Lecture 12 Linear Regression: Test and Confidence Intervals

[Pages:29]Lecture 12 Linear Regression: Test and Confidence Intervals

Fall 2013 Prof. Yao Xie, yao.xie@isye.gatech.edu H. Milton Stewart School of Industrial Systems & Engineering

Georgia Tech

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Outline

? Properties of

^

1

and

^

0

as point estimators ? Hypothesis test on slope and intercept ? Confidence intervals of slope and intercept ? Real example: house prices and taxes

2

Regression analysis

? Step 1: graphical display of data -- scatter plot: sales vs. advertisement cost

! ! ! ! ! ! !

? calculate correlation

3

? Step 2: find the relationship or association between Sales and Advertisement Cost -- Regression

4

Simple linear regression

Based on the scatter diagram, it is probably reasonable to assume that the mean of the random variable Y is related to X by the following simple linear regression model:

Response

Regressor or Predictor

Yi = 0 + 1X i + i i = 1,2,!, n

i

( ) i 0, 2

Intercept

Slope Random error

where the slope and intercept of the line are called regression coefficients. ?The case of simple linear regression considers a single regressor or predictor x and a dependent or response variable Y.

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the adequacy of the fitted model. attoirs oofcEcqausiaotinoanll1y1c-8o.nGveivneinendtattao(gx1iv, ye1)s,p(exc2,iayl2)s,ypm,b(oxlns, yton),thleet numerator and

Regression coefficients uation 11-8. Given data (x1, y1), (x2, y2), p , (xn, nyn), le2t

Sxx

n

n

a

i1

1xi x22

n

n

a

i1

xa2i an

a a xib

i21

xib n

Sxx

a

1xi x22

a

x

2 i

i1

i1

i1

n

(11-10)

(11-10)

n

n

n

n

a a xib a a yib

Sxy

n

a 1yi y2 1xi x2

i1

n

a xi yi n i1

i1 a a xib a

i1

nn

a

yi

b

xy

a

i1

1yi

^y02 1=xi y-x2^1x

a xi yi

i1

i1

i1

n

(11-11)

(11-11)

^1

=

S xy S xx

y^i = ^0 + ^1xi

Fitted (estimated) regression model

Caveat: regression relationship are valid only for values of the regressor variable

within the range the original data. Be careful with extrapolation.

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Estimation of variance

? Using the fitted model, we can estimate value of the response variable for given predictor

!

y^i = ^0 + ^1xi

!

? Residuals: ri = yi - y^i

? Our model:

Yi = 0 + 1Xi + i, i =1,...,n, Var(i) = 2

? Unbiased estimator (MSE: Mean Square Error)

!

n

ri2

^ 2 =

MSE

=

i =1

n-2

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Punchline

? the coefficients

!

^1 and ^0

!

and both calculated from data, and they are subject to

error.

? if the true model is

y

=

1

x

+

0

,

^

1

a

n

d

^

0

are

point estimators for the true coefficients

!

? we can talk about the ``accuracy'' of

^1 and ^0

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