# Determinant Efficiencies in Ill-Conditioned Models

Authors: D. R. Jensen
Journal: Journal of Probability and Statistics
 (2011)

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## Abstract

Thecanonical correlations between subsets of OLS estimators are identified
with design linkage parameters between their regressors. Known collinearity indices are extended to encompass angles between each regressor vector and remaining vectors. One such angle quantifies the collinearity of regressors with the intercept, of concern
in the corruption of all estimates due to ill-conditioning. Matrix identities factorize a determinant in terms of principal subdeterminants and the canonical Vector Alienation Coefficients between subset estimators—by duality, the Alienation Coefficients between
subsets of regressors. These identities figure in the study of D and Ds as determinant efficiencies for estimators and their subsets, specifically, Ds-efficiencies for the constant, linear, pure quadratic, and interactive coefficients in eight known small second-order
designs. Studies on D- and Ds-efficiencies confirm that designs are seldom efficient for both. Determinant identities demonstrate the propensity for Ds-inefficient subsets to be masked through near collinearities in overall D-efficient designs.


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## Body

## 1. Introduction

Given{Y=Xβ+ϵ} of full rank with homogeneous, uncorrelated errors, the OLS estimators β̂ are unbiased with second-moment matrix V(β̂)=σ2(X′X)-1. Such moment matrices pervade experimental design, to include determinants as gauges of D- and Ds-efficiencies for estimators and their subsets. Early references trace to [1–4], and more recently to [5–10] and others. Finding Ds-efficient designs for polynomial models is considered in [11–20], for example. Studies examining the Ds-efficiencies of D-efficient designs confirm that designs are seldom efficient for both; see [13, 21–23]. From those beginnings, the study of D- and Ds-efficiencies continues apace. To wit, a recent key-word search in the Current Index to Statistics shows in excess of 60 listings from 2006 to 2010, and more than 100 from 2001 to 2010. Moreover, these ideas bear fruit in a widening diversity of applications as evidenced in the following.To fix ideas, letD correspond to a polynomial Pc of degree c, namely, g(μ)=∑i=0cβiti. In toxicology studies, a two-stage experiment is proffered in [24], seeking D-efficiency in estimating k=c+1 overall parameters at the first stage, then D1-efficiency at the second stage in estimating a critical “threshold parameter,” using quasilikelihood in nonlinear models. Coupled with this is the Dk-1-efficiency for the remaining k-1 parameters at the second stage. In related work [25], experiments with c chemicals in combination are to be examined along fixed-ratio rays. When restricted to a specified ray, the fundamental hypothesis of noninteracting factors can be rejected when higher-order polynomial terms are required in the total dose-response model g(μ)=β0+β1t+∑i=2cβiti in the linear predictor t. Here D2 refers to [β0,β1] and Dc-1 to Ds-efficiency in the critical estimation of [β2,…,βc], which vanish under the conjectured additivity. Moreover, in [11, 12, 21] D refers to Pk+1, Dk to Pk, and D1 to βk+1, the highest-order term in Pk+1, for example. In short, users often are properly concerned with both D- and Ds-efficiencies, and connections between these basic criteria deserve further study, to be undertaken here.Ill-conditioning, as near-collinearity among the columns ofX, “causes crucial elements of X′X to be large and unstable,” “creating inflated variances,” and estimates that are “very sensitive to small changes in X,” having “degraded numerical accuracy;” see [26–28], for example. Diagnostics include the condition numberc1(X′X), the ratio of largest to smallest eigenvalues; and the Variance Inflation Factors{VIF(β̂j)=vjjwjj;1≤j≤p} with W=X′X and V=(X′X)-1, that is, ratios of actual (vjj) to “ideal” (1/wjj) variances had the columns of X been orthogonal. In models with intercept, “collinearity with the intercept can quite generally corrupt the estimates of all parameters in the model whether or not the intercept is itself of interest and whether or not the data have been (mean) centered,” as noted in [29].To the foregoing list of ills from ill-conditioning, we add that not only are designs seldom efficient for both, butDs-inefficient estimators may be masked in overall D-efficient designs, and conversely. This masking may be quantified in terms of structural dependencies, specifically, through determinant identities linking D- and Ds-efficiencies to various gages of nonorthogonality of the data. The latter include nonvanishing inner products between columns of regressors, Hotelling’s [30] canonical correlations among OLS solutions, and VIFs. An outline follows.Section2 contains supporting material. Details surrounding collinearity diagnostics are topics in Section 3, to include duality of angles between subspaces of the design and parameter spaces, and their connections to VIFs. Section 4 develops basic determinant identities and inequalities of independent interest. Section 5 revisits eight small second-order designs with regard to Ds-efficiencies in estimating the constant, linear, pure quadratic, and interactive coefficients, to include the masking of inefficient estimators. Though in wide usage, with no apparent accounting for collinearity, these designs are seen to exhibit varying degrees of collinearity of regressors with the constant. Since computations proceed from the design matrix itself, an advantage is that prospective designs can be evaluated beforehand in regard to issues studied here, before committing to an actual experiment. Section 6 concludes with a brief summary.


## 2. Preliminaries

### 2.1. Notation

Spaces of note includeℝk as Euclidean k-space; ℝ+k as its positive orthant; 𝔽n×k as the real matrices of order (n×k); 𝕊k as the (k×k) real symmetric matrices; and 𝕊k+ as their positive definite varieties. The transpose, inverse, trace, and determinant of A∈𝕊k+ are A′, A-1, tr(A), and |A|; and A1/2 is its spectral square root. Special arrays include the unit vector 1n′=[1,1,…,1]∈ℝn, the identity In of order (n×n), the block-diagonal matrix Diag(A1,A2)∈𝕊k, the idempotent form Bn=(In-n-11n1n′), and 𝒪(k) as the real orthogonal group of (k×k) matrices. For X(n×p) of rank p≤n, designate a pseudoinverse as X†, its ordered singular values as σ(X)={ξ1≥ξ2≥⋯≥ξp>0}, and by 𝒮p(X)⊂ℝn, the linear span of columns of X. Its condition number is c2(X)=ξ1/ξp, specifically, c2(X)=[c1(X′X)]1/2.The mean, dispersion matrix, and generalized variance for a randomU∈ℝk are designated as E(U)=μ∈ℝk, V(U)=Σ∈𝕊k+, and GV(U)=|Σ|, respectively. To account for dimension, consider G(U)=[GV(U)]1/k=|Σ|1/k as a function homogeneous of unit degree. The class M0:{Y=β01n+Xβ+ϵ}, comprising models with intercept and dispersion V(ϵ)=σ2In, is our principal focus. Unless stated otherwise, we take σ2=1.0, since variance ratios are scale-invariant. A distinction is drawn between centered and uncenteredVIFs, namely, VIFcs and VIFus, the former from columns of X centered to their means. The latter, designated as {VIFu(β̂j);j=0,1,…,k}, are diagonal elements of (X0′X0)-1 divided by reciprocals of diagonals of X0′X0 itself. These are of subsequent interest. Special distributions on ℝ+1 include the Snedecor-Fisher distribution F(·;ν1,ν2,λ) having (ν1,ν2) degrees of freedom and noncentrality λ.


## 2.1. Notation

Spaces of note includeℝk as Euclidean k-space; ℝ+k as its positive orthant; 𝔽n×k as the real matrices of order (n×k); 𝕊k as the (k×k) real symmetric matrices; and 𝕊k+ as their positive definite varieties. The transpose, inverse, trace, and determinant of A∈𝕊k+ are A′, A-1, tr(A), and |A|; and A1/2 is its spectral square root. Special arrays include the unit vector 1n′=[1,1,…,1]∈ℝn, the identity In of order (n×n), the block-diagonal matrix Diag(A1,A2)∈𝕊k, the idempotent form Bn=(In-n-11n1n′), and 𝒪(k) as the real orthogonal group of (k×k) matrices. For X(n×p) of rank p≤n, designate a pseudoinverse as X†, its ordered singular values as σ(X)={ξ1≥ξ2≥⋯≥ξp>0}, and by 𝒮p(X)⊂ℝn, the linear span of columns of X. Its condition number is c2(X)=ξ1/ξp, specifically, c2(X)=[c1(X′X)]1/2.The mean, dispersion matrix, and generalized variance for a randomU∈ℝk are designated as E(U)=μ∈ℝk, V(U)=Σ∈𝕊k+, and GV(U)=|Σ|, respectively. To account for dimension, consider G(U)=[GV(U)]1/k=|Σ|1/k as a function homogeneous of unit degree. The class M0:{Y=β01n+Xβ+ϵ}, comprising models with intercept and dispersion V(ϵ)=σ2In, is our principal focus. Unless stated otherwise, we take σ2=1.0, since variance ratios are scale-invariant. A distinction is drawn between centered and uncenteredVIFs, namely, VIFcs and VIFus, the former from columns of X centered to their means. The latter, designated as {VIFu(β̂j);j=0,1,…,k}, are diagonal elements of (X0′X0)-1 divided by reciprocals of diagonals of X0′X0 itself. These are of subsequent interest. Special distributions on ℝ+1 include the Snedecor-Fisher distribution F(·;ν1,ν2,λ) having (ν1,ν2) degrees of freedom and noncentrality λ.


## 3. Collinearity Diagnostics

Ill-conditioned models{Y=Xβ+ϵ}, burdened with difficulties as cited, trace to nonorthogonality among columns of X. To examine aspects of near collinearity, we first establish duality between design linkage parameters among columns of X, and collinearity among the OLS solutions as quantified by Hotelling's [30] canonical correlations.


