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Single Factor Experiments
Topic:
Comparison of more than 2 groups
One-Way Analysis of Variance
F test
Learning Aims:Understand model parametrization
Carry out an anova
Reason: Multiple t tests wont do!
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Potatoe scab
widespread disease
causes economic loss
known factors: variety, soil condition
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Experiment with different treatments
Compare 7 treatments for effectiveness inreducing scab
Field with 32 plots, 100 potatoes are randomlysampled from each plot
For each potatoe the percentage of the surfacearea affected was recorded. Response variable isthe average of the 100 percentages.
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1-Factor Design
Plots, subjects
Randomisation
Group 1 Group 2 . . . Group I
. . .
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Complete Randomisation
a) number the plots 1, . . ., 32.
b) construct a vector with 8 replicates of 1 and 4replicates of 2 to 7.
c) choose a random permutation and apply it to thevector in b).
in R:
> treatment=factor(c(rep(1,8),rep(2:7,each=4)))
> sample(treatment)
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Exploratory data analysis
Group y y
1 12 10 24 29 30 18 32 26 22.625
2 9 9 16 4 9.5
3 16 10 18 18 15.5
4 10 4 4 5 5.755 30 7 21 9 16.75
6 18 24 12 19 18.25
7 17 7 16 17 14.25
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Graphical display
1 2 3 4 5 6 7
5
10
15
20
25
30
Treatment
y
1 2 3 4 5 6 7
5
10
15
20
25
30
Treatment
y
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Two sample t tests
Group 1 Group 2 : H0:1 =2Group 1 Group 3 : H0:1 =3
Group 1 Group 4 : H0:1 =4Group 1 Group 5 : H0:1 =5Group 1 Group 6 : H0:1 =6
Group 1 Group 7 : H0:1 =7. . .
= 5%,P(Test not significant |H0) = 95%7 groups, 21 independent tests:P(none of the tests sign. |H0) = 0.9521 = 0.34P(at least one test sign. |H0) = 0.66 (more realistic: 0.42)
1 (1 )n
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Bonferroni correction
ChooseTsuch that
1 (1 T)n
=E= 5%
(T =testwise,E= experimentwise)
Since1 (1 n)n , the significance level for asingle test has to be divided by the number of tests.
Overcorrection, not very efficient.
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Analysis of variance
Comparison of more than 2 groups
for more complex designs
global F test
Idea:
total variability
in data=
source of
variation 1+
source of
variation 2+ . . .
Comparison of components
total = treatment + experimental error
total = variability of plots with + variability of plots with
different treatments the same treatment
2
+treatment effect 2
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Definitions
Factor: categorical, explanatory variableLevel: value of a factor
Ex 1: Factor= soil treatment, 7 levels 1 7.= One-way analysis of varianceEx 2: 3 varieties with 4 quantities of fertilizer
= Two-way analysis of varianceTreatment: combination of factor levels
Plot, experimental unit: smallest unit to which a
treatment can be appliedEx: feeding (chicken, chicken-houses), dentalmedicine (families, people, teeth)
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One-way analysis of variance
Model:
response = treatment + error (Plot)
Yij = + Ai + ij(1)
i=1,...,I;j=1,...,Ji
=overall meanAi=ith treatment effectij =random error, N(0, 2)iid.
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Illustration of model (1)
0.
05
0.
10
0.
15
0.
20
0.
25
0.
30
o ooo o oo o oo o o oo ooo oo o oo o ooooo oo o o ooo oo oo o
+A1 +A2 +A3 +A4
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Necessary constraint
Model (1) is overparametrized, a restriction is needed.
usual constraint:JiAi= 0,
Ai= 0ifJi=Jfor alli
Ai denotes the deviation from overall mean.
A1= 0, resp.AI= 0First (or last) group is reference group.
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Decomposition of the deviation of a
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Decomposition of the deviation of a
response from the overall mean
yij y..= yi. y..
+ yij yi.
deviation of deviation from
the group mean the group mean
yi.= 1Ji
jyij mean of groupi,
y..= 1N
i
jyij overall mean,N=
Ji.
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Analysis of variance identity
i
j
(yij y..)2
total variability
=i
j
(yi. y..)2
variability between groups
+i
j
(yij yi.)2
variability within groups
total sum = treatment sum + residual sum
of squares of squares of squares
SStot = SStreat + SSres
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Mean squares
Total mean square: M Stot=SStot/(N 1)Residual mean square: M Sres =SSres/(N I)
SSresN I
=
i(Ji 1)S
2i
i(Ji 1) , S2i =
j(yij yi.)
2
Ji 1
M Sres = 2 = V ar(Yij), E(M Sres) =
2
Treatment mean square: M Streat=SStreat/(I 1)
E(M Streat) =2 +
JiA
2i /(I 1)
dftot=dftreat+ dfres, N 1 =I 1 + N I
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F test
H0: allAi= 0
HA: at least oneAi= 0
Sinceij N(0, 2),F =MStreatMSres
has underH0 anF
distribution withI 1andN Idegrees of freedom.
one-sided test:rejectH0 ifF > F95%,I1,NI
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Chisquare and t distribution
LetZ1, . . . , Z n N(0, 1),iid. Then
X=Z21 + Z
22 + + Z
2n
has a2 distribution withndf,X 2n
LetZ N(0, 1)andX 2
n be independentrandom variables. The distribution of
T = Z
X/nis called thetdistribution withndf,T tn
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F distribution
LetX1 2n andX2 2m be independent random
variables. The distribution of
F = X1/n
X2/m
is called theFdistribution withnandmdf,F Fn,m
Properties: F1,m=t2mE(Fn,m) =
mm2
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R: anova table
> mod1=aov(ytreatment,data=scab)
> summary(mod1)
Df Sum Sq Mean Sq F value Pr(>F)
treatment 6 972.34 162.06 3.608 0.0103 *
Residuals 25 1122.88 44.92
F test is significant, there are significant treatmentdifferences.
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