Agricultural Statistics & Data Science · Design and analysis of experiments · also Design of Experiments III, Block Designs, Combinatorics of Designs
When a block cannot hold every treatment, as with hundreds of breeding lines: the general block design and its information matrix, BIBD and PBIBD, lattices, Youden squares and alpha designs, and how they are built from finite fields, orthogonal Latin squares and difference sets.
In the exams: AICE-JRF Ag Stat Unit 4 (heavy: general block design, BIBD, PBIBD, lattice, Youden named); ASRB NET Ag Stat (heavy)
Incomplete block designs make it possible to test many treatments in small, homogeneous blocks. They are the designs behind modern variety and breeding trials, sensory panels, animal experiments limited by litter size and laboratory work limited by batch size. This subject covers the general block design and its information (C) matrix, connectedness and balance, the intra-block analysis and the recovery of inter-block information, balanced incomplete block designs with their parameters, analysis and construction (finite fields, difference sets, orthogonal Latin squares), partially balanced designs with two associate classes, square, rectangular and cubic lattices, Youden squares and row-column designs, resolvable and alpha designs, augmented designs for unreplicated germplasm, and efficiency factors.
The subject builds on the basic designs subject (blocking, the RCBD and the Latin square, relative efficiency) and the factorial subject (confounding is the factorial counterpart of incomplete blocks: in a confounded factorial, the blocks are incomplete and chosen effects are sacrificed, whereas here all treatment contrasts are kept with reduced precision). Its linear-model theory is that of the linear models subject, and its modern analysis with random blocks is completed in the mixed-models subject.
In examinations, ICAR AICE-JRF/SRF Agricultural Statistics Unit 4 names the analysis of the general block design, balanced incomplete block designs with their construction and analysis, partially balanced incomplete block designs with two associate classes, lattice designs and the Youden square. Typical questions test the parameter relations of a BIBD, Fisher's inequality, the C-matrix and connectedness, adjusted treatment totals and estimates, efficiency factors, the structure of lattices and the degrees of freedom of augmented designs. ASRB NET and ARS papers in agricultural statistics draw on the same material, and the Indian Statistical Service papers include design of experiments (as reported).
The randomised complete block design of the basic designs subject puts every treatment in every block. When the number of treatments is large, as in a breeding trial with 50 to 500 lines, a complete block must cover a large area, and the very heterogeneity that blocking was meant to remove reappears inside the block. Incomplete block designs use blocks smaller than the number of treatments, so each block is homogeneous, and arrange the treatments across blocks so that every treatment comparison can still be estimated. The cost is that treatment and block effects are no longer orthogonal: a treatment's total is affected by which blocks it fell in, and the analysis must adjust for blocks. Incomplete block designs are also natural where block size is fixed by nature: a litter of animals, a pair of twins, the two halves of a leaf, the number of positions in a growth chamber or the number of samples a laboratory can process in a day.
A general block design has v treatments in b blocks. Treatment i is replicated r_i times, block j has k_j plots, and the incidence matrix N (v x b) has element n_ij equal to the number of times treatment i occurs in block j (usually 0 or 1). Let R = diag(r_i), K = diag(k_j), T the vector of treatment totals and B the vector of block totals. The intra-block analysis leads to the reduced normal equations C tau-hat = Q, where C = R - N K^(-1) N' is the information matrix (C-matrix) and Q = T - N K^(-1) B is the vector of adjusted treatment totals. The element Q_i is the treatment total minus the sum, over the blocks containing treatment i, of the block totals divided by the block sizes. The C-matrix is symmetric, its rows sum to zero and its rank is at most v - 1. A design is connected if every treatment contrast is estimable, which happens if and only if rank(C) = v - 1; intuitively, any two treatments can be linked by a chain of treatments that share blocks. A disconnected design splits into groups of treatments that can only be compared within groups and is useless for comparing all treatments.
A connected design is variance balanced if every elementary contrast tau_i - tau_l is estimated with the same variance. For an equireplicate design with equal block sizes this holds if and only if C = theta (I - J/v) for some constant theta, which is the property of the balanced incomplete block design of chapter 3. Other notions exist (efficiency balance, pairwise balance), but variance balance is the one examined most often. Balanced designs treat all treatment pairs equally; partially balanced designs (chapter 5) relax this to two or more classes of pairs, allowing many more parameter combinations.