### 3.1. Duality Results

Partition a genericX∈𝔽n×p as X=[X1,X2] with {X,X1,X2} of orders {(n×p),(n×r),(n×s)}, respectively, having ranks {p,r,s} such that r≤s and r+s=p<n. Accordingly, write {Y=X1β1+X2β2+ϵ}, taking β′=[β1′,β2′], and denoting by 𝒮p(X1) and 𝒮p(X2), the subspaces of ℝn spanned by columns of X1 and X2. We seek a canonical form preserving these subspaces and linkage between (X1,X2), a geometric concept independent of bases for representing 𝒮p(X1) and 𝒮p(X2). Accordingly, let G1=(X1′X1)-1/2P and G2=(X2′X2)-1/2Q, with P∈𝒪(r) and Q∈𝒪(s) to be stipulated. The original model becomes {Y=Z1α1+Z2α2+ϵ} with Z=[Z1,Z2] and α′=[α1′,α2′], such that Z1=X1G1, Z2=X2G2, α1=G1-1β1, and α2=G2-1β2.Following [31], cosines of angles between 𝒮p(X1) and 𝒮p(X2) are found as singular values generated by (X1,X2), to be designated as design linkage parameters{δ1,…,δr}. To these ends, observe that X′X in partitioned form transitions into Z′Z through (3.1)X′X=[X1′X1X1′X2X2′X1X2′X2]⟶[G1′X1′X1G1G1′X1′X2G2G2′X2′X1G1G2′X2′X2G2]=[IrP′RQQ′R′PIs]=[IrDD′Is]=Z′Z.
Here R=(X1′X1)-1/2X1′X2(X2′X2)-1/2; its singular decomposition is R = PDQ′, where D=[Dδ,0]; and elements of Dδ=Diag(δ1,…,δr) comprise the singular values of R. In particular, {ϕj=arccos(δj);1≤j≤r} defines the design linkage angles between 𝒮p(X1) and 𝒮p(X2) as subspaces of ℝn.To continue, partitionV(β̂)=Σ=[Σij] conformably with β′=[β1′,β2′],  β1∈ℝr,β2∈ℝs; designate their inner product space as (ℝr⊕ℝs,(·,·)Σ), where ℝr⊕ℝs is the direct sum and (·,·)Σ their inner product, as in Eaton [32, page 409]. Denote by {ρ1,…,ρr} Hotelling’s [30] canonical correlations. Then by Proposition  10.2 of [32], {ρ1,…,ρr} are cosines of angles between (ℝr,ℝs) as subspaces of (ℝr⊕ℝs,(·,·)Σ). In keeping with earlier usage, identify {𝒮p(β1),𝒮p(β2)} with {ℝr,ℝs}. As Hotelling's canonical correlations are invariant under affine transformations {β1→Aβ1+c1,β2→Bβ2+c2}, parameters may be redefined linearly, preserving subspaces, thus leaving the canonical correlations invariant. Retracing steps leading to the canonical design model embodied in (3.1), but now to preserve {𝒮p(β1),𝒮p(β2)}, it thus suffices to begin with the canonical model {Y=Z1α1+Z2α2+ϵ}, where V(α̂)=σ2(Z′Z)-1 with Z′Z as the rightmost matrix of (3.1).We next establish connections between the design linkage parametersDδ from (3.1), and the corresponding canonical correlations Dρ=Diag(ρ1,…,ρr), as derived eventually from Σ=(X′X)-1. A critical duality result is encoded in the following.Theorem 3.1.
Consider  the design linkage parametersDδ between  {𝒮p(X1),𝒮p(X2)} as subspaces of  ℝn and Hotelling’s [30]  canonical correlations  Dρ between  {𝒮p(β1),𝒮p(β2)} as subspaces of  (ℝr⊕ℝs,(·,·)Σ). Then  Dδ and  Dρ coincide.Proof.
In view of invariance of{ρ1,…,ρr} under nonsingular linear transformations of β1∈ℝr and of β2∈ℝs, canonical correlations between (β̂1,β̂2) proceed as in expression (3.1), but beginning instead on the left with V(α̂) = (Z′Z)-1  in lieu ofX′X. Specifically, with D = [Dδ,0], and using rules for block-partitioned inverses, we have
(3.2)(Z′Z)-1=[Ir[Dδ,0][Dδ,0]′Is]-1=[(Ir-Dδ2)-1-D(Is-D0)-1-D′(Ir-Dδ2)-1(Is-D0)-1]⟶[Ir[Dδ,0][Dδ,0]′Is],
where equality at the first step follows using DD′=Dδ2 and D′D=Diag(Dδ2,0)=D0. The succeeding step utilizes the factors (Ir-Dδ2)1/2 and (Is-D0)1/2, taking the principal diagonal blocks of (Z′Z)-1 into (Ir,Is) as in the rightmost matrix of (3.2), and its off-diagonal block from
(3.3)(Ir-Dδ2)1/2[Dδ,0](Is-D0)-1(Is-D0)1/2=[Dδ,0],
since diagonal matrices commute. But the off-diagonal block is precisely [Dρ,0], the canonical correlations between (β̂1,β̂2), to complete our proof.For subsequent reference, designateδ(X1:X2)=(δ1,…,δr) and ρ(β̂1:β̂2)=(ρ1,…,ρr). Moreover, the foregoing analysis applies for models X0=[1n,X] in M0, where r=1 and s=k as partitioned. In short, we have the following equivalences.Corollary 3.2.
(i) Consider the design linkage parameters{cos(ϕj)=δj;1≤j≤r}, gaging collinearity between  {𝒮p(X1),𝒮p(X2)} as subspaces of  ℝn and the canonical correlations  {cos(ϕj)=ρj;1≤j≤r}, between  {𝒮p(β1),𝒮p(β2)} as subspaces of  (ℝr⊕ℝs,(·,·)Σ). Then angles between these pairs of subspaces correspond one-to-one, that is,  {ϕj=arccos(δj)=arccos(ρj);1≤j≤r}.
(ii) For modelsX0=[1n,X] in M0, the element δ(1n:X)=δ1 generates the angle cos(ϕ1)=δ1 between the regressor vectors and the constant vector. Equivalently, this is given by cos(ϕ1)=ρ1=ρ(β̂0:β̂) from duality.


### 3.2. Collinearity Indices

Stewart [33] reexamined numerical aspects of ill-conditioning, to the following effects for X0=[1n,X]. Taking X0† = (X0′X0)-1X0′ as the pseudoinverse of note, and letting xj† be its jth row, each collinearity index in the collection (3.4){κj=‖xj‖⋅‖xj†‖;j=0,1,…,k}
is constructed to be scale-invariant. Clearly ∥xj†∥2 is found along the principal diagonal of [(X0†)(X0†)′] = (X0′X0)-1. In addition, the conventional VIFus are squares of the collinearity indices, that is, {VIFu(β̂j)=κj2;j=0,1,…,k}. In particular, since x0=1n in X0, we have κ02 = n∥x0†∥2.Transcending Stewart’s analysis, we connect his collinearity indices to angles between subspaces as follows. Choose a typicalxj in X0; rearrange X0 as [xj,X[j]] and similarly β′ as [βj,β[j]′]; and seek elements of (3.5){Qj′(X0′X0)-1Qj=[xj′xjxj′X[j]X[j]′xjX[j]′X[j]]-1;j=0,1,…,k}
as reordered by each permutation matrix Qj. From the clockwise rule, the (1,1) element of each inverse is (3.6){[xj′(In-Pj)xj]-1=[xj′xj-xj′Pjxj]-1=‖xj†‖2;j=0,1,…,k},
where Pj = X[j][X[j]′X[j]]-1X[j]′ is the projection operator onto the subspace 𝒮p(X[j])⊂ℝn. These relationships in turn enable us to connect {κj2;j=0,1,…,k} to the geometry of ill-conditioning as follows.Theorem 3.3.
For models in  M0, let  {VIFu(β̂j)=κj2;j=0,1,…,k} be conventional  VIFus in terms of Stewart’s collinearity indices. These in turn quantify collinearities between subspaces through angles (in  deg) as follows.
(i)
Angles between[xj,X[j]] are given by ϕj=arccos[(1-1/κj2)1/2], in succession for {j=0,1,…,k}.(ii)
Equivalently,{κj2=1/[1-δ2(xj:X[j])]=1/[1-ρ2(β̂j:β̂[j])];j=0,1,…,k}.(iii)
In particular,ϕ0=arccos[(1-1/κ02)1/2] quantifies the degree of collinearity between the regressor vectors and the constant vector.Proof.
From the geometry of the right triangle formed by(xj,Pjxj), the squared lengths satisfy ∥xj∥2=∥Pjxj∥2+RSj, where RSj=∥xj-Pjxj∥2 is the residual sum of squares from the projection. Accordingly, the principal angle between (xj,Pjxj) is given by
(3.7)cos⁡(ϕj)=xj′Pjxj‖xj‖⋅‖Pjxj‖=‖Pjxj‖‖xj‖=(1-RSj‖xj‖2)1/2=(1-1κj2)1/2
for {j=0,1,…,k}, to give conclusion (i) and conclusion (ii) by duality. Conclusion (iii) follows on specializing (x0,P0x0) with x0=1n and P0=X(X′X)-1X′, to complete our proof.Remark 3.4.
The foregoing developments specialize from Section3.1 in that the partition [xj,X[j]] always has r=1 and s=k, giving a single angle ϕj. Rules-of-thumb in common use for problematic VIFs include those exceeding 10, as in [34], or even 4 as in [35], for example. In angular measure, these correspond respectively to ϕj<18.435deg and ϕj<30.0deg.


