section 11 junghoon moon. contents multiple regression one-way anova two-way anova
TRANSCRIPT
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Section 11
Junghoon Moon
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Contents
Multiple Regression One-way Anova Two-way Anova
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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
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Example
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ANOVA
I groups Variation within groups Variation between groups
Variation within groups < variation between groups Group means are different
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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)
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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
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Standard Error Bars http://www.uvm.edu/~jleonard/AGRI85/spring2004/Stan
dard_Error_Bars_in_Excel.html
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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
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Two-way ANOVA table
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Example
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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
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Interaction
Qualitative
QuantitativeNo Interaction
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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