Partial Factorial Design in Practice: 16 Trials to Identify Key Factors — A Case Study of Porosity Rate Improvement in a Die-Casting Company

When process issues involve many candidate factors and the cost of experiments is high, quality professionals often find themselves in a dilemma: full factorial experiments are too expensive, and single-factor trial-and-error is unreliable. Partial factorial design is the solution to this dilemma — it uses a carefully selected subset of experiments to quickly screen out the truly critical influencing factors, making it one of the most cost-effective tools in the Six Sigma DMAIC improvement phase. Today, we will use a real case from an aluminum alloy die-casting company to fully reconstruct the process from 8 candidate factors to 3 key factors.

1. Why Partial Factorial Design is Needed

Let's start with a simple arithmetic problem. Two factors, each with two levels, require only 4 full factorial trials; three factors require 8 trials; and eight factors require 256 trials. In reality, each trial combination often requires repeated sampling, along with time for mold changes, temperature adjustments, and debugging. Under the pressure of mass production scheduling, 256 trials are almost impossible to complete. Therefore, many companies revert to the "single-factor rotation method": changing one factor at a time while keeping others constant. This method seems convenient but has two fatal flaws: first, it misses interactions between factors — Factor A may have no effect on its own but can amplify the effect of Factor B, which the rotation method cannot detect; second, the conclusions can be easily contaminated by noise such as the order of trials and environmental drift.

The approach of partial factorial design is fundamentally different: it does not perform all combinations but only a carefully selected "representative subset." For example, with eight factors, a full factorial design would require 256 trials, but a partial factorial design can achieve this with just 16 trials, reducing the number of trials to one-sixteenth of the original. The trade-off is that effects may become "aliased" — certain effect estimates can be confused with each other. In engineering, this confusion is managed using the concept of resolution. In a resolution III design, main effects are aliased with second-order interactions, making it suitable only for pure screening of main effects. In a resolution IV design, main effects are only aliased with third-order and higher interactions (which are typically negligible in engineering), but second-order interactions may be pairwise aliased. In a resolution V or higher design, both main effects and second-order interactions can be clearly estimated. The most commonly used design in the screening phase is resolution IV: it is sufficient, requires fewer trials, and yields reliable conclusions.

This approach is underpinned by the 80/20 rule: the majority of process output variations are often dominated by a few key factors, while the contributions of the remaining factors are minimal. Since the goal is to "screen," it is not necessary to estimate every effect clearly. First, capture the major contributors, then conduct more detailed experiments on the few key factors. Partial factorial design, in the Six Sigma DMAIC improvement phase, is consistent with the cause-and-effect matrix and failure mode analysis: the former uses experience to narrow down the suspect range, while the latter uses experiments to verify cause-and-effect relationships. One relies on the mind, the other on data, and both are indispensable.

2. Case Background: Excessive Porosity Rate, Eight Candidate Factors on the Table

An aluminum alloy die-casting company supplies motor housing for new energy vehicles. After delivery to the customer, frequent leaks caused by internal porosity were discovered during the machining process, leading to continuous customer complaints. X-ray sampling inspections showed that the average porosity rate in the housing was as high as 8.2%, and the leakage scrap rate after machining was 3.5%. The direct losses, combined with the indirect costs of rework, production changes, and complaint handling, compelled management to list this issue as a key project for the year.

The project team, consisting of process, mold, equipment, quality, and front-line operation experts, conducted a brainstorming session based on the 5M1E (man, machine, material, method, environment) approach, listing 12 candidate factors. To prevent the experiment scale from getting out of control, the team first used a cause-and-effect matrix to score the factors based on their impact and controllability, narrowing down to the 8 most suspicious factors for the experiment, each with two levels:

A: Injection speed: low speed / high speed; B: Pouring temperature: 660°C / 690°C; C: Mold temperature: 180°C / 220°C; D: Injection pressure: 40MPa / 60MPa; E: Vacuum assistance: off / on; F: Coating spray amount: small amount / sufficient amount; G: Slow injection switch position: early / late; H: Holding time: 5 seconds / 8 seconds.

