Is Yates analysis restricted to manual calculation?

No. Software can implement it, and ordinary regression can recover the same contrasts for the appropriate design.

Why do sources sometimes report estimates half as large?

They may report regression coefficients rather than conventional high-minus-low effects. Check the divisor and coding.

Can it repair a missing experimental condition?

No. Missing or aliased information requires a suitable model, assumptions or additional data.

AI Glossary / Statistics and probability / Intermediate

Yates analysis

Definition

Yates analysis is a structured calculation of factor and interaction estimates for suitable factorial experiments, especially two-level designs. Yates's algorithm repeatedly adds and subtracts responses arranged in a prescribed order. The resulting contrasts are scaled into means, effects or regression coefficients according to an explicitly stated convention.

What it means

Factorial experiments vary several inputs together so their separate and combined relationships with an outcome can be studied. A calculation that respects the design can summarize these relationships without repeatedly fitting each comparison by hand.

Yates analysis is a computational method, not a substitute for a valid experimental design. It cannot create information about omitted treatment combinations, supply randomization or remove confounding. Nor is it Yates's continuity correction for a contingency-table chi-square test. The shared name refers to different statistical procedures.

How it works

For a full two-level design, arrange responses in standard order: the first factor changes level fastest, the next changes every two rows, and so on. In each calculation column, put adjacent-pair sums in the upper half and second-minus-first differences in the lower half. Repeat for the number of factors.

The final column contains the total and the factorial contrasts in standard effect order. With one observation at each of 2ᵏ combinations, divide the total by 2ᵏ for the mean. Divide other contrasts by 2ᵏ⁻¹ for conventional high-minus-low factorial effects. Dividing by 2ᵏ instead gives coefficients for predictors coded −1 and +1. Fractional designs require explicit attention to aliasing.

Three practical examples

1. AI benchmarking: Engineers vary two fixed configuration settings at low and high levels. A factorial analysis separates average changes from whether the settings work differently in combination.

2. Manufacturing: A team varies temperature and pressure across planned combinations. Yates calculations provide a compact cross-check of estimated effects from regression software.

3. Workflow design: Researchers compare two interface factors in a balanced experiment. They examine the interaction before assuming the benefit of one feature is the same under both versions of the other feature.

Think of it this way

Think of a carefully organized set of balances that repeatedly groups and contrasts the same measurements. The arrangement ensures each final balance corresponds to a particular factor or combination. If the measurements are placed in the wrong slots, the arithmetic may still look neat, but the labels on the results will no longer mean what you think they mean.

Worked example

Consider an original two-factor experiment with responses in standard order: 10 at A−B−, 14 at A+B−, 12 at A−B+, and 20 at A+B+. The first calculation column is 24, 32, 4 and 8: the two adjacent sums followed by the two adjacent differences.

Repeating the operation gives 56, 12, 8 and 4. The grand mean is 56/4 = 14. Conventional effects are A = 12/2 = 6, B = 8/2 = 4 and AB = 4/2 = 2. The corresponding −1/+1 regression coefficients are three, two and one, giving ŷ = 14 + 3A + 2B + AB.

As a check, at A = B = +1 this predicts 14 + 3 + 2 + 1 = 20. The A change is four at B− and eight at B+, a difference of four between those changes; the conventional AB effect is half that difference. With only these four observations and all four coefficients fitted, no residual degrees of freedom remain for an ordinary error estimate. The calculations alone therefore do not establish statistical significance.

Illustrative case study

In this fictional case, Pine Compute compares two binary system settings across all four combinations. The team randomizes actual run order but reorders the recorded results into standard order for the Yates calculation. A second analyst checks that treatment labels remain correctly attached.

The interaction estimate suggests that one setting's performance change depends on the other. Engineers plan replicated runs to estimate uncertainty and assess reproducibility. They retain the initial results as a screening exercise rather than reporting an exact fit to four observations as proof of a reliable production improvement.

Common misconceptions and mistakes

Standard calculation order is not a recommendation to conduct experiments in that order. Randomizing execution can help protect comparisons from time-related disturbances. Preserve the link between each response and its treatment combination when reordering.

Do not confuse a factorial effect with its −1/+1 regression coefficient; the usual factor-of-two difference changes numerical interpretation. In fractional designs, an estimated contrast may combine aliased effects. The algorithm cannot separate them without additional information or assumptions. An unreplicated saturated fit has no ordinary residual error estimate, so large numerical effects do not automatically come with valid p-values.

Business use

Yates analysis can make small factorial experiments easier to audit and explain. The business value is recognizing interactions and understanding which configuration changes are supported by the design, rather than choosing settings from isolated one-factor comparisons.

Document response units, factor levels, actual run order, calculation order and effect scaling. Use replication or a defensible error-estimation strategy where inference is needed. If results guide deployment of an AI configuration, also test the selected setting on representative workloads; an effect estimated in a narrow experiment may not transfer to a different task mix.

Check your understanding

What is the A effect in the example?

Answer: Six response units, averaging the change from low to high A across the two B levels.

Why is the AB coefficient one while the AB effect is two?

Answer: Under −1/+1 coding, the conventional effect is twice the regression coefficient.

Does fitting all four cell values exactly establish reliability?

Answer: No. The saturated calculation leaves no residual degrees of freedom for ordinary error estimation.

Related terms

Factorial design
Main effect
Contrast
Aliasing (factorial experiments)
Replication (statistics)
standard order
interaction
aliasing
regression coding
replication

Learn next

Next, reproduce the two-factor calculation and verify it with a coded regression. Extend it to three factors, carefully tracking effect labels. Then examine fractional designs and how deliberate aliasing changes which scientific questions can be answered.

Sources

NIST's “Yates Algorithm” explains standard ordering and repeated sums and differences: https://www.itl.nist.gov/div898/handbook/eda/section3/eda35i.htm

Russell Lenth's unrepx documentation explicitly distinguishes conventional factorial effects from −1/+1 regression coefficients: https://search.r-project.org/CRAN/refmans/unrepx/html/yates.html

The four-response calculation and Pine Compute case are original illustrations. Both scaling conventions are stated because the NIST worked table reports coefficient-scaled estimates.

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