### 3.3. Case Study 1

Consider the modelM0:{Yi=β0+β1X1+β2X2+ϵi}, the design X0=[15,X1,X2] of order (5×3), and X0′X0 and its inverse as in (3.8)X0′=[1111110.50.510-11100],X0′X0=[53132.50103],(X0′X0)-1=[0.9375-1.1250-0.3125-1.12501.75000.3750-0.31250.37500.4375].
Note first that VIFu(β̂0)=κ02=0.9375×5=4.6875. Next apply first principles to find (3.9)δ2(15:[X1,X2])=(5)-1[31][2.5003]-1[31]=0.786666,ρ2(β̂0:[β̂1,β̂2])=(0.9375)-1[-1.1250-0.3125][1.75000.37500.37500.4375]-1[-1.1250-0.3125]=0.786666,
both equal to 1-(1/κ02) as in Theorem 3.3(ii). The remaining VIFus are found directly as VIFu(β̂1)=1.7500×2.5=4.3750 and VIFu(β̂2)=0.4375×3=1.3125. Using duality and earlier findings, we further compute (3.10)δ2(X1:[15,X2])=ρ2(β̂1:[β̂0,β̂2])=1-(1κ12)=1-14.3750=0.771429,δ2(X2:[15,X1])=ρ2(β̂2:[β̂0,β̂1])=1-(1κ22)=1-11.3125=0.238095,
thereby preempting the need to undertake singular decompositions as required heretofore.


## 3.1. Duality Results

Partition a genericX∈𝔽n×p as X=[X1,X2] with {X,X1,X2} of orders {(n×p),(n×r),(n×s)}, respectively, having ranks {p,r,s} such that r≤s and r+s=p<n. Accordingly, write {Y=X1β1+X2β2+ϵ}, taking β′=[β1′,β2′], and denoting by 𝒮p(X1) and 𝒮p(X2), the subspaces of ℝn spanned by columns of X1 and X2. We seek a canonical form preserving these subspaces and linkage between (X1,X2), a geometric concept independent of bases for representing 𝒮p(X1) and 𝒮p(X2). Accordingly, let G1=(X1′X1)-1/2P and G2=(X2′X2)-1/2Q, with P∈𝒪(r) and Q∈𝒪(s) to be stipulated. The original model becomes {Y=Z1α1+Z2α2+ϵ} with Z=[Z1,Z2] and α′=[α1′,α2′], such that Z1=X1G1, Z2=X2G2, α1=G1-1β1, and α2=G2-1β2.Following [31], cosines of angles between 𝒮p(X1) and 𝒮p(X2) are found as singular values generated by (X1,X2), to be designated as design linkage parameters{δ1,…,δr}. To these ends, observe that X′X in partitioned form transitions into Z′Z through (3.1)X′X=[X1′X1X1′X2X2′X1X2′X2]⟶[G1′X1′X1G1G1′X1′X2G2G2′X2′X1G1G2′X2′X2G2]=[IrP′RQQ′R′PIs]=[IrDD′Is]=Z′Z.
Here R=(X1′X1)-1/2X1′X2(X2′X2)-1/2; its singular decomposition is R = PDQ′, where D=[Dδ,0]; and elements of Dδ=Diag(δ1,…,δr) comprise the singular values of R. In particular, {ϕj=arccos(δj);1≤j≤r} defines the design linkage angles between 𝒮p(X1) and 𝒮p(X2) as subspaces of ℝn.To continue, partitionV(β̂)=Σ=[Σij] conformably with β′=[β1′,β2′],  β1∈ℝr,β2∈ℝs; designate their inner product space as (ℝr⊕ℝs,(·,·)Σ), where ℝr⊕ℝs is the direct sum and (·,·)Σ their inner product, as in Eaton [32, page 409]. Denote by {ρ1,…,ρr} Hotelling’s [30] canonical correlations. Then by Proposition  10.2 of [32], {ρ1,…,ρr} are cosines of angles between (ℝr,ℝs) as subspaces of (ℝr⊕ℝs,(·,·)Σ). In keeping with earlier usage, identify {𝒮p(β1),𝒮p(β2)} with {ℝr,ℝs}. As Hotelling's canonical correlations are invariant under affine transformations {β1→Aβ1+c1,β2→Bβ2+c2}, parameters may be redefined linearly, preserving subspaces, thus leaving the canonical correlations invariant. Retracing steps leading to the canonical design model embodied in (3.1), but now to preserve {𝒮p(β1),𝒮p(β2)}, it thus suffices to begin with the canonical model {Y=Z1α1+Z2α2+ϵ}, where V(α̂)=σ2(Z′Z)-1 with Z′Z as the rightmost matrix of (3.1).We next establish connections between the design linkage parametersDδ from (3.1), and the corresponding canonical correlations Dρ=Diag(ρ1,…,ρr), as derived eventually from Σ=(X′X)-1. A critical duality result is encoded in the following.Theorem 3.1.
Consider  the design linkage parametersDδ between  {𝒮p(X1),𝒮p(X2)} as subspaces of  ℝn and Hotelling’s [30]  canonical correlations  Dρ between  {𝒮p(β1),𝒮p(β2)} as subspaces of  (ℝr⊕ℝs,(·,·)Σ). Then  Dδ and  Dρ coincide.Proof.
In view of invariance of{ρ1,…,ρr} under nonsingular linear transformations of β1∈ℝr and of β2∈ℝs, canonical correlations between (β̂1,β̂2) proceed as in expression (3.1), but beginning instead on the left with V(α̂) = (Z′Z)-1  in lieu ofX′X. Specifically, with D = [Dδ,0], and using rules for block-partitioned inverses, we have
(3.2)(Z′Z)-1=[Ir[Dδ,0][Dδ,0]′Is]-1=[(Ir-Dδ2)-1-D(Is-D0)-1-D′(Ir-Dδ2)-1(Is-D0)-1]⟶[Ir[Dδ,0][Dδ,0]′Is],
where equality at the first step follows using DD′=Dδ2 and D′D=Diag(Dδ2,0)=D0. The succeeding step utilizes the factors (Ir-Dδ2)1/2 and (Is-D0)1/2, taking the principal diagonal blocks of (Z′Z)-1 into (Ir,Is) as in the rightmost matrix of (3.2), and its off-diagonal block from
(3.3)(Ir-Dδ2)1/2[Dδ,0](Is-D0)-1(Is-D0)1/2=[Dδ,0],
since diagonal matrices commute. But the off-diagonal block is precisely [Dρ,0], the canonical correlations between (β̂1,β̂2), to complete our proof.For subsequent reference, designateδ(X1:X2)=(δ1,…,δr) and ρ(β̂1:β̂2)=(ρ1,…,ρr). Moreover, the foregoing analysis applies for models X0=[1n,X] in M0, where r=1 and s=k as partitioned. In short, we have the following equivalences.Corollary 3.2.
(i) Consider the design linkage parameters{cos(ϕj)=δj;1≤j≤r}, gaging collinearity between  {𝒮p(X1),𝒮p(X2)} as subspaces of  ℝn and the canonical correlations  {cos(ϕj)=ρj;1≤j≤r}, between  {𝒮p(β1),𝒮p(β2)} as subspaces of  (ℝr⊕ℝs,(·,·)Σ). Then angles between these pairs of subspaces correspond one-to-one, that is,  {ϕj=arccos(δj)=arccos(ρj);1≤j≤r}.
(ii) For modelsX0=[1n,X] in M0, the element δ(1n:X)=δ1 generates the angle cos(ϕ1)=δ1 between the regressor vectors and the constant vector. Equivalently, this is given by cos(ϕ1)=ρ1=ρ(β̂0:β̂) from duality.


## 3.2. Collinearity Indices

Stewart [33] reexamined numerical aspects of ill-conditioning, to the following effects for X0=[1n,X]. Taking X0† = (X0′X0)-1X0′ as the pseudoinverse of note, and letting xj† be its jth row, each collinearity index in the collection (3.4){κj=‖xj‖⋅‖xj†‖;j=0,1,…,k}
is constructed to be scale-invariant. Clearly ∥xj†∥2 is found along the principal diagonal of [(X0†)(X0†)′] = (X0′X0)-1. In addition, the conventional VIFus are squares of the collinearity indices, that is, {VIFu(β̂j)=κj2;j=0,1,…,k}. In particular, since x0=1n in X0, we have κ02 = n∥x0†∥2.Transcending Stewart’s analysis, we connect his collinearity indices to angles between subspaces as follows. Choose a typicalxj in X0; rearrange X0 as [xj,X[j]] and similarly β′ as [βj,β[j]′]; and seek elements of (3.5){Qj′(X0′X0)-1Qj=[xj′xjxj′X[j]X[j]′xjX[j]′X[j]]-1;j=0,1,…,k}
as reordered by each permutation matrix Qj. From the clockwise rule, the (1,1) element of each inverse is (3.6){[xj′(In-Pj)xj]-1=[xj′xj-xj′Pjxj]-1=‖xj†‖2;j=0,1,…,k},
where Pj = X[j][X[j]′X[j]]-1X[j]′ is the projection operator onto the subspace 𝒮p(X[j])⊂ℝn. These relationships in turn enable us to connect {κj2;j=0,1,…,k} to the geometry of ill-conditioning as follows.Theorem 3.3.
For models in  M0, let  {VIFu(β̂j)=κj2;j=0,1,…,k} be conventional  VIFus in terms of Stewart’s collinearity indices. These in turn quantify collinearities between subspaces through angles (in  deg) as follows.
(i)
Angles between[xj,X[j]] are given by ϕj=arccos[(1-1/κj2)1/2], in succession for {j=0,1,…,k}.(ii)
Equivalently,{κj2=1/[1-δ2(xj:X[j])]=1/[1-ρ2(β̂j:β̂[j])];j=0,1,…,k}.(iii)
In particular,ϕ0=arccos[(1-1/κ02)1/2] quantifies the degree of collinearity between the regressor vectors and the constant vector.Proof.
From the geometry of the right triangle formed by(xj,Pjxj), the squared lengths satisfy ∥xj∥2=∥Pjxj∥2+RSj, where RSj=∥xj-Pjxj∥2 is the residual sum of squares from the projection. Accordingly, the principal angle between (xj,Pjxj) is given by
(3.7)cos⁡(ϕj)=xj′Pjxj‖xj‖⋅‖Pjxj‖=‖Pjxj‖‖xj‖=(1-RSj‖xj‖2)1/2=(1-1κj2)1/2
for {j=0,1,…,k}, to give conclusion (i) and conclusion (ii) by duality. Conclusion (iii) follows on specializing (x0,P0x0) with x0=1n and P0=X(X′X)-1X′, to complete our proof.Remark 3.4.
The foregoing developments specialize from Section3.1 in that the partition [xj,X[j]] always has r=1 and s=k, giving a single angle ϕj. Rules-of-thumb in common use for problematic VIFs include those exceeding 10, as in [34], or even 4 as in [35], for example. In angular measure, these correspond respectively to ϕj<18.435deg and ϕj<30.0deg.


