A-Optimal Design Calculator

Enter design inputs

Use a symmetric positive definite information matrix. Separate values with commas, spaces, or semicolons, with one row per line.

Number of parameters (p)

Number of runs (n)

Error variance (σ²)

Benchmark trace(M^-1)

Information matrix M = X'X

Example for p = 3: 16, 4, 2 on the first line; 4, 12, 1 on the second; 2, 1, 10 on the third.

Example data table

Use this sample information matrix to test the calculator and confirm the output structure.

Parameter	P1	P2	P3
P1	16	4	2
P2	4	12	1
P3	2	1	10

Example runs

16

Balanced run count for a small demonstration.

Example σ²

1.000000

Simple variance scaling for easy checking.

Benchmark trace

0.300000

Optional comparison value for efficiency percent.

Formula used

Information matrix: M = X'X

A-criterion: A = trace(M^-1)

Scaled A-criterion: A_σ = σ² × trace(M^-1)

Average parameter variance: \(\bar{V}\) = σ² × trace(M^-1) / p

A-score: S_A = p / trace(M^-1)

A-score per run: S_run = p / (n × trace(M^-1))

Benchmark efficiency: E_A = (trace(M^-1)_benchmark / trace(M^-1)_current) × 100

Condition number: κ(M) = λ_max / λ_min

A-optimality seeks to minimize the average variance of parameter estimates. Lower trace(M^-1) means the design distributes information more efficiently across coefficients.

How to use this calculator

Enter the number of parameters represented by your information matrix.
Enter the number of experimental runs used to generate the design.
Provide the model error variance σ² to scale coefficient uncertainty.
Optionally enter a benchmark trace(M^-1) from another design.
Paste the symmetric information matrix, one row per line.
Click Calculate A-optimal design to show results above the form.
Review the A-criterion, A-score, covariance matrix, determinant, and conditioning.
Use the CSV and PDF buttons to save the current result block.

FAQs

1. What does A-optimality measure?

A-optimality measures the average variance of parameter estimates. It uses trace(M^-1) from the information matrix. Smaller values indicate a design that estimates coefficients more precisely on average.

2. Why must the matrix be symmetric?

The information matrix X'X is symmetric by construction. Symmetry is necessary for stable eigenvalue analysis and for interpreting the matrix as a valid representation of experimental information.

3. Why must eigenvalues be positive?

Positive eigenvalues indicate a positive definite matrix. That means the design contains enough information to estimate all parameters and the inverse exists for variance calculations.

4. What does the condition number tell me?

The condition number compares the largest and smallest eigenvalues. Large values suggest unstable estimation, strong parameter dependence, or weak information directions in the design.

5. What is the benchmark efficiency percentage?

It compares your current design against a reference trace(M^-1). Values above 100% mean your current design has a smaller trace and therefore better A-efficiency than the benchmark.

6. Why include the number of runs?

Run count supports fair comparison between designs of different sizes. The calculator reports an A-score per run, helping you judge information efficiency relative to experimentation cost.

7. Can I use this for blocked or constrained designs?

Yes, if you can produce the corresponding information matrix. The calculator evaluates the matrix directly, so blocked, weighted, or constrained designs can still be compared consistently.

8. What does the covariance matrix section show?

It reports σ²M^-1, which contains the estimated coefficient variances on the diagonal and covariances off the diagonal. It helps assess uncertainty and parameter correlation structure.