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Analyses of Variance. Simple Situation Genotype AGenotype B 13534.
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Transcript of Analyses of Variance. Simple Situation Genotype AGenotype B 13534.
Analyses of VarianceAnalyses of Variance
Simple SituationSimple Situation
Genotype A Genotype B135 34
Simple SituationSimple Situation
Genotype A Genotype B135 34115 76102 83110 64
115.5 64.2
Observations and QuestionsObservations and Questions
From the replicated means, genotype A From the replicated means, genotype A is “better” than genotype B.is “better” than genotype B.
What is the probability that this result What is the probability that this result will be repeated if this test were done will be repeated if this test were done say 100 times?say 100 times?
Could this result have occurred by Could this result have occurred by randomrandom chance? chance?
tt-test-test
|x|x11-x-x22| | 2[(2[(11
22++2222)/()/(nn11++nn22)])]
tt = =
ReplicateGenotype
Stephens Lambart1 55 782 66 913 49 974 64 825 70 856 68 77
tt-test-test
ReplicateReplicateGenotypeGenotype
StephensStephens LambartLambart112233445566
555566664949646470706868
787891919797828285857777
xxmean xmean x
3723726262
5105108585
ReplicateReplicateGenotypeGenotype
StephensStephens LambartLambart112233445566
555566664949646470706868
787891919797828285857777
xxmean xmean x
xx22
((x)x)22//nnSS(x)SS(x)
3723726262
23,40223,40223,06423,064
338338
5105108585
43,65243,65243,35043,350
302302
ReplicateReplicateGenotypeGenotype
StephensStephens LambartLambart112233445566
555566664949646470706868
787891919797828285857777
xxmean xmean x
xx22
((x)x)22//nnSS(x)SS(x)
dfdf
3723726262
23,40223,40223,06423,064
33833855
5105108585
43,65243,65243,35043,350
30230255
22 76.676.6 60.460.4
tt-test-test
Stephens = 62 bushelsStephens = 62 bushels
Lambart = 85 bushelsLambart = 85 bushels
Significant?Significant?
tt-test-test
Stephens = 62 bushels Stephens = 62 bushels Lambart = 85 bushelsLambart = 85 bushels
|x|xStephensStephens-x-xLambartLambart| | 2[(2[(StSt
22++LaLa22)/()/(nnStSt++nnLaLa)])]
tt = =
tt = |85-62|/ = |85-62|/2[(60+77)/12] = 4.822[(60+77)/12] = 4.82
cwcw tt10df10df : exceeds 99% table value : exceeds 99% table value
More than two treatmentsMore than two treatments
Rep.Genotype
Brundage Lambert Croft Stephens1 64 78 75 552 72 91 93 663 68 97 78 494 77 82 71 645 56 85 63 706 95 77 76 68
Multiple Multiple tt-tests-tests
Brundage Brundage vv Lambert; Brundage Lambert; Brundage v v Croft; Croft; Brundage Brundage v v Stephens; Lambert Stephens; Lambert v v Croft; Croft; Lambert Lambert v v Stephens; Croft Stephens; Croft v v StephensStephens..
Problems?Problems? If all tests were done at 95% significance If all tests were done at 95% significance
level, and one difference was significant, level, and one difference was significant, we have done 6 tests and would expect 1/20 we have done 6 tests and would expect 1/20 to be significant, at random.to be significant, at random.
More than two treatmentsMore than two treatments
#Genotype
TotalBrundage Lambert Croft Stephens
xx
43272
51085
45676
37262
1,770295
More than two treatmentsMore than two treatments
#Genotype
TotalBrundage Lambert Croft Stephens
xx
43272
51085
45676
37262
1,770295
x2
(x)2/nss(x)
df
31,99431,104
8905
43,65243,350
3025
35,14443,656
4885
23,40223,064
3385
134,192132,174
2,01820
More than two treatmentsMore than two treatments
#Genotype
TotalBrundage Lambert Croft Stephens
xx
43272
51085
45676
37262
1,770295
x2
(x)2/nss(x)
df
31,99431,104
8905
43,65243,350
3025
35,14443,656
4885
23,40223,064
3385
134,192132,174
2,01820
2 178.0 60.4 97.6 76.6 100.9
Analysis of VarianceAnalysis of Variance
From pooled variance we can estimate From pooled variance we can estimate a pooled SED between any two means a pooled SED between any two means = = (2)(100.9)/6 = (2)(100.9)/6 = ++ 5.80, and use this 5.80, and use this in all in all tt-tests.-tests.