## 3.3. Case Study 1

Consider the modelM0:{Yi=β0+β1X1+β2X2+ϵi}, the design X0=[15,X1,X2] of order (5×3), and X0′X0 and its inverse as in (3.8)X0′=[1111110.50.510-11100],X0′X0=[53132.50103],(X0′X0)-1=[0.9375-1.1250-0.3125-1.12501.75000.3750-0.31250.37500.4375].
Note first that VIFu(β̂0)=κ02=0.9375×5=4.6875. Next apply first principles to find (3.9)δ2(15:[X1,X2])=(5)-1[31][2.5003]-1[31]=0.786666,ρ2(β̂0:[β̂1,β̂2])=(0.9375)-1[-1.1250-0.3125][1.75000.37500.37500.4375]-1[-1.1250-0.3125]=0.786666,
both equal to 1-(1/κ02) as in Theorem 3.3(ii). The remaining VIFus are found directly as VIFu(β̂1)=1.7500×2.5=4.3750 and VIFu(β̂2)=0.4375×3=1.3125. Using duality and earlier findings, we further compute (3.10)δ2(X1:[15,X2])=ρ2(β̂1:[β̂0,β̂2])=1-(1κ12)=1-14.3750=0.771429,δ2(X2:[15,X1])=ρ2(β̂2:[β̂0,β̂1])=1-(1κ22)=1-11.3125=0.238095,
thereby preempting the need to undertake singular decompositions as required heretofore.


## 4. Determinant Identities

### 4.1. Background

The generalized variance, as a design criterion for{Y=Xβ+ϵ}, rests in part on the geometry of ellipsoids of the type (4.1)R(β)={β∈Rk:(β̂-β)′X′X(β̂-β)≤c2}.
Choices for c2 in common usage give first (i) a confidence region for β, whose normal-theory confidence coefficient is 1-α on taking c2=S2cα2, with S2 as the residual mean square and cα2 the 100(1-α) percentage point of F(·;k,n-k); and otherwise admitting a lower Chebychev bound as in [36, page  92]. The alternative choice c2=k+2 gives (ii) Cramér’s [37] ellipsoid of concentration for β̂, that is, the measure uniform over R(β) having the same mean and dispersion matrix as β̂. The generalized variance GV(β̂)=|V(β̂)| is proportional to the squared volumes of these ellipsoids, smaller volumes reflecting tighter concentrations.


### 4.2. Factorizations

To continue, let someT(Y)=θ̂∈ℝk be random having E(θ̂)=θ and V(θ̂)=Σ∈𝕊k+; partition θ′=[θ1′,θ2′] and Σ=[Σij] conformably, with θ1∈ℝr and θ2∈ℝs such that r≤s and r+s=k; and let G(θ̂)=|Σ|1/k. The canonical correlations [30], as singular values of Σ11-1/2Σ12Σ22-1/2, are now to be designated as ρ(1:2) = [ρ1,…,ρr],  in lieu ofρ(β̂1:β̂2), and to be ordered as {ρ1≥ρ2≥⋯≥ρr≥0}. Moreover, the quantity γ(1:2)=Πi=1r(1-ρi2) is the Vector Alienation Coefficient of Hotelling [30]. The factorization |Σ|=|Σ11||Σ22| for Σ=Diag(Σ11,Σ22) extends directly as an upper bound for any Σ=[Σij], with further ramifications as follows.Theorem 4.1.
Considerθ̂′=[θ̂1′,θ̂2′]∈ℝk having  E(θ̂′)=[θ1′,θ2′] and  V(θ̂)=[Σij], such that  θ1∈ℝr and  θ2∈ℝs with  r≤s and  r+s=k.
(i)
The determinant ofΣ = [Σij] admits the factorization
(4.2)|𝚺|=|𝚺11||𝚺22|γ(1:2)
such that |Σ|≤|Σ11||Σ22| and γ(1:2)=Πi=1r(1-ρi2)≤1.(ii)
IfΣ=Diag(Σ11,Σ22), then G(θ̂) is the geometric mean G(θ̂)=[G(θ̂1)]r/k[G(θ̂2)]s/k of the quantities G(θ̂1) and G(θ̂2).(iii)
Generally, for anyΣ, the quantity G(θ̂) becomes
(4.3)G(θ̂)=[G(θ̂1)]r/k[G(θ̂2)]s/k[γ(1:2)]1/k
in terms of {G(θ̂1),G(θ̂2),γ(1:2)}.(iv)
Ifθ̂=[θ̂1′,θ̂2′,θ̂3]′ and Σ=[Σij;1≤i,j≤3] are partitioned conformably, with θ̂1∈ℝr,  θ̂2∈ℝs, and θ̂3∈ℝt, such that r+s+t=k, then G(θ̂) admits the factorization
(4.4)G(θ̂)=[G(θ̂1)]r/k[G(θ̂2)]s/k[G(θ̂3)]t/k[γ(1:23)γ(2:3)]1/k,
with γ(1:23) and γ(2:3) as the Vector Alienation Coefficients between {θ̂1,[θ̂2,θ̂3]} and between {θ̂2,θ̂3}, respectively.Proof.
As in Section3.1 with R=Σ11-1/2Σ12Σ22-1/2=PDQ′ and D=[Dρ,0], we have
(4.5)[𝚺11𝚺12𝚺21𝚺22]=[W100W2][IrDD′Is][W1′00W2′]
with W1=Σ111/2P and W2=Σ221/2Q. The middle factor on the right has determinant |Ir-DD′|=Πi=1r(1-ρi2) from the clockwise rule, so that |Σ|=|Σ11||Σ22|Πi=1r(1-ρi2) to give conclusion (i). Conclusion (ii) follows directly from G(θ̂)=[GV(θ̂)]1/k, and conclusion (iii) on combining (i) and (ii). Conclusion (iv) now follows on applying (iii) twice, first on partitioning θ̂ into {θ̂1,[θ̂2,θ̂3]}, whose canonical correlations are ρ(1:23), then [θ̂2,θ̂3] into {θ̂2,θ̂3} having canonical correlations ρ(2:3), to complete our proof.Remark 4.2.
In short, Theorem4.1 links determinants and principal subdeterminants precisely through angles between subspaces. Moreover, arguments leading to conclusion (iv) may be iterated recursively to achieve a hierarchical decomposition for four or more factors, as in the following with k=r+s+t+v, namely,
(4.6)G(θ̂)=[G(θ̂1)]r/k[G(θ̂2)]s/k[G(θ̂3)]t/k[G(θ̂4)]v/k[γ(1:234)γ(2:34)γ(3:4)]1/k.Remark 4.3.
Hotelling’s [30] Vector Alienation Coefficient γ(1:2)=Πi=1r(1-ρi2) is a composite index of linkage between {𝒮p(β1),𝒮p(β2)} as subspaces of (ℝr⊕ℝs,(·,·)Σ), decreasing in each {ρi2;1≤i≤r}. Equivalently, duality asserts that γ(1:2)=Πi=1r(1-δi2) is the identical composite index of linkage between {𝒮p(X1),𝒮p(X2)} as subspaces of ℝn.Theorem4.1 anticipates that Ds-inefficient subset estimators may be masked in a design exhibiting good overall D-efficiency. Conversely, a Ds-inefficient subset may contraindicate, incorrectly, the overall D-efficiency of a design. Details are provided in case studies to follow.


## 4.1. Background

The generalized variance, as a design criterion for{Y=Xβ+ϵ}, rests in part on the geometry of ellipsoids of the type (4.1)R(β)={β∈Rk:(β̂-β)′X′X(β̂-β)≤c2}.
Choices for c2 in common usage give first (i) a confidence region for β, whose normal-theory confidence coefficient is 1-α on taking c2=S2cα2, with S2 as the residual mean square and cα2 the 100(1-α) percentage point of F(·;k,n-k); and otherwise admitting a lower Chebychev bound as in [36, page  92]. The alternative choice c2=k+2 gives (ii) Cramér’s [37] ellipsoid of concentration for β̂, that is, the measure uniform over R(β) having the same mean and dispersion matrix as β̂. The generalized variance GV(β̂)=|V(β̂)| is proportional to the squared volumes of these ellipsoids, smaller volumes reflecting tighter concentrations.


