Transcript SingleFactorANOVA

Single-Factor Analysis of
Variance
Stat 701 Lecture Notes
E. Pena
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Illustrative Example #1
Growing interest in trout farming has prompted a number of experiments
designed to compare various growing conditions. One factor of interest
is the salinity of the water. The effect of salinity on the growth of
rainbow trout (measured by increase in weight) was examined in the
paper “Growth, Training and Swimming Ability of Young Trout
Maintained under Different Salinity Conditions” (Journal of Marine
Biological Association of UK, 1982, 699-708). Full strength seawater
(32% salinity), brackish water (18% salinity), and fresh water (.5%
salinity) were used. The following summary quantities were obtained.
Salinity
Fresh
18%
32%
Number of Fish Mean Weight Gain Sample Standard Deviation
12
8.078
1.786
12
7.863
1.756
8
6.468
1.339
Question: Does the data provide sufficient evidence to conclude that the
mean weight gain is not the same for the three salinity levels?
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Illustrative Example #2
An individual’s critical flicker frequency (cff) is the highest frequency (in
cps) at which the flicker in a flickering light source can be detected. At
frequencies above the cff, the light source appears to be continuous even
though it is actually flickering. An investigation carried out to see if true
average cff depends on iris color yielded the following data. [Source: The
Effect of Iris Color on Critical Flicker Frequency, J. of Gen. Psych., 1973,
91-95.]
Iris Color
Brown
Green
Blue
CFF (Yij)
26.8, 27.9, 23.7, 25.0,
26.3, 24.8, 25.7, 24.5
26.4, 24.2, 28.0, 26.9,
29.1
25.7, 27.2, 29.9, 28.5,
29.4, 28.3
ni
8
Si2
Yi
25.59
1.86
5
26.92
3.40
6
28.17
2.33
Question: Based on this data, what conclusions can be made regarding the
relationship between the cff and iris color?
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Single Factor ANOVA Model I: Assumptions
Data Presentation
Factor Level
(Treatment)
1
2
…
r
Observations
Yij
Y11, Y12 ,..., Y1n1
Y21, Y22 ,..., Y2 n2
…
Yr1 , Yr 2 ,..., Yrnr
Sample Size
ni
n1
n2
…
nr
n
Sample
Mean
Sample
Variance
Y1
S12
Y2
…
S22
Yr
Sr2
Y
S2
…
Assumptions
• Yij  i   ij , j  1,2,..., ni ; i  1,2,..., r;
•  ij are IID Normal(0,  2 )
• error terms are uncorrelated.
This is called the cell-means model. In this fixed effects model,
interest is only on the chosen levels.
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Estimation of Model Parameters
ˆ i  Yi , i  1,2,..., r.
1 n
ˆ  MSE 
( ni  1) Si2 .

n  r i 1
2
Analysis of Variance Table
Source of
Degrees-ofVariation
Freedom
Factor
r-1
(Treatments)
Error
n-r
(Residual)
Corrected
n-1
Total
Sum of
Squares
SSTr
Mean
Squares
MSTr
SSE
MSE
F-Ratio
MSTr/MSE
SSTo
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Formulas
ni
r
r
ni
SSTo   (Yij  Y ) 2   Yij2  CF
i 1 j 1
ni
i 1 j 1
2
Y
SSTr   (Yi  Y ) 2   i  CF
i 1 j 1
i 1 ni
r
r
r
ni
SSE   (Yij  Yi ) 2  SSTo  SSTr
i 1 j 1
(Y.. ) 2
CF 
n
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