Alternatively an analysis of variance Alternatively an analysis of variance could be carried out.could be carried out.
This form of analysis was first This form of analysis was first proposed by Fisher in the 1920’sproposed by Fisher in the 1920’s
Analysis of VarianceAnalysis of Variance
Is an elegant and quicker way to Is an elegant and quicker way to calculate a pooled error term.calculate a pooled error term.
Analysis is simple in simple designs but Analysis is simple in simple designs but can be complicated and lengthy in some can be complicated and lengthy in some designs (i.e. rectangular lattices).designs (i.e. rectangular lattices).
In some experimental designs the In some experimental designs the ANOVA is the only method to estimate ANOVA is the only method to estimate a pooled error term.a pooled error term.
Analysis of VarianceAnalysis of Variance
It can provide an It can provide an FF-test to tests specific -test to tests specific hypotheses. (i.e. to test general hypotheses. (i.e. to test general differences between different differences between different treatments).treatments).
Can be an invaluable Can be an invaluable initial initial contributioncontribution to interpretation of to interpretation of experiments.experiments.
Theory of Analysis of VarianceTheory of Analysis of Variance
Consider a simple CRB design.Consider a simple CRB design.Four treatments (Four treatments (nn = 4). = 4).With all treatments replicated 5 times With all treatments replicated 5 times
(k(k = 5) = 5)..The total experiment would be The total experiment would be nn x x kk = =
20 experimental units.20 experimental units.
Theory of Analysis of VarianceTheory of Analysis of Variance
Rep.Treatment
A B C D1 x11 x21 x31 x41
2 x12 x22 x32 x42
3 x13 x23 x33 x43
4 x14 x24 x34 x44
5 x15 x25 x35 x45
Theory of Analysis of VarianceTheory of Analysis of Variance
Rep.Treatment
MeanA B C D
1 x11 x21 x31 x41 x.1
2 x12 x22 x32 x42 x.2
3 x13 x23 x33 x43 x.3
4 x14 x24 x34 x44 x.4
5 x15 x25 x35 x45 x.5
Theory of Analysis of VarianceTheory of Analysis of Variance
Rep.Treatment
MeanA B C D
1 x11 x21 x31 x41 x.1
2 x12 x22 x32 x42 x.2
3 x13 x23 x33 x43 x.3
4 x14 x24 x34 x44 x.4
5 x15 x25 x35 x45 x.5
Mean x1. x2. x3. x4. x..
Theory of Analysis of VarianceTheory of Analysis of Variance
TSS = TSS = iijj(x(xijij-x-x....))22
TMS = TMS = iijj(x(xijij-x-x....))22/(/(nknk-1)-1)
iijj(x(xijij-x-x....))2 2 = = iijj[(x[(xijij-x-xi.i.) + (x) + (xi.i.-x-x....)])]2 2
iijj[(x[(xijij-x-xi.i.))22+2(x+2(xijij-x-xi.i.)(x)(xi.i.-x-x....)+(x)+(xi.i.-x-x....))22] ]
Theory of Analysis of VarianceTheory of Analysis of Variance
iijj[(x[(xijij-x-xi.i.))22+2(x+2(xijij-x-xi.i.)(x)(xi.i.-x-x....)+(x)+(xi.i.-x-x....))22] ]
22iijj(x(xijij-x-xi.i.)(x)(xi.i.-x-x....))
ii[2[2nn (x (xi.i.-x-x....) ) jj(x(xijij-x-xi.i.)])]
