(a) Let **X** = (X₁ X₂ X₃)' be distributed as N₃ (μ, Σ), where μ = (2 −1 3)' and Σ = 4 & 1 & 0
1 & 2 & 1
0 & 1 & 3 . Find (i) the conditional distribution of (X₁ X₂)' given X₃ = 2. (ii) partial correlation coefficient ρ₁₂.₃ and multiple correlation

Question

(a) Let **X** = (X₁ X₂ X₃)' be distributed as N₃ (μ, Σ), where μ = (2 −1 3)' and Σ = 4 & 1 & 0
 1 & 2 & 1
 0 & 1 & 3 . Find (i) the conditional distribution of (X₁ X₂)' given X₃ = 2. (ii) partial correlation coefficient ρ₁₂.₃ and multiple correlation coefficient R₁.₂₃ (8+7 marks)

(b) (i) Describe the complete analysis of two-way classified data with multiple (but equal) observations per cell, clearly stating the assumptions used. Also state two examples where such type of analysis is used. (ii) Let three mutually independent variables Y₁, Y₂ and Y₃ having common variance σ² and E(Y₁) = β₁ + β₂, E(Y₂) = β₁ + β₃, E(Y₃) = β₁ + β₂ be given. Show that the linear parametric function p₁β₁ + p₂β₂ + p₃β₃ is estimable if and only if p₁ = p₂ + p₃, clearly stating the assumptions used, if any. (5 marks)

(c) (i) State briefly three reasons why an analyst may wish to perform a principal component analysis. (6 marks) (ii) Define canonical correlations and give two examples of their application. Describe the procedure of working out canonical correlations and canonical variates. (9 marks)

UPSC Answer Check · Accepted Answer

(a) (i) Since X ~ N₃(μ,Σ), use the conditional normal formula. Partition X₁ = (X₁ X₂)' and X₂ = X₃. Here μ₁ = (2 −1)', μ₂ = 3,
Σ₁₁ = ((4,1),(1,2)), Σ₁₂ = (0,1)', Σ₂₂ = 3. For X₃ = 2,
E(X₁|X₃=2) = μ₁ + Σ₁₂Σ₂₂⁻¹(2−μ₂)
= (2,−1)' + (0,1)'(1/3)(−1) = (2, −4/3)'.
Cov(X₁|X₃=2) = Σ₁₁ − Σ₁₂Σ₂₂⁻¹Σ₂₁
= ((4,1),(1,2)) − (1/3)((0,0),(0,1)) = ((4,1),(1,5/3)).
Hence
(X₁,X₂)'|X₃=2 ~ N₂((2,−4/3)', ((4,1),(1,5/3))).

(a) (ii) The partial correlation is obtained from the conditional covariance above:
ρ₁₂.₃ = 1/√(4 × 5/3) = √(3/20) = √15/10.
For R₁.₂₃, use Σ₂₃ = ((2,1),(1,3)), cross-covariances (σ₁₂,σ₁₃) = (1,0).
R₁.₂₃² = (1,0)Σ₂₃⁻¹(1,0)' / σ₁₁.
Σ₂₃⁻¹ = (1/5)((3,−1),(−1,2)), so (1,0)Σ₂₃⁻¹(1,0)' = 3/5.
Thus R₁.₂₃² = (3/5)/4 = 3/20, so
ρ₁₂.₃ = √(3/20), R₁.₂₃ = √(3/20).

(b) (i) Let a factor A have a levels, factor B have b levels, and r equal replications per cell. Write y_ijk, i=1,…,a; j=1,…,b; k=1,…,r. The fixed-effects model is
y_ijk = μ + α_i + β_j + γ_ij + ε_ijk,
with Σα_i=0, Σβ_j=0, Σ_iγ_ij=0, Σ_jγ_ij=0, and ε_ijk ~ iid N(0,σ²). Assumptions: independent normal errors, constant variance σ², linear additive model with possible interaction, and fixed factor effects.
Compute total SS = Σ(y_ijk−ȳ_...)², df = abr−1. Then
SSA = brΣ_i(ȳ_i..−ȳ_...)², df = a−1;
SSB = arΣ_j(ȳ_.j.−ȳ_...)², df = b−1;
SSAB = rΣ_iΣ_j(ȳ_ij.−ȳ_i..−ȳ_.j.+ȳ_...)², df = (a−1)(b−1);
SSE = Σ_iΣ_jΣ_k(y_ijk−ȳ_ij.)², df = ab(r−1).
Test H₀A: all α_i=0 by F_A = MSA/MSE ~ F_a−1, ab(r−1); H₀B by F_B = MSB/MSE; H₀AB: all γ_ij=0 by F_AB = MSAB/MSE. If interaction is significant, interpret cell means/simple effects. If not, drop interaction and test main effects in the additive model. MSE estimates σ². Examples: crop yield under fertilizer levels and irrigation levels with replications; tensile strength under machines and operators with repeated measurements.

(b) (ii) Write Y = (Y₁,Y₂,Y₃)' and β=(β₁,β₂,β₃)'. From the mean model,
E(Y) = Xβ, where X = ((1,1,0),(1,0,1),(1,1,0)).
A linear function p'β is estimable iff p is in the column space of X', equivalently p = X'a for some a=(a₁,a₂,a₃)'. Now
E(a'Y) = a₁(β₁+β₂)+a₂(β₁+β₃)+a₃(β₁+β₂)
= (a₁+a₂+a₃)β₁ + (a₁+a₃)β₂ + a₂β₃.
Thus p₁=a₁+a₂+a₃, p₂=a₁+a₃, p₃=a₂, giving
p₁ = (a₁+a₃)+a₂ = p₂+p₃.
Conversely, if p₁=p₂+p₃, choose a₂=p₃, a₁=p₂, a₃=0; then E(a'Y)=p'β. Hence p'β is estimable iff p₁ = p₂ + p₃. Assumptions: linear model, fixed β, independent errors with zero mean and common variance σ² for least-squares inference.

(c) (i) Three reasons for principal component analysis:
To reduce many correlated variables to a few uncorrelated principal components while retaining most variability.
To overcome multicollinearity or instability in regression by using orthogonal components.
To visualize/compress high-dimensional data and separate signal from noise.

(c) (ii) For two random vectors X (p×1) and Y (q×1), canonical correlations are the successive maximum correlations between linear combinations U=a'X and V=b'Y, with later pairs uncorrelated with earlier ones. Let
Σ = ((Σxx,Σxy),(Σyx,Σyy)).
The squared canonical correlations λ₁² ≥ … ≥ λ_k², k=min(p,q), are eigenvalues of Σxx⁻¹ΣxyΣyy⁻¹Σyx. The coefficient vector a satisfies
(Σxx⁻¹ΣxyΣyy⁻¹Σyx − λ²I)a = 0, normalized by a'Σxxa=1.
Then b ∝ Σyy⁻¹Σyxa, normalized by b'Σyyb=1; equivalently b = (1/λ)Σyy⁻¹Σyxa for λ>0. In practice replace Σ by sample covariance S, standardizing variables if scales differ. The canonical variates are U_i=a_i'X and V_i=b_i'Y. Applications: relating academic aptitude/achievement variables to job performance variables; relating economic indicators such as GDP and inflation to social indicators such as health and education.

(a) Let X = (X₁ X₂ X₃)' be distributed as N₃ (μ, Σ), where μ = (2 −1 3)' and Σ = 4 & 1 & 0 1 & 2 & 1 0 & 1 & 3 . Find (i)…

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