statistics-linear regression and correlation analysis

8/3/2019 Statistics-Linear Regression and Correlation Analysis

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Linear Regressionand

Correlation Analysis

Prof Rushen Chahal


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Chapter Goals

To understand the methods for displaying and describingrelationship among two variables

Models

Linear regression

Correlations Frequency tables

Methods for Studying Relationships Graphical

Scatter plots

Line plots 3-D plots


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Two Quantitative Variables

The response variable, also called thedependent variable, is the variable we want

to predict,and is usually denoted by y.

The explanatory variable, also called theindependent variable, is the variable thatattempts to explain the response, and isdenoted byx.


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Scatter Plots and Correlation

A scatter plot (or scatter diagram) is usedto show the relationship between twovariables

Correlation analysis is used to measurestrength of the association (linearrelationship) between two variables

Only concerned with strength of therelationship

No causal effect is implied


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Example

The following graphshows the scatter plotofExam 1 score (x)

and Exam 2 score (y)for 354 students in aclass.

Is there a relationship

between x and y?


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Scatter Plot Examples

y

x

y

x

y

y

x

x

Linear relationships Curvilinear relationships


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Scatter Plot Examples

y

x

y

x

No relationship(continued)


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Correlation Coefficient

The population correlation coefficient (rho) measures the strength of theassociation between the variables

The sample correlation coefficient ris an estimate of and is used tomeasure the strength of the linearrelationship in the sample observations

(continued)


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Features of and r

Unit free

Range between -1 and 1

The closer to -1, the stronger thenegative linear relationship

The closer to 1, the stronger the positivelinear relationship

The closer to 0, the weaker the linearrelationship


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Examples of Approximate r Values

yy

x

y

x

x

y

x

y

x

Tag with appropriate value: -1, -.6, 0, +.3, 1


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Earlier Example

Correlations

.

.

.

.

earson orre a on

g. -ta e )

earson orre at on

g. -ta e )

Exam1

Exam2

orrea ton s sgn cant at t e . eve.


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Questions?

What kind of relationship would you expectin the following situations:

Age (in years) of a car, and its price.

Number of calories consumed per day andweight.

Height and IQ of a person.


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Exercise

Identify the two variables that vary and decidewhich should be the independent variable andwhich should be the dependent variable.

Sketch a graph that you think best representsthe relationship between the two variables.

1. The size of a persons vocabulary over his orher lifetime.

2. The distance from the ceiling to the tip of theminute hand of a clock hung on the wall.


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Introduction to Regression Analysis

Regression analysis is used to:

Predict the value of a dependent variable basedon the value of at least one independent variable.

Explain the impact of changes in an independentvariable on the dependent variable.

Dependent variable: the variable we wish to

explain.Independent variable: the variable used to

explain the dependent variable.


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Simple Linear Regression

Model Only one independent variable, x.

Relationship between x and y isdescribed by a linear function.

Changes in y are assumed to be

caused by changes in x.


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Types of Regression Models

Positive Linear Relationship

Negative Linear Relationship

Relationship NOT

Linear

No Relationship


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ni ,...,2,1; !! ii10i XY

Linear component

Population Linear Regression

The population regression model:

Populationy intercept

PopulationSlopeCoefficient

RandomError term,

or residual

Dependent

Variable

IndependentVariable

Random Errorcomponent


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Linear Regression Assumptions

The Error terms i, i=1, 2. , n areindependent and i ~ Normal (0,

2).

The Error terms i, i=1, 2. , n have constant

variance 2

. The underlying relationship between the X

variable and the Y variable is linear.


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Population Linear Regression(continued)

Random Errorfor this xi value

Y

X

Observed Valueof y

i

for xi

Predicted Valueof yi for xi

ii10i XY !

xi

Slope = 1

Y-Intercept = 0

i


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i10i XbbY !

The sample regression line provides an estimateof the population regression function.

Estimated Regression Model

Estimate ofthe regressiony-intercept

Estimate of theregression slopeEstimated

(or predicted)y value

Independent

variable

The individual random error terms ei have a mean of zero

The Regression Function: iiii XEXYE 1010 )()( FFIFF !!


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Earlier Example


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ResidualA residualis the difference between the observed

response yiand the predicted response i. Thus, foreach pair of observations (xi,yi), the i

th residual is

ei= yi i= yi (b0 + b1xi)

Least Squares Criterion b0 and b1 are obtained by finding the

values of b0 and b1 that minimize the sum

of the squared residuals.