## 4.2. Factorizations

To continue, let someT(Y)=θ̂∈ℝk be random having E(θ̂)=θ and V(θ̂)=Σ∈𝕊k+; partition θ′=[θ1′,θ2′] and Σ=[Σij] conformably, with θ1∈ℝr and θ2∈ℝs such that r≤s and r+s=k; and let G(θ̂)=|Σ|1/k. The canonical correlations [30], as singular values of Σ11-1/2Σ12Σ22-1/2, are now to be designated as ρ(1:2) = [ρ1,…,ρr],  in lieu ofρ(β̂1:β̂2), and to be ordered as {ρ1≥ρ2≥⋯≥ρr≥0}. Moreover, the quantity γ(1:2)=Πi=1r(1-ρi2) is the Vector Alienation Coefficient of Hotelling [30]. The factorization |Σ|=|Σ11||Σ22| for Σ=Diag(Σ11,Σ22) extends directly as an upper bound for any Σ=[Σij], with further ramifications as follows.Theorem 4.1.
Considerθ̂′=[θ̂1′,θ̂2′]∈ℝk having  E(θ̂′)=[θ1′,θ2′] and  V(θ̂)=[Σij], such that  θ1∈ℝr and  θ2∈ℝs with  r≤s and  r+s=k.
(i)
The determinant ofΣ = [Σij] admits the factorization
(4.2)|𝚺|=|𝚺11||𝚺22|γ(1:2)
such that |Σ|≤|Σ11||Σ22| and γ(1:2)=Πi=1r(1-ρi2)≤1.(ii)
IfΣ=Diag(Σ11,Σ22), then G(θ̂) is the geometric mean G(θ̂)=[G(θ̂1)]r/k[G(θ̂2)]s/k of the quantities G(θ̂1) and G(θ̂2).(iii)
Generally, for anyΣ, the quantity G(θ̂) becomes
(4.3)G(θ̂)=[G(θ̂1)]r/k[G(θ̂2)]s/k[γ(1:2)]1/k
in terms of {G(θ̂1),G(θ̂2),γ(1:2)}.(iv)
Ifθ̂=[θ̂1′,θ̂2′,θ̂3]′ and Σ=[Σij;1≤i,j≤3] are partitioned conformably, with θ̂1∈ℝr,  θ̂2∈ℝs, and θ̂3∈ℝt, such that r+s+t=k, then G(θ̂) admits the factorization
(4.4)G(θ̂)=[G(θ̂1)]r/k[G(θ̂2)]s/k[G(θ̂3)]t/k[γ(1:23)γ(2:3)]1/k,
with γ(1:23) and γ(2:3) as the Vector Alienation Coefficients between {θ̂1,[θ̂2,θ̂3]} and between {θ̂2,θ̂3}, respectively.Proof.
As in Section3.1 with R=Σ11-1/2Σ12Σ22-1/2=PDQ′ and D=[Dρ,0], we have
(4.5)[𝚺11𝚺12𝚺21𝚺22]=[W100W2][IrDD′Is][W1′00W2′]
with W1=Σ111/2P and W2=Σ221/2Q. The middle factor on the right has determinant |Ir-DD′|=Πi=1r(1-ρi2) from the clockwise rule, so that |Σ|=|Σ11||Σ22|Πi=1r(1-ρi2) to give conclusion (i). Conclusion (ii) follows directly from G(θ̂)=[GV(θ̂)]1/k, and conclusion (iii) on combining (i) and (ii). Conclusion (iv) now follows on applying (iii) twice, first on partitioning θ̂ into {θ̂1,[θ̂2,θ̂3]}, whose canonical correlations are ρ(1:23), then [θ̂2,θ̂3] into {θ̂2,θ̂3} having canonical correlations ρ(2:3), to complete our proof.Remark 4.2.
In short, Theorem4.1 links determinants and principal subdeterminants precisely through angles between subspaces. Moreover, arguments leading to conclusion (iv) may be iterated recursively to achieve a hierarchical decomposition for four or more factors, as in the following with k=r+s+t+v, namely,
(4.6)G(θ̂)=[G(θ̂1)]r/k[G(θ̂2)]s/k[G(θ̂3)]t/k[G(θ̂4)]v/k[γ(1:234)γ(2:34)γ(3:4)]1/k.Remark 4.3.
Hotelling’s [30] Vector Alienation Coefficient γ(1:2)=Πi=1r(1-ρi2) is a composite index of linkage between {𝒮p(β1),𝒮p(β2)} as subspaces of (ℝr⊕ℝs,(·,·)Σ), decreasing in each {ρi2;1≤i≤r}. Equivalently, duality asserts that γ(1:2)=Πi=1r(1-δi2) is the identical composite index of linkage between {𝒮p(X1),𝒮p(X2)} as subspaces of ℝn.Theorem4.1 anticipates that Ds-inefficient subset estimators may be masked in a design exhibiting good overall D-efficiency. Conversely, a Ds-inefficient subset may contraindicate, incorrectly, the overall D-efficiency of a design. Details are provided in case studies to follow.


## 5. Case Studies

### 5.1. The Setting

Our tools are informative in input-output studies. In particular, specify{Y=X0β+ϵ} as a second-order model Y(x1,x2,x3) in three regressors and p=10 parameters, namely, (5.1){Yi=β0+β1xi1+β2xi2+β3xi3+β11xi12+β22xi22+β33xi32+β12xi1xi2+β13xi1xi3+β23xi2xi3+ϵi;i=1,2,…,n}.
Next partition β′=[β0,βL′,βQ′,βI′] with βL′=[β1,β2,β3] as slopes; βQ′=[β11,β22,β33] as pure quadratic terms reflecting diminishing (-) or increasing (+) returns to inputs; and βI′=[β12,β13,β23] as interactive terms reflecting synergistic (+) or antagonistic (-) effects for pairs of regressors in combination. Further let βM′=[βL′,βQ′,βI′] exclusive of β0, the latter a base line for Y(0,0,0). We proceed under conventional homogeneous and uncorrelated errors, the minimizing solution β̂=(X0′X0)-1X0′Y being unbiased with V(β̂)=σ2(X0′X0)-1. We take σ2, although unknown, to reflect natural variability in experimental materials and protocol, and thus applicable in a given setting independently of the choice of design. Accordingly, for present purposes we may standardize to σ2=1.0 for reasons cited earlier.


### 5.2. The Designs

Early polynomial response designs made use of factorial experiments, setting levels as needed to meet the required degree. For example, the second-order model (5.1) in three regressors would require 33=27 runs. However, in the early 1950s such designs were seen to be excessive, in carrying redundant interactions beyond the pairs required in the model (5.1). In industrial and other settings where parsimony is desired, several small second-order designs have evolved, often on appending a few additional runs to two-level factorials or fractions thereof.Eight such small designs of note here are the hybrids(H310,H311B) of [38], the small composite SCD [39], the BBD [40], the central composite rotatable design CCD [41], and designs ND [42], HD [43], and BDD [44]. The designs [H310,H311B,SCD,BBD,CCD,ND,HD,BDD] have numbers of runs as [11,11,11,13,15,11,11,11], respectively. These follow on adding a center run to all but design ND, rendering all as unsaturated having at least one degree of freedom for error. Specifically, the design ND of [42] already has 11 runs and is unsaturated. All designs have been scaled to span the same range for each regressor; and none strictly dominates another under the positive definite dispersion ordering. All determinants as listed derive from the respective V(β̂) = (X0′X0)-1 and its submatrices. Subset efficiencies for {βL,βQ,βI} were examined in [45] for selected designs using criteria other than D- and Ds-efficiencies. Our usage here, as elsewhere in the literature, considers GV(β̂) and G(β̂) to be efficiency indices for β̂ specific to a particular design, to include subsets {β̂i;i∈I}, and smaller values reflect greater efficiencies through smaller volumes of concentration ellipsoids. On the other hand, the comparative efficiencies of two designs for estimating β or {βi;i∈I} are found as ratios of these quantities.


### 5.3. Numerical Studies

Details for these designs are listed in the accompanying tables. Table1 gives values G(·)=[GV(·)]1/dim for β̂ and selected subsets, with dim as the order of the determinant. Also listed are angles ϕ(1n:X)deg between regressors and the constant, to be noted subsequently. Table 2 displays the squared canonical correlations ρ2(β̂i:β̂j) between designated subsets, and Table 3 the corresponding Vector Alienation Coefficient γ(β̂i:β̂j)=Πi=1r(1-ρi2), for specified pairs. Here {0L,QI} refers to the pair {[β̂0,β̂L],[β̂Q,β̂I]}, for example. Moreover, values of the composite indices γ(β̂i:β̂j) = γ(Xi:Xj), if much less than unity, serve to alert the user as to potential problems with ill-conditioning.Table 1
RootsG(β̂) and G(β̂i) of generalized variances for β̂ and subset estimators {β̂i∈ℝt}, and angles ϕ(1n:X)deg between regressors and the constant, against reference values ϕ**<18.44 and ϕ*<30.00, in eight small second-order designs.
DesignG(β̂)G(β̂0)G(β̂L)G(β̂Q)G(β̂I)G(β̂Q,β̂I)G(β̂M)ϕ(1n:X)H3101.499165.04771.188212.080261.610451.830341.5848425.11*H311B1.2805010.00001.00001.176801.999971.534131.3301717.55**SCD1.5387610.00001.666671.042104.166672.083761.6314217.55**BBD1.1053310.00000.833331.645831.111111.352291.1507716.10**CCD0.9372510.00000.714291.033001.250001.136330.9734114.96**ND2.030638.75001.250007.800311.250003.122562.2203018.80*HD2.332585.00001.911896.613131.952753.574442.5900425.24*BDD2.218354.64771.588094.966252.257523.226732.4495026.25*Table 2
Squared canonical correlations between designated subsets(β̂i,β̂j) of estimators for eight small second-order designs.
SubsetsH310H311BSCDBBDCCDNDHDBDD{0L,QI}0.819900.909090.909090.923080.933330.893330.794760.813040.400000.111110.542610.117210.400000.111110.111110.117210.400000.111110.03524{L,Q}0.111110.111110.064610.111110.111110.014130.083330.006310.01413{L,QI}0.400000.111110.544560.117210.400000.111110.111110.117210.400000.083330.111110.09890{Q,I}0.031530.104180.104180.00196{0,L}0.107140.117900.02281{0,Q}0.819900.909090.909090.923080.933330.892860.641030.79815{0,I}0.049180.00220{0,LQI}0.819900.909090.909090.923080.933330.896100.818180.80440Table 3
Vector Alienation Coefficientsγ(β̂i:β̂j) between subsets (β̂i,β̂j) of estimators, and the factor abc=[γ(0:LQI)γ(L:QI)γ(Q:I)]1/10, for each of eight small second-order designs.
SubsetsH310H311BSCDBBDCCDNDHDBDD{0L,QI}0.180100.090910.019640.076920.066670.084280.074170.14057{L,Q}1.000001.000001.000001.000001.000000.724280.785140.90914{L,QI}1.000001.000000.216001.000001.000000.724280.359860.70225{Q,I}1.000001.000001.000001.000001.000001.000000.968470.80092{0,L}1.000001.000001.000001.000001.000000.892860.882100.97719{0,Q}0.180100.090910.090910.076920.066670.107140.358970.20185{0,I}1.000001.000001.000001.000001.000001.000000.950820.99780{0,LQI}0.180100.090910.090910.076920.066670.103900.181820.19566abc0.842460.786790.675000.773760.762770.772060.758900.80197


