Experimental design and optimal information

For a fixed full-rank linear Gaussian model with known common error variance, Fisher information for the coefficient vector is the Gram matrix divided by that variance. Even without Gaussianity, the same inverse Gram matrix controls the least-squares covariance under spherical errors. Thus selecting predictor rows changes which coefficient directions are measured precisely. Repeated observations at a setting add information under independent errors; highly correlated repeats can add much less, requiring a covariance-aware information matrix instead.

A design criterion compares feasible matrices subject to an explicit run budget and factor restrictions. Rank must be checked first: a singular information matrix leaves some parameter directions unidentified. Scaling matters when criteria compare particular coordinate uncertainties. Changing the model basis consistently leaves determinant comparisons unchanged up to a common factor, but a trace criterion can change under rescaling because it sums coordinate-specific variances. State the parameterization and target rather than calling one design intrinsically informative without context.

D-optimality maximizes the determinant of the information matrix, equivalently minimizing the determinant of coefficient covariance for fixed variance. In a Gaussian confidence ellipsoid, volume is proportional to the square root of that covariance determinant, with confidence level and residual scale held fixed. A-optimality minimizes the trace of inverse information, which averages coordinate variances in the chosen scaling. A c-optimal design instead minimizes uncertainty in one prespecified contrast c. These criteria need not select the same measurements.

A large determinant is a product-of-directions criterion, not a promise that every direction is individually as precise as possible. Subject to a bounded predictor region and a simple line model, endpoint measurements can be very efficient for slope estimation. If a quadratic term is scientifically plausible, endpoints alone cannot distinguish its effect from the intercept when squared predictor values are constant. A design optimized for a misspecified model can therefore be precise about an inadequate representation. Reserve opportunities to assess that representation.

Randomizing run order helps prevent treatment settings from being systematically aligned with uncontrolled time trends. Blocking can account for known nuisance groups, while replication provides information about variability. Center points or additional factor levels can expose curvature that endpoint-only designs conceal. These choices should be planned alongside precision criteria, not treated as cosmetic adjustments after an optimizer has consumed every run. Randomization supports a design-based argument but does not automatically make measurement errors Gaussian or remove every form of interference.

Computer searches over a finite candidate set often return a good design rather than a proved global optimum. Record the candidate set, restrictions, model terms, optimization criterion, and any optimality certificate or approximation claim. For nonlinear models or GLMs, information may depend on unknown coefficients, making a locally optimal design conditional on a working parameter value. Sensitivity analysis, sequential updating, or robust design criteria can address that dependence, but the original local optimum should not be advertised as universally optimal.