Useful Block Designs in Biostatistics

Abstract

Randomized Complete Block Designs (RCBD), Balanced Incomplete Block Designs (BIBD) and the so-called Generalized Lattice Designs as Alpha Designs are useful designs in Biostatistics. A complete table of BIBDs with the smallest number b of blocks with at most v = 25 treatments and block sizes k for $2<k\le \frac{v}{2}$ is presented. Such a table did not exist until now. The analysis of the different block designs (Randomized Complete Block design, BIBDs and Alpha Design) is here not done with the commercial statistical packages SAS or SPSS. These packages can now only be hired for a year and are quite expensive. We used the package of R for the analysis, which is free of charge and it is now used in most Universities.

1. Introduction

Experiments in Biostatistics to compare treatments need homogeneous conditions. R.A. Fisher, a statistician at Rothamsted Experimental Station in Hertfordshire in England, published in 1926 an article “The arrangement of field experiments” [1]. Within 10 and a half pages Fisher gives all principles of experimental designs: replication, randomization and blocking. In agriculture with variety trials, the experimental field was often laid down next to a ditch. Plots parallel to the ditch have the same growing conditions, but plots farther away from the ditch have other growing conditions than plots next to the ditch. When one wants to investigate v varieties Fisher proposed starting with the v plots next to the ditch. These v plots form then a block of plots with the same growing conditions. The varieties are placed in this first block in a randomized order. The plots adjacent to the first block, farther away from the ditch form a new block of v plots. In this second block, the v varieties are placed again in a random order. This design is called a randomized complete block design for v varieties with b = 2 blocks. The complete blocks are also called replications, because all the v varieties are present in this block. The statistical model for the yield in such a randomized complete block design with v varieties and b blocks is that of a two-fold analysis of variance and is

$${\mathit{y}}_{\mathit{ij}}=\mu +{\alpha }_i+{\beta }_j+{\mathit{e}}_{\mathit{ij}},i=1,\dots ,v;j=1,\dots ,b;$$E1

In (1), y_ij is the random model of the observed yield y_ij, μ is the general mean, α_i is the varietal effect, β_j is the block effect. Further, we have the side conditions ${\sum }_{i=1}^v\alpha$_i = 0 and ${\sum }_{j=1}^b\beta$_j = 0. Furthermore, the e_ij terms are independently distributed random errors with a normal distribution, each having an expectation of 0 and a variance of σ². If the blocks of a block design are randomly selected from a huge set of blocks available, we have a mixed model with random β_j. Due to the rarity of this scenario in Biostatistics, we will not be discussing it in this chapter.

If all pairs of varietal mean differences α_p – α_q are by Least Squares Method estimated by $\overline{y}$_p – $\overline{y}$_q for p$\ne$q = 1,…, v and where $\overline{y}$_p = ${\sum }_{j=1}^{b}y$_pj/b and $\overline{y}$_q = ${\sum }_{j=1}^{b}y$_qj/b it can be shown that all pairs have the same variance $\frac{2{\sigma }^2}{b}$. This is a nice property of a randomized complete block design.

In Biostatistics, Complete Block Designs are very useful. The researcher must only be looking for the same experimental conditions for his v treatments. Often b blocks with the same number k experimental units per block are used.

We present the analysis with R of a randomized complete Block Designs in Section 2.

But unfortunately, the Biostatistician often comes in the situation that his number of treatments v is larger than the block size k. In this case, the Balanced Incomplete Block Design (BIBD) for the investigation of his v treatments is a good alternative design. A BIBD consists of b blocks each with k experimental units but k < v. The number of times a treatment is used in a BIBD is r, the number of replicates. Further, each pair of treatments occurs in a BIBD λ times together in all the blocks; λ = r·(k − 1)/(v − 1) is called the number of concurrences. In a BIBD, we observe the property of equal variance for all treatment effects and treatment effect differences estimated by the Least Squares Method. Specifically, the variance for a treatment effect is given by (σ²/(r·v))·(1 + k·r·(v − 1)/(λ·v)) and the variance for a treatment effect difference is given by σ² (2 k/λv).

The BIBDs are discussed in Section 3, their analysis with R in Section 4.

Often the number of blocks in a BIBD is very large and the design is not useful in field trials or in other Biostatistics trials. Section 5 introduces an alternative method known as Alpha Designs and their analysis with R is presented in Section 6.

The analysis of the different block designs (Randomized Complete Block design, BIBDs and Alpha Design) is here not done with the commercial statistical packages SAS or SPSS. These packages can now only be hired for a year and are quite expensive. We used the package of R for the analysis, which is free of charge and it is now used in most Universities.

2. Analysis of a randomized complete block design with R

Kuehl [2] gives Example 8.1 of a randomized complete block design on pages 257–258. Current nitrogen fertilization recommendations for wheat included applications of specified amounts at specified stages of plant growth. The recommendations were developed through the use of periodic stem tissue analysis of nitrate content of the plants. Stem tissue analysis was thought to be very effective to monitor the nitrogen status of the crop and provide a basis for predicting required nitrogen for optimum production. In certain situations, however, the stem nitrate tests were found to over-predict the required nitrate amounts. Consequently, the researcher wanted to evaluate the effect of several different fertilization timing schedules on the stem tissue nitrate amounts and wheat production to refine the recommendation procedure. The treatment design included six different nitrogen application timing and rate schedules that were thought to provide the range of conditions necessary to evaluate the process. For the purpose of comparison, a control treatment with no nitrogen, treatment (1), was included, as well as the current standard recommendation.