A block design is binary if no treatment occurs more than once in a block, equireplicate if all r_i are equal, and proper if all blocks have the same size k. Most designs in this subject are binary, equireplicate and proper, which simplifies the C-matrix to rI - (1/k) N N'. Designs that break these properties (unequal block sizes from field shape, extra replication of checks) are analysed by the same general method but lose the simple formulas.
Take v = 3 treatments in b = 3 blocks of k = 2: {1, 2}, {2, 3} and {1, 3}, so each treatment is replicated r = 2 times and each pair occurs together once. Then N N' has 2 on the diagonal and 1 elsewhere, and C = R - N K^(-1) N' = 2I - (1/2) N N' has diagonal elements 1 and off-diagonal elements -0.5. This equals 1.5 (I - J/3), so the design is variance balanced, with theta = 1.5, which is lambda v / k = 1 x 3 / 2 for this BIBD. The rank of C is 2 = v - 1, so the design is connected. By contrast, four treatments in the blocks {1, 2}, {1, 2}, {3, 4}, {3, 4} give a C-matrix of rank 2, not 3: treatments 1 and 2 never share a block with 3 or 4, so the contrast between the two groups cannot be estimated within blocks. Such a design is disconnected and fails its purpose, however neat it looks.
In the intra-block analysis block effects are treated as fixed and eliminated, so treatment comparisons use only differences within blocks. The ANOVA has: blocks unadjusted for treatments (b - 1 degrees of freedom, sum of the squared block totals over block sizes minus the correction factor); treatments adjusted for blocks (v - 1 degrees of freedom, sum of squares tau-hat'Q); error by subtraction (n - b - v + 1 degrees of freedom, with n the number of plots); and total (n - 1). Adjusted treatment means are the general mean plus tau-hat_i. The order matters: because treatments and blocks are not orthogonal, the treatment sum of squares adjusted for blocks differs from the unadjusted one, and it is the adjusted one that tests treatments. Modern software fits the linear model directly (blocks then treatments) and reports the same sums of squares.
Block totals also carry information about treatments: a block containing high-yielding treatments tends to have a high total. If block effects are random with variance sigma_b squared, block totals provide a second, inter-block estimate of treatment effects, with error variance sigma squared + k sigma_b squared per plot (for block size k), larger than the intra-block error variance sigma squared. Yates (1939, 1940) showed how to combine the two: each estimate is weighted by the reciprocal of its variance, with weights w = 1/sigma squared and w' = 1/(sigma squared + k sigma_b squared), estimated from the error and adjusted-block mean squares. Recovery of inter-block information gains most when blocks are not very different (sigma_b squared small relative to sigma squared) and in designs with small blocks and few replications. When blocks differ greatly, w' is small and the combined analysis is close to the intra-block one. Today the combined analysis is obtained by fitting blocks as random effects by REML in a mixed model, which is the standard analysis of alpha and lattice designs in breeding programmes.
For a well-replicated trial with clearly different blocks, the intra-block analysis is safe and simple. For trials with many small blocks and few replicates, such as alpha and lattice designs in breeding, the combined analysis by REML is preferred and is now the default in most software. Whichever is used, the report should state it, because adjusted means and their standard errors differ between the two.
Suppose the intra-block error variance per plot is estimated as 10 and the variance per plot relevant to inter-block comparisons, sigma squared + k sigma_b squared, as 40. The weights are then w = 1/10 = 0.10 and w' = 1/40 = 0.025, so per unit of information the intra-block estimate counts four times as much as the inter-block one. If blocks had differed hardly at all, the two variances would be close and the inter-block information would add substantially to precision; if blocks had differed enormously, w' would approach zero and the combined analysis would reduce to the intra-block one. The weights are estimated from the data, so the combined estimates are only approximately optimal, but in moderate and large experiments the gain is real.
