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eduKateSG · Why Mathematics? · D-Optimal Designs: Information Matrices, Determinants and Constrained Experiments
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 classical experimental designs cannot fit the allowed runs? D-optimal design selects points from a feasible candidate set to maximise the determinant of the information matrix for a prespecified model. The determinant measures joint parameter information, making constraints, model dependence and trade-offs visible before data are collected.
Did You Know? “Optimal” never means best for every possible question. A D-optimal design is optimal, or approximately so, for its chosen model, candidate set, run count and determinant criterion. Change any of those ingredients and a different design may be preferred.
The practical value is not the label “optimal” but the audit trail. A reader can see the feasible settings, the columns of the intended model, the selected rows and the criterion used to compare alternatives. That trail makes constraints testable and reveals what the design does not protect. The tiny quadratic example below shows the complete chain from candidate points to determinant before the article moves to algorithms, efficiency, validation and model risk. Every later claim remains tied to that visible linear-algebra contract. That makes the method understandable without treating the optimisation software as an authority that cannot be questioned.
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
Factorial, fractional factorial and response-surface designs provide strong balance and familiar analysis. They may fail when the run budget is irregular, some combinations are impossible, factor levels differ, safety removes corners or the intended regression model is unusual. Computer-aided optimal design starts with the actual feasible candidates and selects a subset. This is more principled than deleting inconvenient runs from a classical design after the fact.
The candidate set is the list of treatment combinations the experiment could genuinely execute. It may be a grid, a finite set of recipes, allowable classroom schedules or safe operating points. Candidate quality is a design decision. If a useful region is omitted, the algorithm cannot select it. If an impossible point is included, mathematical efficiency may produce an unusable plan. Document how the set was constructed.
Choose the regression terms the experiment should estimate: perhaps an intercept and main effects, selected interactions or a full quadratic. The model converts each candidate into a row vector x. D-optimality judges information for those columns only. An omitted interaction receives no protection. A needlessly large model can consume runs and reduce precision. Use science, hierarchy and earlier experiments to specify terms before seeing new responses.
Stack selected candidate rows to form X. For a one-factor quadratic, each row is [1,x,x squared]. With x equal to −1, 0 and 1, the rows are [1,−1,1], [1,0,0] and [1,1,1]. The columns correspond to intercept, linear and quadratic coefficients. Full column rank is essential; otherwise at least one stated coefficient combination cannot be estimated.
Under independent equal-variance errors, X transpose X is proportional to the Fisher information for the coefficients. Large information means small coefficient uncertainty. The matrix also shows correlation among estimates: orthogonal columns produce zero off-diagonal entries, while constrained designs may be nonorthogonal. If weights or unequal variances are planned, the relevant information becomes X transpose W X rather than the unweighted form.
| x | 1 | x | x squared |
|---|---|---|---|
| -1 | 1 | -1 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 1 | 1 | 1 |
D-optimality maximises det(X transpose X). The inverse information matrix is proportional to coefficient covariance, so maximising the determinant minimises the determinant of that covariance. Geometrically it shrinks the confidence-ellipsoid volume for all model coefficients together. The criterion is invariant to nonsingular reparameterisation after appropriate scaling, but reported raw determinant values depend on coding and are not directly comparable across differently scaled models.
For x=−1,0,1, X transpose X equals [[3,0,2],[0,2,0],[2,0,2]]. Its determinant is 4. The calculation can be checked by expanding along the second row: 2 times (3×2−2×2)=4. The positive determinant confirms full rank. Repeating one row while omitting a distinct point in a three-run saturated quadratic makes the matrix rank deficient and the determinant zero.
A saturated design has exactly as many runs as model parameters. It can estimate the coefficients when full rank, but it leaves no pure-error degrees of freedom and is fragile to a lost run or model misspecification. D-optimal selection can identify a good saturated set, yet practical experiments often add replication, lack-of-fit points or validation runs. Maximum algebraic efficiency is not always maximum scientific reliability.
| Row | Col 1 | Col 2 | Col 3 |
|---|---|---|---|
| Col 1 | 3 | 0 | 2 |
| Col 2 | 0 | 2 | 0 |
| Col 3 | 2 | 0 | 2 |
NIST explains that D-optimal algorithms select runs from a candidate set to maximise the determinant of X transpose X. The designs are useful under constraints, usually nonorthogonal and model dependent; an exchange algorithm may find a strong design without guaranteeing the global optimum. Read the NIST handbook.