The response variable was set as the porosity rate: for each trial combination, 3 products were X-ray inspected, and the porosity rate was averaged. Prior to the experiment, the team conducted a measurement system analysis to ensure the consistency of X-ray readings — inaccurate measurements would render all subsequent analyses meaningless.

3. Experimental Design: How 16 Trials Represent 256

The team chose a 2^(8-4) resolution IV partial factorial design, which involves 16 trials. The key to this design lies in the "generators": 4 main factors are arranged in a full factorial manner, while the other 4 factors are generated by the interactions of the main factors, such as E=ABC, F=ABD, G=ACD, H=BCD. This way, the main effect of each generated factor is aliased with the corresponding third-order interaction, which is almost negligible in engineering; second-order interactions may be pairwise aliased, so this stage focuses on accurately screening the main effects.

Before implementing the trials, the team did two things: first, randomization — the order of the 16 combinations was randomized to avoid systematic contamination of a particular level by factors such as equipment warming and operator fatigue; second, blocking — the experiment was conducted across two shifts, with the shift as a blocking variable included in the analysis to separate shift differences from experimental errors. The environmental temperature and equipment status were recorded before each trial to ensure data traceability.

It's worth noting a detail: partial factorial experiments typically include center points to check for curvature in the response, but in this case, vacuum assistance is a qualitative factor with "on/off" levels, making it impossible to take a middle level. Therefore, the team abandoned the center points and instead used three repeated samples per combination to estimate experimental errors. There is no standard answer in design; a good design is one that fits the actual process constraints.

Why can 16 trials be trusted? The key lies in "balance": the high and low levels of each factor appear an equal number of times across all 16 trials, and the interference from other factors is designed to cancel out. This way, when calculating the main effect of a factor, the influence of other factors is "washed away" on average, leaving only the contribution of that factor. This orthogonality is the foundation of partial factorial design's confidence in drawing conclusions from a small number of trials.

4. Implementation and Data Analysis: Three Key Factors Emerge

The 16 combinations were completed in random order over two days, and the data was summarized and imported into statistical software for analysis, which proceeded in four steps.

Step 1: View the Pareto chart of standardized effects. The horizontal axis shows the absolute values of the standardized effects of each factor and interaction term, while the vertical line is the significance reference line. Factors A (injection speed), D (injection pressure), and E (vacuum assistance) significantly exceeded the reference line, forming the first tier; Factor C (mold temperature) was close to the reference line, indicating "marginal significance."

Step 2: View the normal probability plot. The effect points roughly fell on a straight line, indicating no outliers in the data and that the errors were approximately normal, confirming the validity of the model assumptions.

Step 3: Perform ANOVA. The model was overall significant, with a goodness-of-fit R² of 0.87 and a non-significant lack-of-fit term — indicating that the model did not miss any important structural information, and the 16 trials were a competent representation.

Step 4: Interpret the main effects plot. The porosity rate was significantly reduced at high injection speed, high injection pressure, vacuum assistance on, and higher mold temperature, aligning perfectly with die-casting process theory — high-speed filling reduces oxidation and inclusions, high injection pressure ensures proper compaction, and vacuum assistance expels gases from the mold cavity.

Conclusion: Factors A, D, and E are the key factors for porosity rate, and Factor C is retained as a factor to be optimized. Factors F, G, and H were not significant, which is actually good news — they can be set at the levels that are least costly and most convenient, such as a holding time of 5 seconds to shorten the cycle. The value of DOE lies not only in identifying the factors that need to be managed but also in informing us which factors can be ignored, thus concentrating management resources where they are most needed.