A balanced incomplete block design (BIBD) with parameters (v, b, r, k, lambda) arranges v treatments in b blocks of k plots (k < v) so that each treatment occurs at most once in a block, each treatment is replicated r times, and every pair of treatments occurs together in exactly lambda blocks. The parameters satisfy vr = bk (both count the plots) and lambda(v - 1) = r(k - 1) (both count, for a given treatment, the plots shared with other treatments in its blocks). Fisher's inequality states b >= v (equivalently r >= k). A BIBD with b = v (and so r = k) is called symmetric; in a symmetric BIBD any two blocks have exactly lambda treatments in common. These relations decide whether a design can exist: v = 9, k = 3, lambda = 1 gives r = 4 and b = 12, a valid parameter set (and the design exists); v = 6, k = 3, lambda = 2 gives r = 5 and b = 10. Parameters that give a non-integer r or b, or violate Fisher's inequality, are impossible; satisfying the conditions is necessary but not sufficient for existence.
For a BIBD the C-matrix is (lambda v / k)(I - J/v), so the reduced normal equations solve simply: tau-hat_i = k Q_i / (lambda v). The adjusted treatment sum of squares is (k / (lambda v)) times the sum of Q_i squared, with v - 1 degrees of freedom; the error has bk - b - v + 1 degrees of freedom. The variance of the estimated difference between two treatments is 2k sigma squared / (lambda v), the same for every pair: this is variance balance. As an example, in a BIBD with v = 7, b = 7, r = 3, k = 3, lambda = 1, a treatment with total 30 that appears in blocks with totals 24, 27 and 33 has Q = 30 - (24 + 27 + 33)/3 = 2 and tau-hat = 3 x 2 / 7 = 0.857. The error has 21 - 7 - 7 + 1 = 8 degrees of freedom.
Compared with an RCBD with the same number of replications r and the same error variance per plot, the variance of a treatment difference in a BIBD is larger by the factor 1/E, where the efficiency factor E = lambda v / (r k), which equals v(k - 1) / [k(v - 1)]. For v = 7, k = 3, E = 14/18 = 0.78; for v = 4, k = 2 (all six pairs), E = 4/6 = 0.67. The efficiency factor is always below 1 because some information is lost to blocks. The design pays when the smaller blocks reduce the error variance per plot by more than the factor E, which is the usual situation when many treatments make complete blocks large and heterogeneous.
BIBDs suit experiments with a moderate number of treatments and a natural small block: sensory evaluation, where a taster can judge only three or four samples before fatigue; animal experiments with litters or pens of fixed size; laboratory assays with a fixed number of positions per run; and comparisons of methods by several analysts each of whom can handle only a few samples. In field variety trials with many entries, BIBDs usually need too many replicates, which is why lattices and alpha designs took over there. When a BIBD with the required parameters does not exist, a PBIBD or a cyclic or computer-generated design with an efficiency factor near the bound is used instead.
In the BIBD with v = 7, b = 7, r = 3, k = 3, lambda = 1, suppose the sum of squared adjusted treatment totals is 14. The adjusted treatment sum of squares is k/(lambda v) x 14 = 3/7 x 14 = 6, with 6 degrees of freedom, a mean square of 1. If the error mean square on 8 degrees of freedom is 0.5, F = 2.0 on 6 and 8 degrees of freedom, below the 5% critical value of about 3.58, so treatment differences are not significant. With error variance 12 per plot, the standard error of a difference between two adjusted treatment means is the square root of (2 x 3 x 12 / 7) = 3.21, compared with the square root of (2 x 12 / 3) = 2.83 in an RCBD with three replicates and the same error variance; the ratio of the variances is 1/E = 1.29.
The unreduced (irreducible) BIBD takes all C(v, k) combinations of k treatments as blocks; it always exists but quickly becomes large (for v = 8, k = 4 it has 70 blocks). The complement of a BIBD, replacing each block by the treatments not in it, is again a BIBD with block size v - k. Taking a resolvable design and adding treatments, or taking the residual or derived design of a symmetric BIBD, gives further designs. Tables of BIBDs, such as those of Fisher and Yates and of Cochran and Cox, list designs for practical parameter ranges.