The experiment was conducted in an irrigated field with a water gradient along one direction of the experimental area. Since plant responses are affected by variability in the amount of available moisture, the field plots were grouped into blocks of six plots such that each block occurred in the same part of the water gradient. Thus, any differences in plant responses caused by the water gradient could be associated with the blocks. The resulting design was a randomized complete block design with four blocks of six field plots to which the nitrogen treatments were randomly allocated.

The layout of the experimental plots in the field is shown below. The observed NO₃ nitrogen content (ppm × 10⁻²) from a sample of wheat stems is shown for each plot with the treatment number in parentheses before it.

Block 1	(2) 40.89	(5) 37.99	(4) 37.18	(1) 34.98	(6) 34.89	(3) 42.07	Irrigation
Block 2	(1) 41.22	(3) 49.42	(4) 45.85	(6) 50.15	(5) 41.99	(2) 46.69	Gradient
Block 3	(6) 44.57	(3) 52.68	(5) 37.61	(1) 36.94	(2) 46.65	(4) 40.23	↓
Block 4	(2) 41.90	(4) 39.20	(6) 43.29	(5) 40.45	(3) 42.91	(1) 39.97

We now bring the data in the R-package using the following R commands.

Note: The coefficient Block2 represents the estimated difference in effect between Block 2 and Block 1.

The coefficient Treatment2 represents the estimated difference in effect between Treatment 2 and Treatment 1. For a Randomized Complete Block Design, the estimate with the Least Squares Method of a treatment is equal to the mean of the treatment in the experiment.

The estimate of the standard error of the differences of the least squares mean of two treatment means for a Randomized Complete Block Design uses the same estimate for the variance σ², it is the square of the residual standard error s = 2.683, hence s² = 7.1985 with 15 degrees of freedom. The estimate of the standard error of the differences of the least squares means of two treatment means is given by √ (s² (2/4)) = √3.5993 = 1.897.

Note: The estimate of the standard error of a Least Square Mean of a treatment is √ s²/4 = √ 7.1985/4 = √1.7996 = 1.342.

But the investigator mentioned now that treatment 4 was the standard fertilizer recommendation for wheat. The nitrate nitrogen in the stem of the wheat plant measured throughout the growing season is used to assess nitrogen requirements for optimum wheat yields. The investigator would be interested in differences between any of the individual nitrogen timing treatments and the current recommendation of each stage of growth. The Dunnett’s test can be used to compare the standard recommendation to each of the other timing treatments including the no nitrogen control treatment. The no nitrogen control provides a means of evaluating the nitrogen available without fertilization in these particular plots.

In the Dunnett’s test table, as described by Dunnett [3, 4], for a degree of freedom (df) of 15 and k = 5 (the number of treatments to compare with the control), with a significance level (α) of 0.05 (two-sided), the critical value is 2.82. Therefore we must compare the absolute value of the difference in means between treatment i and 4 with 2.82× SE($\overline{y}$_i – $\overline{y}$₄) = 2.82 × √ 3.5993 = 5.35.

Only the absolute difference of treatment 3 and 4, 6.16, is larger than 5.35.

$$\mid {\overline{y}}_1–{\overline{y}}_4\mid =2.34<5.35$$E2

$$\mid {\overline{y}}_2–{\overline{y}}_4\mid =3.42<5.35$$E3

$$\mid {\overline{y}}_3–{\overline{y}}_4\mid =\mathbf{6.16}\mathbf{>}\mathbf{5.35}$$E4

$$\mid {\overline{y}}_5–{\overline{y}}_4\mid =1.11<5.35$$E5

$$\mid {\overline{y}}_6–{\overline{y}}_4\mid =2.61<5.35$$E6

With the R- package “nCDunnett”, we can find the quantile of the Dunnett’s test.

This package can be used for the central and non-central Dunnett’s test.

The degrees of freedom is nu = 15. We have 5 treatments which are compared with the control hence, the correlation coefficient is 0.5 for two comparisons, this is given for all 5 treatment comparisons by the vector rho = c(0.5, 0.5, 0.5, 0.5, 0.5).

For the test, the non-centrality parameter is δ = 0. This is indicated by the vector delta = c(0,0,0,0,0) for the 5 treatment comparisons. We want to use the significance level α = 0.05 for a two-sided test. We indicate this by the confidence coefficient p = 1 – α = 0.95 and, in the command, we use the indication two-sided = TRUE. The computation is done with 32 points of the Gaussian quadrature method. The R- commands are then:

This quantile point 2.82 is also given by the table of Dunnett [4].

With the R-package “multcomp” for multi-comparisons, we can find the significance of the Dunnett’s test where control is Treatment 4.

Only the difference of Treatment 3 and Treatment 4 has a significant p-value Pr(>| t|) = 0.022 < 0.05.

3. Balanced incomplete block designs

Balanced Incomplete Block designs consist of b blocks of size k experimental units each but k < v. The number of times a treatment is used in a BIBD is r, the number of replicates. Further, each pair of treatments occurs in a BIBD together in all the blocks λ times; λ = r·(k − 1)/(v − 1) is called the number of concurrences.

