Gets stronger as power p moves up away from 1 P is less than 1 = - Push out left tail & pull in right tail - Gets stronger as power p moves down away from 1

21. Country’s GDB vs. Life Expectancy

22. P=

23. P=

24. P= Use

26. How do you know what transformation will make the scatterplot straight? ** DO NOT just push buttons!! ** • We will develop methods of selection 1. Logarithmic Transformation 2. Power Transformation

27. 1. Logarithmic Transformations

28. Exponential Growth A variable grows… Linearly: Exponentially:

29. The King’s Chess Board… King’s Offer: 1,000,000 grains - 30 days Wise Man: 1 grain per day and double for 30 days

30. Cell Phone Growth

31. Suspect Exponential Growth… • Calculate Ratios of Consecutive Terms - IF approximately the same… continue

32. Suspect Exponential Growth… 2. Apply a Transformation that: a. Transforms exponential growth into linear growth b. Transforms non-exponential growth into non-linear growth

33. Logarithm Review… • log(AB)= • log(A/B)= • logXp =

34. The Transformation… We hypothesize an exponential model of the form y=abx To gain linearity, use the (x, log(y)) transformation Form? –

35. When our data is growing exponentially… if we plot the log of y versus x, we should observe a straight line for the transformed data!

37. LOG (Y) = -263 + 0.134 (year) R-sq = 98.2%

38. Eliminate first 4 years & perform regression

39. LOG (New Y) = -189 + 0.0970(New X) R-sq = 99.99%

40. Predictions in Exponential Growth Model • Regression is often used for predictions • In exponential growth, ________ rather than actual values follow a linear pattern • To make a prediction of Exp. Growth we must thus “undo” the logarithmic transformation. • The inverse operation of a logarithm is _____________________

41. LOG (New Y) = -189 + 0.0970(New X) R-sq = 99.99% Predict the number of cell phone users in 2000.

42. If a variable grows exponentially… its ___________ grow linearly! In other words… if (x, y) is exponential, then (x, log(y)) is linear!

43. Read and do Technology Toolbox- Page 210-211 on your own!!

44. 2. Power Transformations

45. Power Law Model Example: Pizza Shop- order pizza by diameter 10 inch 12 inch 14 inch Amount you get depends on the area of the pizza Area circle = pi times the square of the radius Power Law Model

46. Power Laws • We expect area to go up with the square of dimension • We expect volume to go up with the cube of a dimension Real Examples: Many Characteristics of Living Things Kleiber’s Law- The rate at which animals use energy goes up as the ¾ power of their body weight (works from bacteria to whales).

47. Power Laws Become Linear • Exponential growth becomes linear when we apply the logarithm to the response variable (y). • Power Laws become linear when we apply the logarithm transformation to BOTH variables.

48. To Achieve Linearity… • The power law model is • Take the logarithm of both sides of equation (this straightens scatterplot) • Power p in the power law becomes the slope of the straight line that links log(y) to log(x) • Undo transformation to make prediction

49. Fish Example… Read Example 4.9 page 216 Model: weight = a x length3

50. Log (weight) = log a + [3x log(length)] • Yes appears very linear- perform LSRL on [log(length), log(weight)]