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Understanding Experimental Design: Latin Squares, Lattice Squares, and Rectangular Lattices

Learn about the importance of appropriate experimental design, including Latin squares, lattice squares, and rectangular lattices in research. Discover advantages, disadvantages, and key features of each design for accurate and reliable results.

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Understanding Experimental Design: Latin Squares, Lattice Squares, and Rectangular Lattices

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  1. Experimental Design

  2. Last Class • Need for appropriate experimental design. • Single replicate designs. • Completely randomized block designs. • Randomized block designs. • Block arrangement.

  3. Non-randomized Fertility Gradient Randomized

  4. Completely Randomized Block

  5. Randomized Complete Block

  6. Models Yij =  + gi + eij Yijk =  + ri +gj + eijk

  7. a. I II III IV I II b. III IV I III c. II IV

  8. Latin Square Lattice Square Rectangular Lattices

  9. Latin Square • Latin square designs have blocking in two directions at right angles to each other. • The blocks are referred to as rows and columns. • If there are n treatments to be tested there will be n x n experimental units (i.e. plots).

  10. Latin Square I II III Rows IV V I II III IV V Columns

  11. Latin Square I II III Rows IV V I II III IV V Columns

  12. Latin Square I II III Rows IV V I II III IV V Columns

  13. Latin Square I II III Rows IV V I II III IV V Columns

  14. Latin Square I II III Rows IV V I II III IV V Columns

  15. Latin Square I II III Rows IV V I II III IV V Columns

  16. Latin Square I II III Rows IV V I II III IV V Columns

  17. Latin Square I II III Rows IV V I II III IV V Columns

  18. Latin Square I II III Rows IV V I II III IV V Columns

  19. Latin Square I II III Rows IV V I II III IV V Columns

  20. Latin Square I II III Rows IV V I II III IV V Columns

  21. Latin Square I II III Rows IV V I II III IV V Columns

  22. Latin Square I II III Rows IV V I II III IV V Columns

  23. Latin Square I II III Rows IV V I II III IV V Columns

  24. Latin Square I II III Rows IV V I II III IV V Columns

  25. Latin Square I II III Rows IV V I II III IV V Columns

  26. Latin Square I II III Rows IV V I II III IV V Columns

  27. Latin Square I II III Rows IV V I II III IV V Columns

  28. Latin Square I II III Rows IV V I II III IV V Columns

  29. Latin Square I II III Rows IV V I II III IV V Columns

  30. Latin Square Yijk =  +gi + rj + ck + eijk Where Yijk is the performance of the ith genotype in the jth row and kth column;  in the overall mean; gi is the effect of the ith genotype; rj is the effect of the jth row; ck is the effect of the kth column; and eijk is the error term.

  31. Latin Square • Advantage of latin square designs is their accuracy and ability to remove gradients in two directions. • Disadvantage is that they require large levels of replication. A 10 entry experiment would require 100 experimental units. • Latin square analyses are intolerant to missing values.

  32. Lattice Square • Lattice squares look a little like Latin squares. • Lattice squares must consist of test entries that are the square of a whole number (4, 9, 16, 25, …), termed n x n lattices. • The number of replicates is determined by the number of entries; where the number of replicates is n +1.

  33. Lattice Square I II III IV

  34. Lattice Square I II III IV

  35. Lattice Square I II III IV

  36. Lattice Square I II III IV

  37. Lattice Square I II III IV

  38. Lattice Square I II III IV

  39. Lattice Square I II III IV

  40. Lattice Square I II III IV

  41. Lattice Square I II III IV

  42. Lattice Square I II III IV

  43. Lattice Square I II III IV

  44. Lattice Square I II III IV

  45. Lattice Square Yijk =  +gai + bak + rj + eijk Where Yijk is the performance of the ith genotype in the jth replicate and kth sub-block;  in the overall mean; gai is the effect of the ith genotype adjusted according to sub-blocks; bak is the effect of the kth sub-block adjusted according to the entries in that block; rj is the effect of the jth replicate; and eijk is the error term.

  46. Lattice Square • Lattice squares are usually more effective than RCB’s. • Have restraints on the number of entries and replicates.

  47. Lattice Square • Lattice squares are resolvable. • However, they are not trulyrandomized. • Errors in plot arrangement (i.e. planting) renders them useless.

  48. Rectangular Lattice • Lattice squares must have n x n, rectangular lattices have m x n entries. • Every entry in the rectangular lattice can appear in the same sub-block with any other entry in the test only once [(0,1) designs].

  49. Rectangular Lattice I II

  50. Rectangular Lattice I II

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