In a BIBD, we observe the property of equal variance for all treatment effects and treatment effect differences estimated by the Least Squares Method. Specifically, the variance for a treatment effect is given by (σ²/(r·v))·(1 + k·r·(v − 1)/(λ·v)) and the variance for a treatment effect difference is given by σ² (2 k/λv).

3.1 Theoretical background

We give at first an introduction into the theory behind BIBDs.

We consider experiments that include a treatment factor as well as disturbance factor that affects the experimental result. Block designs are experimental designs to eliminate the influence of that. Restricting ourselves to one treatment factor is not a loss of generality because, in the case of multiple disturbance factors, we can incorporate all factor level combinations as treatments of a new factor. We assume that the treatment factor has v levels, called treatments and the ith treatment occurs $r_i\left(i=1,\dots ,v\right)$ times then the $r_i$ are called replications. The block may have b levels, called blocks. Block $b_j$ has $k_{j}j=1,\dots ,b$ elements which are called block sizes.

If each treatment occurs equally often, say r times, in a block design, then the design is called to be equireplicate and if all blocks have the same size k, the design is called proper.

Any block design can be represented by the matrix

$$\left(\begin{array}{cc}D\left(r_1.\dots ,r_v\right)& \mathcal{N}\\ {\mathcal{N}}^T& D\left(k_1,\dots ,k_b\right)\end{array}\right)$$E7

which includes the diagonal matrices $D\left(r_1.\dots ,r_v\right)$ and $D\left(k_1,\dots ,k_b\right)$ of replications and block sizes, respectively. Additionally, it includes the matrix $\mathcal{N},$ which is called incidence matrix. The elements of the incidence matrix $\mathcal{N}=\left(n_{\mathit{ij}}\right)$ with v rows and b columns show how often the ith treatment (representing the ith row) occurs in the jth block (representing the jth column). If all n_ij is either 0 or 1, the incidence matrix and the corresponding block design are called binary.

The elements of the incidence matrix $\mathcal{N}=\left(n_{\mathit{ij}}\right)$ lead to the two diagonal matrices because$\mathcal{N}{e}_b=D\left(r_1,\dots ,r_v\right)\text{and}{\mathcal{N}}^T{e}_v=D\left(k_1,\dots ,k_b\right)$; e.g. $e_b$ is a column vector of length b of ones.

The levels of the block factor are called blocks. The b column sums k_j of the incidence matrix are the elements of $D\left(k_1,\dots ,k_b\right)$and are called block sizes. The v row sums r_i of the incidence matrix are the elements of $D\left(r_1,\dots ,r_v\right)$ and are called replications. A block design is complete, if the elements of the incidence matrix are all positive (n_ij ≥ 1). A block design is incomplete, if the incidence matrix contains at least one zero. Blocks are called incomplete, if in the corresponding column of the incidence matrix there is at least one zero.

In block designs, the randomization has to be done as follows: the experimental units in each block are randomly assigned to the treatments, occurring in this block. This randomization is done for each block separately.

In complete block designs with v plots per block, where each of them is assigned to exactly one of the v treatments, the randomization is completed. If k < v, (incomplete block designs) the abstract blocks, obtained by the mathematical construction have to be randomly assigned to the real blocks.

For incomplete binary block designs in place of the incidence matrix, often a shorter writing is in use. Each block is represented by a bracket including the symbols (numbers) of the treatments, contained in the block.

A block design with a symmetric incidence matrix is a symmetric block design.

It can easily be seen that the sum of the replications r_i as well as the sum of all block sizes k_j equals the Number N of the experimental units of a block design. Therefore, for each block design, we have:

$$\sum _{i=1}^v{r}_i=\sum _{j=1}^b{k}_j=N,$$E8

Especially for equireplicate and proper block designs (r_i = r and k_j = k) this gives:

$$\mathit{vr}=\mathit{bk}$$E9

A (completely) balanced incomplete block design (BIBD) is a proper and equireplicate incomplete block design with the additional property that each pair of treatments occurs in equally many, say in λ, blocks. A BIBD with v treatments with r replications in b blocks of size k < v, is called a B(v, k, λ)-design. A BIBD for a pair (v, k) is called elementary, if it cannot be decomposed in at least two BIBDs for this pair (v, k). A BIBD for a pair (v, k) is a minimum BIBD for this pair (v, k), if r (and by this also b and λ) is minimum. For a BIBD besides (9) we receive the relation for the number of concurrences λ:

$$\lambda \left(v-1\right)=r\left(k-1\right)$$E10

The Eqs. (9) and (10) are necessary but not sufficient conditions for the existence of a BIBD. For instance, the quintuple v = 16, r = 3, b = 8, k = 6, λ = 1 fulfills the necessary conditions but no BIBD with these parameters exists. The reason is that Fisher’s inequality is violated, which as Fisher [5] showed is also a necessary condition.

$$b\ge v$$E11

Hence the design with v = 16, r = 3, b = 8, k = 6, λ = 1 is not a BIBD.

A researcher likes to have the experiment in replications, which means that a set of blocks forms a replication where all the treatments are present once. Such an incomplete block design is called resolvable. For a resolvable BIBD, Bose [6] showed that a necessary condition is:

$$b\ge r+v–1$$E12

But even if (9), (10) and (11) are valid, a BIBD not necessarily exists. Cases for this are v = 22, k = 8, b = 33, r = 12, λ = 4 and v = 34, r = 12, b = 34, k = 12, λ = 4. The minimum BIBD for v = 22 and k = 8 is v = 22, k = 8, b = 66, r = 24, λ = 8.

