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1
Two-factor Experiments
Two factors (inputs) A, B
Separate total variation in output values into: Effect due to
A
Effect due to B
Effect due to interaction of A and B (I)
Experimental error
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2
Two-factor ANOVA
Factor A Cinput levels
Factor B Rinput levels
n measurements for each input combinationRCn total
measurements
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3
Two Factors, n Replications
Factor A
1 2 j C
FactorB
1
2
i
R
CELLS(With n
observations)
R
OW
T
R
E
AT
M
E
N
T
COLUMN TREATMENT
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4
Sum-of-Squares
As before, use sum-of-squares identity
SST = SSC + SSR + SSI + SSE
Degrees of freedom df(SSC) = C 1
df(SSR) = R 1 df(SSI) = (C 1)(R 1)
df(SSE) = CR(n 1)
df(SST) = CRn - 1
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FORMULAS FOR SUM OF
SQUARES
5
= = =
= = =
= =
=
=
=
=
+=
=
=
R
i
C
j
n
k
ijk
R
i
C
j
n
K
ijijk
R
i
C
j
jiij
C
j
j
R
i
i
xxSST
xxSSE
xxxxnSSI
xxnRSSC
xxnCSSR
1 1 1
2
1 1 1
2
1 1
2
1
2
1
)(
)(
)(
)(
)(
meangrandx
meancolumnx
meanrowxmeancellx
nobservatioindividualx
membercellk
levelatmentcolumn trejleveltreatmentrowi
rowofno.R
columnofno.C
cellperobs.ofno.n
j
i
ij
ijk
=
=
=
=
=
=
=
=
=
=
=
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Two-Factor ANOVA
1)]CR(n1),1)(R(C;[11)]CR(n1),(R;[11)]CR(n1),(C;[1TAB
2
e
2
II
2
e
2
RR
2
e
2
CCCOMP
2
e
2
I
2
R
2
C
FFFF
ssFssFssFF
1)][CR(nSSEs1)]1)(R[(CSSIs1)(RSSRs1)(CSSCsMS
1)CR(n1)1)(R(C1R1CdfSSESSISSRSSCSS
ErrorIBA
===
====
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Need for Replications
If n=1 Only one measurement of each configuration
Can then be shown that SSI = SST SSC SSR
Since SSE = SST SSC SSR SSI
We have SSE = 0
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Need for Replications
Thus, when n=1 SSE = 0
No information about measurement errors
Cannot separate effect due to interactions
from measurement noise
Must replicate each experiment at least twice
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Step 1. State the Hypothesis
Null hypothesis has 3 parts, e.g., Row means all are equal.
Column means all are equal.
The interaction effect is zero.
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Step 2. Select Statistical Test
and Significance Level
Normally use same -level for testing
all 3 F ratios.
10
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Step 3. Select Samples and
Collect Data
Strive for a balanced design
Ideally, randomly sample
11
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Step 4. Find Regions of
Rejection
Generally have 3 different critical values for each F test
12
Numerator
Denominator
T 1Tdf N=
1)]CR(n1),1)(R(C;[11)]CR(n1),(R;[11)]CR(n1),(C;[1TAB
2
e
2
II
2
e
2
RR
2
e
2
CCCOMP
2
e
2
I
2
R
2
C
FFFF
ssFssFssFF
1)][CR(nSSEs1)]1)(R[(CSSIs1)(RSSRs1)(CSSCsMS
1)CR(n1)1)(R(C1R1CdfSSESSISSRSSCSS
ErrorIBA
===
====
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ExampleB (Mbytes)
A 32 64 128
1 0.25 0.21 0.15
0.28 0.19 0.112 0.52 0.45 0.36
0.48 0.49 0.30
3 0.81 0.66 0.50
0.76 0.59 0.61
4 1.50 1.45 0.70
1.61 1.32 0.68
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Example
00.389.349.3Tabulated
5.295.1052.460Computed
0024.00720.02576.01238.1squareMean
12623freedomDeg0293.04317.05152.03714.3squaresofSum
ErrorABBA
]12,6;95.0[]12,2;95.0[]12,3;95.0[ === FFFF
F
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Conclusions From the
Example
77.6% (SSA/SST) of all variation in response
time due to degree ofmultiprogramming
11.8% (SSB/SST) due to memory size
9.9% (SSAB/SST) due to interaction
0.7% due to measurement error
95% confident that all effects and interactionsare statistically
significant