A-optimality minimises the trace of the inverse information matrix, targeting average coefficient variance. G-optimality minimises the maximum prediction variance over a specified space. V- or I-type criteria target average prediction variance. A design that is excellent for joint coefficient estimation may not best protect worst-case prediction. State the question first, then choose the criterion; do not select a letter because software makes it easy.
D-efficiency scales the determinant so designs with the same model, run count and parameter count can be compared. Exact formulas differ by convention and benchmark. Report the definition used rather than presenting a percentage without context. A lower-efficiency design may still be preferred if it adds replication, easier randomisation, safer settings or better diagnostics. Efficiency is one input to a design decision, not an automatic command.
A common algorithm begins with a candidate subset, evaluates swaps between selected and unselected points, and keeps improvements until none is found. Different starts can reach different local optima. Use multiple starts, record the seed and compare final determinants. Some algorithms return exact designs with integer run counts; approximate-design theory may return weights that must be rounded carefully. Reproducibility includes algorithm settings, not only the chosen rows.
Suppose the design is optimised for a straight line but the true response curves. Endpoints may estimate slope well yet leave the centre behaviour uncertain. Adding a quadratic term changes X and can make the old design singular. Robust practice compares plausible model sets, includes lack-of-fit or centre points and validates predictions. “D-optimal” is always shorthand for “D-optimal under this stated model and space.”
Centre and scale quantitative factors before building polynomial terms. Coding improves numerical conditioning and makes coefficients and candidate geometry easier to compare. Categorical factors need appropriate contrasts. The determinant can change dramatically under careless units, although the underlying fitted space may be equivalent. Save the coding transformation so physical settings, model rows and predictions remain connected.
Safety, physics or ethics may make some factor combinations impossible. Remove them from the candidate set before optimisation and explain why. Do not generate a rectangular design and delete forbidden points afterward, because deletion changes rank and efficiency. When restrictions create disconnected regions, verify prediction behaviour across each region and avoid interpolating through physically impossible space as if it were observed continuity.
Mixture factors sum to one and occupy a simplex rather than a cube. Geographic, engineering and timetable constraints can produce other irregular spaces. D-optimal algorithms are useful precisely because candidate sets can represent these shapes. The regression basis must still respect the science and identifiability. A convenient polynomial on an irregular boundary can extrapolate badly inside gaps, so map prediction variance and validate at meaningful points.
Replicating selected points estimates pure error, supports equipment checks and protects against a failed run. D-optimality may spread runs instead of replicating because unique points increase coefficient information. Add replication deliberately when noise estimation matters. If the response variance differs across candidates, allocate replication or use weights accordingly. Never label repeated software output from the same physical run as experimental replication.
Experiments may span days, sites or batches. Include block terms or constrain allocation so important treatment information is not confounded with blocks. A design optimal without blocks can lose rank after an operational schedule is imposed. Generate the design with the real block structure whenever possible. Check within-block estimability, randomise run order inside allowed groups and retain block identifiers in the analysis.
Begin with a defensible initial design, fit the model, inspect uncertainty and add points that improve the remaining information. Sequential augmentation can be more efficient than choosing all runs at once, but response-adaptive point selection changes the inferential plan. Prespecify decision rules where confirmatory claims matter. Recalculate the information matrix from all selected points and separate exploratory model changes from confirmatory validation.
At candidate vector x, the scaled prediction variance is x transpose (X transpose X) inverse x. Map it across the design space. D-optimality controls the joint coefficient volume but can leave some locations less precise. High leverage or high variance near constraints signals where confirmation may be valuable. Prediction intervals also include residual variance and, for future observations, new observation noise.
A school can test only a limited set of timetable, group-size and feedback combinations because some schedules are infeasible. A candidate set lists feasible programmes. A model might include main effects and selected interactions. D-optimal selection chooses an informative subset, but student clustering, teacher effects, ethics and outcome validity still matter. The algorithm chooses settings; it does not randomise students or guarantee the model captures learning.
An engineer must fit a quadratic response with 13 runs, avoid unsafe high-temperature and high-pressure combinations, and include two material categories. Classical tables may not match. A D-optimal design can select from the safe candidate grid. The team should then inspect coefficient correlations, prediction variance, run order, replication and sensitivity to model terms before committing expensive equipment time.
Compute budgets may allow only selected combinations of model size, learning rate, data mixture and regularisation. Candidate points encode memory and time limits. D-optimality can inform a regression surrogate, but stochastic training requires seed replication and outcome definitions. Failed jobs are not random missingness by default. A surrogate optimised over an old hardware or data regime may not generalise to a changed system.