#### 5.3.1. An Overview

To fix ideas, observe for theCCD that G(β̂Q,β̂I)=1.13633, G(β̂Q)=1.03300, and G(β̂I)=1.25000 from Table 1. These not only are comparable in magnitude, but are commensurate, in having been adjusted for dimensions and thus homogeneous of unit degree, as are all entries in Table 1. Moreover, since (β̂Q,β̂I) are uncorrelated and γ(β̂Q,β̂I) = 1.0 for the CCD from Table 3, G(β̂Q,β̂I) is the geometric mean 1.13633=(1.03300)3/6(1.25000)3/6 from Theorem 4.1(ii). A further rough spot check of Table 1 may be summarized as follows.Summary Properties
(P1)
Compared withG(β̂), values for G(β̂0) appear excessive throughout.(P2)
Values forG(β̂L) are roughly comparable across designs.(P3)
The eight designs sort essentially into two groups.(P4)
Designs{H310,H311B,SCD,BBD,CCD} overall are comparatively D- and Ds-efficient, with the noted exception being G(β̂I)=4.16667 for the SCD.(P5)
The designs{ND,HD,BDD} are considerably less D-efficient, with their generalized variances GV(β̂) being {1192.09,4768.37,2886.03}, respectively, in comparison with {57.342,11.852,74.422,2.722,0.523} for the remaining designs; and each of the former is burdened by unequivocal Ds-inefficiency for βQ, to be treated subsequently.


#### 5.3.2. Further Details

We next examine Hartley’s [39] SCD in some detail, first in terms of generalized variances. Values for GV(β̂),  GV(β̂0), and GV(β̂M) appear in the first row of Table 4, along with γ(β̂0:β̂M)=(1.0-0.909090)=0.090909 using ρ2(0:M)=0.909090 from Table 2. Theorem 4.1 (i) now asserts that GV(β̂)=GV(β̂0)GV(β̂M)γ(β̂0:β̂M), as verified numerically through 74.4216=10.00(81.8638)(0.090909). In a similar manner, β̂M partitions into {β̂L,[β̂Q,β̂I]}, where GV(β̂L)=4.6296 and GV([β̂Q,β̂I])=81.8638 from Table 4. The squared canonical correlations between {β̂L,[β̂Q,β̂I]} are ρ2(L:QI)=[0.4000,0.4000,0.4000]′ from Table 2, so that γ(L:QI)=0.21600 as in Table 3. Theorem 4.1 (i) again recovers GV(β̂M) as 81.8638=4.6926(81.8638)(0.21600) since G(β̂L) and γ(L:QI) are reciprocals in this instance. Moreover, GV(β̂Q,β̂I)=GV(β̂Q)GV(β̂I)γ(Q:I) translates into 81.8638=1.1317(72.3379)(1.0), where γ(Q:I)=1.0 since elements of {β̂Q,β̂I} are mutually uncorrelated from Table 2. In summary, the value G(β̂) for the SCD admits the factorization of (4.6), on identifying {θ̂1,θ̂2,θ̂3,θ̂4} with {β̂0,β̂L,β̂Q,β̂I}, respectively, given numerically from Tables 1 and 3 as (5.2)1.53876=(10.00)1/10(1.66667)3/10(1.04210)3/10(4.16667)3/10[(0.090909)(0.21600)(1.0)]1/10.Table 4
Generalized variances and Vector Alienation Coefficients between designated subsets for Hartley’s [39] SCD in k=3 regressors.
GV(β̂)GV(β̂0)GV(β̂M)γ(0:LQI))=(1-ρ12)74.421610.000081.86380.090909GV(β̂M)GV(β̂L)GV(β̂Q,β̂I)γ(L:QI)=Πi=1r(1-ρi2)81.86384.629681.86380.216000GV(β̂Q,β̂I)GV(β̂Q)GV(β̂I)γ(Q:I)=Πi=1r(1-ρi2)81.86381.131772.33791.000000Corresponding factorizations proceed similarly for other designs. Details are left to the reader, but values for[γ(0:LQI)γ(L:QI)γ(Q:I)]1/10, the rightmost factor of (5.2), are supplied for each design as the final row of Table 3. Although the tables, together with Theorem 4.1, support other factorizations, the one featured here seems most natural in terms of the parameters {β0,βL,βQ,βI}, together with their central roles in identifying noteworthy treatment effects in second-order models.


### 5.4. Masking

TheD-efficiency index of the SCD, at GV(β̂)=74.4216, is larger but roughly comparable to that of H310 at GV(β̂)=57.3418. What cannot be anticipated from these facts alone, however, is that the (3×3) determinant GV(β̂I)=72.3379 for the SCD is comparable to its (10×10) determinant GV(β̂)=74.4216, despite their disparate dimensions. Adjusting for dimensions gives G(β̂)=(74.4216)1/10=1.53876 and G(β̂I)=(72.3379)1/3=4.16667 for the SCD. This illustrates the masking of a remarkably inefficient estimator for βI, despite the value G(β̂)=1.53876 in estimating all parameters. This masking stems from the nonorthogonality of subset estimators as reflected in their canonical correlations and Vector Alienation Coefficients. In contrast are the corresponding commensurate values for the H310 design, namely, G(β̂)=(57.3418)1/10=1.49916 and G(β̂I)=(4.1768)1/3=1.61045. It may be noted that the condition number c1(X0′X0) is 21.59 for H310, with the somewhat larger value 54.01 for the SCD.We next examine theDs-inefficiencies of ND and HD for βQ as noted earlier, with G(β̂Q) taking values 7.80031 and 6.61313, respectively. Our reference for masking is G(β̂L,β̂Q). These values are not listed in Table 1, but may be recovered from Tables 2 and 3 as follows. Specifically, for ND we have (5.3)[GV(β̂L)GV(β̂Q)γ(L:Q)]1/6=[GV(β̂L,β̂Q)]1/6=G(β̂L,β̂Q),(1.25000)3/6(7.80031)3/6(0.72428)1/6=2.95912,
where neither G(β̂L)=1.25000 nor G(β̂L,β̂Q)=2.95912 appears excessive. In consequence, that G(β̂Q)=7.80031 is excessive would be masked on examining G(β̂L) and G(β̂L,β̂Q) only. Parallel steps for HD give the factorization (1.91189)3/6(6.61313)3/6(0.78514)1/6 = 3.41528, with similar conclusions in regard to masking.


### 5.5. Collinearity with the Constant

Advocates for these and other small designs have focused onD,Ds, and other efficiency criteria, as well as the parsimony of small designs and their advantage in industrial experiments. To the knowledge of this observer, none has considered prospects for ill-conditioning and its consequences, despite the fact that columns of X are necessarily inter-linked as a consequence of second-order from first-order effects. Nonetheless, from Section 3.1 and Corollary 3.2, we may compute angles between the constant vector and the span of the regressors using duality together with the information at hand. This may prove to be critical in view of the admonition [29] that “collinearity with the intercept can quite generally corrupt the estimates of all parameters in the model.” As noted in Remark 3.4, rules-of-thumb for problematic VIFs include those exceeding 10 or 4 or, in angular measure, ϕ**<18.435deg and ϕ*<30.00deg. From the last row of Table 2, the angles ϕ(1n:X) have been computed for each of the eight designs, as listed in the final column of Table 1. For example, arccos(0.81990)=25.1116deg for H310. It is seen that all designs are flagged as potentially problematic using rules-of-thumb as cited. This adds yet another layer of concerns, heretofore unrecognized, in seeking further to implement these designs already in wide usage.


## 5.1. The Setting

Our tools are informative in input-output studies. In particular, specify{Y=X0β+ϵ} as a second-order model Y(x1,x2,x3) in three regressors and p=10 parameters, namely, (5.1){Yi=β0+β1xi1+β2xi2+β3xi3+β11xi12+β22xi22+β33xi32+β12xi1xi2+β13xi1xi3+β23xi2xi3+ϵi;i=1,2,…,n}.
Next partition β′=[β0,βL′,βQ′,βI′] with βL′=[β1,β2,β3] as slopes; βQ′=[β11,β22,β33] as pure quadratic terms reflecting diminishing (-) or increasing (+) returns to inputs; and βI′=[β12,β13,β23] as interactive terms reflecting synergistic (+) or antagonistic (-) effects for pairs of regressors in combination. Further let βM′=[βL′,βQ′,βI′] exclusive of β0, the latter a base line for Y(0,0,0). We proceed under conventional homogeneous and uncorrelated errors, the minimizing solution β̂=(X0′X0)-1X0′Y being unbiased with V(β̂)=σ2(X0′X0)-1. We take σ2, although unknown, to reflect natural variability in experimental materials and protocol, and thus applicable in a given setting independently of the choice of design. Accordingly, for present purposes we may standardize to σ2=1.0 for reasons cited earlier.


