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eduKateSG · Why Mathematics? · Balanced Incomplete Block Designs: Incidence Matrices, Pairwise Balance and Connected Comparisons
Start with the question, audit the worked example, or jump to the assumptions and practice. Every chapter stays open for uninterrupted reading.
Understand the question
Define the quantity and follow the mechanism.Route 2 · Chapters 9–16
Reproduce the example
Audit the numbers one step at a time.Route 3 · Chapters 17–24
Check assumptions
See where design and calibration matter.Route 4 · Chapters 25–32
Explore applications
Connect the method to a concrete investigation.Route 5 · Chapters 33–42
Study and verify
Practise, inspect failure cases and read further.
Full chapter index · Practice and checking · Mathematics Learning Hub
Why is mathematics important when every block is too small to hold every treatment? A balanced incomplete block design, or BIBD, deliberately places only some treatments in each block while keeping treatment replications and pairwise meetings balanced. Incidence matrices, concurrence counts and connected contrasts turn a practical capacity limit into a transparent experimental design.
Did You Know? Seven treatments can be compared in seven blocks of size three so that every treatment appears three times and every pair appears together exactly once. The 21 planned observations form the celebrated order-seven projective-plane pattern, often pictured as the Fano plane.
Full chapter index
Chapters 1–8 · Understand the question
Chapters 9–16 · Reproduce the example
Chapters 17–24 · Check assumptions
Chapters 25–32 · Explore applications
Chapters 33–42 · Study and verify
A complete block is ideal when every local group can receive every treatment. Real laboratories, classrooms, farms and manufacturing batches often have limited capacity. A session may fit three learning activities, a batch may hold four formulations, or a participant may tolerate only a few conditions. An incomplete block accepts that limit before data collection. The mathematical task is to preserve enough overlap that all treatment contrasts remain connected and comparably precise.
Balance has three parts in a BIBD. Each block contains the same number k of treatments. Each treatment appears in the same number r of blocks. Every unordered pair of treatments occurs together in exactly lambda blocks. These are design properties, not observations produced by chance. They must be checked from the final executed incidence pattern, because a missing run or substituted treatment can destroy balance even when the original plan was perfect.
Write v for the number of treatments, b for blocks, r for replications of each treatment, k for treatments per block and lambda for pair concurrence. The tuple (v,b,r,k,lambda) is a compact design description. It does not state the outcome model, the experimental unit or the randomisation, so a complete protocol still needs those facts. Parameters that look plausible are not automatically feasible; they must satisfy counting identities and admit an actual block construction.
Use blocks 123, 145, 167, 246, 257, 347 and 356. Treatment 1 occurs in the first three blocks; each other treatment also occurs three times. Pair 1–2 meets in block 123, pair 1–4 in 145 and pair 6–7 in 167. Checking every pair reveals exactly one shared block. The pattern is efficient because each small block contributes three pairwise meetings while the whole schedule distributes those meetings evenly.
| Block | Treatments |
|---|---|
| B1 | 1, 2, 3 |
| B2 | 1, 4, 5 |
| B3 | 1, 6, 7 |
| B4 | 2, 4, 6 |
| B5 | 2, 5, 7 |
| B6 | 3, 4, 7 |
| B7 | 3, 5, 6 |
Count treatment appearances in two ways. There are b blocks and k entries per block, so the schedule has bk treatment slots. There are v treatments and each repeats r times, so it also has vr slots. Therefore bk=vr. In the example, 7×3=7×3=21. Failure of this equality proves the proposed parameter set cannot describe a BIBD, although satisfying it alone does not guarantee a construction exists.
Fix one treatment. It appears in r blocks and meets k−1 partners each time, producing r(k−1) partner meetings. It must meet each of the other v−1 treatments lambda times, producing lambda(v−1) meetings. Hence lambda(v−1)=r(k−1). For (7,7,3,3,1), both sides equal 6. This identity explains why pairwise balance is tied to replication rather than being a separate decorative property.