Before running, verify full rank, determinant, condition number, variance-inflation patterns, leverage, prediction-variance map, treatment replication and block confounding. Compare several starting seeds and criteria. Inspect whether every factor range has meaningful coverage. A high determinant with extreme coefficient correlation or an unprotected scientific contrast can still be a poor practical design. Pre-data diagnostics are one of optimal design’s greatest strengths.
After fitting, inspect residuals, run order, lack of fit, influential points and prediction performance at confirmation settings. Revisit whether the prespecified model remains credible, but label model changes as exploratory. A design can be mathematically optimal yet reveal unexpected curvature or heteroscedasticity. The response data test the model assumptions; the determinant only assessed planned information under them.
Construct the three candidate rows, calculate X transpose X and determinant 4, then repeat a point and verify determinant zero. Add a fourth centre replicate and inspect pure-error degrees of freedom. Compare D-, A- and G-oriented choices on a slightly larger candidate set. Finally hold out a feasible point, fit the model and check whether its prediction interval covers a simulated observation.
D-optimality depends on the model, candidate set, criterion, variance assumptions, run count and algorithm. It does not ensure global optimality, orthogonality, model truth, safe execution, causal inference without randomisation or external validity. Determinants can be numerically unstable without coding. A design with little replication may be fragile. These limits are reasons to document the process, not reasons to abandon optimisation.
Publish the candidate set or generation rule, forbidden regions, model formula and coding, number of runs, criterion, algorithm, starts and seed, selected points and replicates, determinant or defined efficiency, block and randomisation plan, prediction-variance diagnostics and confirmation results. Another analyst should be able to regenerate X and test whether the reported selection is feasible and full rank.
Use D-optimal design when classical designs cannot respect the real constraints and the model is clear enough to guide selection. Prefer a classical design when it fits the question and offers simpler balance, randomisation and interpretation. Add replication and validation when reliability matters. Call the result optimal only within its stated mathematical contract, then test that contract against new observations.
List candidate points in one table and selected runs in another. Calculate coded model columns, selection counts, X transpose X, determinant, inverse, coefficient correlations and prediction variances. Use flags for forbidden points, blocks, replicates and confirmation runs. Keep formulas auditable and avoid comparing raw determinants across different scaling or model dimensions without an explicit normalisation.
Generate the candidate matrix, apply coding, verify rank and run an exchange algorithm from multiple random starts. Save the selected indices, determinant trajectory and seed. Compute A, D and G diagnostics, plot prediction variance and simulate model misspecification. Unit tests should catch duplicate saturated rows, forbidden selections and any mismatch between physical settings and model rows.
Industrial statisticians, chemical engineers, quality scientists, data scientists and operations researchers use optimal designs under real constraints. The transferable skill is expressing a scientific goal as a model, candidate space and information criterion, then auditing the trade-off. Professional work also requires safety, domain knowledge, measurement design, software validation and communication; a high determinant is never a complete qualification.
Build a candidate set of paper-airplane wing angle and paper-weight settings, excluding combinations that cannot be folded safely. Prespecify a quadratic or simpler model, select a small D-optimal subset and compare it with an ad hoc subset. Run randomised trials, reserve one confirmation setting and map prediction uncertainty. Explain how the preferred design changes when the model loses its interaction term.
Optimal does not mean universally best. The largest determinant does not validate the model. A nonorthogonal design is not automatically flawed. A candidate set is not a random sample of the world. Algorithm convergence is not proof of the global optimum. Replicated computer evaluations are not experimental replicates unless the stochastic process is rerun independently.
The fractional-factorial guide gives a classical way to reduce runs during screening. The response-surface guide explains quadratic optimisation. This article handles irregular budgets and constrained candidate spaces when those classical structures do not fit.
Students should first calculate a tiny determinant by hand, then use code for larger candidate sets. Parents can ask whether an educational optimisation study listed infeasible schedules and validated the chosen model. A useful sequence is define the question, write the model, build candidates, select points, randomise execution, validate predictions and explain what would change the design.
- Build the three-row quadratic model matrix.
- Calculate determinant 4.
- Create a singular selection.
- Change to a linear model.
- Exclude a candidate and redesign.
- Compare coefficient and prediction criteria.
### What does D stand for? It refers to the determinant criterion.
### Is every D-optimal design unique? No; ties, algorithms and candidate symmetries can produce alternatives.
### Why specify the model first? The information matrix columns come from that model.
### Must the design be randomised? Yes, within real execution constraints, for causal protection against uncontrolled order effects.
