# What is the meaning of "each treatment " in BIBD?

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 ?

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)

• should this be in math overflow? Mar 22, 2017 at 13:25
• @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) Mar 22, 2017 at 15:07

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.