Q7
(a) (i) What is confounding in factorial experiments ? (ii) A $2^6$ factorial experiment is conducted in blocks of size $2^3$. Write the confounded effects such that no main effect or two factor interaction are confounded. Give the list of independent and generalised interactions confounded along with the elements of key block only. (iii) Give the break-up of degrees of freedom for a $2^n$ factorial experiment in $2^k$ blocks. (b) What are principal components ? Describe how to compute the principal components of the vectors X₁ = $\begin{bmatrix} 1 \\ 0 \\ -1 \end{bmatrix}$ and X₂ = $\begin{bmatrix} -1 \\ 1 \\ 0 \end{bmatrix}$. Give X₁ and X₂ in terms of the principal components. (c) Define Regression estimator. Show bias = – Cov ($\bar{x}$, b). Under what conditions is bias negligible ? Find the mean square error of the estimator to first degree of approximation. Give comparison of Regression estimator with Ratio estimator.
हिंदी में प्रश्न पढ़ें
(a) (i) बहु-उपादानी प्रयोगों में संकरण क्या है ? (ii) एक $2^6$ बहु-उपादानी प्रयोग $2^3$ आकार के खंडकों में संचालित किया गया। संकीर्ण प्रभावों को लिखिए जिसमें कोई भी मुख्य उपादान या दो घटक अन्योन्यक्रिया संकीर्ण न हों। संकीर्ण होने वाले स्वतंत्र व व्यापकीकृत अन्योन्यक्रियाओं की सूची लिखिए, साथ ही केवल प्रमुख खंडक के अवयव लिखिए। (iii) $2^k$ खंडकों में $2^n$ बहु-उपादानी प्रयोग के लिए स्वातंत्र्य कोटियों का विभाजन दीजिए। (b) मुख्य घटक क्या हैं ? सदिश X₁ = $\begin{bmatrix} 1 \\ 0 \\ -1 \end{bmatrix}$ और X₂ = $\begin{bmatrix} -1 \\ 1 \\ 0 \end{bmatrix}$ के मुख्य घटकों के परिकलन का विवरण दीजिए । X₁ और X₂ को मुख्य घटकों के रूप में लिखिए । (c) समाश्रयण आकलक परिभाषित कीजिए । दर्शाइए अभिनति = – सहप्रसरण ($\bar{x}$, b) । किन प्रतिबंधों के अंतर्गत अभिनति नगण्य होती है ? प्रथम घात के सन्निकट आकलक की त्रुटि वर्ग माध्य ज्ञात कीजिए । समाश्रयण आकलक की अनुपात आकलक के साथ तुलना कीजिए ।
Directive word: Derive
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How this answer will be evaluated
Approach
Derive the required expressions systematically across all five sub-parts. For (a)(i)-(iii), allocate ~35% time covering confounding definition, the specific 2^6 in 2^3 blocks construction with ABC, DEF, ABCDEF as confounded effects, and the general df breakdown. For (b), spend ~25% time on PCA computation: construct data matrix, find covariance, eigenvalues (3, 1, 0), eigenvectors, and express X₁, X₂ in PC terms. For (c), allocate ~40% time deriving regression estimator bias, MSE approximation, and comparison with ratio estimator via Cochran's approach. Begin with definitions, proceed through step-by-step derivations, and conclude with clear interpretations.
Key points expected
- (a)(i) Define confounding as mixing of treatment effects with block effects; distinguish complete vs partial confounding
- (a)(ii) Identify confounded effects: ABC, DEF, and their generalized interaction ABCDEF; verify no main effect or 2-factor interaction is confounded; construct key block with I, AD, BE, CF, ABDE, ACDF, BCEF, ABCDEF
- (a)(iii) State df breakdown: blocks (2^k - 1), treatments (2^n - 1), error (2^n - 2^k - n + nk - k), total (2^n - 1)
- (b) Define PCs as uncorrelated linear combinations maximizing variance; compute covariance matrix [2 -1; -1 2], eigenvalues λ₁=3, λ₂=1, eigenvectors [1/√2, -1/√2]ᵀ and [1/√2, 1/√2]ᵀ; express X₁ = (1/√2)PC₁ + (1/√2)PC₂, X₂ = (-1/√2)PC₁ + (1/√2)PC₂
- (c) Define regression estimator Ŷ_reg = Ȳ + b(X̄ - x̄); derive bias = -Cov(x̄, b) using E(b) = β + O(1/n); state negligible bias when n is large or ρ ≈ 0; derive MSE ≈ S²_y(1-ρ²)(1/n + 1/N); compare: regression has smaller MSE when |ρ| > 1/2 C_x/C_y
Evaluation rubric
| Dimension | Weight | Max marks | Excellent | Average | Poor |
|---|---|---|---|---|---|
| Setup correctness | 20% | 10 | Correctly identifies confounding structure for (a)(ii) with proper generalized interaction; sets up covariance matrix correctly for PCA in (b); defines regression estimator with appropriate population/sample distinctions in (c) | Basic definitions correct but misses generalized interaction in (a)(ii); minor errors in covariance matrix setup; incomplete regression estimator definition | Fundamental misunderstanding of confounding; wrong matrix dimensions for PCA; confuses regression with ratio estimator definition |
| Method choice | 20% | 10 | Uses Yates' method or modular arithmetic for key block in (a)(ii); applies spectral decomposition correctly for PCA in (b); employs Taylor series approximation for bias/MSE derivation in (c) | Correct general approach but inefficient methods; skips verification steps; uses direct rather than elegant derivation methods | Incorrect method for finding confounded effects; attempts PCA without eigenvalue decomposition; fails to use approximation for large-sample results |
| Computation accuracy | 20% | 10 | Precise arithmetic: correct eigenvalues (3, 1, 0) and normalized eigenvectors for (b); accurate key block elements for (a)(ii); correct bias expression and MSE formula with proper order terms in (c) | Minor arithmetic slips in eigenvector normalization or block construction; correct final formulas with intermediate calculation errors | Major computational errors: wrong eigenvalues, incorrect key block membership, wrong sign in bias derivation, missing (1-ρ²) factor in MSE |
| Interpretation | 20% | 10 | Explains why ABC, DEF choice preserves main effects; interprets PC directions as variance-maximizing orthogonal axes; explains when regression beats ratio estimator using efficiency conditions | States results without explaining 'why'; limited physical/statistical interpretation of principal components; superficial comparison of estimators | No interpretation provided; fails to explain significance of confounding pattern; treats PCA as mechanical exercise without geometric insight |
| Final answer & units | 20% | 10 | Complete enumeration: all three confounded effects with key block elements for (a)(ii); explicit PC expressions for X₁, X₂ in (b); final MSE formula and clear comparison conditions for (c) | Most elements present but incomplete key block or missing one PC expression; comparison stated without conditions | Missing critical final answers: no generalized interaction, no PC expressions, no MSE formula, or no comparison conclusion |
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