Report the estimand, candidate space, restrictions, model and coding, criterion, algorithm and seed, selected runs, rank, determinant or defined efficiency, blocks, replication, randomisation, variance assumptions, prediction diagnostics, model checks, confirmation evidence and scope. Preserve the exact candidate and selection files for reproducibility.
The rows of X span the coefficient directions the experiment can learn. D-optimality enlarges their joint information volume, while singularity reveals a missing direction. This geometry connects linear algebra directly to experimental choice.
Recreate the candidate set, coding, model matrix and selection; verify feasibility, rank, determinant, algorithm restarts, randomisation and held-out validation. If the model or safe region changed after data collection, distinguish the exploratory redesign from the original confirmatory plan.
Reproduce the selection independently. Starting from the saved candidate set and model formula, rebuild X, rerun the algorithm from several seeds and compare determinants. If several designs tie or nearly tie, evaluate operational simplicity, replication, block balance and prediction variance before choosing among them. A tiny numerical gain may not justify a schedule that is harder to randomise or more vulnerable to one failed run. Report the tolerance used to call designs equivalent.
Perform a model-perturbation audit. Add one scientifically plausible interaction or quadratic term, remove one doubtful term and recalculate rank and efficiency. The original design need not remain optimal, but it should not become catastrophically blind to a credible alternative. If it does, add protective points or narrow the claim. This is not permission to choose the model after observing a convenient response; it is a pre-data robustness check for the design contract.
Perform a candidate-space audit as well. Plot selected and unselected points in physical units, mark forbidden regions and inspect boundary coverage. Verify that rounding coded settings back to machine controls does not merge distinct candidates. Check whether the selected points can be executed in random order or whether setup constraints introduce blocks. When practical constraints force a different order, regenerate or augment the design using those constraints rather than editing the schedule informally.
After data arrive, separate information geometry from model fit. The determinant describes planned coefficient precision under assumptions; residual plots, lack-of-fit evidence and confirmation runs assess whether the response behaves accordingly. A high determinant cannot rescue a biased measurement, an omitted mechanism or a drifting process. Preserve a holdout or run new confirmation points where the decision will actually operate, especially near constraints or predicted optima.
Teach the method with the saturated quadratic example before using larger algorithms. Learners can see that three distinct rows span three coefficient directions, while a repeated row collapses the determinant to zero. From there, the exchange algorithm is no longer a black box: it searches for rows whose geometry expands the joint information volume. That mental model supports better software use and clearer explanations to decision makers.
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: Write the candidate set
Setup: Use x values −1, 0 and 1 for a quadratic model.
Task: Create rows [1,x,x squared].
Checking target: Confirm every permitted setting appears once in the candidate list.
Practice case 2: Calculate the determinant
Setup: Use the three unique candidate rows.
Task: Compute X transpose X and its determinant.
Checking target: The determinant should be 4 for the saturated quadratic example.
Practice case 3: Create a singular choice
Setup: Repeat one candidate and omit another.
Task: Recalculate the matrix rank and determinant.
Checking target: A zero determinant means the stated quadratic coefficients are not all estimable.
Practice case 4: Change the model
Setup: Remove x squared and fit a straight line.
Task: Rebuild X and compare the best design points.
Checking target: Optimality changes because the model changed.
Practice case 5: Add a forbidden region
Setup: Exclude one unsafe candidate combination in two factors.
Task: Regenerate the candidate set before selection.
Checking target: Do not select a point that cannot be executed.
Practice case 6: Compare A and D criteria
Setup: Use one design matrix with unequal coefficient variances.
Task: Calculate determinant and trace of the inverse.
Checking target: Explain why a D-best design need not be A-best.
Practice case 7: Inspect prediction variance
Setup: Calculate x transpose (X transpose X) inverse x over the candidate space.
Task: Find the worst predicted point.
Checking target: Relate the maximum to G-optimality without calling it a proof of global optimality.
Practice case 8: Restart the exchange algorithm
Setup: Use several initial subsets.
Task: Run exchanges until no improving swap is found.
Checking target: Compare determinants and retain the best reproducible result.
Practice case 9: Plan confirmation
Setup: Reserve feasible points not used to fit the model.
Task: Compare observed and predicted responses.
Checking target: Validation tests the model, not the determinant alone.
Practice case 10: Write the conclusion
Setup: State model, candidate set, criterion and selected runs.
Task: Report determinant or efficiency plus diagnostic limits.
Checking target: Name algorithm settings, randomisation and confirmation evidence.