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I'm looking at Latin Squares. I've seen standard Latin squares, and Latin squares when the rows are randomised, then columns are randomised.

When would one be used over the other?

It seems as though the randomised would always be better as the ordering of the rows is less consistent. For example, in a 5x5 Latin square the pattern D,E would appear in the rows 4 times. I'm not sure how to justify that though, or if it is even true.

edit - standard and randomised Latin square, experiment

Often there is initially a standard cyclic square as follows :

    Experiment : 
    ------------ 

         Di = ith Driver
         Cj = jth Car
    A,B,C,D = different tyres
    reponse = fuel efficiency

    ------------ 

    standard latin square

        [C1] [C2] [C3] [C4]
    [D1] "A"  "B"  "C"  "D" 
    [D2] "B"  "C"  "D"  "A" 
    [D3] "C"  "D"  "A"  "B" 
    [D4] "D"  "A"  "B"  "C" 

The rows and columns of this can be randomised to give

    randomised latin square

        [C1] [C2] [C3] [C4]
    [D1] "D"  "B"  "A"  "C" 
    [D2] "C"  "A"  "D"  "B" 
    [D3] "A"  "C"  "B"  "D" 
    [D4] "B"  "D"  "C"  "A" 

edit 2 - experiment across different days with the same Latin square

In this approach the experiments have the same tyres for each row and column

    Experiment : 
    ------------ 
         Di = ith Driver
         Cj = jth Car
    A,B,C,D = different tyres
    reponse = fuel efficiency
    ------------ 

    Day 1

        [C1] [C2] [C3] [C4]
    [D1] "D"  "B"  "A"  "C" 
    [D2] "C"  "A"  "D"  "B" 
    [D3] "A"  "C"  "B"  "D" 
    [D4] "B"  "D"  "C"  "A" 

    Day 2

        [C1] [C2] [C3] [C4]
    [D1] "D"  "B"  "A"  "C" 
    [D2] "C"  "A"  "D"  "B" 
    [D3] "A"  "C"  "B"  "D" 
    [D4] "B"  "D"  "C"  "A" 

edit 3 - experiment across different days with different same Latin squares

Here there are two different squares, but note that (D1, C2) is the same for each of them.

    Day 1

        [C1] [C2] [C3] [C4]
    [D1] "A"  "B"  "C"  "D" 
    [D2] "B"  "C"  "D"  "A" 
    [D3] "C"  "D"  "A"  "B" 
    [D4] "D"  "A"  "B"  "C" 

    Day 2

        [C1] [C2] [C3] [C4]
    [D1] "D"  "B"  "A"  "C" 
    [D2] "C"  "A"  "D"  "B" 
    [D3] "A"  "C"  "B"  "D" 
    [D4] "B"  "D"  "C"  "A" 

edit 4 - experiment across different days, ensuring that the same tyre isn't used in the same row/column position across days

Here the standard square has been used as a base.

In order to remove bias between days the rows have been shifted, so

row 1 sent to row 4
row 2 sent to row 1
row 3 sent to row 2
row 4 sent to row 4

Here is the latin square experiment. Latin square to reduce bias within the square, and altered rows to reduce bias between the experiments

Day 1

    [C1] [C2] [C3] [C4]
[D1] "A"  "B"  "C"  "D" 
[D2] "B"  "C"  "D"  "A" 
[D3] "C"  "D"  "A"  "B" 
[D4] "D"  "A"  "B"  "C" 

Day 2

    [C1] [C2] [C3] [C4]
[D1] "B"  "C"  "D"  "A" 
[D2] "C"  "D"  "A"  "B" 
[D3] "D"  "A"  "B"  "C" 
[D4] "A"  "B"  "C"  "D" 

Summary

So the reason that we don't use the same square (like in edit 2) for each day (experiment) is that there may be bias introduced from driver D1 using car C1 with tyres B. Therefore we need to reorder the square between experiments in a non random manner.

In edit 3 I've used the standard square and the randomised square but this isn't enough (presumably) because there are some squares which have the same tyre for each experiment. So I have made edit 4 based on the standard square, with the rows shifted between days (experiments) to ensure that the same tyre doesn't appear for the same car/driver.

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A Latin square is just a mathematical phenomenon that has been described well before its use in experimental design, so it need not be randomized to qualify as such.

However, in experimental design you always want to randomize assignment. Namely, while the fact that you're using a Latin square mitigates spatial dependencies within that experiment, there may still be bias between experiments if you were to always use the same Latin square.

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  • $\begingroup$ Thanks - but in this case why are standard sometimes used over randomised ? In that I've seen literature which doesn't first randomise the square prior to carrying out an ANOVA on it. And when you say that "there may still be bias between experiments if you were to always use the same latin square", what is the context of this? The standard square (that I've added to my original post) has the pattern CD three times in the rows, whereas the randomised doesn't. Is this what you're referring to? $\endgroup$ – baxx Oct 30 '18 at 15:11
  • $\begingroup$ I'm not sure I understand what you mean. There is no fundamental difference between what you call the standard Latin square and a randomized Latin square. For example, in your randomized version, B is horizontally adjacent to D, and A to C, each 4 times. The point is not to have the same treatments at the same locations across experiments. $\endgroup$ – Frans Rodenburg Oct 31 '18 at 1:50
  • $\begingroup$ I've edited the original post with a contrived experiment, to make referencing easier. Please see the summary I've added, I couldn't quite fit it into a comment. I shall clean up the post tomorrow. Thanks $\endgroup$ – baxx Oct 31 '18 at 2:37
  • $\begingroup$ I wouldn't recommend trying to construct a Latin square that seems more random/unbiased. Instead, if you randomly generate two Latin squares in your example, the chance that exactly the pattern that arises in those two Latin squares also happens to be a combination that is somehow interacting is minimal. $\endgroup$ – Frans Rodenburg Oct 31 '18 at 10:17
  • $\begingroup$ Ah ok - so as long as there isn't exactly the same pattern we're alright. So a few squares can match, but that won't create enough bias to distort the results. $\endgroup$ – baxx Oct 31 '18 at 18:49

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