Statistics and Data Analysis

Part 16: Linear Regression16-1/46

Statistics and Data Analysis

Professor William GreeneStern School of Business

IOMS DepartmentDepartment of Economics


Statistics and Data Analysis

Part 16 – Regression


Sales Population - semilogIncome Demographics

- Box Jenkins


A Regression Analysis that People Really Cared

About

The Year 2000 World Health Report by WHO

http://www.who.int/whr/2000/en


Health Care System Performance


New York Times, Page 1, June 21, 2000


That Number 37 Ranking

What is the source? What is it? Ranking of what? And why are we looking at it in our

class on Statistics and Data Analysis? Interesting It’s an application of regression

analysis.


The Source Behind the News

http://www.who.int/entity/healthinfo/paper30.pdf


What Did They Study?


The standard measure of health care success is Disability

Adjusted Life Expectancy,

DALE


The WHO Researchers

Were Interested in

a Broader Measure

These are the items listed in the NYT editorial.


They Created a Measure COMP = Composite Index

“In order to assess overall efficiency, the first step was to combine the individualattainments on all five goals of the health system into a single number, which we call the composite index. The composite index is a weighted average of the five component goals specified above. First, country attainment on all five indicators (i.e., health, health inequality, responsiveness-level, responsiveness-distribution, and fair-financing) were rescaled restricting them to the [0,1] interval. Then the following weights were used to construct the overall composite measure: 25% for health (DALE), 25% for health inequality, 12.5% for the level of responsiveness, 12.5% for the distribution of responsiveness, and 25% for fairness in financing. These weights are based on a survey carried out by WHO to elicit stated preferences of individuals in their relative valuations of the goals of the health system.”

(From the WHO Technical Report)


Did They Rank Countries by

COMP? Yes, but that was not what

produced the number 37

ranking!


So, What is Going On?

A Model: Health Care Output = a function of Health Care Inputs

OUTPUT = COMP

INPUTS = Health Care Spending and Education of the Population


The WHO COMP Equation

1

22 3

log = = α+β log

+β log +β (log ) i =1,...,191 countries

i i i

i

i i i

COMP Maximum Attainable - InefficiencyHealthExp

Educ Educ + e


Estimated Model

β1

β2

β3

α


The Best a Country Could Do vs. What They Actually Do


The US Ranked 37th!

Countries were ranked by overall efficiency


Linear Regression Correlation (and vs. causality) Examining correlation

Descriptive: Relationship between variables Predictive: Use values of one variable to predict

another. Control: Should a firm increase R&D? Understanding: What is the elasticity of demand

for our product? (Should we raise our price?) The regression relationship


Positive Correlation and Regression

0 1 2 Financial Cases

2.4 -2.3 -2.2 -2.1 -2.0 -1.9 -

Expected Number of Real Estate Cases Given Number of Financial Cases

The “regression of R on F”


Correlation of Home Prices with Other Factors

What explains the pattern? Is the distribution of average listing prices random?


IncomePC

Listin

g

3250030000275002500022500200001750015000

900000

800000

700000

600000

500000

400000

300000

200000

100000

Scatterplot of Listing vs IncomePC


Regression

Modeling and understanding correlation “Change in y” is associated with “change

in x” How do we know this? What can we infer from the observation? Causality and correlation

http://en.wikipedia.org/wiki/Causality and see, esp. “Probabilistic Causation” about halfway down the article.


Correlation – Education and Life Expectancy

EDUC

DALE

121086420

80

70

60

50

40

30

20

01

OECD

Scatterplot of DALE vs EDUC

Causality? Correlation? Does more education make people live longer? A hidden driver of both? (GDPC)

Graph Scatterplots With Groups/ Categorical variable is OECD.


Useful Description(?)

Scatter plot of box office revenues vs. number of “Can’t Wait To See It” votes on Fandango for 62 movies. What do we learn from the figure? Is the “relationship” convincing? Valid? (Real?)


More Movie Madness

Domestic

Over

seas

6005004003002001000

1400

1200

1000

800

600

400

200

0

Scatterplot of Overseas vs Domestic

Domestic

Over

seas

5004003002001000

700

600

500

400

300

200

100

0

Scatterplot of Overseas vs Domestic

Did domestic box office success help to predict foreign box office success?

499 biggest movies up to 2003500 biggest movies up to 2003

Note the influence of an outlier.

Movies.mtp


Average Box Office by Internet Buzz Index

= Average Box Office for Buzz in Interval


Correlation Is there a conditional expectation?

The data suggest that the average of Box Office increases as Buzz increases.

Average Box Office = f(Buzz) is the “Regression of Box Office on Buzz”


Is There Really a Relationship?

BoxOffice is obviously not equal to f(Buzz) for some function. But, they do appear to be “related,” perhaps statistically – that is, stochastically. There is a correlation. The linear regression summarizes it.

A predictor would be Box Office = a + b Buzz. Is b really > 0? What would be implied by b > 0?