A difference set modulo v is a set of k residues such that every non-zero residue occurs exactly lambda times among the differences of its elements. Developing the set cyclically, adding 0, 1, ..., v - 1 to each element modulo v, gives a symmetric BIBD. For example {0, 1, 3} modulo 7 has differences 1, 3, 2, 6, 4, 5 (each non-zero residue once), so its seven developments {0, 1, 3}, {1, 2, 4}, {2, 3, 5}, ... form the BIBD with v = b = 7, r = k = 3, lambda = 1. Bose's method of symmetrically repeated differences extends the idea to several initial blocks and to non-symmetric designs. Cyclic designs are easy to lay out and randomise and are widely used even when not exactly balanced.
When s is a prime or a power of a prime, the finite (Galois) field GF(s) exists and yields a complete set of s - 1 mutually orthogonal Latin squares (MOLS) of order s. From them one builds the affine resolvable BIBD with v = s squared treatments in blocks of k = s: the rows, the columns and the letter classes of each square give s + 1 replicates, so r = s + 1, b = s(s + 1) and lambda = 1. This is the balanced lattice of chapter 6. Finite projective geometries PG(2, s) give symmetric BIBDs with v = b = s squared + s + 1, r = k = s + 1 and lambda = 1, of which v = 7 (s = 2) is the smallest. Because no pair of orthogonal Latin squares of order 6 exists, the balanced lattice for 36 treatments does not exist.
To check whether a set is a difference set, list all ordered differences and count them. The set {0, 1, 3} modulo 7 gives each non-zero residue exactly once, so it generates a BIBD with lambda = 1. The set {0, 1, 2} modulo 7, by contrast, gives differences 1, 2, 1, 6, 5, 6: residues 1 and 6 appear twice and 3 and 4 not at all, so its developments would put some pairs together twice and others never, giving an unbalanced cyclic design rather than a BIBD. For v = 13, the set {0, 1, 3, 9} gives each non-zero residue once and generates the symmetric BIBD with v = b = 13, k = r = 4, lambda = 1, which is the projective plane of order 3.
BIBDs exist only for limited parameter sets and often need many replications. Bose and Nair introduced partially balanced incomplete block designs (PBIBDs), in which pairs of treatments fall into association classes and pairs in the same class occur together in the same number of blocks. In a PBIBD with two associate classes, each treatment has n1 first associates, with which it occurs in lambda1 blocks, and n2 second associates, with which it occurs in lambda2 blocks. The parameters satisfy n1 + n2 = v - 1, lambda1 n1 + lambda2 n2 = r(k - 1), and vr = bk, together with the association scheme parameters p^i_jk, the number of treatments that are j-th associates of one and k-th associates of another treatment of an i-th associate pair. Elementary contrasts between first associates and between second associates are estimated with two different variances, so the design is partially balanced. Analysis follows the general block design; with modern software the C-matrix need not be inverted by hand.
In a group divisible design, v = mn treatments are divided into m groups of n; treatments in the same group are first associates (n1 = n - 1) and those in different groups second associates (n2 = n(m - 1)). In a triangular design, v = q(q - 1)/2 treatments are placed in the off-diagonal cells of a symmetric q x q array; treatments in the same row or column are first associates (n1 = 2(q - 2)). In a Latin-square type design, v = s squared treatments are placed in an s x s array, and treatments in the same row or column (or in the same letter class of a superimposed square) are first associates. Cyclic PBIBDs come from developing initial blocks modulo v. The simple square lattice is a Latin-square type PBIBD with lambda1 = 1 and lambda2 = 0.
The intra-block analysis of a two-class PBIBD follows the general route: compute adjusted totals Q, solve C tau-hat = Q (closed-form solutions using the association parameters exist, but software inverts C directly), obtain the adjusted treatment sum of squares and two average variances, one for first-associate and one for second-associate comparisons. The average variance over all pairs, weighted by n1 and n2, gives the efficiency factor. Recovery of inter-block information proceeds as for BIBDs.
With v = 8 treatments in m = 4 groups of n = 2, each treatment has n1 = 1 first associate (its group partner) and n2 = 6 second associates. If the design has r = 3 and k = 4, the relation lambda1 n1 + lambda2 n2 = r(k - 1) = 9 must hold, for instance lambda1 = 3 and lambda2 = 1 (3 + 6 = 9). Comparisons between group partners, which share three blocks, are then more precise than comparisons between treatments in different groups, which share one. Group divisible designs suit situations where some comparisons, such as between closely related lines, matter more or are naturally made more often.
