What is orthogonal Latin square design?

In combinatorics, two Latin squares of the same size (order) are said to be orthogonal if when superimposed the ordered paired entries in the positions are all distinct. A set of Latin squares, all of the same order, all pairs of which are orthogonal is called a set of mutually orthogonal Latin squares.

How do you make Latin squares?

Step 1: Make the first row using the formula: row1 = 1,2,n,3,n-1,n-2. Step 2: Fill in the first column sequentially. Step 2: Continue filling in the columns sequentially until the square is completed. A completed balanced square design with an even number of conditions.

What is a symmetric Latin square?

A Latin square has six conjugate Latin squares obtained by uniformly permuting its (row, column, symbol) triples. We say that a Latin square has conjugate symmetry if at least two of its six conjugates are equal.

How many 4×4 Latin squares are there?

576 possible
This example illustrates one of the 576 possible Latin squares for a 4-by-4 layout; larger squares have many orders of magnitude more combinations (e.g., 161,280 for a 5-by-5 layout).

What is meant by orthogonal array?

In mathematics, an orthogonal array is a “table” (array) whose entries come from a fixed finite set of symbols (typically, {1,2,…,n}), arranged in such a way that there is an integer t so that for every selection of t columns of the table, all ordered t-tuples of the symbols, formed by taking the entries in each row …

What’s the difference between a Latin square and Graeco-Latin square?

We write the Latin square first then each of the Greek letters occurs alongside each of the Latin letters. A Graeco-Latin square is a set of two orthogonal Latin squares where each of the Greek and Latin letters is a Latin square and the Latin square is orthogonal to the Greek square.

Why a 2 * 2 Latin square is not possible?

The number of rows and columns has to correspond to the number of treatment levels. N = t 2 (the number of rows times the number of columns) and t is the number of treatments. Note that a Latin Square is an incomplete design, which means that it does not include observations for all possible combinations of i, j and k.

What is the purpose of constructing a Latin square?

We denote by Roman characters the treatments. Therefore the design is called a Latin square design. This kind of design is used to reduce systematic error due to rows (treatments) and columns.

Is a Latin square a magic square?

They were discovered by Euler a few centuries later, who saw them as a new type of magic square, and it’s thanks to him that we call them Latin squares. Latin squares are grids filled with numbers, letters or symbols, in such a way that no number appears twice in the same row or column.

What is the strength of an orthogonal array?

Definition. An Orthogonal Array of strength t with N rows, k columns (k ≥ t) and. based on s symbols is an N × k array with entries 0,1,…,s − 1, say, so that every N × t subarray contains each of the st possible t-tuples. equally often as a row (say λ times)

Why orthogonal array is used in Lean?

When an orthogonal array (OA) of n rows is used as the design matrix in an experiment, n is the number of runs. A lean design is a design matrix formed by deleting some rows and columns of an OA, which still allows efficient estimation of the effects of the factors corresponding to the remaining columns of the OA.

What are the disadvantages of a Latin square design?

Disadvantages of latin square designs 1. Number of treatments is limited to the number of replicates which seldom exceeds 10. 2. If have less than 5 treatments, the df for controlling random variation is relatively large and the df for error is small.