## 5.2. The Designs

Early polynomial response designs made use of factorial experiments, setting levels as needed to meet the required degree. For example, the second-order model (5.1) in three regressors would require 33=27 runs. However, in the early 1950s such designs were seen to be excessive, in carrying redundant interactions beyond the pairs required in the model (5.1). In industrial and other settings where parsimony is desired, several small second-order designs have evolved, often on appending a few additional runs to two-level factorials or fractions thereof.Eight such small designs of note here are the hybrids(H310,H311B) of [38], the small composite SCD [39], the BBD [40], the central composite rotatable design CCD [41], and designs ND [42], HD [43], and BDD [44]. The designs [H310,H311B,SCD,BBD,CCD,ND,HD,BDD] have numbers of runs as [11,11,11,13,15,11,11,11], respectively. These follow on adding a center run to all but design ND, rendering all as unsaturated having at least one degree of freedom for error. Specifically, the design ND of [42] already has 11 runs and is unsaturated. All designs have been scaled to span the same range for each regressor; and none strictly dominates another under the positive definite dispersion ordering. All determinants as listed derive from the respective V(β̂) = (X0′X0)-1 and its submatrices. Subset efficiencies for {βL,βQ,βI} were examined in [45] for selected designs using criteria other than D- and Ds-efficiencies. Our usage here, as elsewhere in the literature, considers GV(β̂) and G(β̂) to be efficiency indices for β̂ specific to a particular design, to include subsets {β̂i;i∈I}, and smaller values reflect greater efficiencies through smaller volumes of concentration ellipsoids. On the other hand, the comparative efficiencies of two designs for estimating β or {βi;i∈I} are found as ratios of these quantities.


## 5.3. Numerical Studies

Details for these designs are listed in the accompanying tables. Table1 gives values G(·)=[GV(·)]1/dim for β̂ and selected subsets, with dim as the order of the determinant. Also listed are angles ϕ(1n:X)deg between regressors and the constant, to be noted subsequently. Table 2 displays the squared canonical correlations ρ2(β̂i:β̂j) between designated subsets, and Table 3 the corresponding Vector Alienation Coefficient γ(β̂i:β̂j)=Πi=1r(1-ρi2), for specified pairs. Here {0L,QI} refers to the pair {[β̂0,β̂L],[β̂Q,β̂I]}, for example. Moreover, values of the composite indices γ(β̂i:β̂j) = γ(Xi:Xj), if much less than unity, serve to alert the user as to potential problems with ill-conditioning.Table 1
RootsG(β̂) and G(β̂i) of generalized variances for β̂ and subset estimators {β̂i∈ℝt}, and angles ϕ(1n:X)deg between regressors and the constant, against reference values ϕ**<18.44 and ϕ*<30.00, in eight small second-order designs.
DesignG(β̂)G(β̂0)G(β̂L)G(β̂Q)G(β̂I)G(β̂Q,β̂I)G(β̂M)ϕ(1n:X)H3101.499165.04771.188212.080261.610451.830341.5848425.11*H311B1.2805010.00001.00001.176801.999971.534131.3301717.55**SCD1.5387610.00001.666671.042104.166672.083761.6314217.55**BBD1.1053310.00000.833331.645831.111111.352291.1507716.10**CCD0.9372510.00000.714291.033001.250001.136330.9734114.96**ND2.030638.75001.250007.800311.250003.122562.2203018.80*HD2.332585.00001.911896.613131.952753.574442.5900425.24*BDD2.218354.64771.588094.966252.257523.226732.4495026.25*Table 2
Squared canonical correlations between designated subsets(β̂i,β̂j) of estimators for eight small second-order designs.
SubsetsH310H311BSCDBBDCCDNDHDBDD{0L,QI}0.819900.909090.909090.923080.933330.893330.794760.813040.400000.111110.542610.117210.400000.111110.111110.117210.400000.111110.03524{L,Q}0.111110.111110.064610.111110.111110.014130.083330.006310.01413{L,QI}0.400000.111110.544560.117210.400000.111110.111110.117210.400000.083330.111110.09890{Q,I}0.031530.104180.104180.00196{0,L}0.107140.117900.02281{0,Q}0.819900.909090.909090.923080.933330.892860.641030.79815{0,I}0.049180.00220{0,LQI}0.819900.909090.909090.923080.933330.896100.818180.80440Table 3
Vector Alienation Coefficientsγ(β̂i:β̂j) between subsets (β̂i,β̂j) of estimators, and the factor abc=[γ(0:LQI)γ(L:QI)γ(Q:I)]1/10, for each of eight small second-order designs.
SubsetsH310H311BSCDBBDCCDNDHDBDD{0L,QI}0.180100.090910.019640.076920.066670.084280.074170.14057{L,Q}1.000001.000001.000001.000001.000000.724280.785140.90914{L,QI}1.000001.000000.216001.000001.000000.724280.359860.70225{Q,I}1.000001.000001.000001.000001.000001.000000.968470.80092{0,L}1.000001.000001.000001.000001.000000.892860.882100.97719{0,Q}0.180100.090910.090910.076920.066670.107140.358970.20185{0,I}1.000001.000001.000001.000001.000001.000000.950820.99780{0,LQI}0.180100.090910.090910.076920.066670.103900.181820.19566abc0.842460.786790.675000.773760.762770.772060.758900.80197


### 5.3.1. An Overview

To fix ideas, observe for theCCD that G(β̂Q,β̂I)=1.13633, G(β̂Q)=1.03300, and G(β̂I)=1.25000 from Table 1. These not only are comparable in magnitude, but are commensurate, in having been adjusted for dimensions and thus homogeneous of unit degree, as are all entries in Table 1. Moreover, since (β̂Q,β̂I) are uncorrelated and γ(β̂Q,β̂I) = 1.0 for the CCD from Table 3, G(β̂Q,β̂I) is the geometric mean 1.13633=(1.03300)3/6(1.25000)3/6 from Theorem 4.1(ii). A further rough spot check of Table 1 may be summarized as follows.Summary Properties
(P1)
Compared withG(β̂), values for G(β̂0) appear excessive throughout.(P2)
Values forG(β̂L) are roughly comparable across designs.(P3)
The eight designs sort essentially into two groups.(P4)
Designs{H310,H311B,SCD,BBD,CCD} overall are comparatively D- and Ds-efficient, with the noted exception being G(β̂I)=4.16667 for the SCD.(P5)
The designs{ND,HD,BDD} are considerably less D-efficient, with their generalized variances GV(β̂) being {1192.09,4768.37,2886.03}, respectively, in comparison with {57.342,11.852,74.422,2.722,0.523} for the remaining designs; and each of the former is burdened by unequivocal Ds-inefficiency for βQ, to be treated subsequently.


### 5.3.2. Further Details

We next examine Hartley’s [39] SCD in some detail, first in terms of generalized variances. Values for GV(β̂),  GV(β̂0), and GV(β̂M) appear in the first row of Table 4, along with γ(β̂0:β̂M)=(1.0-0.909090)=0.090909 using ρ2(0:M)=0.909090 from Table 2. Theorem 4.1 (i) now asserts that GV(β̂)=GV(β̂0)GV(β̂M)γ(β̂0:β̂M), as verified numerically through 74.4216=10.00(81.8638)(0.090909). In a similar manner, β̂M partitions into {β̂L,[β̂Q,β̂I]}, where GV(β̂L)=4.6296 and GV([β̂Q,β̂I])=81.8638 from Table 4. The squared canonical correlations between {β̂L,[β̂Q,β̂I]} are ρ2(L:QI)=[0.4000,0.4000,0.4000]′ from Table 2, so that γ(L:QI)=0.21600 as in Table 3. Theorem 4.1 (i) again recovers GV(β̂M) as 81.8638=4.6926(81.8638)(0.21600) since G(β̂L) and γ(L:QI) are reciprocals in this instance. Moreover, GV(β̂Q,β̂I)=GV(β̂Q)GV(β̂I)γ(Q:I) translates into 81.8638=1.1317(72.3379)(1.0), where γ(Q:I)=1.0 since elements of {β̂Q,β̂I} are mutually uncorrelated from Table 2. In summary, the value G(β̂) for the SCD admits the factorization of (4.6), on identifying {θ̂1,θ̂2,θ̂3,θ̂4} with {β̂0,β̂L,β̂Q,β̂I}, respectively, given numerically from Tables 1 and 3 as (5.2)1.53876=(10.00)1/10(1.66667)3/10(1.04210)3/10(4.16667)3/10[(0.090909)(0.21600)(1.0)]1/10.Table 4
Generalized variances and Vector Alienation Coefficients between designated subsets for Hartley’s [39] SCD in k=3 regressors.
GV(β̂)GV(β̂0)GV(β̂M)γ(0:LQI))=(1-ρ12)74.421610.000081.86380.090909GV(β̂M)GV(β̂L)GV(β̂Q,β̂I)γ(L:QI)=Πi=1r(1-ρi2)81.86384.629681.86380.216000GV(β̂Q,β̂I)GV(β̂Q)GV(β̂I)γ(Q:I)=Πi=1r(1-ρi2)81.86381.131772.33791.000000Corresponding factorizations proceed similarly for other designs. Details are left to the reader, but values for[γ(0:LQI)γ(L:QI)γ(Q:I)]1/10, the rightmost factor of (5.2), are supplied for each design as the final row of Table 3. Although the tables, together with Theorem 4.1, support other factorizations, the one featured here seems most natural in terms of the parameters {β0,βL,βQ,βI}, together with their central roles in identifying noteworthy treatment effects in second-order models.