Lattice designs, introduced by Yates for variety trials, arrange v = k squared treatments in blocks of k. The treatments are written in a k x k array. In one replicate the rows of the array form the blocks; in a second replicate the columns form the blocks; further replicates use the letter classes of orthogonal Latin squares superimposed on the array. A simple lattice uses two such groupings (two replicates), a triple lattice three, and a balanced lattice all k + 1 groupings, which requires a complete set of k - 1 MOLS and therefore exists when k is a prime or prime power. The balanced lattice is a BIBD with lambda = 1 (r = k + 1, b = k(k + 1)); simple and triple lattices are PBIBDs. For 25 varieties, k = 5: a simple lattice has 2 replicates of 5 blocks of 5; a balanced lattice has 6 replicates and 30 blocks. Groupings can be repeated to give more replicates (for example a simple lattice repeated twice gives four replicates).
Rectangular lattices accommodate v = k(k + 1) treatments in blocks of k, for example 20 treatments in blocks of 4 or 30 in blocks of 5, filling the gaps between square numbers. Cubic lattices arrange v = k cubed treatments in a three-dimensional array with blocks of k plots, the three groupings corresponding to the three directions of the array, which suits very large numbers of entries with small blocks. All lattices are resolvable: each replicate contains every treatment once, so if the incomplete blocks turn out to be unnecessary the experiment can be analysed as an RCBD with replicates as blocks, and field operations can be organised replicate by replicate. Lattices were the standard design for national variety trials for decades before alpha designs.
Lattices are analysed by the intra-block method or, better, with recovery of inter-block information, which in lattices is straightforward because of their structure. The efficiency of a lattice relative to an RCBD depends on how much the small blocks reduce error: the gain is large in heterogeneous fields and small in uniform ones. Because the design is resolvable, the relative efficiency can be computed after the trial by comparing the effective error with the error of the RCBD analysis using replicates as blocks.
For a square lattice with k squared entries: each replicate has k blocks of k plots; a simple lattice has 2k blocks in total, a triple lattice 3k, and a balanced lattice k(k + 1). For 49 entries (k = 7), a balanced lattice needs 8 replicates and 56 blocks, possible because 7 is prime; for 36 entries (k = 6) only simple and triple lattices are available, because a complete set of orthogonal Latin squares of order 6 does not exist. For 100 entries, a 10 x 10 simple or triple lattice is common, though alpha designs now offer more flexibility in the number of entries.
For 9 entries (k = 3), write the entries in a 3 x 3 array: 1 2 3 / 4 5 6 / 7 8 9. Replicate I uses the rows as blocks: {1, 2, 3}, {4, 5, 6}, {7, 8, 9}. Replicate II uses the columns: {1, 4, 7}, {2, 5, 8}, {3, 6, 9}. Each entry has 2(k - 1) = 4 first associates (the entries in its row or column), with which it shares one block, and (k - 1) squared = 4 second associates, with which it never shares a block. In the field, the entries are first assigned to array positions at random, then blocks are randomised within replicates and entries within blocks. A triple lattice would add a third replicate based on the letters of a 3 x 3 Latin square, and the balanced lattice a fourth from the orthogonal square.
A Youden square is an incomplete Latin square: a k x v array in which each of the v treatments occurs once in each of the k rows, and the v columns, each containing k treatments, form the blocks of a symmetric BIBD. It controls two sources of variation, like a Latin square, when one of them (the columns) can only take k < v treatments. The blocks of any symmetric BIBD can be arranged as a Youden square. For example the BIBD with v = b = 7, k = 3, lambda = 1 gives a 3 x 7 Youden square: three rows (such as three positions on a greenhouse bench or three times of day) and seven columns (blocks). The analysis removes row effects (orthogonal to treatments, since every treatment appears once per row) and adjusts treatments for columns as in a BIBD, so the error has bk - b - v + 1 - (k - 1) degrees of freedom.
In field trials with many entries, plots form a grid of field rows and columns, and fertility can trend in both directions. Row-column designs arrange treatments so that both rows and columns act as incomplete blocks; resolvable row-column designs (latinized designs, row-column alpha designs) are produced by design-generation software and analysed by mixed models with random row and column effects, often combined with spatial modelling of residual correlation. Their aim is the same as the Latin square's, but for numbers of treatments far larger than any square could hold.