Using Regression to Predict

Domestic

Over

seas

6005004003002001000

1250

1000

750

500

250

0

S 73.0041R-Sq 52.2%R-Sq(adj) 52.1%

Regression95% PI

Regression of Foreign Box Office on DomesticOverseas = 6.693 + 1.051 Domestic

Predictor: Overseas = a + b Domestic. The prediction will not be perfect. We construct a range of “uncertainty.”

Stat Regression Fitted Line PlotOptions: Display Prediction Interval

The equation would not predict Titanic.


Effect of an Outlier is to Twist the Regression Line

DomesticBox

Fore

ignB

ox

5004003002001000

700

600

500

400

300

200

100

0

S 66.9303R-Sq 47.4%R-Sq(adj) 47.3%

Regression of Foreign Box Office on DomesticForeignBox = 20.78 + 0.9202 DomesticBox

Domestic

Over

seas

6005004003002001000

1400

1200

1000

800

600

400

200

0

S 73.0041R-Sq 52.2%R-Sq(adj) 52.1%


Without Titanic, slope = 0.9202

With Titanic, slope = 1.051


Least Squares Regression


ab

How to compute the y intercept, a, and the slope, b, in y = a + bx.


Fitting a Line to a Set of Points

Income

PerC

apita

G

27000260002500024000230002200021000

6.4

6.3

6.2

6.1

6.0

5.9

5.8

5.7

5.6

Scatterplot of PerCapitaG vs Income

Choose a and b tominimize the sum of squared residuals

Gauss’s methodof least squares.

N N N2 2 2

i i i i ii 1 i 1 i 1SS [y - a - bx ] [y - (a + bx )] e

Residuals i i i

i i

e y (a bx )ˆ y y

Yi

Xi

Predictionsa + bxi


Computing the Least Squares Parameters a and b

N Ni ii 1 i 1

N2 2x ii 1

Nxy i ii 1

1 1y = y = 20.721 x = x = 0.48242N N

1Var(x) = s = (x x) = 0.02453N-1

1Cov(x,y) = s = (x x)(y y) = 1.784N-1

4 numbers are needed :

xy2x

s 1.784b 72.7181s 0.02453

a y - bx = 20.721- (72.7181)(0.48242) = -14.36


Least Squares Uses Calculus

N 21i iN-1 i=1

2N i i1

N-1 i=1

N1i iN-1 i=1

2N i i1

N-1 i=1

N1i i iN-1 i=1

SS = (y - a - bx )

(y - a -bx )SS =a a

= 2(y - a -bx )(-1) = 0

(y - a -bx )SS =b b

= 2(y - a -bx )(-x ) = 0

N1i=1 i iN-1

N 21i=1 iN-1

The solution is a = y - bx where

Σ (x - x)(y - y)b =

Σ (x - x)


b Measures Covariationb is related to the correlation of x and y.

Predictor Box Office = a + b Buzz.

xyxy

x y

y

x

Cov(x,y)b = Var(x)

Note the numerator of b isthe covariance of x and y.If Cov(x,y) = 0, then b = 0.

Also, since the correlationsCov(x,y)is r ,

s sVar(x)Var(y)s

b Correlation of x and y.s


Is There Really a Statistically Valid Relationship?

We reframe the question.If b = 0, then there is no (linear) relationship. How can we find out if the regression relationship is just a fluke due to a particular observed set of points? To be studied later in the course.

BoxOffice = a + b Cntwait3. Is b really > 0?


Interpreting the Function

EDUC

DALE

121086420

80

70

60

50

40

30

20

S 7.87034R-Sq 59.2%R-Sq(adj) 59.0%

Fitted Line PlotDALE = 35.16 + 3.611 EDUC

ab

a = the life expectancy associated with 0 years of education. No country has 0 average years of education. The regression only applies in the range of experience.b = the increase in life expectancy associated with each additional year of average education.

The range of experience (education)


Correlation and Causality

EDUC

DALE

121086420

80

70

60

50

40

30

20

S 7.87034R-Sq 59.2%R-Sq(adj) 59.0%

Fitted Line PlotDALE = 35.16 + 3.611 EDUC

Does more education make you live longer (on average)?


Causality?

Height (inches) and Income ($/mo.) in first post-MBA Job (men). WSJ, 12/30/86.Ht. Inc. Ht. Inc. Ht. Inc.70 2990 68 2910 75 3150 67 2870 66 2840 68 2860 69 2950 71 3180 69 2930 70 3140 68 3020 76 3210 65 2790 73 3220 71 3180 73 3230 73 3370 66 2670 64 2880 70 3180 69 3050 70 3140 71 3340 65 2750 69 3000 69 2970 67 2960 73 3170 73 3240 70 3050

Estimated Income = -451 + 50.2 Height

Correlation = 0.84 (!)


Using Regression to Predict

Domestic

Over

seas

6005004003002001000

1250

1000

750

500

250

0

S 73.0041R-Sq 52.2%R-Sq(adj) 52.1%

Regression95% PI



Summary Using scatter plots to examine data The linear regression

Description Predict Control Understand

Linear regression computation Computation of slope and constant term Prediction Covariation vs. Causality

Interpretation of the regression line as a conditional expectation

Statistics and Data Analysis

Documents

Transcript of Statistics and Data Analysis