Balanced Incomplete Block Design Properties

A Balanced Incomplete Block Design (BIBD) is a type of experimental design used in statistics. It involves v treatments and b blocks. Each block contains k plots, where k < v. Each treatment appears in r blocks, and every pair of distinct treatments appears together in λ blocks. The incidence matrix N is a v × b matrix where N_ij = 1 if treatment i is in block j, and 0 otherwise.

There are fundamental relationships that hold true for a BIBD:

• The total number of plots in all blocks equals the total number of times treatments appear across all plots: $bk = vr$.
• The relationship between parameters considering pairs of treatments: $\lambda (v-1) = r(k-1)$.

Let's analyze each given statement based on these properties and the properties of the incidence matrix N.

Analyzing Statement 1: $\lambda(k - 1) = r(v - 1)$

The standard fundamental relationship for a BIBD is $\lambda(v-1) = r(k-1)$. The given statement is $\lambda(k - 1) = r(v - 1)$. This is different from the standard identity. Unless specific parameter values coincidentally satisfy this equation, this statement is generally incorrect for any BIBD parameters (v, b, r, k, $\lambda$).

Analyzing Statement 2: $b \ge v$

This statement represents Fisher's inequality for BIBDs. For a standard BIBD where r > $\lambda$ (which implies k < v), it can be proven that the number of blocks (b) must be greater than or equal to the number of treatments (v). This is a necessary condition for the existence of a BIBD (except for trivial cases or specific degenerate cases). Therefore, this statement is generally considered correct for non-trivial BIBDs.

Analyzing Statement 3: rank (NN') = v

The matrix NN' is a v × v matrix. Its elements (NN')_ij represent the number of blocks containing both treatment i and treatment j. For i = j, (NN')_ii is the number of blocks containing treatment i, which is r. For i $\ne$ j, (NN')_ij is the number of blocks containing both treatment i and treatment j, which is $\lambda$.

So, $NN' = (r - \lambda)I + \lambda J$, where I is the v × v identity matrix and J is the v × v matrix of all ones.

For a connected BIBD (a standard assumption), this matrix NN' is non-singular, meaning its determinant is non-zero. The eigenvalues of $(r - \lambda)I + \lambda J$ are $r - \lambda$ (with multiplicity v-1) and $r - \lambda + v\lambda$ (with multiplicity 1). Using the identity $\lambda(v-1) = r(k-1)$, we can simplify $r - \lambda + v\lambda = r + \lambda(v-1) = r + r(k-1) = r + rk - r = rk$.

The eigenvalues are $r - \lambda$ (v-1 times) and $rk$ (1 time). For a proper BIBD (k < v, r > $\lambda$), $r - \lambda > 0$ and $rk > 0$. Since all eigenvalues are positive, the matrix NN' is positive definite and hence its rank is v. Therefore, this statement is correct for a connected BIBD.

Analyzing Statement 4: trace (NN') = bk

The trace of a matrix is the sum of its diagonal elements. As determined above, the diagonal elements of NN' are (NN')_ii = r for all i = 1, ..., v.

So, trace(NN') = $\sum_{i=1}^{v} (NN')_{ii} = \sum_{i=1}^{v} r = vr$.

From the fundamental relationship of a BIBD, we know that $bk = vr$.

Substituting $vr$ with $bk$, we get trace(NN') = $bk$. Therefore, this statement is correct.

Based on the analysis, the correct statements are b ≥ v, rank (NN') = v, and trace (NN') = bk.