Seven treatments compared in a growth chamber with only three shelf positions per run, and with shelf position thought to matter, fit a 3 x 7 Youden square: seven runs (columns), each holding three treatments forming the blocks of the (7, 7, 3, 3, 1) BIBD, arranged so that each treatment occupies each shelf position exactly once across the runs. Shelf effects are then removed orthogonally and run effects by the BIBD adjustment. The same structure suits tasting panels where each taster scores three of seven samples in three serving positions.
A design is resolvable if its blocks can be grouped into replicates, each containing every treatment exactly once. Resolvability is valuable in practice: each replicate can be managed (sown, sprayed, harvested) as a unit, a whole replicate can be dropped if damaged, the experiment can be analysed as an RCBD if blocking turns out not to matter, and replicates can be spread over sites or times. Lattices and many BIBDs (such as the affine resolvable designs) are resolvable, but lattices require v to be a square or k(k + 1).
Patterson and Williams (1976) introduced alpha designs, a flexible class of resolvable incomplete block designs for any number of treatments v that can be written as v = sk (s blocks of size k per replicate), with block sizes differing by at most one when v is not a multiple of k. They are generated cyclically from a small array (the alpha array), and good arrays give efficiency factors close to the theoretical upper bound. Alpha designs (and their successors generated by computer search) are now the standard design for variety and breeding trials in national programmes and international centres, typically with two or three replicates and blocks of 4 to 10 plots. Alpha designs are analysed with blocks within replicates as random effects by REML, which recovers inter-block information automatically, and spatial terms can be added. For example, 24 entries in blocks of 4 give s = 6 blocks per replicate; with three replicates there are 18 blocks.
Partially replicated (p-rep) designs, developed by Cullis and colleagues for early-stage testing, replicate a fraction of the new entries (often 20 to 30%) instead of relying only on checks. Replicated entries contribute to the error estimate and to modelling spatial trends, while most entries still occupy one plot. They are analysed with mixed models that include spatial terms and, increasingly, genetic relationships among entries, and they are a modern successor to augmented designs in many breeding programmes.
Block size in an alpha design is a compromise. Smaller blocks follow field heterogeneity more closely but lose more information to blocks (lower efficiency factor) and give more block effects to estimate; larger blocks are more efficient in the combinatorial sense but less homogeneous. Practical guidance from variety-testing programmes is to choose blocks roughly matching the scale of fertility patches observed on the station, often 4 to 10 plots, and to check after each season how much the blocks reduced error compared with complete replicates. Because alpha designs are resolvable, that comparison is always available.
In early-generation breeding and germplasm evaluation there may be hundreds or thousands of new entries with seed for only one plot each. Federer's augmented designs (1956) handle this by replicating a few standard check varieties and augmenting each block with unreplicated new entries. In an augmented RCBD, c checks appear in each of b blocks, and the new entries are distributed among the blocks, one plot each. The checks provide the error estimate and the block effects, and each new entry's yield is adjusted for the effect of the block it is in. With c checks and b blocks, the error degrees of freedom are (b - 1)(c - 1), so enough checks and blocks are needed for a usable error (for example 4 checks in 5 blocks give 12 degrees of freedom). Block size is c plus the number of new entries per block; with 4 checks, 5 blocks and 30 new entries, each block has 4 + 6 = 10 plots.
The analysis estimates block effects from the checks, adjusts each new entry by subtracting its block effect, and compares entries using standard errors that differ by comparison type: between two checks, between a check and a new entry, between two new entries in the same block and between two new entries in different blocks, the last being the least precise. Augmented designs can also be built on Latin squares or incomplete blocks, and modern analyses use mixed models, sometimes with spatial adjustment and with relationship information among entries, as in partially replicated (p-rep) designs where a fraction of new entries is replicated.
Checks in augmented designs should be stable, well-known varieties that represent the standard against which new entries are judged, and there should be enough of them, and enough blocks, to give a usable error estimate: a design with 2 checks in 4 blocks has only 3 error degrees of freedom. Spreading the blocks across the field and keeping blocks compact makes the block adjustment meaningful; very long blocks with checks bunched at one end adjust poorly for fertility variation within the block.
