Chapter 11

Simple Linear Regression and Correlation

Learning Objectives• Use simple linear regression for building empirical

models • Estimate the parameters in a linear regression model• Determine if the regression model is an adequate fit

to the data• Test statistical hypotheses and construct confidence

intervals • Prediction of a future observation• Use simple transformations to achieve a linear

regression model• understand the correlation

Regression analysis• Relationships between two or more variables• Useful for these types of problems• Predict a new observation• Sometimes a regression model will arise from a

theoretical relationship• At other times no theoretical knowledge• Choice of the model is based on inspection of a

scatter diagram• Empirical model

Regression Model• Mean of the random variable Y is related to x

E[Y|x]=Y|x=0+ 1x 0 and 1 called regression coefficients• Appropriate way to generalize this to a probabilistic

model • Assume that the expected value of Y is a linear

function of x• Actual value of Y is determined by the mean value

function plus a random error termY=0+ 1x+

• Where called random error term

1 and • Suppose that the mean and variance of

are 0 and 2 • Slope, 1, can be interpreted as the change

in the mean of Y • Height of the line at any value of x is just

the expected value of Y for that x• Variability of Y at a particular value of x is

determined by the error variance 2

• Implies that there is a distribution of Y-values at each x

Graph of the Variability

• Distribution of Y for any given value of x

• Values of x are fixed, and Y is a random variable with the following mean and varianceMean: Y|x=0+ 1x

Variance 2

True regression line

Y

x

Simple Linear Regression• Values of the intercept, slope and the

error variance will not be known• Must be estimated from sample data• Fitted model is used in prediction of future

observations of Y at a particular level of x

Method of Least Squares• True relationship

between Y and x is a straight line

• Assume n pairs of observations

• Estimates of 0 and 1 result in a line that is a “best fit” to the data

• Called method of least squares

Least Squares Method• Assuming the n observations in the sample• Sum of the squares of the deviations of the

observations from the true regression line

• Taking the partial derivatives

210

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Least Squares Method-Cont.• Simplifying

• Results are

• Fitted or estimated regression line is

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Using Special Symbols• Convenient to use special symbols

• Numerator

• Denominator

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Residual Error

• Describes the error in the fit of the model to the ith observation yi

• Each pair of observations satisfies

• Denoted by ei

iioi exy 1̂ˆ

iii yye ˆ

Estimating 2

• Another unknown parameter,2, the variance of the error term

• Residuals ei are used to obtain an estimate of 2

• Sum of squares of the residuals, often called the error sum of squares

• A more convenient computing formula

• SST is the total sum of squares

2ˆ 2

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Example• Regression methods were used to analyze the data from

a study investigating the relationship between roadway surface temperature (x) and pavement deflection ( y).

• Summary quantities were as follows

1478 ,86.8 ,75.12 ,20 2 iii xyyn

67.1083 and 8.215,1432iii yxx

Questions(a) Calculate the least squares estimates of the slope

and intercept. Graph the regression line.(b) Use the equation of the fitted line to predict what

pavement deflection would be observed when the surface temperature is 85F.

(c) What is the mean pavement deflection when the surface temperature is 90F?

(d) What change in mean pavement deflection would be expected for a 1F change in surface temperature?

Solution• Need to have

• Hence, the slope and intercept

• Regression line

6.339918.143215 2014782 xxS

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Solution-Cont.

• Graph of the regression line

00.10.20.30.40.50.60.70.8

-50 0 50 100 150

x

y

Solution-Cont.

• Pavement deflection

• Mean pavement deflection

• Change in mean pavement deflection

6836.0)85(0041612.03299892.0ˆ y

7045.0)90(0041612.03299892.0ˆ y

00416.01̂

Properties of the Least Estimators• Assumed that the error term in the model is a

random variable • Estimators will be viewed as random variables• Properties of the slope

• Properties of the intercept

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Analysis of Variance Approach

• Used to test for significance of regression• Partitions the total variability in the response

variable into two components

• First term is called error sum of squares• Second term is called regression sum of squares• Symbolically

SST=SSR+SSE

• SST is the total corrected sum of squares

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Analysis of Variance

• SST, SSR, and SSE has n-1, 1, and n-2 d.o.f, respectively

• SSR= β1Sxy and SSE=SST- β1Sxy

• Divide by its d.o.f MSR=[SSR/1] and MSE=[SSE/n-2]

• Then F=MSR/MSE follows F1,n-2 distribution

Hypothesis Tests for Slope• Adequacy of a linear regression model• Appropriate hypotheses for slope are

H0: β1=β1,0

H1: β1#β1,0 • Test Statistic

• Follows the F 1,n-2 distribution• Reject H0 if f0>f,1,n-2

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Analysis of Variance for Testing Significance of Regression

Example• Consider the data from the previous example on

x=roadway surface temperature and y=pavement deflection.

