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Consider the following Lattice design using ee ee, ee ee ee as the standard array with s=3 a ee eee eee ee po ee ee ee ee, a, a a ee ee po a a Consider the following statements:
(A) . As the Lattice design has been constructed by using the complete set of (s-1 = 2) Mutually Orthogonal Latin Squares
(MOLS) possible, it is a balanced Lattice
(B) .The Lattice design constructed is a BIBD with parameters v=9, b=12, r =4, k=3, A=1
(C) .The blocks X, XI and XII have been constructed by superimposing the following Latin Square arrangement on the
standard array:
a eee eee
re ee eee,
a ee eee
(D) . For the Lattice design, the number of treatments v = 3 = 9 is a perfect square and its block size is 3, i. the square root
of this number v.
Choose the correct statements from the options given below:
EXPLANATION
Incomplete Block Designs
- Balanced Lattice Design Construction and Properties:
- Lattice Design is an incomplete block design where the number of treatments $v$ is a perfect square ($v = s^2$). For $s = 3$, $v = 3^2 = 9$ treatments in blocks of size $k = s = 3$.
- Statement (A): By utilizing the complete set of $s - 1 = 2$ Mutually Orthogonal Latin Squares (MOLS), a complete set of $s + 1 = 4$ replicates is constructed, ensuring that every pair of treatments occurs together in the same block exactly $\lambda = 1$ time (Balanced Lattice). (True)
- Statement (B): The parameters of this balanced lattice are $v = 9, b = 12, r = 4, k = 3, \lambda = 1$, which satisfies all BIBD conditions: $vr = bk = 36$ and $\lambda(v - 1) = r(k - 1) = 8$. (True)
- Statement (C): Blocks X, XI, and XII are generated by superimposing Latin squares on the standard array. (True)
- Statement (D): The number of treatments $v = 9$ is a square and block size $k = 3 = \sqrt{v}$. (True)
- Conclusion: (A), (C) and (D) only (Option 3).
Correct option: (3) (A), (C) and (D) only.
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