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My doubt is regarding Balanced Incomplete Block Design (BIBD)

Consider the following design with 9 treatments and 12 blocks each of size 3, is this design BIBD or not ?

Image/table of a block design

in my opinion it's not BIBD (but I am not sure whether my logic is correct).

This is what my logic is: since we have 9 treatments we must have $9\choose 2$ treatment pairs. If it would have been BIBD it should include all $9\choose2$ treatment pairs, but it does not include all $9\choose2$ treatment pairs, e.g. there is no block having treatment pairs {2,3},{2,5},{3,5},{3,8}, so its not BIBD.

Is my logic correct? (and I can not ask more clear than this)

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  • $\begingroup$ should this be in math overflow? $\endgroup$
    – Haitao Du
    Mar 22, 2017 at 13:25
  • $\begingroup$ @hxd1011: No, this belongs here! BIBD (Balanced Incomplete Block Designs) is an important topic in Design of Experiments. (and, at mathowerflow, this would be closed immeadiately as not research level) $\endgroup$ Mar 22, 2017 at 15:07

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Yes, your conclusion and argument is correct. To make this clearer you could try to draw the graph of the design as in examples of connected designs in DOE ( you should be able to adapt the code from that post)

Since the design contains $12 \cdot 3 =36$ points, and $\binom{9}{2}=36$, could accomodate a BIBD with the pair cooccurrence constant $\lambda=1$, but some pairs are lacking, as you said, and some, like 28 occurs multiple times. Some examples: 28 in blocks 1,6,9,11, 56 in blocks 5,8 and more.

To construct a BIBD with your parameters you can try, in R:

library(crossdes)
> D  <-  find.BIB(9,12,3,iter=100)
> D
      [,1] [,2] [,3]
 [1,]    3    6    7
 [2,]    1    5    9
 [3,]    1    2    6
 [4,]    6    8    9
 [5,]    1    7    8
 [6,]    2    4    8
 [7,]    3    5    8
 [8,]    4    5    6
 [9,]    2    3    9
[10,]    2    5    7
[11,]    4    7    9
[12,]    1    3    4
> isGYD(D)

[1] The design is a balanced incomplete block design w.r.t. rows.

For more information, you can have a look at https://www.r-bloggers.com/generating-balanced-incomplete-block-designs-bibd/

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    $\begingroup$ find.BIBD .,., thats amazing .,., thank you so much $\endgroup$
    – ANUJ NAIN
    Mar 28, 2017 at 15:34

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