(a) (i) What is confounding in factorial experiments ? (ii) A 2^6factorial experiment is conducted in blocks of size2³. Write…

(a) Confounding in a 2⁶ experiment in blocks of 8

(i) Confounding in factorial experiments means deliberately arranging treatment combinations in blocks so that one or more treatment contrasts, usually higher-order interactions, become indistinguishable from block differences. The information on the confounded effects cannot be separated from block effects.

(ii) Let the six factors be A, B, C, D, E, F. In a 2⁶ experiment in blocks of size 2³, there are 2³ = 8 blocks, so 8 − 1 = 7 treatment degrees of freedom are confounded.

Choose independent defining contrasts of order 3: ABC, ADE, BDF.

Their generalised interactions are: ABC·ADE = BCDE, ABC·BDF = ACDF, ADE·BDF = ABEF, ABC·ADE·BDF = CEF.

Thus the confounded effects are: ABC, ADE, BDF, BCDE, ACDF, ABEF, CEF. No main effect or two-factor interaction is confounded.

Taking the key block as the block containing (1), its elements are: (1), ABD, ABEF, ACDF, ACE, BCDE, BCF, DEF.

(iii) For a 2ⁿ factorial in 2ᵏ blocks of size 2ⁿ⁻ᵏ, for one replication:

• Total df = 2ⁿ − 1.
• Blocks df = 2ᵏ − 1.
• Confounded treatment contrasts df = 2ᵏ − 1.
• Unconfounded treatment df = (2ⁿ − 1) − (2ᵏ − 1) = 2ⁿ − 2ᵏ.

If there are r replications:

• Replication df = r − 1.
• Blocks within replications df = r(2ᵏ − 1).
• Unconfounded treatments df = 2ⁿ − 2ᵏ.
• Error df = (r − 1)(2ⁿ − 2ᵏ).
• Total df = r2ⁿ − 1.

(This assumes the same effects are confounded in every replicate. For r = 1 there is no error df unless higher-order interactions are taken as negligible.)

(b) Principal components of X₁ and X₂

Principal components are uncorrelated linear combinations of the original variables chosen successively to have maximum variance, obtained from the eigenvectors of the covariance or correlation matrix.

Treat X₁ and X₂ as two variables observed on three units: X₁ = [1, 0, −1]ᵗ, X₂ = [−1, 1, 0]ᵗ.

Their means are X̄₁ = 0 and X̄₂ = 0. The sample covariance matrix, using divisor n − 1 = 2, is S = [ [1, −1/2], [−1/2, 1] ].

Eigenvalues satisfy |S − λI| = 0: (1 − λ)² − 1/4 = 0 ⇒ λ₁ = 3/2, λ₂ = 1/2.

For λ₁ = 3/2, eigenvector is (1, −1)ᵗ/√2. For λ₂ = 1/2, eigenvector is (1, 1)ᵗ/√2.

Hence the principal components are Y₁ = (X₁ − X₂)/√2, Y₂ = (X₁ + X₂)/√2.

Numerically, Y₁ = [2, −1, −1]ᵗ/√2, Y₂ = [0, 1, −1]ᵗ/√2.

Inverting, X₁ = (Y₁ + Y₂)/√2, X₂ = (Y₂ − Y₁)/√2.

(c) Regression estimator in sample surveys

Let y be the study variable and x an auxiliary variable with known population mean X̄. From a simple random sample of size n, compute ȳ, x̄ and the sample regression coefficient b = sₓᵧ/sₓ². The regression estimator of the population mean Ȳ is T_reg = ȳ + b(X̄ − x̄).

For its bias, E(T_reg) = E(ȳ) + X̄ E(b) − E(b x̄). Since E(ȳ) = Ȳ, Bias(T_reg) = X̄ E(b) − E(b x̄) = −[E(b x̄) − E(b)E(x̄)] = −Cov(x̄, b).

The bias is negligible when n is large, because x̄ − X̄ and b − β are both of order n⁻¹ᐟ², so Cov(x̄, b) is of order n⁻¹, while the standard error is of order n⁻¹ᐟ². It is exactly zero if b is a preassigned constant or independent of x̄.

To first degree of approximation, T_reg − Ȳ ≈ (ȳ − Ȳ) − β(x̄ − X̄), where β = Sₓᵧ/Sₓ². Therefore MSE(T_reg) ≈ Var(ȳ) + β²Var(x̄) − 2β Cov(ȳ, x̄) = (1 − f)/n [Sᵧ² + β²Sₓ² − 2βSₓᵧ] = (1 − f)/n [Sᵧ² − Sₓᵧ²/Sₓ²] = (1 − f)/n Sᵧ²(1 − ρ²), where f = n/N and ρ is the population correlation between y and x.

For the ratio estimator T_R = ȳ X̄/x̄, MSE(T_R) ≈ (1 − f)/n [Sᵧ² + R²Sₓ² − 2R Sₓᵧ], R = Ȳ/X̄. Thus MSE(T_R) − MSE(T_reg) ≈ (1 − f)/n Sₓ²(R − β)² ≥ 0. So the regression estimator is asymptotically at least as efficient as the ratio estimator; they coincide when the regression line passes through the origin, i.e. R = β.