The so-called unreduced or trivial BIBD can, for any pair (v,k) with positive integers v and k, k < v, always be constructed by forming all k-combinations of the v numbers.

Hence $=\left(\begin{array}{c}v\\ k\end{array}\right),r=\left(\begin{array}{c}v-1\\ k-1\end{array}\right),\lambda =\left(\begin{array}{c}v-2\\ k-2\end{array}\right)$. Often a BIBD can be found as a

part of such an unreduced BIBD and this is a reduced BIBD.

One case for which such a reduction is not possible is that with v = 8 and k = 3.

There is no other case for $v\le 25$ and $2<k<v-1$ where no unreduced BIBD exists; see for more cases with v > 25 Rasch et al. [7] and Section 3.3.

In addition to Completely Balanced Incomplete Block Designs (BIBDs), Partially Balanced Incomplete Block Designs (PBIBDs) are also known, where not only one number of concurrences but two may occur. The consequence of this is that estimators of some treatment differences have two different variances. In this chapter, we do not consider these designs.

Example 3.1 (from Rasch and Herrendörfer [8, 9]).

For v = 7 and k = 3 the trivial BIBD is:

$$\begin{array}{c}\left(1,2,3\right)\left(1,3,6\right)\left(1,6,7\right)\left(2,4,7\right)\left(3,5,6\right)\\ \left(\mathit{1},\mathit{2},\mathit{4}\right)\left(\mathit{1},\mathit{3},\mathit{7}\right)\left(2,3,4\right)\left(2,5,6\right)\left(3,5,7\right)\\ \left(1,2,5\right)\left(1,4,5\right)\left(2,3,5\right)\left(2,5,7\right)\left(3,6,7\right)\\ \left(1,2,6\right)\left(1,4,6\right)\left(2,3,6\right)\left(\mathit{2},\mathit{6},\mathit{7}\right)\left(4,5,6\right)\\ \left(1,2,7\right)\left(1,4,7\right)\left(2,3,7\right)\left(3,4,5\right)\left(\mathit{4},\mathit{5},\mathit{7}\right)\\ \left(1,3,4\right)\left(\mathit{1},\mathit{5},\mathit{6}\right)\left(2,4,5\right)\left(\mathit{3},\mathit{4},\mathit{6}\right)\left(4,6,7\right)\\ \left(1,3,5\right)\left(1,5,7\right)\left(2,4,6\right)\left(3,4,7\right)\left(5,6,7\right)\end{array}$$E13

There are three elementary BIBD, two of them with parameters b = 7, r = 3, λ = 1 and blocks{(1,2,4), (1,3,7), (1,5,6), (2,4,5), (2,6,7), (4,6,7), (3,4,6)}—they are in the trivial BIBD bold and italic. The incidence matrix is

$$\left(\begin{array}{ccc}1& 1& 10\begin{array}{ccc}0& 0& 0\end{array}\\ 1& 0& \begin{array}{ccc}0& 1& \begin{array}{ccc}1& 0& 0\end{array}\end{array}\\ \begin{array}{c}0\\ 1\\ \begin{array}{c}0\\ 0\\ 0\end{array}\end{array}& \begin{array}{c}1\\ 0\\ \begin{array}{c}0\\ 0\\ 1\end{array}\end{array}& \begin{array}{ccc}\begin{array}{c}0\\ 0\\ \begin{array}{c}1\\ 1\\ 0\end{array}\end{array}& \begin{array}{c}1\\ 0\\ \begin{array}{c}1\\ 0\\ 0\end{array}\end{array}& \begin{array}{ccc}\begin{array}{c}0\\ 0\\ \begin{array}{c}0\\ 1\\ 1\end{array}\end{array}& \begin{array}{c}0\\ 1\\ \begin{array}{c}1\\ 0\\ 1\end{array}\end{array}& \begin{array}{c}1\\ 1\\ \begin{array}{c}0\\ 1\\ 0\end{array}\end{array}\end{array}\end{array}\end{array}\right)$$E14

A further BIBD with parameters b = 7, r = 3, λ = 1 is the septuple {(1,2,6), (1,3,4), (1,5,7), (2,3,7), (2,4,5), (4,6,7), (3,5,6)}—its blocks are in the trivial BIBD italic but not bold.

The set of the residual 21 of the 35 blocks is a further elementary {(1,2,4), (1,3,7), (1,5,6), (2,3,5), (2,6,7), (4,5,7), (3,4,6)} block design—its blocks are in the trivial BIBD neither bold nor italic. Rasch and Herrendörfer [8, 9] showed that this set is an elementary BIBD.

Let N be the incidence matrix of a BIBD for the pair (v,k). If we replace all Zeros of N by Ones and all Ones by Zero, we obtain the incidence matrix of a new BIBD which is called the complementary BIBD of the original one. If the original BIBD has the parameters v, r, b, k, λ, then the complementary BIBD has the parameters

$$v_c=v;b_c=b,r_c=v-r,k_c=v-k\text{and}{\lambda }_{\mathit{ck}}=b-2r+\lambda .$$E15

We consider the blocks {(1,2,4), (1,3,7), (1,5,6), (2,4,5), (2,6,7), (4,6,7), (3,4,6)} of the first elementary BIBD from above. By replacing each of the blocks by the corresponding block with the treatments not in the original block, we obtain the complementary block design {(3,5,6,7), (2,4,5,6), (2,3,4,7), (1,4,6,7), (1,3,4,5), (1,2,3,6), (1,2,5,7)}.

