A complement to Luis' excellent response.

One of the reasons for using incomplete block desings is tha the smaller the block, the better the control of the environment. There are textbooks on experimental designs. I still have an old Cochran & Cox book, 1950 edition! There must be much more up to date books.

What I want to underline is that it is good to take advice from experienced statisticians or experimenter BEFORE making your experiment. Consider this experiment with 7 treatments numbered 1 to 7.

This very compact design is an incomplete latin square.

Block 1 : 1, 2, 4

Block 2 : 2, 3, 5

Block 3 : 3, 4, 6

Block 4 : 4, 5, 7

Block 5 : 5, 6, 1

Block 6 : 6, 7, 2

Block 7 : 7, 1, 3

Note that each treatment is compared exactly once with each other. It makes comparisons between traitments easier. It would not have been the case if this apparently more straightforward design had been adopted:

Block 1 : 1, 2, 3

Block 2 : 2, 3, 4

Block 3 : 3, 4, 5

etc.

thank you Luis

A complement to Luis' excellent response.

One of the reasons for using incomplete block desings is tha the smaller the block, the better the control of the environment. There are textbooks on experimental designs. I still have an old Cochran & Cox book, 1950 edition! There must be much more up to date books.

What I want to underline is that it is good to take advice from experienced statisticians or experimenter BEFORE making your experiment. Consider this experiment with 7 treatments numbered 1 to 7.

This very compact design is an incomplete latin square.

Block 1 : 1, 2, 4

Block 2 : 2, 3, 5

Block 3 : 3, 4, 6

Block 4 : 4, 5, 7

Block 5 : 5, 6, 1

Block 6 : 6, 7, 2

Block 7 : 7, 1, 3

Note that each treatment is compared exactly once with each other. It makes comparisons between traitments easier. It would not have been the case if this apparently more straightforward design had been adopted:

Block 1 : 1, 2, 3

Block 2 : 2, 3, 4

Block 3 : 3, 4, 5

etc.

Thanks Luc Baudouin for a detailed and easy-to-understand post