But!But! jj(x(xijij-x-xi.i.) = 0) = 0
22iijj(x(xijij-x-xi.i.)(x)(xi.i.-x-x....) = 0) = 0
Theory of Analysis of VarianceTheory of Analysis of Variance
iijj[(x[(xijij-x-xi.i.))22++2(x2(xijij-x-xi.i.)(x)(xi.i.-x-x....))+(x+(xi.i.-x-x....))22] ]
iijj[(x[(xijij-x-xi.i.))22 + (x + (xi.i.-x-x....))22] ]
iijj(x(xijij-x-x....))22 = = iijj(x(xijij-x-xi.i.))22++kkii(x(xi.i.-x-x....))22] ]
kkii(x(xi.i.-x-x....))22 = Between Treatment SS = Between Treatment SS
iijj(x(xijij-x-xi.i.))2 2 = Within Treatment SS= Within Treatment SS
Theory of Analysis of VarianceTheory of Analysis of Variance
df [WTSS] = df [WTSS] = nknk--n n : df [BTSS] = : df [BTSS] = nn-1-1
MS = SS/dfMS = SS/df
WTMS ~WTMS ~ 22nk-n dfnk-n df : B : BTMS ~TMS ~ 22
n-1 dfn-1 df
kkii(x(xi.i.-x-x....))22 = Between Treatment SS = Between Treatment SS
iijj(x(xijij-x-xi.i.))2 2 = Within Treatment SS= Within Treatment SS
Theory of Analysis of VarianceTheory of Analysis of Variance
WTMS ~WTMS ~ 22nk-n dfnk-n df : B : BTMS ~TMS ~ 22
n-1 dfn-1 df
YYijij = = + g + gii + e + eijij
ggii = BTMS : e = BTMS : eij ij = WTMS= WTMS
Assumption is homogeneity of error Assumption is homogeneity of error variance between treatments.variance between treatments.
Theory of Analysis of VarianceTheory of Analysis of Variance
Source of variation df EMS
Between treatments n-1 e2 + kt
2
Within treatments nk-n e2
Total nk-1
[e2 + kt
2]/e2 = 1, if kt
2 = 0
Analysis of Variance of CRBAnalysis of Variance of CRB
Source df SS
Between treatments k-1 [G1
2/n1 + G22/n2 … Gk
2/nk] - CF
Within treatments jk-k By difference
Total jk-1 [x112 + x12
2 + … + xjk2] - CF
CF = [CF = [xxijij]]22//jkjk
More than two treatmentsMore than two treatments
Rep.Genotype
Brundage Lambert Croft Stephens1 64 78 75 552 72 91 93 663 68 97 78 494 77 82 71 645 56 85 63 706 95 77 76 68
More than two treatmentsMore than two treatments
#Genotype
TotalBrundage Lambert Croft Stephens
xx
43272
51085
45676
37262
1,770295
TSS = TSS = ∑∑(64(6422 + 72 + 7222 + 68 + 6822 + .... + 68 + .... + 6822) - CF) - CF
CF = CF = ∑∑(64 + 72 + 68 + .... + 68)(64 + 72 + 68 + .... + 68)22/24/24
BSS = BSS = ∑∑(432(43222/6 + 510/6 + 51022/6 + 456/6 + 45622/6 + 372/6 + 37222/6) - CF/6) - CF
WSS = TSS - BSSWSS = TSS - BSS
More than two treatmentsMore than two treatments
#Genotype
TotalBrundage Lambert Croft Stephens
xx
43272
51085
45676
37262
1,770295
TSS = TSS = ∑∑(64(6422 + 72 + 7222 + 68 + 6822 + .... + 68 + .... + 6822) - CF) - CF
CF = CF = ∑∑(64 + 72 + 68 + .... + 68)(64 + 72 + 68 + .... + 68)22/24/24
BSS = BSS = ∑∑(432(43222 + 510 + 51022 + 456 + 45622 + 372 + 37222)/6 - CF)/6 - CF
WSS = TSS - BSSWSS = TSS - BSS
More than two treatmentsMore than two treatments
#Genotype
TotalBrundage Lambert Croft Stephens
xx
43272
51085
45676
37262
1,770295
TSS = TSS = ∑∑(64(6422 + 72 + 7222 + 68 + 6822 + .... + 68 + .... + 6822) - CF) - CF
CF = CF = ∑∑(64 + 72 + 68 + .... + 68)(64 + 72 + 68 + .... + 68)22/24/24
BSS = BSS = ∑∑(432(43222/6 + 510/6 + 51022/6 + 456/6 + 45622/6 + 372/6 + 37222/6) - CF/6) - CF
WSS = TSS - BSSWSS = TSS - BSS