It is sufficient if a table of BIBDs contains only the designs with $\left\{v,k\le \frac{v}{2}\right\}$ because we can for the case $k>\frac{v}{2}$ easily construct all designs using the complementary BIBDs.

Since October 2022, the R Package ibd by B. N. Mandal has the function

bibd(v,b,r,k,lambda,ntrial,pbar=FALSE)

that generates a balanced incomplete block design with given number of treatments (v), number of blocks (b), number of replications (r), block size (k) and number of concurrences (lambda); ntrial is the number of trials (default is one) and pbar is the logical value indicating whether progress bar will be displayed or not (default is FALSE). The function works best for most values of treatments (v) up to 30 and block size (k) up to 10. However, for block size (k) up to 3, much larger number of treatments (v) may be used.

In the output of the function bibd() of Mandal his package ibd gives besides the design is also given NNP, where N is the incidence (v × b)-matrix (n_ij) and NP is N′ (the transpose of N). For a BIBD, the matrix NNP is a (v × v) symmetric matrix with r on the diagonal and lambda in the rest of the matrix. Also it gives Aeff for the lower bound to the A-efficiency of the generated design and Deff for the lower bound to the D-efficiency of the generated design. For a BIBD, the Aeff and Deff must be 1 or 0.999999.

3.2 A table of all smallest b for BIBD for $v\le 25$

To construct a BIBD is often not easy and needs methods of combinatorics (finite geometries and others) which are described in Rasch and Herrendörfer [8, 9]. Therefore, we present a link to a website of Springer https://doi.org/10.1007/978-3-662-67078-1_9 belonging to Rasch and Verdooren [10] for a complete table of BIBDs with the smallest number b of blocks with at most v = 25 treatments and block sizes k for $2<k\le \frac{v}{2}$. Such a table did not exist until now. Fisher and Yates [11] published an incomplete table of BIBDs.

Below, we give all 110 Balanced Incomplete Block Designs (BIBDs) with v < 26 and k≤ v/2 and smallest number b of blocks of size. The BIBDs for k>v/2 are the complementary BIBDs, which can be easily be obtained by replacing the treatments given in the blocks of the original BIBD by the treatments not occurring in the original blocks.

Example 3.1.

Design 2 (German “Plan” = Design, “Behandlungen” = Treatments) below.

Plan 2

v	k	b	r	λ
7	3	7	3	1

Design 2 has the complementary BIBD with the parameters v_c=7, b_c=7, k_c = 7–3 = 4, r_c = 7–3 = 4, λ_c = 7–2·3 + 1 = 2 and the blocks are now:

In Rasch and Verdooren [10], we find the following table of smallest b for BIBDs. The designs of these 110 BIBDs can be found on the website of Springer https://doi.org/10.1007/978-3-662-67078-1_9.

If the R program ibd on a PC with 64 bits processor did not give a solution after 5 minutes, then we mentioned this design as not constructible by ibd. For Design 21, ibd gives a design with a NNP matrix that does not belong to a BIBD and also Aeff and Deff was not 0.9999999.

Table 1 provides a comprehensive list of 110 designs of BIBD that can be used by experimenters to find the desired design, given that $\mathit{v}\mathit{\le }\mathbf{25}$. These designs can be downloaded on the website of Springer https://doi.org/10.1007/978-3-662-67078-1_9 belonging to Rasch and Verdooren [10]. Be aware that the website uses the German words “Plan” and “BUBD” for Design and BIBD respectively.

