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Yates Analysis for Designed Experiments

NIST/SEMATECH Section 1.3.5.18 Yates Analysis for Designed Experiments

What It Is

Yates analysis is an efficient algorithm for computing all main effects and interaction effects in a two-level full factorial (2^k) experiment. It systematically applies pairwise sums and differences to the ordered response totals to extract every factor effect in a single pass.

When to Use It

Use Yates analysis when analyzing data from a 2^k factorial experiment to efficiently compute all main effects and interactions without constructing the full ANOVA table. The algorithm systematically organizes the calculations through k column operations on the 2^k response totals, providing effect estimates that can be directly compared to assess factor importance. It is a classical tool in industrial experimentation and process optimization.

How to Interpret

After applying the Yates algorithm, the resulting contrasts are ordered by the standard Yates convention: grand mean, factor A, factor B, AB interaction, factor C, AC, BC, ABC, and so on. The magnitude of each effect estimate indicates the practical importance of that factor or interaction. Effects can be ranked and plotted on a normal probability plot of effects (Daniel plot) to identify which are statistically significant -- significant effects will deviate from the line formed by the inactive effects. The sum of squares for each effect can be used to construct an ANOVA table and compute F-statistics.

Assumptions and Limitations

Yates analysis assumes a complete 2^k factorial design with a fixed number of replicates per treatment combination. It requires that the factors have exactly two levels (low and high). For fractional factorials or designs with more than two levels, modified approaches are needed. The standard analysis assumes normally distributed errors with constant variance across all treatment combinations.

Reference: NIST/SEMATECH e-Handbook, Section 1.3.5.18

Formulas

Yates Algorithm (Column Operation)

yi′=y2i−1+y2ii=1,…,2k−1 (first half)yi+2k−1′=y2i−y2i−1i=1,…,2k−1 (second half)\begin{aligned} y_i' &= y_{2i-1} + y_{2i} \quad &i = 1, \ldots, 2^{k-1} \text{ (first half)} \\ y_{i+2^{k-1}}' &= y_{2i} - y_{2i-1} \quad &i = 1, \ldots, 2^{k-1} \text{ (second half)} \end{aligned}

At each of k column steps, the responses are grouped into adjacent pairs. The first half of the new column contains pairwise sums and the second half contains pairwise differences (second minus first). After k steps, the first entry is the grand total.

Effect Estimate

Effect=Contrastn⋅2k−1\text{Effect} = \frac{\text{Contrast}}{n \cdot 2^{k-1}}

Each contrast from the Yates algorithm is divided by the number of replicates times 2^(k-1) to obtain the estimated effect, representing the average change in response when that factor moves from its low to high level.

Sum of Squares for an Effect

SS=(Contrast)2n⋅2kSS = \frac{(\text{Contrast})^2}{n \cdot 2^k}

The sum of squares for each factor or interaction, used to assess statistical significance via the F-test.