Create a v-by-b matrix N with treatments as rows and blocks as columns. Enter 1 when a treatment appears in a block and 0 otherwise. Every row sum equals r and every column sum equals k. The product N N transpose has diagonal entries r and off-diagonal entries equal to pair concurrences. For the seven-treatment plan, its diagonal is 3 and every off-diagonal is 1, so one matrix multiplication verifies all pair meetings at once.
Treat treatments as vertices and draw an edge weighted by the number of blocks in which each pair meets. A BIBD produces a complete graph with equal edge weights lambda. More general incomplete block designs may have unequal weights yet remain connected. Connectivity matters because a path of shared blocks can still support indirect comparison. If the graph splits into components, contrasts between components are not estimable no matter how sophisticated the software appears.
For equal block sizes, the treatment information matrix after adjusting for block effects is C=R−N N transpose divided by k, where R contains treatment replications on its diagonal. In the example R=3I. The resulting matrix has diagonal 2, off-diagonal −1/3 and row sums zero. The zero direction represents adding the same constant to every treatment effect; treatment contrasts live in the remaining six-dimensional space.
| Identity | Left side | Right side |
|---|---|---|
| bk=vr | 7×3=21 | 7×3=21 |
| lambda(v−1)=r(k−1) | 1×6=6 | 3×2=6 |
The information matrix has one zero eigenvalue and six equal nonzero eigenvalues of 7/3. Equal nonzero eigenvalues express the symmetry of the design: every elementary treatment difference has the same information. If residual variance is sigma squared, the variance of an adjusted difference is 6 sigma squared divided by 7 in this example. That calculation assumes the additive block model and homoscedastic errors; it is not a universal guarantee for every outcome process.
Raw treatment means can be misleading in an incomplete design because treatments do not appear in every block. Adjusted treatment estimates compare outcomes after accounting for block shifts. In a perfectly balanced design, symmetry makes the algebra clean, but the adjustment is still conceptually important. A high raw mean may reflect unusually favourable blocks. Report adjusted contrasts with uncertainty and retain block identifiers in the data so readers can reproduce the calculation. NIST’s Durbin-test documentation states the defining equal-block, equal-replication and equal-pair properties for a balanced incomplete design.
A common model is response=overall mean+treatment effect+block effect+error. It assumes treatment differences do not systematically depend on blocks. With one observation per treatment-block cell, treatment-by-block interaction is not separately estimated from residual noise. If interaction is scientifically likely, replicate the structure, broaden the model or narrow the claim. Balance controls allocation; it does not make an additive scientific assumption true.
The block list fixes which treatments meet, but execution still needs randomisation. Randomise block order when time can drift. Randomise treatment positions or experimental units within each block. Randomly relabel a mathematically equivalent construction when treatment labels carry no restrictions. Document constraints such as washout or safety. Randomisation protects against systematic bias; it does not change the required incidence counts.
A block should group experimental units that share an important nuisance condition before treatment assignment. Examples include batch, field neighbourhood, day, machine or matched participant set. Grouping observations after seeing outcomes can manufacture apparent precision. A session is not automatically a useful block if its units are heterogeneous or if treatments influence one another inside it. Define the experimental unit and interference boundary first.
Suppose one planned response is lost. The original treatment assignment still occurred, but the observed incidence matrix is no longer balanced for analysis. Do not insert the treatment mean merely to restore a tidy table. Fit the actual observed pattern, inspect rank and calculate the changed standard errors. If the loss depends on outcome or treatment, a missing-data assumption is also required. Report planned and realised designs separately.
If a treatment cannot be administered, replacing it with another changes treatment replication and pair concurrence. The design may remain connected but is no longer the advertised BIBD. Recalculate N, N N transpose and C before analysis. When feasible, reschedule the missed treatment in a new compatible block. Any repair should be planned without using outcome values, otherwise the repair itself can adapt to favourable results.