v	k	b	r	λ	Design	Constructible by ibd
6	3	10	5	2	1	Yes
7	3	7	3	1	2	Yes
8	3	56	21	6	3	Yes
	4	14	7	3	4	Yes
9	3	12	4	1	5	Yes
	4	18	8	3	6	Yes
10	3	30	9	2	7	Yes
	4	15	6	2	8	Yes
	5	18	9	4	9	Yes
11	3	55	15	3	10	Yes
	4	55	20	6	11	Yes
	5	11	5	2	12	Yes
12	3	44	11	2	13	Yes
	4	33	11	3	14	Yes
	5	132	55	20	15	Yes
	6	22	11	5	16	Yes
13	3	26	6	1	17	Yes
	4	13	4	1	18	Yes
	5	39	15	5	19	Yes
	6	26	12	5	20	Yes
14	3	182	39	6	21	Yes
	4	91	26	6	22	Yes
	5	182	65	20	23	Yes
	6	91	39	15	24	Yes
	7	26	13	6	25	Yes
15	3	35	7	1	26	Yes
	4	105	28	6	27	Yes
	5	42	14	4	28	Yes
	6	35	14	5	29	Yes
	7	15	7	3	30	Yes
16	3	80	15	2	31	Yes
	4	20	5	1	32	Yes
	5	48	15	4	33	Yes
	6	16	6	2	34	Yes
	7	80	35	14	35	Yes
	8	30	15	7	36	Yes
17	3	136	24	3	37	No
	4	68	16	3	38	Yes
	5	68	20	5	39	No
	6	136	48	15	40	No
	7	136	56	21	41	No
	8	34	16	7	42	Yes
18	3	102	17	2	43	Yes
	4	153	34	6	44	Yes
	5	306	85	20	45	No
	6	51	17	5	46	Yes
	7	306	119	42	47	No
	8	153	68	28	48	No
	9	34	17	8	49	No
19	3	57	9	1	50	Yes
	4	57	12	2	51	Yes
	5	171	45	10	52	No
	6	57	18	5	53	Yes
	7	57	21	7	54	Yes
	8	171	72	28	55	No
	9	19	9	4	56	Yes
20	3	380	57	6	57	Yes
	4	95	19	3	58	Yes
	5	76	19	4	59	Yes
	6	190	57	15	60	No
	7	380	133	42	61	No
	8	95	38	14	62	No
	9	380	171	72	63	No
	10	38	19	9	64	No
21	3	70	10	1	65	Yes
	4	105	20	3	66	Yes
	5	21	5	1	67	No
	6	42	12	3	68	No
	7	30	10	3	69	Yes
	8	105	40	14	70	Yes
	9	35	15	6	71	No
	10	42	20	9	72	No
22	3	154	21	2	73	Yes
	4	77	14	2	74	Yes
	5	462	105	20	75	No
	6	77	21	5	76	Yes
	7	44	14	4	77	Yes
	8	66	24	8	78	Yes
	9	154	63	24	79	No
	10	77	35	15	80	Yes
	11	42	21	10	81	No
23	3	253	33	3	82	Yes
	4	253	44	6	83	Yes
	5	253	55	10	84	Yes
	6	253	66	15	85	No
	7	253	77	21	86	No
	8	253	88	28	87	No
	9	253	99	36	88	No
	10	253	110	45	89	No
	11	23	11	5	90	Yes
24	3	184	23	2	91	Yes
	4	138	23	3	92	No
	5	552	115	20	93	No
	6	92	23	5	94	No
	7	552	161	42	95	No
	8	69	23	7	96	No
	9	184	69	24	97	No
	10	276	115	45	98	No
	11	552	253	110	99	No
	12	46	23	11	100	No
25	3	100	12	1	101	Yes
	4	50	8	1	102	No
	5	30	6	1	103	Yes
	6	100	24	5	104	No
	7	100	28	7	105	No
	8	75	24	7	106	No
	9	25	9	3	107	Yes
	10	40	16	6	108	No
	11	300	132	55	109	No
	12	50	24	11	110	No

Table 1.

BIBDs with smallest b for $v\le 25$ and $\left\{v,2<k\le \frac{v}{2.}\right\}$.

But we know that designs with more than 100 blocks will be applied less frequently.

In place of using such a design, we recommend to decrease the number of plots k if the number of blocks b is then smaller. Another possibility is, to increase v by a placebo treatment and use the design with this v with smaller b.

Example 3.2.

We consider design 87 with parameters v = 23, b = 253, r = 88, k = 8 and λ = 28. If we can use k = 7 then the design 77 with parameters

v = 23, b = 44, r = 14, k = 7 and λ = 4 needs much less effort.

Example 3.3.

We consider design 14 with parameters v = 12, b = 33, r = 11, k = 4 and λ = 3. If we can use v = 13 (with a placebo treatment) then the design 18 with parameters v = 13, b = 13, r = 4, k = 4 and λ = 1 needs much less effort.

3.3 A conjecture about trivial BIBD

The following conjecture was already published in several articles—see for instance, Rasch and Herrendörfer [8, 9, 12, 13, 14], Rasch et al. [7].

For v = 8 and k = 3, the elementary and smallest BIBD (v,b,r,k,λ) is the trivial one with b = 56, r = 21, λ = 6. This can be shown by verifying that there is no quintuple (8,b,r,3,λ) with a number of blocks b < 56 fulfilling Eqs. (2), (3) and (4). Certainly, the complementary BIBD of this design is both elementary and trivial. The conjecture is as follows:

Conjecture For $3\le k\le \frac{v}{2}$ the case (v, k) = (8, 3) (Design 3 in Table 1) is the only one where the trivial BIBD is elementary.

If we look at Table 1, we see that the conjecture is true for $v\le 25$. In Rasch et al. [7] and Teuscher and Rasch [15], many cases are given where the conjecture is true. A counter-example could not be shown.

3.4 BIBD with too large number of blocks b

It is true that a BIBD needs often a large number b of blocks. The Biostatistician who cannot use the large number b of blocks for the BIBD can use a type of Incomplete Block Designs as Lattice Designs. In Cochran and Cox [16], Lattice Designs are given in Chapter 10. But these Lattice Designs exist only when the number of treatments v is an exact sqares. An extension of these Lattice Designs are the generalized lattice designs as Alpha Designs by Patterson and Williams [17].

The variance of the estimator of treatment differences of these designs is not the same for all treatment differences but often about the same order. A researcher likes to have the experiment in replications, which means that a set of blocks forms a replication where all the treatments are present once. Such an incomplete block design is called resolvable. If v is equal to the product of b and k, resolvable Alpha Designs are possible. We describe them in Section 5.

