Math 558 Lecture #8
Orthogonality
Orthogonality is important in Design of Experiments because it corresponds to zero correlation among the estimators of interest. Experimental analysis of an orthogonal design is usually straightforward because you can estimate each main effect and interaction independently.Therefore many procedures in the analysis of experimental designs are based on the decomposition of data vectors into orthogonal components.
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More Terminology
The function T : Ω → τ is called the treatment factor. Examples:
T = time in minutes is a treatment factor from the set of experimental units Ω to τ, the set of treatment levels. This means that under T a treatment is assigned to each experimental unit. We can associate an N−dimensional vector space V with Ω. This vector space has the vectors of the form v = {v1, v2, …vN }. For the bread rise experiment, a vector in 12 − dimensional vector space will look like v = {v1, v2….v12}
More Terminology
The treatment subspace of V is the set generated by the vectors which are constant on each treatment. As we have denoted the treatment factor by T, the treatment subspace will be denoted by VT.
A vector v in V is treatment vector if it belongs to VT . It is called a contrast if ∑ω∈Ω vω = 0
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
10 B 0 0 1 0 1 0 τ τˆ
11 A 1 1 0 1 0 0 τ τˆ
12 C 0 -1 4
-1 0 0 1 τ τˆ CC
Table: Caption
Terminology
For each treatment i let ui be the vector whose value on plot ω is
1 if T(ω)=i
0 otherwise
Note that every vector in VT is a unique combination of u1, u2, …..ut,. In our example these vectors are uA, uB, uc,
Terminology
Scalar product of two vectors: The scalar product of two vectors v and w is given by
v.w= ∑ω∈Ω vω.wω = vTw Also
v.v= ∑ω∈Ω vω.vω = vTv
The quantity ∑ω∈Ω v2ω is called sum of squares of v. It is also known as squared length of v and can be written as ||v||2.
We say that two vectors v and w are orthogonal if v.w = 0. The notation is v ⊥ w.
Terminology
From the previous table we can see that uA.uB = 0 = uA.uC = uB.uC.
Also uA.uA = 4 = uB.uB = uC.uC. Generally ui.ui = ri
Ifi̸=jthenui ⊥uj and{ui :i∈τ}isanorthogonalbasisforVT. Note that the vectors τ and τˆ are in the treatment space as well. We can show that if there are t treatments then dim(VT ) = t.
Orthogonal Projection
If W is the subspace of V then the orthogonal complement of W denoted by W⊥ (W perp) is the set
{v∈V :visorthogonaltowforallw∈W}
Theorem .1
Let W be the subspace of V. Then the following hold. i W⊥ is also the subspace of V.
ii (W⊥)⊥ = W
iii dim(W⊥)= dim V – dim W
iv V is the internal direct sum W ⊕ W⊥. This means that given any vector v in V, there is a unique vector x in W and a unique vector z in W⊥ such that v = x + z. We call x the orthogonal projection of v onto W and denote it by x = PW (v). Orthogonal projection is a linear transformation, so PW is an N × N matrix. We can write PW(v) = PWv.
Theorem 1 continued
Theorem 1 continued
Orthogonality in Experimental designs
Two contrasts are said to be orthogonal is if the sum of products of their corresponding coefficients is zero. That is if
lm = am1τ1 +am2τ2 +..amkτt
andln =an1τ1 +an2τ2 +..antτk
are two contrasts, then they are orthogonal iff
am1 .an1 + am2 .an2 + ……amk .ank = 0
Orthogonality in the contrasts ensures that the two contrasts are independently(independent of each other) distributed.
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