Math 558 Lecture #11
Estimation continued
Now from treatment effect estimates let us move to the linear combination of the treatment effects. We can be interested in the estimates of treatment contrasts as well as other linear combinations of different treatment effects. Generally speaking we will be interested in estimating
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λ1τ1 + λ2τ2 + ……. + λtτt
Some examples of the above combinations can be τ1 − τ2,
τ1 + τ2 + τ3 − τ4 etc. For the first case λ1 = 1 and λ2 = −1. For the secondcaseλ1 =λ2 =λ3 =1andλ4 =−1.Inallsuchlinear combinations λi′s are always known as the statistician decides what contrasts and linear combinations are of interest. Whereas, τi′s are the parameters to be estimated.
Estimation
LetxisavectorinVT suchthatx=∑ti=1 λiui ri
x.Y = ∑ r (ui.Y)
=∑r (SUM)T=i =∑λiYi./ri i=1 i i=1
E(x.Y) =x.E(Y)
=x.τ = ∑ r ui.τ = ∑λiτi
as ui.τ = riτi (From our running example uA.τ = 4τA)
tλi t i=1 i i=1
Estimation
So x.Y is linear unbiased estimator of
λ1τ1 + λ2τ2 + ……. + λtτt = ∑ti=1 λiτi. We can show that x.Y has the smallest variance among all other estimators. Hence it is a Linear Unbiased Estimator of ∑ti=1 λiτi.
Let us consider some special cases for the values of λi.
CASE1:λi =1andλj =0fori̸=j.Thelinearcombination ∑ti=1 λiτi reduces to τi. From the last slide the BLUE of τi is y ̄i..
CASE2.λi =1andλj =−1andandλk =0forkdifferentfromi
and j. The linear combination ∑ti=1 λiτi reduces to τi − τj. The
BLUE of τ − τ is y ̄ − y ̄ . i j i. j.
CASE 3: λ = ri for i = 1, 2, …t.. Then ∑t λ τ = ∑t ri τ = τ ̄ i N i=1ii i=1Ni
EStimation
BLUE for τ ̄ is
∑t λiyi.=∑t ri/Nyi. i=1ri i=1 ri
=∑t 1yi. i=1 N
EStimation
Variance of the estimator x.Y
Var(x.Y) =xT.Var(Y)x =xTσ2x
=σ2xTx =σ2 ||x||2
tλi tλi ||x||2 =x.x=∑r ui.∑r ui
i=1 i i=1 i =∑t λ2iui.ui
=∑t λ2i i=1 ri
Estimation
Variance of the estimator x.Y
t λ2 Var(x.Y) = σ2||x||2 = σ2 ∑ i
i=1 ri Now consider all three cases discussed in slide 4.
CASE1:λ =1andλ =0fori̸=j.TheBLUEof∑t
i j i=1ii i.
with the variance σ2 . ri
CASE2.λi =1andλj =−1andandλk =0forkdifferentfromi
and j. The linear combination ∑ti=1 λiτi reduces to τi − τj. The
BLUEofτ −τ isy ̄ −y ̄ withthevarianceσ21 + 1
CASE 3: λi = ri for i = 1, 2, …t.. The variance of the estimator of
N τ ̄ is σ2/N
Replication and Variance
While designing an experiment we plan to have estimators of the contrats with minimum variance. One way to reduce the variance of the estimators is to have equal replication for the treatments. The variance of estimator of the contrast τi − τj is
σ21 + 1 ri rj
Replication and Variance
Proposition: If the positive numbers r1, r2, …rt have a fixed sum N then ∑1 minimizedwhenr1 =r2 =….rt =N/t.Thatiswhywetryto
achieve as much equal replication as possible.
Table from lecture 8 and 9 (for reference)
1 C 0 −1
in VT Orthogonal basis forVT P FV
-1 0 0 1 τ τˆ CC
-1 0 0 1 τ τˆ CC
-1 0 0 1 τ τˆ CC
Some vectors 4
2 C 0 −1 4
3 C 0 −1 4
4 B -1 0 1 0 1 0 τ τˆ BB
5 B -1 0 1 0 1 0 τ τˆ BB
6 A 1 1 0 1 0 0 τ τˆ
7 B -1 0 1 0 1 0 τ τˆ
1 0 1 0 0 τ τˆ
1 0 1 0 0 τ τˆ
11 A 1 1 0 1 0 0 τ τˆ
10 B 0 0 1 0 1 0 τ τˆ
12 C 0 -1 -1 0 0 1 τ τˆ
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