QE Capability Enhancement (17) | Partial Factorial Experiments: Finding the Main Causes with the Fewest Trials

1. Why the Same Data Leads to Different Conclusions in Confirmation Experiments

A manufacturing company was working on improving the dimensional variation of injection-molded parts. Seven candidate factors were identified: material temperature, mold temperature, holding pressure, holding time, injection speed, cooling time, and moisture content of the raw material. The production line only allowed a two-day window, and a full factorial design of 128 trials was impractical. The quality engineer (QE) chose a 16-trial partial factorial design (seven factors, third-order aliasing, resolution IV), with random order execution and a center point inserted between every two trials.

The analysis of variance (ANOVA) results were very "clean": the p-values for material temperature and holding pressure were both less than 0.01, and their effect sizes were the top two. Based on this, the report recommended "increasing the material temperature by 8°C and the holding pressure by 6MPa." However, after two weeks of mass production, the mean dimension only moved half of the expected amount, and the variation remained unchanged.

Upon reviewing the data, two issues were identified. First, in this design, the main effect of "material temperature" was aliased with the third-order interaction "mold temperature × injection speed × cooling time," making it statistically impossible to distinguish which one was actually causing the effect. Second, the true second-order interaction "holding pressure × cooling time" was aliased with another second-order interaction, and their effects canceled each other out, so it did not appear in the results. The data was not wrong; the design itself had bundled the information the QE wanted to see with the information they did not want to see, and the report did not mention any "unresolved" effects.

2. Principle: Confounding is Not an Accident, but a Cost That Has Been Priced

The approach in partial factorial design is to conduct only 2^(k-p) trials out of the 2^k possible level combinations. The p factors that are not tested are not deleted but are written as higher-order interactions using "generators," forming what is known as the alias structure. For example, in a 16-trial design with seven factors, selecting half of the combinations requires accepting certain conventions (such as D=ABC, E=AB, F=AC, G=BC). These generators are multiplied to form the defining relation I=ABCD=..., and each main effect or interaction can find its "shadow term" in the defining relation, with identical values that cannot be separated.

Resolution is the measure of this cost: the word length of the shortest generator in the defining relation is the resolution of the design.

• Resolution III: Main effects are confounded with second-order interactions. It can only answer "whether this factor has an effect," and it must accept that "a second-order interaction may masquerade as a main effect."
• Resolution IV: Main effects are clear, confounded only with third-order and higher interactions, which are typically considered negligible in engineering; however, second-order interactions are confounded with each other and cannot be estimated separately.
• Resolution V: Both main effects and second-order interactions can be estimated clearly, suitable for situations where combination recommendations are needed after screening.

Whether a design can be used depends on two things: whether the resolution covers the conclusions you want to draw, and whether the number of trials leaves enough degrees of freedom for error.

The underlying support is the principle of effect sparsity: in real systems, usually only a few effects are significant, and most third-order and higher interactions can be ignored. Partial factorial design is a bet on this empirical rule—using higher-order interactions as "stand-ins" to exponentially reduce the number of trials. This bet comes with a cost, and the cost must be written into the design and the report.

3. Five Practical Steps: From Selecting a Design to Resolving Confounding

Step 1: Reduce the Candidate Factors to 6-15 and Fix Two Levels. Use a cause-and-effect matrix or fishbone diagram to score and reduce the factors, ensuring no more than 15 factors enter the experiment. Criteria: the distance between two levels should be ≥ 3 times the equipment resolution; otherwise, the set values themselves cannot be distinguished. Levels should be within the range achievable on-site, not at the equipment limits. If more than 15 factors are identified, do not force a 2^(k-p) design; instead, use Plackett-Burman or D-optimal designs for the first cut, focusing only on screening main effects.

Step 2: Select the Resolution Based on the Conclusions You Want to Draw, Not by Habit. Common combinations are as follows:

Criteria: if you only want to lock down 2-3 key factors and plan to conduct optimization experiments later, a resolution of R≥IV is sufficient; if you need to provide factor combination recommendations in the screening stage, you must choose R≥V, or plan a fold-over design for R=IV results. In any case, do not accept a design with a resolution lower than III.

Step 3: Write the Alias Chains into the Experimental Plan Beforehand. List the alias terms for each main effect and label them as "resolvable" or "unresolvable." Criteria: "unresolvable" effects in the report must not be written as "insignificant"; effects on the same alias chain must not be attributed to a single factor. This step should be done beforehand, at a cost of about half an hour; doing it afterward results in a flawed report.