Perfect pairwise balance is sometimes impossible because of capacity, eligibility or safety restrictions. A connected incomplete block design can still estimate all treatment contrasts. Precision will vary across pairs, so publish the concurrence matrix or contrast variances. Optimal-design algorithms can choose a good feasible schedule, but “good” must be tied to a stated criterion. Calling an unbalanced design a BIBD would conceal an important difference.
A BIBD is resolvable when its blocks can be grouped into complete replicates that partition all treatments. Resolvability helps when a day, site or round can host several small blocks and every treatment should appear once per round. The seven-treatment, block-size-three example is not resolvable because seven is not divisible by three. Operational requirements must be checked alongside the five BIBD identities.
Increasing blocks can improve precision, estimate residual variation and protect against losses. Yet repeating exactly the same block pattern under the same hidden condition may add less information than expected. Plan independent experimental units, not duplicated data rows. Degrees of freedom come from the executed randomisation and model. A large number of observations inside a few blocks does not automatically provide many independent blocks.
Imagine seven feedback prompts but only three can be tested in one tutorial session. Sessions are blocks; individual student-task encounters might be experimental units if interference is controlled. A BIBD can make every prompt appear equally often and every pair share a session once. The outcome could be later transfer, not immediate completion alone. Teacher, topic and student clustering still need modelling, and one design cannot prove a prompt works equally for every learner.
Seven crop varieties may be compared in field blocks that accommodate only three plots because land conditions change rapidly. Pairwise concurrence ensures every variety shares a local neighbourhood with every other variety once. Randomise varieties within blocks and map plot positions. Spatial residual patterns can reveal that the block definition was too coarse. Weather seasons and locations define the population to which the variety contrasts can reasonably generalise.
A participant may taste only a few products before fatigue changes ratings. Products are treatments and participants are incomplete blocks. Balance spreads each product and pair across participants. Presentation order must also be randomised or counterbalanced, and carryover needs washout. Because ratings from one participant are correlated, analysis should recognise the participant block. A design that ignores fatigue may be balanced on paper yet biased in experience.
Incomplete blocks also appear when servers, devices or users can test only subsets of algorithms. The same incidence logic distributes pairwise comparisons. Network interference, time drift and shared infrastructure can violate the simple additive model, so log placement, time and version. A balanced assignment is valuable because it separates comparison geometry from outcome noise, but it cannot replace monitoring of the system that generates the measurements.
Power depends on residual variance, effect size, the contrast information from C, alpha and the number of independent blocks or replicates. Use pilot variance cautiously and simulate the planned analysis when clustering or missingness is complex. Increasing k creates more within-block comparisons but may make blocks less homogeneous. Increasing b adds cost. Planning is a trade-off between statistical information and the operational conditions needed to execute valid comparisons.
Plot residuals by block, treatment, run order and relevant covariates. Inspect adjusted versus raw treatment means, influence and pairwise standard errors. Recompute the incidence and concurrence matrices from realised assignments rather than trusting the protocol file. Examine whether missingness or variance differs by treatment. Diagnostics can reveal model strain; they cannot retrospectively create randomisation or identify an unrecorded interference path.
Rebuild the seven blocks, verify bk=vr and lambda(v−1)=r(k−1), then calculate N N transpose. Form C and confirm one zero plus six eigenvalues of 7/3. Simulate block shifts and treatment effects, fit the additive model and compare adjusted with raw means. Delete one observation and repeat. This sequence validates the arithmetic, the software implementation and the interpretation under a controlled failure.
A BIBD needs equal replications, block sizes and pair concurrences. It does not estimate every treatment-by-block interaction, remove measurement bias, guarantee external validity or cure informative missingness. Some parameter sets satisfy counting identities but have no construction. Complex eligibility constraints may require a different connected design. State these limits so efficiency is understood as a planned exchange of assumptions and resources.
Report v, b, r, k and lambda; list or provide the realised blocks; identify experimental units and block meaning; explain randomisation; state missing observations and repairs; publish the model, adjusted contrasts and uncertainty. If the design is only approximately balanced, say so. A reader should be able to reconstruct N from the report and determine which pairs were compared directly.