2

10

22

x))b(b(y

)y(ye

!

!


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b0 is the estimated average value of

y when the value of x is zero.

b1 is the estimated change in the

average value of y as a result of a

one-unit change in x.

Interpretation of theSlope and the Intercept


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The Least Squares Equation

The formulas for b1 and b0 are:

algebraic equivalent:

!

nxx

n

yxxy

b2

2

1

)(

!

21 )(

))((

xx

yyxxb

xbyb 10 !and


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Finding the Least Squares Equation

The coefficients b0 and b1 willusually be found using computersoftware, such as Excel, Minitab, or

SPSS.

Other regression measures will alsobe computed as part of computer-based regression analysis.


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Simple Linear RegressionExample

A real estate agent wishes to examine therelationship between the selling price of a

home and its size (measured in square feet)

A random sample of 10 houses is selected

Dependent variable (y) = house price in$1000s

Independent variable (x) = square feet


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Sample Data for House PriceModel

House Price in $1000s(y)

Square Feet(x)

245 1400

312 1600279 1700

308 1875

199 1100

219 1550405 2350

324 2450

319 1425

255 1700


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Model Summary

. a . . .

Adjusted

Std. Error of

re c ors: ons an , quare ee.

Coefficientsa

. . . .

. . . . .

ons an

quare ee

1

.

Unstandardized Standardized

.

epen en ar a e: ouse r ce.

SPSS Output

The regression equation is:

feet)(square0.11098.248pricehouse !


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0

50

100

150

200

250

300

350

400

450

0 500 1000 1500 2000 2500 3000

Square Feet

HousePrice($1000s)

Graphical Presentation

House price model: scatter plot andregression line


Slope= 0.110

Intercept

= 98.248


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Interpretation of the Intercept, b0

b0 is the estimated average value of Y when

the value of X is zero (if x = 0 is in the rangeof observed x values)

Here, no houses had 0 square feet, so

b0

= 98.24833 just indicates that, for houses

within the range of sizes observed,

$98,248.33 is the portion of the house price

not explained by square feet



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Interpretation of theSlope Coefficient, b1

b1 measures the estimated change in

the average value of Y as a result of

a one-unit change in X

Here, b1 = .10977

tells us that the averagevalue of a house increases by .10977($1000)

= $109.77, on average, for each additional

one square foot of size



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Least Squares Regression Properties

The sum of the residuals from the leastsquares regression line is 0 ( ).

The sum of the squared residuals is a

minimum (minimized ). The simple regression line always passes

through the mean of the y variable and the

mean of the x variable.

The least squares coefficients b0 and b1 are

unbiased estimates of 0 and 1.

0)( ! yy

2)( yy


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Exercise

The growth of children from early childhood throughadolescence generally follows a linear pattern. Data onthe heights of female Americans during childhood, fromfour to nine years old, were compiled and the leastsquares regression line was obtained as = 32 + 2.4xwhere is the predicted height in inches, and xis age in

years. Interpret the value of the estimated slope b1 = 2. 4. Would interpretation of the value of the estimated

y-intercept, b0 = 32, make sense here? What would you predict the height to be for a female

American at 8 years old? What would you predict the height to be for a femaleAmerican at 25 years old? How does the quality of thisanswer compare to the previous question?


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The coefficient of determination is theportion of the total variation in thedependent variable that is explained byvariation in the independent variable.

The coefficient of determination is alsocalled R-squared and is denoted as R2

Coefficient of Determination, R2

1R02ee


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Coefficient of Determination, R2

(continued)

Note: In the single independent variable case, the coefficientof determination is

where:R2 = Coefficient of determinationr = Simple correlation coefficient

22

rR !


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Examples of ApproximateR2 Values

y

x

y

x

y

x

y

x


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Examples of ApproximateR2 Values

R2 = 0

No linear relationshipbetween x and y:

The value of Y does not

depend on x. (None of thevariation in y is explainedby variation in x)

y

xR2 = 0


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SPSS OutputModel Summary

. a . . .1

Adjusted

Std. Error of

re c ors: ons an , quare ee.

ANOVAb

. . . . a

. .

.

egress on

es ua

ota

1

Sum of

.

re c ors: ons an , quare eea.

e en en ar a e: ouse r ce.

Coefficientsa

. . . .