4. Analysis of a BIBD with R

In a BIBD, the treatment effects are usually estimated with the intrablock analysis and the Least Squares Method using matrix solution of the normal equations with b = (X’X) ⁻¹X’y and the variance of a varietal contrast p’b is found as σ² p’(X’X)⁻¹p. However, now BIBD trials are analyzed with recovery of interblock analysis using a mixed model where the replications and treatments are fixed factors, the incomplete block effects in the replications and the experimental error are random effects. With the REML (Restricted Maximum Likelihood method) from Patterson and Thompson [18], the variance components are estimated. These estimated variance components are then inserted into the variance-covariance matrix V. The practical best linear unbiased estimator for b is calculated as b = (X’V ⁻¹ X)^{− 1}X’ V ⁻¹ y.

In R, the intrablock analysis and the recovery of interblock analysis information can be done. We demonstrate this with the following example of Cochran and Cox [16] page 443–444; with the following example of an experiment in a resolvable Balanced Incomplete Block Design with v = 6 treatments, k = 2 number of units per block, b = 15 number of blocks, r = 5 number of replications and λ = r·(k – 1)/(v – 1) = 5·(2–1)/(6–1) = 1.

The objective of this experiment was to compare the effects of length of cold storage on the tenderness and flavor of beef roasts. The treatments were six periods of storage (0, 1, 2, 4, 9 and 18 days); these are denoted by the treatment symbols 1, 2, 3, 4, 5, 6, respectively. Thirty roasts from the round of an animal were used. Four muscles are provided 6 roasts, while 3 muscles each provided 2 roasts. The roasts from any muscle group naturally pair up, as each roast on the left side of an animal corresponds to a roast on the right side. From previous experience, it was believed that the 2 roasts in any pair would give the same results, hence these two roasts form a block. Variation among different pairs from the same muscle was expected to be somewhat larger, and variation among muscles to be still larger.

These options prompted the use of an incomplete block design in blocks of k = 2, each block comprising the left and right roasts in a pair. When grouping the blocks into replications, it was natural to put roasts from the same muscle to the same replicate. In this case, the first 4 muscles could be allocated to separate replications, allowing for a distinct replicate to be formed for each muscle. The remaining replication consisted of the 3 smaller muscles.

The treatments in a block were randomized. The order of the blocks from the design of this resolvable BIBD was also randomized per replication. Scoring for tenderness was done by 4 judges, each marking on a scale from 0 to 10. The scores shown are their total (out of 40). A high score indicating very tender beef.

The plan of this experiment with the treatment in parentheses and the scores for tenderness are given below.

Rep I	Rep II	Rep III	Rep IV	Rep V
Block 1 (1); 7 (2) 17	Block 4 (1); 17 (3) 27	Block 7 (1); 10 (4) 25	Block 10 (1); 25 (5) 40	Block 13 (1); 11 (6) 27
Block 2 (3); 26 (4) 25	Block 5 (2); 23 (5) 27	Block 8(2); 26 (6) 37	Block 11 (2); 25 (4) 34	Block 14 (2); 24 (3) 21
Block 3 (5); 33 (6) 29	Block 6 (4); 29 (6) 30	Block 9 (3); 24 (5) 26	Block 12 (3); 34 (6) 32	Block 15 (4); 26 (5) 32

The analysis with R is as follows for the BIBD from Cochran and Cox [16], page 443–444.

Note: The coefficient rep2 represents the estimated difference in effect between rep 2 and rep 1.

The coefficient treat2 represents the estimated difference in effect between treat 2 – treat 1.

The estimate of the Standard error for the differences between the least squares means of two treatment means is the same in the case of a BIBD the same. Estimate for the variance σ² is square of residual standard errors = 2.271, hence s² = 7.734 with 10 degrees of freedom. The estimate of the Standard error for the differences between the least squares means of two treatment means is √[2·k·s²/ (λ v)] = √[2·2·7.734/(1·6)] = 2.271, where λ = r·(k – 1)/(v – 1) = 5·(2–1)/(6–1) = 1.

Note: The estimate of the Standard error for the least squares estimate of a treatment mean is same in the case of a BIBD the same. Estimate for the variance σ² is square of residual standard error s = 2.271, hence s² = 7.734 with 10 degrees of freedom. The estimate of the Standard error for the least squares estimate of treatment mean is with λ = r·(k – 1)/(v – 1) = 5·(2 – 1)/(6 – 1) = 1 given by √[(s²/(r·v))·(1 + (k·r·(v-1)/(λ·v))] = √[(7.734/(5·6))))·(1 + (2·5·(6 – 1)/(1.6))] = √2.40613 = 1.551.

Note: To get the interblock estimates, we must use the mixed model with the random effect of blocks in replications. We use, therefore, the R-package lme4.

Note: To get the estimates for the fixed effects rep and treat, we use the R-package lmerTest.

5. Alpha designs

Patterson and Williams [17] introduced the concept of Alpha designs for variety trials in binary connected incomplete block designs with a block size of k. They start with a rectangular array with column lengths k (the size of the incomplete blocks) of the sequence of 1, …,v varieties and shift the columns according to an array. For many combinations of varieties v and block sizes k, they give a procedure to construct Alpha designs. The name “Alpha” comes from the first letter in the Greek alphabet that was used to construct the design. John and Williams [19] made more Alpha designs based on cyclic designs; see for the definition of cyclic design chapter 3 of their book. Tables of cyclic designs are given by John et al. [20] and Lamacraft and Hall [21]. There is a computer program CycDesigN [22] available to generate incomplete block designs as alpha designs and cyclic designs (see the website of VSN-international: http://www.vsni.co.uk/software/cycdesign/). This package has general algorithms for generating incomplete block and row-column designs, which give better results than the alpha and cyclic construction methods. In variety testing trials one wants to use resolvable incomplete block designs where the design can be divided into r groups (= replications) such that each group contains each of the v crosses exactly once.