Choose a BIBD when blocks genuinely cannot contain all treatments, pairwise comparisons are the main goal and equal precision is worth the scheduling constraint. Use a complete block when capacity permits. Use a connected or optimal incomplete design when balance is infeasible. The best design is the one that answers the estimand honestly and can be executed without hidden substitutions.
Create one sheet for the block list and one for outcomes. Use a binary incidence table, row and column totals, a concurrence matrix and a realised-assignment audit. Keep treatment labels, block IDs, randomisation order, outcome units and missingness reasons in separate fields. Formula checks should flag any row sum not equal to r, column sum not equal to k or off-diagonal concurrence not equal to lambda.
Generate the incidence matrix from a tidy block table, calculate N N transpose and C, inspect rank and eigenvalues, then fit treatment and block effects. Unit tests should confirm the two parameter identities, equal pair counts and invariant results under harmless relabelling. Simulate block effects and missing observations to show when raw means diverge from adjusted estimates.
Agronomists, product testers, manufacturing engineers, clinical researchers and education researchers use incomplete blocks when local capacity is smaller than the treatment set. The transferable mathematical skill is to turn operational limits into an incidence structure whose estimable comparisons are visible. Careers also require subject knowledge, ethical judgement, measurement skills and communication; the design alone does not grant access to a profession.
Create seven paper-airplane configurations but allow each testing session to fly only three. Use the seven-block plan, randomise flight order and measure distance under a consistent protocol. Record wind or room conditions as block information. Compare raw and block-adjusted means, then deliberately omit one flight and show how concurrence and uncertainty change. Keep the investigation small, safe and reproducible.
A balanced design does not mean outcomes will be equal. Incomplete does not mean defective. Pairwise balance does not estimate treatment-by-block interaction. A BIBD is not created by giving every treatment the same total replication while ignoring pair meetings. Connectedness guarantees estimability of contrasts under the model, not equal precision, causality without randomisation or generalisation beyond studied blocks.
The randomised-block guide explains complete blocks and within-block comparisons. The Latin-square guide controls two crossed nuisance factors. Read those destinations as foundations, then use the present article for the capacity-limited case.
Students can learn the design by counting slots and pairs before touching software. Parents can ask whether a school experiment compared methods within comparable sessions and whether every method had fair direct overlap. A useful learning sequence is draw blocks, build N, check identities, calculate one contrast and explain one limitation. Accuracy matters more than memorising parameter letters.
- Verify the seven-treatment block list.
- Prove both BIBD counting identities.
- Build N and N N transpose.
- Explain the zero eigenvalue of C.
- Remove one observation and inspect precision.
- Design a connected alternative when one pair is forbidden.
### Why are blocks incomplete? Capacity prevents every treatment from appearing in every block.
### What makes the design balanced? Equal block size, equal treatment replication and equal pair concurrence.
### Must a useful design be perfectly balanced? No, but it must be connected for all desired contrasts.
### Can raw means be compared? Use block-adjusted estimates because treatment exposure to blocks differs.
Report the estimand, experimental unit, block definition, BIBD parameters, planned and realised block lists, incidence and concurrence checks, randomisation, missingness, model, adjusted contrasts, uncertainty, diagnostics and scope. Preserve enough information for another analyst to reproduce the design matrix and verify every claimed comparison.
Each shared block creates a direct local comparison. Equal concurrence spreads that direct evidence evenly across all pairs, while the connected network also supports indirect comparisons. The design therefore links simple counting to the geometry of estimation.
Before publication, confirm the executed treatment slots, row and column totals, pair concurrences, connectivity, model rank, residual degrees of freedom and every internal calculation. If execution changed the plan, describe the realised design by its actual properties rather than its intended name.