Example of Analysis of VarianceExample of Analysis of Variance
Source df SS MS F
Between genotypes 3 1636.5 545.5 5.41**
Within genotypes 20 2018.0 100.9
Total 23 3654.5
** = 0.01 > P > 0.001
Rep.Treatment
A B C D1 x11 x21 x31 x41
2 x12 x22 x32 x42
3 x13 x23 x33 x43
4 - x24 x34 x44
5 - - x35 -
Analysis of Variance of CRBAnalysis of Variance of CRB
Rep.Treatment
A B C D1 x11 x21 x31 x41
2 x12 x22 x32 x42
3 x13 x23 x33 x43
4 - x24 x34 x44
5 - - x35 -Total G1 G2 G3 G4
Analysis of Variance of CRBAnalysis of Variance of CRB
Rep.Treatment
A B C D1 x11 x21 x31 x41
2 x12 x22 x32 x42
3 x13 x23 x33 x43
4 - x24 x34 x44
5 - - x35 -Total G1 G2 G3 G4
Analysis of Variance of CRBAnalysis of Variance of CRB
Analysis of Variance of CRBAnalysis of Variance of CRB
Source df SS
Between treatments k-1 [G1
2/n1 + G22/n2 … Gk
2/nk] - CF
Within treatments jk-k By difference
Total jk-1 [x112 + x12
2 + … + xjk2] - CF
CF = [CF = [xxijij]]22//jkjk
Assumptions behind the ANOVAAssumptions behind the ANOVA
Assumption of data being normally Assumption of data being normally distributed.distributed.
Homogeneity of error variance.Homogeneity of error variance.Additivity of variance effects.Additivity of variance effects.Data collected from a properly Data collected from a properly
randomized experiment.randomized experiment.
Dealing with Wrongful DataDealing with Wrongful Data
It is usually assumed that the data It is usually assumed that the data collected is collected is correctcorrect!.!.
Why would data not be Why would data not be correct?correct?Mis-recording, mis-classification, Mis-recording, mis-classification,
transcription errors, errors in data transcription errors, errors in data entry.entry.
Outliers.Outliers.
Dealing with Wrongful DataDealing with Wrongful Data
What things can help?What things can help?Keep detailed records, on each Keep detailed records, on each
experimental unit.experimental unit.Decide beforehand what values Decide beforehand what values
would arouse suspision.would arouse suspision.
Dealing with Wrongful DataDealing with Wrongful Data
What do you do with suspicios data?What do you do with suspicios data?If correct, and it is discarded, then If correct, and it is discarded, then
valuable information is lost. This valuable information is lost. This will bias the results.will bias the results.
If wrong and included, will bias If wrong and included, will bias results and may have extreme results and may have extreme consequences.consequences.
Checking ANOVA AccurucyChecking ANOVA Accurucy
Coefficient of variation: [Coefficient of variation: [ee//]x100.]x100. CV=(CV=(√100.9/73.75)*100=13.6%√100.9/73.75)*100=13.6%
RR22 value = {[TSS-ESS]/TSS}x100. value = {[TSS-ESS]/TSS}x100.RR22 = (1654/3654)*100 = 44.7%. = (1654/3654)*100 = 44.7%.
Compare the effect of blocking or Compare the effect of blocking or sub-blocking (discussed later).sub-blocking (discussed later).
Next ClassNext ClassANOVA of RCB ANOVA of RCB
DesignsDesigns
ANOVA of Latin ANOVA of Latin Square DesignsSquare Designs