Oneway Classification
One Treatment (Factor) in a CRD; may be unequal replications
yij =μ+αi
μi
+ij i=1,2,...,t,
j=1,...,n
i,
ij ∼iid N(0,σ
2)
equivalent to assuming yij ∼N(μi,σ
2),j=1,...,n
ifor each treatment i.
Also involves the assumption of homogeneity of variance i.e., same variance
in each population.
Estimation
ˆμi=¯yi.=(
jyij)/ni,i=1,...,t
ˆσ2=s2=ij(yij −¯yi. )2
N−t,N=ini
αp−αq=¯yp. −¯yq.,p=q
SE(¯yp. −¯yq.)=sd=s1
np
+1
nq
A(1−α)100% C.I. for αp−αq(or μp−μq.)is
(¯yp. −¯yq.)±tα/2(ν)·sd
where
tα/2(ν) = upper α/2 percentage point of the t-distribution with νd.f.
ν=N−t
Testing Hypotheses
AoV Table
SV d.f. SS MS Fp-value
Trt t−1MS
Trt Fc=MS
Trt /MSE Pr(F>F
c)
Error N−tMSE(= s2)
Tota l N−1
The F-statistic tests
H0:μ1=μ2=···μtvs. Ha: at least one ineq.
or equivalently
H0:α1=α2=···=αtvs. Ha: at least one ineq.
Testing H0:μp=μqvs. Ha:μp=μqor equivalently
H0:αp=αqvs. Ha:αp=αq
Use the t-statistic
tc=|¯yp.−¯yq.|
sd
Rej. H0iff tc>t
α/2(ν)ν=N−t
Contrasts (or Comparisons)
iciμiis said to be a contrast of means μ1,μ
2,...,μ
tif c1,c
2,...,c
tare
constants such that ici= 0. Examples
μ1−μ2,2μ1−μ2−μ3,μ
1−1
3μ2−1
3μ4−1
3μ5
Test for Preplanned Comparisons (Equal Sample Size Case i.e., n1=n2=
···=n)
H0:iciμi=0 vs. Ha:iciμi=0
tc=|ici¯yi.|
s(c2
i
n)1
2
Rej. H0:iftc>t
α/2(N−t)
or
Fc=n(ci¯yi.)2/(c2
i)
s2Rej. H0:ifFc>F
a(1,N−t)
Pairwise Comparison of Means
Individual Comparisons:
* By the t-test of H0:μp=μq
* By the C.I.’s for μp−μq
* Equivalently,using the Least Significance Difference (LSD) when sample sizes are equal.
t-test for H0:μp−μq=0gives Rej:H0if
|¯yp.−¯yq.|>t
α/2(N−t)·s·2/n
LSDα
,n = sample size
Multiple Comparisons:
* Tukey’s procedure for all possible pairwise comparisons simultaneously
(HSD).
* Scheffe’sprocedure for several comparisons (contrasts of the type ciμi)
simultaneously.
Oneway Analysis of Covariance
One factor experiment in a CRD; a single covariateis also measured. Assume
equal replication.
yij =μ+τi+β(xij −¯x..)+ij
i=1,...,t
j=1,...,n ij ∼iid N (0,σ
2)
⇐⇒ Assuming straight line regressions for each treatment with the same slope β
Treatment 1: y1j=α1+βx1j+1j,j=1,...,n
Treatment 2: y2j=α2+βx2j+2j,j=1,...,n
.
.
..
.
.
Treatment t: ytj =αt+βxtj +tj ,j=1,...,n
where αi=μ+τi−¯x.. Estimation
ˆμi=¯yi.(Adj.) = ¯yi−b(¯xi. −¯x..)
‘Adjusted Treatment Means’
b=Exy
Exx
Exy =ij(xij −¯xi.)(yij −¯yi. )
Exx =ij(xij −¯xi.)2
¯yi. =jyij
n¯xi. =jxij
n¯x.. =ijxij
nt
ˆσ2=s2MS Error from the ‘Adjusted AoV’
A(1−α) 100% C.I. for μp−μqis
(¯yp.(Adj.)−¯yq.(Adj.))±tα/2(ν)sd
where
sd=s2
n+(¯xp. −¯xq.)
Exx
21/2
and
ν=t(n−1) −1
Testing Hypotheses
An analysis of covariance table
SV df SS MS F
Trt t−1SSTrt MSTrt MSTrt/MSEUnadj.
Error(Unadj.) t(n−1) SSEUnadj. MSEUnadj.
Regression 1 SSReg MSReg MSReg/MSE
Error(Adj.) t(n−1) −1SSE MSE(= s2)
Tota l tn −1SSTot
Trt(Adj.) t−1SSTrt MSTrt MSTrt/MSE
Error(Adj.) t(n−1) −1SSE MSE(= s2)
The F-statistic for Trt tests the hypothesis
H0:μ1=μ2=···=μtversus Ha: at least one inequality
when the covariateis not present in the model. TheF-statistic for Regression
tests the hypothesis
H0:β=0 versus Ha:β=0
The F-statistic for Trt(Adj.) tests the hypothesis
H0:τ1=τ2=···=τtversus Ha: at least one inequality
when βis not zero. This test is equivalent to comparing the intercepts of the
regression lines i.e.,
H0:α1=α2=···=αtversus Ha: at least one inequality
If this hypothesis is rejected, then at least one pair of treatment effects (equiv-
alently, adjusted treatment means) is different.
