Homework 1. I have a field of 36 plots, six rows and six columns. I want to lay out 6 treatments (herbicides) in a Latin square but I am concerned about effects of neighboring treatments on any given plot because there may be some unintended drifting of the herbicides onto neightboring plots when spraying or drift due to water flow during rains. Below are some potential designs. While none of these would completely eliminate such effects from treatment comparisons, they would likely all be preferable to, say, a completely randomized design. Verbally describe, as you would to a soil scientist, the layout of the designs and the situations in which each has advantages (that is, what would motivate you to use any one of these designs) Finally, which one would you suggest for the situation described above? Latin Square Complete Latin Square Row Complete Latin Square Column Complete Latin Square 2. Consider this row of numbers as the treatments (in order) in the first row of a Latin Square where the treatments are numbered 0,1 ..., 5. 0 5 1 4 2 3 Now add 1 to each number and reduce it modulo 6 (i.e. write down the remainder when the number is divided by 6) That is the second row. Using this second row, do the same thing to create a third row and so on. (A) Write down the array you just found. Find all the pairs (next to each other) in the rows where one element in the pair is 4. Write all pairs in each row to the right of that row in your array. Here (a,b) is to be considered different from (b,a). (B) Going through all the columns in the same way, list all the neighbor pairs that include 4. Note any that are missing. (C) Does your computation in (A) [only what was required] prove that the design is row complete? Does it show that the design is not row complete? Explain briefly. (D) Does your computation in (B) [only what was required] prove that the design is column complete? Does it show that the design is not column complete? Explain briefly. (E) Rearrange the rows (i.e. row i becomes row j) so that the first column looks the same as the row of numbers at the beginning of this question (question 2). Redo part (B). 3. Try making a similar array for an 8x8 Latin Square (including the arrangement of the rows so that the order of numbers in the first row matches that in the first column) and just write out the resulting 8x8 display of treatment assignments to the rows and columns (no need to check the pairs, just write the design). I will step you through the method as given in section 6.5.2 of the book. To get your first starting row use the smallest and largest numbers 0 7 then for each subsequent position (odd or even) add (for odd) and subtract (for even) 1 from the previous odd or even position entry. In position 3 you would have 0+1 because 0 is in position 1 and for postion 4 you have 7-1 = 6. At this point we have 0 7 1 6 You take it from here, completing the first row, then the square, then the rearranged square. For this particular square if you look at pairs that involve the number 2 and get row complete then (I am telling you just to save you time) the whole square will be row complete. So ... is it row complete? The same is true (again this is not a general statement) for the columns of this particular square so again using the pairs with 2, is this square column complete? [note, the properties you found here will hold in general for an even number of treatments if you produce the array using these steps. As an ungraded challenge, you might try writing a SAS program (or R or S+) to generate such a square from just the initial knowledge of n (assuming n is even) ] 4. Work problem 6.10 on page 135 but don't bother showing every pair is orthogonal. Write down all four squares but just show that the first 2 form an orthogonal pair (in other words together they form a Graeco-Latin square). This problem uses modulo arithmetic which means you have the remainder of the first integer when divided by the second. For example, 9 modulo 4 is 1 (9 = 2*4 +1), 12 modulo 5 is 2, 7 modulo 5 is also 2 and 3 modulo 5 is 3. The problem gives a clever algorithm for constructing Graeco-Latin squares when the number of treatments t is odd.