. . . . .

onstant

quare eet

1.


.



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Standard Error of Estimate

The standard deviation of the variation ofobservations around the regression line iscalled the standard error of estimate

The standard error of the regressionslope coefficient (b1) is given by sb1

Is


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Model Summary

. a . . .1

Adjusted

Std. Error of

re c ors: ons an , quare eea.

Coefficientsa

. . . .

. . . . .

(Constant)

quare ee

1

.


.


SPSS Output

41.33032s !

0.03297s1b

!


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Comparing Standard Errors

y

y y

x

x

x

y

x

1bssmall

1bslarge

Issmall

Islarge

Variation of observed y valuesfrom the regression line

Variation in the slope of regressionlines from different possible samples


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Inference about the Slope: t-Test t-test for a population slope

Is there a linear relationship between X and Y? Null and alternate hypotheses

H0: 1 = 0 (no linear relationship)

H1: 1 { 0 (linear relationship does exist) Test statistic:

Degree of Freedom:

1b

11

s

bt

!

2nd.f. !

where:

b1 = Sample regression slopecoefficient

1 = Hypothesized slopesb1 = Estimator of the standard

error of the slope


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House Pricein $1000s

(y)

Square Feet(x)

245 1400

312 1600

279 1700

308 1875

199 1100

219 1550

405 2350

324 2450

319 1425

255 1700

(sq.ft.)0.109898.25pricehouse !

Estimated Regression Equation:

The slope of this model is 0.1098

Does square footage of the houseaffect its sales price?

Example: Inference about the Slope:t Test

(continued)


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Inferences about the Slope:t Test Example - Continue

H0: 1 = 0

HA: 1 { 0

Test Statistic: t = 3.329

There is sufficient evidencethat square footage affects

house price

From Excel output:

Coefficients Standard Error t Stat P-value

Intercept 98.24833 58.03348 1.69296 0.12892

Square Feet 0.10977 0.03297 3.32938 0.01039

1bs t*

b1

Decision: Reject H0

Conclusion:

Reject H0Reject H0

E/2=.025

-t/2Do not reject H0

0t(1-/2)

E/2=.025

-2.3060 2.3060 3.329

d.f. = 10-2 = 8


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Confidence Interval for the Slope

Confidence Interval Estimate of the Slope:

Excel Printout for House Prices:

At 95% level of confidence, the confidence interval forthe slope is (0.0337, 0.1858)

1b/211sb Es t

Coefficients Standard Error t Stat P-value Lower 95% Upper 95%

Intercept 98.24833 58.03348 1.69296 0.12892 -35.57720 232.07386

Square Feet 0.10977 0.03297 3.32938 0.01039 0.03374 0.18580

d.f. = n - 2


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Confidence Interval for the Slope

Since the units of the house price variable is$1000s, we are 95% confident that the averageimpact on sales price is between $33.70 and$185.80 per square foot of house size

Coefficient

s

Standard

Error t Stat P-value Lower 95%

Upper

95%

Intercept 98.24833 58.03348 1.69296 0.12892 -35.57720 232.07386

Square Feet 0.10977 0.03297 3.32938 0.01039 0.03374 0.18580

This 95% confidence interval does not include 0.

Conclusion: There is a significant relationship betweenhouse price and square feet at the .05 level of significance


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Residual Analysis

PurposesExamine for linearity assumption

Examine for constant variance for alllevels of x

Evaluate normal distribution assumption

Graphical Analysis of Residuals

Can plot residuals vs. xCan create histogram of residuals tocheck for normality


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Residual Analysis for Linearity

Not Linear Linear

x

residua

ls

x

y

x

y

x

residua

ls


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Residual Analysis for Constant Variance

Non-constant variance Constant variance

x x

y

x x

y

resid

uals

resid

uals


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House Price Model Residual Plot

-60

-40

-20

0

20

40

60

80

0 1000 2000 3000

Square Feet

Residuals

Example: Residual Output

RESIDUAL OUTPUT

Predicted

House

Price Residuals

1 251.92316 -6.923162

2 273.87671 38.12329

3 284.85348 -5.853484

4 304.06284 3.937162

5 218.99284 -19.99284

6 268.38832 -49.38832

7 356.20251 48.79749

8 367.17929 -43.17929

9 254.6674 64.33264

10 284.85348 -29.85348

statistics-linear regression and correlation analysis

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