Q8
(a) Differentiate between randomised block design and balanced incomplete block design. In usual notations, for a balanced incomplete block design, prove that (i) bk = vr (ii) λ(v – 1) = r(k – 1) and (iii) b ≥ v. 20 (b) Explain the concept of confounding in design of experiment. In an experiment with three factors A, B and C, each at two levels, three replicates are divided in two blocks, each of four units. How will you confound ABC in the first, AC in the second and BC in the third replication ? 15 (c) Differentiate among fixed, random and mixed effect models with examples. How are the three basic principles of design fulfilled in randomised block design ? Explain. 15
हिंदी में प्रश्न पढ़ें
(a) यादृच्छिक खंड अभिकल्पना तथा संतुलित अपूर्ण खंडक अभिकल्पना में अंतर बताइए । सामान्य प्रयुक्त संकेताक्षरों में सिद्ध कीजिए कि संतुलित अपूर्ण खंडक अभिकल्पना में (i) bk = vr (ii) λ(v – 1) = r(k – 1) तथा (iii) b ≥ v. 20 (b) प्रयोगात्मक अभिकल्पना में संकरण के सिद्धांत की व्याख्या कीजिए । किसी प्रयोग में जिसमें तीन उपादान A, B तथा C जिनमें प्रत्येक दो स्तरों पर हैं, तीन पुनरावृत्त चार इकाइयों के दो खंडों में विभाजित हैं । आप ABC को पहले, AC को दूसरे तथा BC को तीसरे पुनरावृत्त में किस प्रकार संकीर्ण करेंगे ? 15 (c) नियत, यादृच्छिक एवं मिश्रित प्रभाव मॉडलों में उदाहरणों सहित विभेद कीजिए । यादृच्छिक खण्डक अभिकल्पना में अभिकल्पना के तीन मूलभूत सिद्धान्तों का समावेश कैसे होता है ? स्पष्ट कीजिए । 15
Directive word: Differentiate
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How this answer will be evaluated
Approach
Begin with a structured comparison of RBD vs BIBD in part (a), then rigorously prove all three BIBD parameters using standard notation with clear algebraic steps. For part (b), first define confounding with factorial design context, then explicitly construct the three replication schemes showing which treatment combinations go to which block. For part (c), use tabular comparison for model types with agricultural/industrial examples, then explain how RBD satisfies randomization, replication, and local control. Allocate approximately 40% time to part (a) given its 20 marks and proof demands, 30% each to parts (b) and (c).
Key points expected
- Part (a): Clear distinction between RBD (complete blocks, all treatments per block) and BIBD (incomplete blocks, not all treatments appear in each block) with structural conditions
- Part (a): Correct proofs of bk = vr, λ(v–1) = r(k–1), and b ≥ v using incidence matrix properties or combinatorial counting with λ defined as pairwise concurrence
- Part (b): Accurate definition of confounding as sacrificing higher-order interaction information to achieve block homogeneity, with distinction between complete and partial confounding
- Part (b): Correct construction of three replications: Rep I confounds ABC (assign +++ and +–– to Block 1, ++– and +–+ to Block 2, etc.), Rep II confounds AC, Rep III confounds BC using Yates notation
- Part (c): Precise differentiation of fixed (levels specifically chosen, inference only to those levels), random (levels random sample from population, variance component estimation), and mixed models with appropriate examples like crop varieties vs fertilizer doses
- Part (c): Explanation of how RBD achieves randomization (random allocation within blocks), replication (multiple blocks), and local control (homogeneous blocks reducing experimental error)
Evaluation rubric
| Dimension | Weight | Max marks | Excellent | Average | Poor |
|---|---|---|---|---|---|
| Setup correctness | 20% | 10 | Correctly defines all parameters (v, b, r, k, λ) for BIBD; properly sets up 2³ factorial with 3 replicates and 2 blocks of 4 units; accurately identifies model assumptions for fixed/random/mixed effects | Defines most parameters but confuses r with k or misstates block structure; partial understanding of model distinctions | Incorrect parameter definitions; wrong experimental setup (e.g., wrong number of treatment combinations); conflates fixed and random effects |
| Method choice | 20% | 10 | Uses incidence matrix or combinatorial counting for BIBD proofs; applies Yates algorithm/ABCD rule for confounding; employs ANOVA framework appropriately for model comparison | Attempts correct methods but with gaps in logical flow; partial application of confounding rules; incomplete ANOVA specification | Wrong method for proofs (e.g., assumes complete design); incorrect confounding assignment; no systematic approach to model differentiation |
| Computation accuracy | 20% | 10 | All three BIBD relations derived correctly with proper algebraic manipulation; accurate assignment of all 8 treatment combinations across three replications with correct confounding patterns | Two of three proofs correct; minor errors in treatment combination assignments; some computational slips in variance expressions | Major errors in proofs (e.g., circular reasoning); wrong confounding patterns; incorrect variance component expressions |
| Interpretation | 20% | 10 | Explains why b ≥ v ensures estimability; interprets confounding as information sacrifice with recovery through partial confounding; clarifies when each model type is appropriate with Indian examples (e.g., IARI wheat varieties as fixed, soil types as random) | Partial interpretation of design efficiency; limited explanation of confounding consequences; generic examples without context | No interpretation of why relations matter; fails to explain confounding consequences; no practical examples or irrelevant ones |
| Final answer & units | 20% | 10 | Clear statement of all proved relations; explicit block compositions for all three replications shown in table format; concise summary table for model comparison; proper conclusion on design principles | Most relations stated but presentation unclear; replications described but not tabulated; incomplete summary | Missing final statements; no tabular presentation; disorganized conclusion; no synthesis across parts |
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