Balance Conditions for Incomplete-Block Designs

What Balance Means in an Incomplete-Block Design

In a complete block design, every block contains every treatment. In an incomplete-block design, each block contains only some of the treatments. This can be useful when the experimental units in a block are limited, but it also raises a design question: are treatments represented evenly, and are treatment pairs compared equally often? Balance gives precise answers to both questions.

For example, if one treatment appears in many more blocks than another, the two treatments do not have equal replication. Even if every treatment appears equally often, some pairs might share blocks more often than others. A balanced incomplete-block design controls both features through a small set of integer parameters.

Definition: A balanced incomplete-block design (BIBD) has \(v\) treatments arranged in \(b\) blocks, with \(k\) distinct treatments in each block, where \(k<v\). Each treatment appears in exactly \(r\) blocks, and every pair of distinct treatments appears together in exactly \(\lambda\) blocks. The positive integers \(v,b,k,r,\lambda\) are the design parameters. Here \(r\) is the replication number and \(\lambda\) is the pairwise concurrence number.

The word incomplete refers to the fact that \(k<v\): a block does not contain all treatments. The word balanced refers to equal replication and equal pairwise concurrence. A design with equal-sized blocks alone is not necessarily balanced. Nor is equal replication alone enough: the numbers of blocks shared by treatment pairs must also be equal.

A useful way to record the layout is with an incidence matrix. Its rows correspond to treatments and its columns to blocks; an entry is 1 when that treatment occurs in that block and 0 otherwise. Each column then has \(k\) ones, each row has \(r\) ones, and the overlap of any two distinct rows has \(\lambda\) ones. This gives a compact way to express the balance conditions.

The Two Counting Identities

The parameters cannot be chosen independently. Counting the treatment appearances in two different ways gives the first identity. There are \(b\) blocks with \(k\) treatment places each, so there are \(bk\) treatment appearances in total. Alternatively, there are \(v\) treatments, each appearing \(r\) times, giving \(vr\) appearances.

$$ bk=vr $$

The second identity follows by fixing one treatment. It shares a block with \(k-1\) other treatments in each of its \(r\) blocks, for \(r(k-1)\) treatment-pair encounters. On the other hand, there are \(v-1\) other treatments, and the fixed treatment occurs with each of them in exactly \(\lambda\) blocks. That count is \(\lambda(v-1)\).

r(k-1)=\lambda(v-1)

Together, these identities are often called the basic BIBD parameter identities. They are necessary conditions: every BIBD must satisfy them. They also provide ways to solve for an unknown parameter, or to rule out a proposed parameter set when the equations fail.

Key result: For a BIBD, \(bk=vr\) and \(r(k-1)=\lambda(v-1)\). Consequently, \(r=bk/v\) and \(\lambda=r(k-1)/(v-1)\) when the denominators are nonzero. These calculated values must be positive integers for a conventional BIBD. Satisfying the identities is necessary, but by itself does not prove that a design with those parameters exists.

Worked Examples of the Balance Conditions

Worked Example: Verifying a Design With Four Treatments

Suppose there are four treatments, labelled A, B, C, and D, and the blocks are AB, AC, AD, BC, BD, and CD. Verify that this is a BIBD and identify its parameters.

There are \(v=4\) treatments and \(b=6\) blocks. Each block contains \(k=2\) distinct treatments, and \(k<v\), so the blocks are incomplete. Treatment A appears in AB, AC, and AD; the same count holds for B, C, and D. Thus \(r=3\).

Each block in this design is one of the six possible pairs of distinct treatments. Therefore every pair occurs together exactly once, so \(\lambda=1\). The parameter identities check:

bk=6(2)=12,\qquad vr=4(3)=12

r(k-1)=3(2-1)=3,\qquad \lambda(v-1)=1(4-1)=3

Both conditions hold, and the direct checks of replication and concurrence establish that the listed blocks form a BIBD with parameters \((v,b,r,k,\lambda)=(4,6,3,2,1)\).

Worked Example: Checking Pairwise Concurrence in a Seven-Treatment Design

Consider seven treatments, labelled 1 through 7, in these seven blocks:

123, 145, 167, 246, 257, 347, and 356.

Each block contains \(k=3\) treatments, and there are \(v=7\) treatments and \(b=7\) blocks. Counting occurrences in the list, every treatment appears in three blocks, so \(r=3\). To check pairwise concurrence, list the pairs contributed by each block:

123 contributes 12, 13, 23; 145 contributes 14, 15, 45; 167 contributes 16, 17, 67; 246 contributes 24, 26, 46; 257 contributes 25, 27, 57; 347 contributes 34, 37, 47; and 356 contributes 35, 36, 56.

