1 Chapter 4: More on Two-Variable Data 4.1Transforming Relationships 4.2Cautions 4.3Relations in Categorical Data

2 Example Year Cell Phone Users (thousands) 5,283 16,009 24,134 33,786 44,043 55,312 69,209 86,047

3 Scatterplot for Cell Phone Example

4 Residuals Plot

5 What’s going on here? Do the data (y) increase by a constant amount each year? –This would suggest a linear model. Or, do the data increase by a fixed percentage each year? That is, can you multiply the y-value by a fixed number to get the next year’s number, and then multiply that number by the fixed number to get the following year’s number? –This would suggest an exponential model.

6 Transformation of the Variables The next step is to apply a mathematical transformation that changes exponential growth into linear growth. –The transformation that can help here is to take the logarithm of the y-variable, then re-plot and re-calculate the LSR.

7 New LSR, with Transformed y Residuals Plot

8 We are dealing with a transformed y-value! Model: In order to use the model for prediction, we must “undo” the logarithm transformation to return to the original units of measurement. –How do we do this? Now use the new model to predict cell phone subscribers for 2000.

9 How do we predict for year 2000?

10 Plotting our original data vs. our exponential model …

11 Problem 4.6, p. 212 Problem 4.11, p. 213 Reading: pp Homework

12 Power Law Models General form of a power law model: Biologists have found that many characteristics of living things are described quite closely by power laws. –For example, the rate at which animals use energy goes up as the ¾ power of their body weight (Kleiber’s Law).

13 LSR and Power Law Models As we saw in the last section, exponential growth models become linear when we apply the logarithm transformation to the response variable y. Power law models become linear when we apply the logarithm transformation to both variables, x and y.

14 Log Transformations for Power Law Models Looking carefully at the last equation, the power (p) becomes the slope of the straight line that links log y to log x. –We can estimate what power (p) the law involves by regressing log y on log x and using the slope of the regression line to estimate the power.

15 Problem 4.13, p 219

16 Problem 4.13, p. 219 Log of Both Variables

17 Residuals Analysis (Transformed Data)

18 Undoing the Transformation Let’s do the math to see what we need:

19 Predicting Lifespan for Humans

20 HW Problem: –4.14, p. 220

21 Warm-Up Problem 4.25, pp Create appropriate model Predict seed count for tree with seed weight of 1,000 mg.

Log of both L1 and L2 Axes off to see trend Y2 vs. original data I. II. III. IV. V.

Cautions about Correlation and Regression The correlation (r) and the LSR line are not resistant. As we have seen, extrapolation is often dangerous. –Predicting past the x-variable for which the model was developed.

24 The French Paradox The paradox refers to the fact that the French have long had low rates of heart disease (Japan is the only developed country with a lower rate), despite a diet relatively rich in saturated animal fats. The French propensity to drink wine the way some Americans guzzle soft drinks has been cited as a likely explanation of the paradox, since numerous studies have indicated that alcohol consumed in moderation helps to prevent atherosclerosis, or accumulation of fatty deposits in arteries, which is the underlying cause of most heart attacks. + from NY Times article

25 Lurking Variables As we discussed in the example of amount of wine consumed vs. number of incidents of heart disease, there can be other variables not measured in a correlation study that may influence the interpretation of relationships among those variables. –Lurking Variables It is possible to show, for example, that there is a high correlation between shoe size and intelligence for a group of children varying in age from, say, 4 to 15. –What is the lurking variable? To control for age, we can calculate the correlation between shoe size and IQ for each of the different ages. –Age 4, 5, 6, …

26 Correlation Between Shoe Size and IQ? (Common Response) Age Shoe Size IQ

27 See Figure 4.18, p. 227

28 Lurking Variables That Change Over Time Many lurking variables change systematically over time. One useful method for detecting lurking variables is to plot both the response variable and the regression residuals against the time order of the observations (whenever the time order is available). See Example 4.12, p. 228

29

30 Using Averaged Data Be careful when applying the results of a study that uses averages to individuals. Problem 4.31, p. 231

31 Causation Simply put, a strong correlation between two variables says nothing about one variable causing the other. One variable may in fact cause the other to change, but a correlation or LSR line cannot tell us that. –More investigation is needed! A designed study with proper experimental controls should be used.

32 Figure 4.22, p. 232 Causation Common Response Confounding

33 Confounding The effects of two variables on a response variable are said to be confounded when they cannot be distinguished from one another. –Definition: Two or more variables that might have caused an effect were simultaneously present, so that we do not know to which to attribute the effect. –See 1, Example 4.13 (p. 232), and explanation, p. 233, top of p Does this mean that we cannot ever suggest causation? –Read the two paragraphs on p. 235 (establishing causation).

34 Causation Example 4.14, p. 232 –Numbers 1 and 2 (p. 233)

35 Common Response Example 4.15, p. 233

36 Homework Reading through p. 240

37 Problems Problems on p. 237: –4.33, 4.34, , p.257

38 Problem 4.73, p. 257 Power law model might best fit, so take log of L1 and L2. Plot below of L3 and L4.

, cont. The pendulum period is proportional to the square root of its length.

Relations in Categorical Variables There are many relationships of interest to us that cannot be described by using correlation and LSR techniques. –Recall that correlation and LSR require both variables to be quantitative. Often, we want to study the relationship between two variables that are inherently categorical.

41 Two-Way Table (Ex. 4.19, p. 241) Age Group Education25 to 3435 to 5455+Total Did not complete HS 4,4749,15514,22427,853 Complete HS11,54626,48120,06058, yrs college 10,70022,61811,12744, yrs college11,06623,18310,59644,845 Total37,78681,43556,008175,230 cell

42 Two-Way Table The row variable is level of education. –In this study, is level of education the explanatory or response variable? The column variable is age. –Explanatory or response? Marginal distributions: –The distributions of education alone and age alone are called marginal distributions because their totals are in the margins: Education at the right, and age at the bottom.

43 Marginal Distributions It is often advantageous to display the marginal distribution in percents instead of raw numbers.

44 Conditional Distributions The previous graph looked at the breakdown of education levels for the entire population. Many times, however, we are looking for breakdowns (i.e., distributions) for a certain group within the population. –For example, of those people with 4+ years of college, look at the distribution across age groups. –Let’s complete a bar graph for this comparison. –This is a conditional distribution.

45 One Conditional Distribution for Example 4.19

46 Different Question What proportion of each age group received 4+ years of college education?

47 Read paragraph at the bottom of page 248.

48 One set of conditional distributions: Figure 4.27, p. 248

49 Problems 4.53, p , p. 251

50 Graph for Problem 4.59

51 Homework Read through the end of the chapter. Be sure you understand “Simpson’s Paradox.” Problem: –4.62, p. 253

52 Simpson’s Paradox Problem 4.60, p. 251 Statement of the Paradox: –Simpson’s paradox refers to the reversal of the direction of a comparison or association when data from several groups are combined to form a single group.

53 Practice/Review Problems Problem: –4.68, p. 254 –4.72 (parts a-c), p. 257

54