Step 4: Schedule the Trials, Collect Data, and Ensure Sufficient Error Degrees of Freedom. The rules for randomization, blocking, and center points are the same as in full factorial designs. Criteria: at least 3 center points (3-4 recommended for a 16-trial design with seven factors), pure error degrees of freedom ≥ 3, aiming for ≥ 6; statistical power for the target effect size (generally 1.5-2 times the process standard deviation or an improvement of 10% in tolerance) should be ≥ 0.8; if power is insufficient, add replicates, do not explain "not detected" after the fact. The response should first pass the measurement system: GR&R (P/TV) ≤ 10% can be used directly for screening, 10%-30% only for identifying large effects, and > 30% requires repairing the measuring instrument.

Step 5: Analyze According to the Criteria and Resolve Confounding According to the Criteria. The analysis order is: first use the principle of effect sparsity and half-normal plots, normal probability plots for initial screening; then use Lenth's PSE method for robust pseudo-standard error estimation (especially suitable for screening designs with no or few replicates); finally, confirm the few significant effects with ANOVA. Significant criteria must meet all three conditions: p < 0.05 (0.01 for key characteristics), effect size ≥ 1.5 times the measurement system uncertainty, and effect size reaching the engineering significance threshold (e.g., ≥ 10% tolerance).

Resolving confounding is determined by the observed results:

• If the largest second-order interaction effect size is ≥ half of the largest main effect, or if the alias chain of the main effect shows anomalies—conduct a fold-over design: add the same number of trials as the first round, mirroring the initial design, to increase the resolution from IV to V or higher, directly resolving the second-order interactions. A fold-over for an eight-trial design requires only eight more trials, much cheaper than starting over.
• If multiple factors are deemed insignificant—use projection: select a subset from the retained significant factors, and the original design automatically degrades to a higher resolution or full factorial, allowing the estimation of interactions without additional trials.
• If the center point mean significantly differs from the corner point mean (p < 0.05)—the response surface is already curved, and two levels are insufficient; switch to a response surface design.
• Finally, conduct 3-5 confirmation trials with the preferred combination, and the measured mean must fall within the 95% prediction interval of the model; otherwise, return to Step 2 to reselect the design.

4. Five Common Misunderstandings

Misunderstanding 1: Reading "Unresolvable" as "No Effect." In designs with resolution III or IV, an effect being insignificant may be due to it being offset by other effects on the same alias chain, not because it does not exist. The report must distinguish between "insignificant" and "unresolvable."

Misunderstanding 2: Using an Eight-Trial Resolution III Design to Provide Parameter Combination Recommendations. In such designs, any "significant main effect" could be a second-order interaction masquerading as a main effect. Setting parameters based on this is equivalent to treating the alias as a causal relationship.

Misunderstanding 3: Starting Without Center Points or Replicates. When the error degrees of freedom are zero, the half-normal plot, p-values, and effect size ranking lose their reference points. Three center points are the minimum configuration for screening designs.

Misunderstanding 4: Using Plackett-Burman Design to Estimate Interactions. PB design has a resolution of III, with main effects confounded with many second-order interactions. It is only suitable for the first cut, reducing the number of factors from a dozen to four or five, not for finalizing the design.

Misunderstanding 5: Jumping to Production Parameters Directly from Screening Conclusions. Screening only answers "whether this factor is worth keeping"; the optimal point still requires confirmation with a full factorial or response surface design. Skipping the confirmation trials is treating screening as verification.

5. Self-Check List

• Candidate factors have been reduced to 6-15, with a level distance ≥ 3 times the equipment resolution, and levels taken from the on-site achievable range.
• The resolution matches the conclusions you want to draw: screen main effects with R≥IV; provide combination recommendations with R≥V, or a fold-over design has been planned.
• The defining relation and alias terms for each main effect have been written into the experimental plan; "unresolvable" effects in the report are not written as "insignificant."
• Center points ≥ 3, pure error degrees of freedom ≥ 3; power for the target effect size ≥ 0.8; randomization of order, and blocking if necessary.
• If the largest second-order interaction is ≥ half of the largest main effect, fold-over or projection has been used to resolve confounding, and 3-5 confirmation trials have been completed.

Screening saves the number of trials, but confounding is the bill.

Knowledge code: 6.4.1

Version: v20260927

Author: QTank QTank is dedicated to providing systematic professional knowledge, methodologies, and practical tools for quality management practitioners, helping companies continuously improve their quality capabilities.