These are 21 distinct pairs. The total number of distinct pairs among seven treatments is \(7(6)/2=21\), so every possible pair occurs exactly once. Hence \(\lambda=1\). The identities agree with the block list:

bk=7(3)=21,\qquad vr=7(3)=21

r(k-1)=3(3-1)=6,\qquad \lambda(v-1)=1(7-1)=6

This is a BIBD with parameters \((7,7,3,3,1)\). Notice that the identity check supports the verification, while the block list itself establishes that every treatment has the stated replication and every pair the stated concurrence.

Worked Example: Rejecting an Impossible Parameter Set

A proposed design has \(v=8\) treatments, \(k=4\) treatments per block, and \(r=6\) appearances per treatment. Could these parameters belong to a BIBD?

The first identity would require the number of blocks to be

b=\frac{vr}{k}=\frac{8(6)}{4}=12.

The second identity would require

\lambda=\frac{r(k-1)}{v-1} =\frac{6(4-1)}{8-1} =\frac{18}{7} \approx 2.571.

But \(\lambda\) counts blocks and must be an integer. Therefore no BIBD can have these specified values of \(v\), \(k\), and \(r\). The calculation rules out the proposed parameter set without needing to construct or inspect any blocks. Conversely, if the calculation had produced an integer \(\lambda\), that would only have passed this necessary arithmetic check; it would not, by itself, prove that the blocks can be arranged as required.

A Counting Justification From the Incidence Matrix

The same identities can be derived by counting incidences and treatment pairs. This view helps explain what is being counted and provides a bridge to the incidence-matrix representation used for incomplete-block designs.

Let \(N\) be the \(v\)-by-\(b\) incidence matrix, with \(N_{ij}=1\) if treatment \(i\) is in block \(j\), and \(N_{ij}=0\) otherwise. Summing all entries column by column gives \(bk\), because each of the \(b\) columns contains \(k\) ones. Summing row by row gives \(vr\), because each of the \(v\) rows contains \(r\) ones. Since both sums count exactly the same entries, \(bk=vr\).

For the second identity, count ordered pairs of distinct treatments that occur in a common block. Each block has \(k\) choices for the first treatment and \(k-1\) choices for the second, so it contributes \(k(k-1)\) ordered pairs. Across all blocks, the count is \(bk(k-1)\). Alternatively, each treatment occurs in \(r\) blocks and is paired in each such block with \(k-1\) other treatments. Across all \(v\) treatments, this gives \(vr(k-1)\). Using \(bk=vr\), both expressions agree.

Balance gives a more specific count: there are \(v(v-1)\) ordered pairs of distinct treatments, and each ordered pair occurs together in exactly \(\lambda\) blocks. The total is therefore \(\lambda v(v-1)\). Equating the two counts and dividing by \(v\) gives the second identity:

bk(k-1)=\lambda v(v-1) \quad\Longrightarrow\quad r(k-1)=\lambda(v-1),

where the final step uses \(bk=vr\). Counting unordered pairs instead gives the same result: each block contributes \(k(k-1)/2\) pairs, while the design as a whole contributes \(\lambda v(v-1)/2\).

How to Use the Identities

When some design parameters are known, the identities can determine others. For instance, if \(v\), \(k\), and \(r\) are specified, then

b=\frac{vr}{k}, \qquad \lambda=\frac{r(k-1)}{v-1}.

Both results must be positive integers for a conventional BIBD. If either is not an integer, the proposed values cannot describe such a design. If both are integers, the parameter set is arithmetically compatible with these two identities, but further construction or verification is still needed to establish existence.

When a complete block list is available, check the design directly: every block must contain \(k\) distinct treatments; each treatment must occur in \(r\) blocks; and every distinct pair must occur in \(\lambda\) blocks. The identities are useful cross-checks, but they do not replace these conditions when verifying a specific list.

Common pitfall: “All treatments appear equally often, so the design is balanced.” Equal replication verifies \(r\), but not pairwise balance. For example, two pairs could occur together more often than other pairs even when every treatment appears in the same number of blocks. Check that every pair has the same concurrence \(\lambda\), as well as checking equal block size and equal replication.

These conditions describe the arrangement of treatments in blocks. They do not, on their own, specify a response model, establish experimental assumptions, or determine how treatment effects should be estimated. Their role here is structural: they define balance and constrain which combinations of design parameters are possible. That structure matters because it makes the treatment representation even across blocks and treatment pairs, a property used in later analysis of incomplete-block data.

Check Your Understanding

Use the definitions and identities to answer the following questions.

In a BIBD, what does \(r\) count, and what does \(\lambda\) count?
Explain why counting treatment appearances in two ways gives \(bk=vr\).
A proposed design has \(v=6\), \(k=3\), and \(r=4\). What values must \(b\) and \(\lambda\) have according to the basic identities?
Why does an integer value of \(\lambda=r(k-1)/(v-1)\) not prove that a BIBD with the proposed parameters exists?
Can equal replication alone establish pairwise balance? State what additional condition must be checked.