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Linear Regression - Topics. Basics of Linear Regression Variation in Linear Regression Linear Regression Analysis Goodness of Fit Standard Error terms for Linear Regression Hypothesis testing. Regression - Types. Linear Regression.

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slide1

Linear Regression - Topics

  • Basics of Linear Regression
  • Variation in Linear Regression
  • Linear Regression Analysis
  • Goodness of Fit
  • Standard Error terms for Linear Regression
  • Hypothesis testing
slide3

Linear Regression

  • A statistical technique that uses a single, independent variable (X) to estimate a single dependent variable (Y).
  • Based on the equation for a line:

Y = b + mX

slide4

e

=

b b

+

Y

X

i

0 1

Linear Regression - Model

Y

? (the actual value of Yi)

Yi

X

Xi

slide5

Linear Regression - Model

Population

Regression Coefficients for a . . .

ˆ

Sample

Y = b0 + b1Xi + e

ˆ

Y = b0 + b1Xi

slide6

ANOVA CRD - Variation

SST is a measure of the total variation of observations. A measure of the differences in observations.

SSTR

Due to treatments.

SST

SSE

SST = SSTR + SSE

Random/unexplained.

slide7

Linear Regression - Variation

Ice Cream Example

Y

Y

= 2.53

slide8

Linear Regression - Variation

Ice Cream Example

Sample Regression Line

Regression Model

slide9

Linear Regression - Variation

SSR

Due to regression.

SST

SSE

SST = SSR + SSE

Random/unexplained.

slide10

Linear Regression - Variation

Y

SSE =(Yi-Yi )2

_

SST =(Yi-Y)2

_

SSR = (Yi -Y)2

_

Y

X

Xi

slide11

Determining the Regression Line/Model

  • Use Excel (or any other popular statistical software)
    • Select Tools, Data Analysis, Regression
    • Provide the X range
    • Provide the Y range
    • Output the analysis to a new sheet
  • Manual Calculations
slide13

_

SSy =(Yi - Y)2

_

_

SSxy =(Xi - X)(Xi - Y)

SSE

=

S

YX

n-2

Determining the Regression Line/Model Manual Calculations

_

_

SSE =(Yi-Yi )2

SSR = (Yi -Y)2

SST =(Yi-Y)2

_

b1=SSxy/SSx

SSx =(Xi - X)2

_

_

b0 = Y – b1X

MSE = SSE / df

MSR = SSR / df

R2 = SSR/SST

t-test = b1 / Sb1

measures of model goodness
Measures of Model Goodness
  • R2 – Coefficient of Determination
  • F-test > F-crit or p-value less than alpha
  • Standard Error
  • t-test
hypothesis testing for
Hypothesis testing for
  • Testing to see if the linear relationship between X and Y is significant at the population level.
  • t-test
  • Follow the 5-step process
    • H0:

HA:

    • t-crit, alpha or alpha/2, n-2 df
standard error terms in linear regression
Standard Error Terms in Linear Regression
  • Se(standard error of the estimate)

A measure of variation around the regression line

If the Se is small…

Standard deviation Of the Errors

  • Sb1(standard error of the the sampling distribution of b1)

Standard deviation of the slopes

A measure of the variation of the slopes from different samples If the Sb1 is small…our b1 estimate is probably very accurate

Estimates of …

b1

b1

b1

linear regression example
Linear Regression Example
  • Petfood, Estimate Sales based on Shelf Space
  • Two sets of samples, 12 observations each
  • Perform a Regression Analysis on both sets of data

Sample1

Sample2