The Practice of StatisticsThird Edition

Chapter 4:More about Relationships

between Two Variables

Copyright © 2008 by W. H. Freeman & Company

Daniel S. Yates

Section 4.1 Modeling Nonlinear Data

Linear relationship

x y y/x

0 2.0 5.1

1 7.14.7

2 11.85.3

3 17.15.1

4 22.2

Constant value y/x indicate linear relationship

y = a + bx

1.501

0.21.7

12

12

x

y

xx

yy

Section 4.1 Modeling Nonlinear Data

Exponential relationship

x y yn/yn-1

0 1.0

2.101 2.12.05

2 4.31.84

3 7.92.03

4 16.0

Constant value yn/yn-1 called common ratio indicates exponential relationship

y = abx

0.1

1.2

1

2

y

y

Section 4.1 Modeling Nonlinear Data

Power relationship

x y yn/yn-1 y/x

0 0.0

2.11 2.17.7 14.1

2 16.23.3 37.3

3 53.52.38 73.9

4 127.4

Neither yn/yn-1 or y/x are constant indicates possible power relationship

y = axb

Section 4.1 Modeling Nonlinear Data

•Many important real world situations exhibit exponential or power relationships.

•Exponential and power relationships can be transformed into linear forms so linear regression analysis can be utilized.

• Linear regression only works for linear models. (That sounds obvious, but when you fit a regression, you can’t take it for granted.)

• A curved relationship between two variables might not be apparent when looking at a scatterplot alone, but will be more obvious in a plot of the residuals. – Remember, we want to see “nothing” in a plot of the

residuals.

• No regression analysis is complete without a display of the residuals to check that the linear model is reasonable.

• Residuals often reveal subtleties that were not clear from a plot of the original data.

Section 4.1 Modeling Nonlinear Data

• For exponential relationship - logy is linear with respect to x

• For power relationship - logy is linear with respect to logx

Transforming Exponential Data

Year Cell Phone Users

1986 503

1987 890

1988 1545

1989 2701

1990 4734

1991 8345

1992 14356

1993 25019

1994 45673

Steps

1) Graph data

2) Check common ratio if you suspect exponential relationship

3) Create new list with log of the y values

4) Graph data. X vs log Y

5) Perform linear regression on the transformed data. Store equ. in Y1

6) Transform back by taking raising 10 to both sides of the equation.

7) Graph data to check

XY 2434186.0727179.480 10*10ˆ

• Linear models give a predicted value for each case in the data.

• We cannot assume that a linear relationship in the data exists beyond the range of the data.

• Once we venture into new x territory, such a prediction is called an extrapolation.

Section 4.2 Interpreting Correlation and Regression

• r and LSRL describe only linear relationships

• r and LSRL are strongly influenced by a few extreme observations – influential points

• Always plot your data

• The use of a regression line to predict outside the domain of values of the explanatory variable x is called extrapolation and cannot be trusted.

Lurking Variables and Causation• No matter how strong the association, no matter how large the

R2 value, no matter how straight the line, there is no way to conclude from a regression alone that one variable causes the other.– There’s always the possibility that some third variable is

driving both of the variables you have observed.• With observational data, as opposed to data from a designed

experiment, there is no way to be sure that a lurking variable is not the cause of any apparent association.

Section 4.2 Interpreting Correlation and Regression

• Lurking variables are variables that can influence the relationship of two variables.

• Lurking variables are not measured or even considered.

• Lurking variables can falsely suggest a strong relationship between two variables or even hide a relationship.

Lurking Variables and Causation (cont.)

• The following scatterplot shows that the average life expectancy for a country is related to the number of doctors per person in that country:

Lurking Variables and Causation (cont.)

• This new scatterplot shows that the average life expectancy for a country is related to the number of televisions per person in that country:

Lurking Variables and Causation (cont.)• Since televisions are cheaper than doctors, send TVs to

countries with low life expectancies in order to extend lifetimes. Right?

• How about considering a lurking variable? That makes more sense…– Countries with higher standards of living have both longer

life expectancies and more doctors (and TVs!).– If higher living standards cause changes in these other

variables, improving living standards might be expected to prolong lives and increase the numbers of doctors and TVs.

New England Patriotsy = 0.9716x + 194.48

R2 = 0.3658

0

50

100

150

200

250

300

350

400

0 20 40 60 80 100 120

Jersey Number

Wei

gh

t (lb

s.)

• Strong association of variables x and y can reflect any of the following underlying relationships– Causation - changes in x cause changes in y

– ex. Consuming more calories with no change in physical activity causes weight gain.

– Common response – both x and y respond to some unobserved variable or variables.

– ex. There may be perceived cause and effect between SAT scores and undergrad GPA but both variables are likely responding to student knowledge and ability

– Confounding – the effect on y of the explanatory variable x is mixed up with the effects on y of other lurking variables.

– ex. Minority students have lower ave. SAT scores than whites; but minorities on average grew up in poorer households and attended poorer schools. These socioeconomic variables make cause and effect suspect.

Strong Association

Strong Association

Strong Association

• A carefully designed experiment is the best way to get evidence that x causes y. Lurking variables must be kept under control.

Section 4.3 Relations in Categorical Data

• Categorical data may be inherently categorical such as; sex,race and occupation.

• Categorical data may be created by grouping quantitative data.

• Two way tables – hold categorical data

Income 25-34 35-54 55 + Total

0-19,999

4,506 2,738 3,400 10,644

20,000-39,999

8,724 5,622 4,789 19,135

40,000-49,999

12,643 16,893 7,642 37,178

Total 25,873 25,253 15,831 66,957

Age Groupexample

Row variable – Income Column variable - Age

• The totals of the rows and column are called marginal distributions.

• The totals may be off from the table data due to rounding error.

• The data may also be represented by percents.

• Relationships between categorical data may be calculated from the two way table.

• Data may be represented by a bar chart.

• Conditional distributions satisfy a certain condition on the table.– Ex. Distribution of income level for 25-34 year

olds.– Ex. Distribution of age for people making

$20,000 - $39,999

Example Outcome Hospital A

Hospital B

Total

Died 63

(3%)

16

(2%)

79

Survived 2037

(97%)

784

(98%)

2,821

Total 2,100 800 2,900

Outcome Hospital A

Hospital B

Hospital A

Hospital B

Died 6

(1%)

8

(1.3%)

57

(3.8%)

8

(4%)

Survived 594

(99%)

592

(98.7%)

1,443

(96.2%)

192

(96%)

Total 600 600 1,500 200

Good Condition Poor Condition

Lurking Variable