• (a) Test for significance of regression using α = 0.05. What conclusions can you draw?• (b) Estimate the standard errors of the slope and

intercept.

Solution• Use the steps in hypotheses testing1) Parameter of interest is slope of the regression line 1

2)3)4) = 0.055) The test statistic is

6) Reject H0 if f0 > f,1,18 where f0.05,1,18 = 4.416

0: 10 H

0: 11 H

)2/(1/

0

nSSSS

MSMSf

E

R

E

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Solution7) Using the results from the previous example

• Hence, the test statistic

8) Since 73.95 > 4.416, reject H0 and conclude the model specifies a useful relationship at = 0.05

• Standard error

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SSSSS

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ˆ 2

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Confidence Intervals on the Slope and Intercept• Interested to obtain C.I. estimates of the

parameters• Width of these C.I. is a measure of the overall

quality of the regression line• 100(1-α)% C.I. on the slope β1

• 100(1-α)% C.I. on the intercept β0

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Confidence Interval on the Mean Response• Constructed on the mean response at a specified

value of x, say, x0

• Called a C.I. about the regression line• C.I. about the mean response at the value of x=x0

• Applies only to the interval

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Prediction of New Observations• An important application of a regression model• New observation is independent of the observations

used to develop the regression model• C.I. for Y|x is inappropriate• Prediction interval on a future observation at the value

x0

• Always wider than the C.I. at x0• Depends on both the error from the fitted model and

the error associated with future observations

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Example

• The first example presented data on roadway surface temperature x and pavement deflection y

• Find a 99% confidence interval on each of the following:

• (a) Slope • (b) Intercept• (c) Mean deflection when temperature x=85o F• (d) Find a 99% prediction interval on pavement

deflection when the temperature is 90oF.

Solutiona) Confidence interval on the slope

• Critical value t/2,n-2 = t0.005,18 = 2.878• Hence

b) Confidence interval on the intercept

xxn

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tS

t2

2,2/11

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2,2/1ˆˆˆˆ

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1

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0

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Solutionc) 99% confidence interval on when x=85 F

d) 99% prediction interval when x=90 F.

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Residual Analysis• Helpful in checking the

errors are approximately normally distributed with constant variance

• Useful in determining whether additional terms in the model are required

• Construct normal probability plot of residuals

• Patterns of residual plots

Coefficient of Determination(R2)• Judge the adequacy of a regression model• Coefficient of determination

• Referred as the amount of variability in the data explained by the regression model and

• SSR is that portion of SST that is explained by the use of the regression model

• SSE is that portion of SST that is not explained by the use of the regression model

SSTSSESST

SSTSSRR

2

Transformation of Data Points• Inappropriateness of straight-line regression model• Scatter diagram• Consider the exponential function Y=β0eβ1x transformed to a straight line• By a logarithmic transformation

ln Y=lnβ0 +β1 x + ln

• Another intrinsically linear function is Y=β0+β1(1/x)+

• By using the reciprocal transformation z=1/xY=β0 +β1 z +

• Transformed error terms are normally distributed

Correlation• Assumed that x is a mathematical variable and

that Y is a random variable• Many applications involve situations in which both

X and Y are random variables• Suppose observations are jointly distributed

random variables• Measures the strength of linear association

between two variables and denoted by • Shows how closely the points in a scatter diagram

are spread around the regression line

Hypothesis Tests • Useful to test the hypotheses

H0: =0

H1: #0 • Appropriate test statistic

• Follows the t distribution with n-2 degrees of freedom

• Reject the null hypothesis if

212RnRTo

2, no tt

Next Agenda

• Chapters 13 deals with designing and conducting engineering experiments

• ANOVA in designing single factor experiments will be emphasized