Section 11

Junghoon Moon

Contents

Multiple Regression One-way Anova Two-way Anova

ANOVA table for multiple regression

MSE is estimate of σ2

Source DF Sum of Squares

Mean Square

F

Model

Error

P

N-p-1

SSM

SSE

SSM/DFM

SSE/DFE

F=MSM/MSE

Total N-1 SST SST/DFT

Example

ANOVA

I groups Variation within groups Variation between groups

Variation within groups < variation between groups Group means are different

One-way ANOVA Means μ1,μ2,μ3,…..,μI

SD’s σ1,σ2,σ3,….,σI

Hypothesis H0 : μ1=μ2=μ3=…..=μI

Ha : not all μi are equal. Assumptions

I independent samples Populations are normally distributed. Common SD: σ1=σ2=σ3=….=σI

Xij = μi + εij

S2p=s2

w (MS within groups = mean square error)

One-way ANOVA table

Source DF Sum of Squares

Mean Square

F

Groups

Error

I-1

N-I

SSG

SSE

SSM/DFM

SSE/DFE

F=MSM/MSE

Total N-1 SST SST/DFT

Standard Error Bars http://www.uvm.edu/~jleonard/AGRI85/spring2004/Stan

dard_Error_Bars_in_Excel.html

Contrasts

2

0

/ 2. . ( )

, 0

: 0. . : 0

, ( )

i i i

i i

ic p

i

a

d f N Ic

a a

c a x

aSE s

n

H vs H

ct tSE

Two-way ANOVA table

Example

More example Recycling rate: 36 towns in eastern Massachusetts. Region (northeast, northwest, southeast or southwest) “Pay as you throw” program (Pay; Yes or No). a) Complete ANOVA table

Interaction

Qualitative

QuantitativeNo Interaction

Multiple Regression and ANOVA

One-way ANOVA Multiple Regression (ANOVA

table)0 1 2 3 ij*lessons_Computer+ *lessons_None+ *lessons_Piano+Score

Analysis of Variance

Source DF SS MS F PRegression 3 207.281 69.094 9.24 0.000Residual Error 74 553.437 7.479Total 77 760.718