The resolvable incomplete block designs, and particularly the so-called generalized lattice (GL) or Alpha designs, have become most suitable for crop variety trials; because they make it easier to find designs for a large number of varieties and different (even small) sizes of incomplete blocks, see Williams [23] and Patterson et al. [24].

The program CycDesigN [22] gives such resolvable incomplete block designs. All these above-mentioned designs are connected. In a connected incomplete block design, one can estimate all differences between the varieties. Patterson and Silvey [25] indicate about 70% saving in use of land and labor for variety trials, if certain incomplete block designs are adopted rather than complete block designs. Partially replicated designs are now very popular for large variety trials: see e.g. Cullis et al. [26] and Williams et al. [27].

One well-known other block design procedure is the OPTEX procedure from the SAS package but numerous other packages are available including a number of open-source R packages.

In oil palm breeding trials, the alpha designs are very useful to make connected partial diallel or incomplete diallel crossing scheme of the female parent dura and the male parent pisifera to produce the wanted tenera hybrids. Because the tenera palms are planted at the corners of equilateral triangles with side lengths of 9 m, the plots with 6 × 6 palms are quite large. The Alpha designs of Patterson and Williams are then used to find resolvable incomplete block designs with the computer program CycDesigN. This is described in Verdooren [28]. See further Verdooren et al. [29] where the analysis of oil palm breeding trials in incomplete block designs is given for estimating the General Combining Ability of the parents using mixed models.

The varietal effects are usually estimated with the intrablock analysis and the Least Squares Method using matrix solution of the normal equations with b = (X’X^{)− 1}X’y and the variance of a varietal contrast p’b is found as σ² p’(X’X^{)− 1}p. However, now varietal trials are analyzed with recovery of interblock analysis information using a mixed model where the replications and varieties are fixed factors, the incomplete block effects in the replications and the experimental error are random effects.

With the REML (Restricted Maximum Likelihood) method of Patterson and Thompson [15], the variance components are estimated. These estimated variance components are then inserted into the variance-covariance matrix V. The practical best linear unbiased estimator for b is calculated as b = (X’V ⁻¹ X^{)− 1}X’ V ⁻¹ y. In R, the intrablock analysis and the recovery of information with interblock analysis can be done.

6. Analysis of an Alpha Design with R

Kuehl [2] gives the following exercise 10.4. A variety trial was conducted in an alpha design α(0,1,2) resolvable design; α(0,1,2) means an alpha design with: 0 some varieties are not together in a block, 1 some varieties are one times together in a block; 2 some varieties are twice together in a block. There were v = 18 varieties in r = 4 replicate groups. Hence, this is a resolvable design. There were 3 blocks with 6 varieties in each replicate. Hence, the block size is k = 6 and the number of blocks b = 4·3 = 12.

Varieties 1 and 5 are control varieties. The table gives first the yield y in kg/plot and then in parentheses the variety number.

	Replicate I
Block
1	88.2 (5)	82.5 (10)	84.3 (15)	87.0 (6)	84.5 (12)	88.9 (8)
2	82.4 (1)	82.9 (14)	83.1 (3)	84.7 (13)	83.3 (16)	89.0 (4)
3	93.1 (2)	82.7 (11)	88.9 (17)	88.6 (18)	84.1 (9)	87.5 (7)
	Replicate II
Block
4	85.4 (4)	73.0 (11)	84.2 (7)	80.3 (14)	79.6 (10)	86.0 (6)
5	87.9 (8)	85.1 (9)	79.4 (18)	80.7 (13)	89.3 (5)	81.5 (3)
6	82.4 (1)	88.5 (2)	87.0 (12)	85.4 (17)	85.9 (15)	79.1 (16)
	Replicate III
Block
7	83.6 (6)	79.4 (17)	81.3 (4)	80.5 (9)	80.9 (8)	79.3 (1)
8	80.4 (7)	88.2 (5)	82.3 (14)	88.0 (12)	90.0 (2)	83.6 (3)
9	81.4 (18)	84.8 (15)	81.0 (10)	81.2 (13)	79.1 (11)	83.8 (16)
	Replicate IV
Block
10	80.5 (16)	77.1 (11)	84.4 (17)	90.4 (6)	82.9 (14)	83.0 (12)
11	87.9 (8)	78.9 (18)	81.4 (1)	83.5 (2)	82.2 (15)	79.0 (3)
12	84.2 (7)	83.0 (10)	87.6 (9)	81.7 (13)	91.3 (5)	87.4 (4)

The analysis with R is as follows for the Alpha Design from Kuehl [2], Exercise 10.4.

Note: To get the estimate of a treatment mean with the Least Squares Method, we use the R-package lsmeans.

Note: Interblock analysis with recovery of information.

Model with replicate and variety is fixed and blocks in the replicates are random.

We use the R-packages lme4 and lmerTest.