## 5.3.1. An Overview

To fix ideas, observe for theCCD that G(β̂Q,β̂I)=1.13633, G(β̂Q)=1.03300, and G(β̂I)=1.25000 from Table 1. These not only are comparable in magnitude, but are commensurate, in having been adjusted for dimensions and thus homogeneous of unit degree, as are all entries in Table 1. Moreover, since (β̂Q,β̂I) are uncorrelated and γ(β̂Q,β̂I) = 1.0 for the CCD from Table 3, G(β̂Q,β̂I) is the geometric mean 1.13633=(1.03300)3/6(1.25000)3/6 from Theorem 4.1(ii). A further rough spot check of Table 1 may be summarized as follows.Summary Properties
(P1)
Compared withG(β̂), values for G(β̂0) appear excessive throughout.(P2)
Values forG(β̂L) are roughly comparable across designs.(P3)
The eight designs sort essentially into two groups.(P4)
Designs{H310,H311B,SCD,BBD,CCD} overall are comparatively D- and Ds-efficient, with the noted exception being G(β̂I)=4.16667 for the SCD.(P5)
The designs{ND,HD,BDD} are considerably less D-efficient, with their generalized variances GV(β̂) being {1192.09,4768.37,2886.03}, respectively, in comparison with {57.342,11.852,74.422,2.722,0.523} for the remaining designs; and each of the former is burdened by unequivocal Ds-inefficiency for βQ, to be treated subsequently.


## 5.3.2. Further Details

We next examine Hartley’s [39] SCD in some detail, first in terms of generalized variances. Values for GV(β̂),  GV(β̂0), and GV(β̂M) appear in the first row of Table 4, along with γ(β̂0:β̂M)=(1.0-0.909090)=0.090909 using ρ2(0:M)=0.909090 from Table 2. Theorem 4.1 (i) now asserts that GV(β̂)=GV(β̂0)GV(β̂M)γ(β̂0:β̂M), as verified numerically through 74.4216=10.00(81.8638)(0.090909). In a similar manner, β̂M partitions into {β̂L,[β̂Q,β̂I]}, where GV(β̂L)=4.6296 and GV([β̂Q,β̂I])=81.8638 from Table 4. The squared canonical correlations between {β̂L,[β̂Q,β̂I]} are ρ2(L:QI)=[0.4000,0.4000,0.4000]′ from Table 2, so that γ(L:QI)=0.21600 as in Table 3. Theorem 4.1 (i) again recovers GV(β̂M) as 81.8638=4.6926(81.8638)(0.21600) since G(β̂L) and γ(L:QI) are reciprocals in this instance. Moreover, GV(β̂Q,β̂I)=GV(β̂Q)GV(β̂I)γ(Q:I) translates into 81.8638=1.1317(72.3379)(1.0), where γ(Q:I)=1.0 since elements of {β̂Q,β̂I} are mutually uncorrelated from Table 2. In summary, the value G(β̂) for the SCD admits the factorization of (4.6), on identifying {θ̂1,θ̂2,θ̂3,θ̂4} with {β̂0,β̂L,β̂Q,β̂I}, respectively, given numerically from Tables 1 and 3 as (5.2)1.53876=(10.00)1/10(1.66667)3/10(1.04210)3/10(4.16667)3/10[(0.090909)(0.21600)(1.0)]1/10.Table 4
Generalized variances and Vector Alienation Coefficients between designated subsets for Hartley’s [39] SCD in k=3 regressors.
GV(β̂)GV(β̂0)GV(β̂M)γ(0:LQI))=(1-ρ12)74.421610.000081.86380.090909GV(β̂M)GV(β̂L)GV(β̂Q,β̂I)γ(L:QI)=Πi=1r(1-ρi2)81.86384.629681.86380.216000GV(β̂Q,β̂I)GV(β̂Q)GV(β̂I)γ(Q:I)=Πi=1r(1-ρi2)81.86381.131772.33791.000000Corresponding factorizations proceed similarly for other designs. Details are left to the reader, but values for[γ(0:LQI)γ(L:QI)γ(Q:I)]1/10, the rightmost factor of (5.2), are supplied for each design as the final row of Table 3. Although the tables, together with Theorem 4.1, support other factorizations, the one featured here seems most natural in terms of the parameters {β0,βL,βQ,βI}, together with their central roles in identifying noteworthy treatment effects in second-order models.


## 5.4. Masking

TheD-efficiency index of the SCD, at GV(β̂)=74.4216, is larger but roughly comparable to that of H310 at GV(β̂)=57.3418. What cannot be anticipated from these facts alone, however, is that the (3×3) determinant GV(β̂I)=72.3379 for the SCD is comparable to its (10×10) determinant GV(β̂)=74.4216, despite their disparate dimensions. Adjusting for dimensions gives G(β̂)=(74.4216)1/10=1.53876 and G(β̂I)=(72.3379)1/3=4.16667 for the SCD. This illustrates the masking of a remarkably inefficient estimator for βI, despite the value G(β̂)=1.53876 in estimating all parameters. This masking stems from the nonorthogonality of subset estimators as reflected in their canonical correlations and Vector Alienation Coefficients. In contrast are the corresponding commensurate values for the H310 design, namely, G(β̂)=(57.3418)1/10=1.49916 and G(β̂I)=(4.1768)1/3=1.61045. It may be noted that the condition number c1(X0′X0) is 21.59 for H310, with the somewhat larger value 54.01 for the SCD.We next examine theDs-inefficiencies of ND and HD for βQ as noted earlier, with G(β̂Q) taking values 7.80031 and 6.61313, respectively. Our reference for masking is G(β̂L,β̂Q). These values are not listed in Table 1, but may be recovered from Tables 2 and 3 as follows. Specifically, for ND we have (5.3)[GV(β̂L)GV(β̂Q)γ(L:Q)]1/6=[GV(β̂L,β̂Q)]1/6=G(β̂L,β̂Q),(1.25000)3/6(7.80031)3/6(0.72428)1/6=2.95912,
where neither G(β̂L)=1.25000 nor G(β̂L,β̂Q)=2.95912 appears excessive. In consequence, that G(β̂Q)=7.80031 is excessive would be masked on examining G(β̂L) and G(β̂L,β̂Q) only. Parallel steps for HD give the factorization (1.91189)3/6(6.61313)3/6(0.78514)1/6 = 3.41528, with similar conclusions in regard to masking.


## 5.5. Collinearity with the Constant

Advocates for these and other small designs have focused onD,Ds, and other efficiency criteria, as well as the parsimony of small designs and their advantage in industrial experiments. To the knowledge of this observer, none has considered prospects for ill-conditioning and its consequences, despite the fact that columns of X are necessarily inter-linked as a consequence of second-order from first-order effects. Nonetheless, from Section 3.1 and Corollary 3.2, we may compute angles between the constant vector and the span of the regressors using duality together with the information at hand. This may prove to be critical in view of the admonition [29] that “collinearity with the intercept can quite generally corrupt the estimates of all parameters in the model.” As noted in Remark 3.4, rules-of-thumb for problematic VIFs include those exceeding 10 or 4 or, in angular measure, ϕ**<18.435deg and ϕ*<30.00deg. From the last row of Table 2, the angles ϕ(1n:X) have been computed for each of the eight designs, as listed in the final column of Table 1. For example, arccos(0.81990)=25.1116deg for H310. It is seen that all designs are flagged as potentially problematic using rules-of-thumb as cited. This adds yet another layer of concerns, heretofore unrecognized, in seeking further to implement these designs already in wide usage.


## 6. Conclusions

Duality of (i) Hotelling’s [30] canonical correlations{ρ1,…,ρr} between the OLS estimators {β̂1,β̂2} and (ii) the design linkage parameters{δ1,…,δr} between {X1,X2} is established at the outset. Stewart’s [33] collinearity indices are then extended to encompass angles {ϕ0,ϕ1,…,ϕk} between each column of X0=[1n,X1,…,Xk] and remaining columns. In particular, ϕ0 quantifies numerically the collinearity of regressors with the intercept, of concern in the prospective corruption of all estimates due to ill-conditioning.Matrix identities factorize a determinant in terms of principal subdeterminants and theVector Alienation Coefficients of [30] between {β̂1,β̂2}. By duality, the latter also are Alienation Coefficients between {X1,X2}. These identities in turn are applied in the study of Ds-efficiencies for the parameters {β0,βL,βQ,βI} in eight small second-order designs from the literature. Studies on Ds- and D-efficiencies, as cited in our opening paragraph, confirm that designs are seldom efficient for both. Our determinant identities support a rational explanation. In particular, these identities unmask the propensity for Ds-inefficient subset estimators to be masked through near collinearities in overall D-efficient designs.Finally, the evidence suggests that all eight designs are vulnerable, to varying degrees, to the corruption of all estimates due to ill-conditioning. In short, we have exposed quantitatively the structural origins of masking through Hotelling’s [30] canonical correlations, and their equivalent design linkage parameters. This analysis in turn proceeds from the design matrix itself rather than empirical estimates, so that any design can be evaluated beforehand with regard to masking and possible subset inefficiencies, rather than retrospectively after having committed to a given design in a particular experiment.



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*Source: 182049-2011-10-05.xml*