A strong audit also follows one treatment contrast from assignment to interpretation. Identify the blocks containing each treatment, show the direct shared-block evidence, then show how the fitted model combines direct and indirect paths. Compare its standard error with another pair. In a true BIBD the elementary pair variances agree; disagreement signals imbalance, unequal variance, weights or a different estimand. This worked trace prevents the word “balanced” from becoming a substitute for an actual calculation.
Next, test design resilience before outcomes are known. Remove each planned observation in turn and recalculate rank, connectivity and the largest contrast variance. This leave-one-slot exercise identifies fragile positions and informs contingency planning. It does not authorise selective removal after seeing results. If one slot is operationally vulnerable, add a prespecified repair block or replication. Record whether the repair changes b, r, k or lambda, and describe the final design accurately.
Finally, connect the mathematics to the decision. List the priority contrasts, smallest useful effect, feasible block capacity, expected variance and population of blocks. Check that equal precision is genuinely valuable; sometimes a few comparisons matter more and a tailored optimal incomplete design is better. Explain why the chosen block definition should reduce nuisance variation and why interference inside blocks is acceptable. A technically valid incidence matrix attached to a poor experimental unit cannot support the intended causal claim.
For teaching, ask the learner to prove the two identities with words before symbols, construct N from the block list, recover pair counts from N N transpose and explain the zero eigenvalue as an unidentifiable common shift. Then use a small simulated outcome table to compare marginal and adjusted means. The goal is transfer: capacity constraints become combinatorics, combinatorics becomes linear algebra, and linear algebra becomes honest uncertainty.
Continue through the Mathematics Learning Hub. Rebuild the central design, verify its identities, inspect one failure case and explain which conclusion the evidence can and cannot support. The most useful next step is a small reproducible calculation, followed by a plain-language account of the assumptions.
Extended practice laboratory
Practice case 1: Verify the Fano plan
Setup: Use the seven listed three-treatment blocks.
Task: Count each treatment and every unordered pair.
Checking target: Each treatment must occur three times and each pair exactly once.
Practice case 2: Break pairwise balance
Setup: Replace treatment 7 with 6 in one block.
Task: Recalculate treatment replications and pair concurrences.
Checking target: Identify which contrasts lose precision and whether the design remains connected.
Practice case 3: Build the incidence matrix
Setup: Write treatments as rows and blocks as columns.
Task: Enter 1 for presence and 0 for absence, then multiply N by its transpose.
Checking target: The diagonal should be 3 and every off-diagonal entry 1.
Practice case 4: Check the information matrix
Setup: Use C=R−N N transpose divided by k.
Task: Calculate row sums and eigenvalues.
Checking target: One eigenvalue should be zero and six should equal 7/3.
Practice case 5: Compare complete and incomplete blocks
Setup: Imagine all seven treatments fit in every block.
Task: Explain the run-cost and within-block comparison differences.
Checking target: Keep the estimand fixed while naming what the incomplete design sacrifices.
Practice case 6: Audit a missing observation
Setup: Remove one response from one planned block.
Task: Refit the actual incidence pattern instead of filling a mean.
Checking target: Report changed replication, concurrence and estimability.
Practice case 7: Randomise execution
Setup: Start from the valid block list.
Task: Randomise block order, units within blocks and treatment labels where scientifically allowed.
Checking target: Verify that randomisation does not alter the incidence properties.
Practice case 8: Estimate a contrast
Setup: Assign responses to the 21 planned units.
Task: Fit block and treatment effects and estimate treatment 1 minus treatment 2.
Checking target: Use a block-adjusted standard error rather than raw marginal means.
Practice case 9: Design an education example
Setup: Treat seven feedback prompts as treatments and tutorial sessions as blocks of size three.
Task: State the experimental unit, assignment and outcome.
Checking target: Check contamination, capacity and the meaning of a session block.
Practice case 10: Write the conclusion
Setup: Use the fitted contrast and its uncertainty.
Task: State what balance supports and what the unreplicated assumptions do not settle.
Checking target: Name the target population, missingness and one confirmation step.