Suppose four checks are sown in every block, the overall mean of all check plots is 48 and the mean of the checks in block 3 is 52. Block 3 is estimated to be 4 units better than average, so a new entry with an observed yield of 60 in block 3 has an adjusted yield of 56. An entry with the same observed yield in a block whose checks averaged 45 (3 below average) would be adjusted to 63. The adjustment is only as good as the check-based block estimates, which is why the number of checks per block and their spread across the field matter.
The efficiency factor of an incomplete block design compares the average variance of elementary treatment contrasts with that of an RCBD with the same replication and the same error variance per plot. For an equireplicate design, the average variance of tau-hat_i - tau-hat_l is 2 sigma squared / (r E), where E is the efficiency factor; in an RCBD the variance is 2 sigma squared / r, so E <= 1. E is the harmonic mean of the canonical efficiency factors, the non-zero eigenvalues of C/r. For a BIBD all canonical efficiency factors are equal to lambda v/(rk) and E takes that value.
For an equireplicate design with v treatments and blocks of size k < v, the efficiency factor cannot exceed v(k - 1) / [k(v - 1)], the value attained by a BIBD when one exists. This bound lets designers judge how good an available design is: an alpha design with E within a few per cent of the bound is nearly as good as a BIBD would be. Smaller blocks have lower efficiency factors (more information lost to blocks) but usually smaller error variance per plot; the net precision relative to an RCBD is E times the ratio of the RCBD error variance to the incomplete block error variance. For example, with v = 9 and k = 3 the bound is 9 x 2 / (3 x 8) = 0.75. If blocks of 3 plots have an error variance of 60% of that of complete blocks of 9, the incomplete block design has relative precision 0.75 / 0.60 = 1.25 compared with an RCBD with the same replication.
Incomplete block designs spend more degrees of freedom on blocks than complete block designs do, so their error has fewer degrees of freedom. For large trials this matters little, but for small experiments the comparison of designs should also consider error degrees of freedom, as with the adjustment factor discussed for relative efficiency in the basic designs subject.
Two candidate designs for 30 entries in blocks of 5 with three replicates can be compared by their efficiency factors before any field work: a design with E = 0.79 loses less information to blocks than one with E = 0.74, against an upper bound of 30 x 4 / (5 x 29) = 0.83. Design software reports E (or the related A-optimality criterion), and good alpha designs come within one or two per cent of the bound. The efficiency factor says nothing about how much error the blocks will remove; that depends on the field and is known only after the trial.
Students meet incomplete block designs in examinations (parameter relations, Fisher's inequality, C-matrices, BIBD estimates and efficiency) and in breeding or sensory theses where many entries must be compared. Knowing why an alpha design is analysed with random blocks, and how augmented designs adjust new entries, lets them analyse their own trials and read published ones critically.
A five-year statistician in a breeding or variety-testing programme generates alpha and row-column designs for dozens of trials a season, chooses block sizes to match field heterogeneity, sets up augmented and partially replicated designs for early-generation material, and analyses them with mixed models, reporting adjusted means, standard errors and the efficiency of blocking. They also design incomplete block layouts for sensory panels and laboratory batches.
A senior head of trial design and analysis sets standards for a national or company testing network: default design families, block sizes, numbers of checks and replicates, analysis pipelines with recovery of inter-block information and spatial adjustment, and how efficiency is monitored across sites and years. They decide when to move from lattices to alpha or p-rep designs and how to combine trials.
A PhD with long experience constructs new designs (optimal incomplete block, row-column and partially replicated designs), develops their combinatorial and statistical theory, and advises breeding programmes on design strategy. As a reviewer, they judge whether a published trial's design and analysis justify its variety comparisons.
Proofs of the BIBD relations, Fisher's inequality and the combined-estimator formulas are outlined, not given. The detailed analysis of PBIBDs (formulas involving the p^i_jk) and of lattices with recovery of inter-block information is described rather than worked. Construction of alpha arrays, row-column and p-rep designs is by software and only described. Optimality theory of block designs is in the advanced design subject.
Original practice questions written to the syllabus and pattern of the exams named in each question's examTags. They are not past papers and do not reproduce any board's questions. Every numerical answer was computed or checked with a script; the values needed are stated in the question.
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