At the end of the analysis, the team conducted two routine checks: first, residual diagnostics to confirm that the residuals were randomly distributed and did not show a funnel shape with increasing fitted values, indicating stable errors and no missing structure; second, verifying the direction of significant effects using process knowledge — the reduction in porosity rate at high injection speed is physically plausible, and the conclusions can be confidently presented. The mutual verification of data and mechanisms is a double safeguard for the implementation of statistical conclusions.

5. Confirmation Trials and Implementation Results

Using the optimal combination — high injection speed, 60MPa injection pressure, vacuum assistance on, and mold temperature of 220°C — the team conducted five batches of confirmation trials. The results were encouraging: the average porosity rate dropped from 8.2% to 0.9% to 1.1%, and the leakage scrap rate after machining decreased from 3.5% to 0.4%. Confirmation trials are essential because there are differences between experimental conditions and mass production environments, and batch verification is the final safeguard for implementing conclusions.

Subsequently, the project team solidified the key factor parameters into work instructions: injection speed and pressure were written into the process card and included in SPC monitoring, vacuum pressure was incorporated into the daily inspection, and mold temperature was automatically controlled by the mold temperature machine with alarm limits set. Three months of tracking data showed that the porosity rate stabilized around 1%, with no rebound. In total, the project's experimental and testing costs were less than 100,000 yuan, saving approximately 1.8 million yuan annually in rework, scrap, and complaint handling costs, a highly favorable return on investment.

More importantly, the team's capabilities were enhanced: previously, they relied on experienced masters to "test parameters," but now they have learned to use a scientific path of design, data, and verification. Optimizing the mold venting channels was listed as the next improvement topic — quality improvement is never a one-time deal but a cyclical, upward spiral.

This practice also had an unexpected benefit: the methodology was replicated. The die-casting workshop standardized the "cause-and-effect matrix to screen factors + partial factorial experiments to determine key factors + confirmation trials to verify" approach, and in the following two months, the same method was used to solve two adjacent issues, shrinkage porosity and mold sticking, both of which were resolved in less than two weeks. The value of a method lies not in solving one problem but in becoming a reusable organizational capability.

6. Five Common Misconceptions

Misconception 1: Directly using full factorial when there are many factors. 256 trials are impractical for mass production scheduling, and projects often stall halfway. In the screening phase, using a resolution IV partial factorial design, 16 trials can clearly screen the main effects. Once a few key factors are identified, full factorial or response surface methods can be used for detailed optimization — this is the correct progression.

Misconception 2: Selecting a low resolution to analyze interactions. In a resolution III design, main effects are aliased with second-order interactions, making it like "opening a blind box" to interpret interactions. If subsequent confirmation of the importance of interactions is needed, a fold-over design can be used to reverse the signs and conduct another round of trials to resolve the aliasing.

Misconception 3: Ignoring randomization and blocking. Conducting trials in a convenient order allows systematic drifts such as shift changes, temperature increases, and material batches to sneak into the conclusions, leading to mistaking noise for effects. Randomization is the cornerstone of DOE and cannot be skipped.

Misconception 4: Scaling up without confirmation trials. Trials provide "small sample" conclusions, and independent batch confirmation trials must be conducted before scaling up to avoid a single failed scale-up eating up all the benefits.

Misconception 5: Focusing only on significant factors and ignoring the cost of non-significant factors. Non-significant means "any level will do," so choose the cheapest, most stable, and most time-saving levels — the hidden benefits of DOE lie precisely in these "non-managed" factors.

The essence of partial factorial design is to use statistical ingenuity to save on experimental resources. It is not about cutting corners but about spending the limited number of trials where the information content is highest: screen accurately first, then optimize. For every quality professional trapped by "too many factors, trials too expensive," the value of these 16 trials goes beyond just saving 240 trials; it transforms improvement from a "luck-based" process into a "data-driven" one.

Fewer trials, but no ambiguity in conclusions — screen accurately first, then optimize, making the most reliable decisions with the fewest trials.

Knowledge code: 6.4.1

Version: v20260806

Author: Quality Think Tank

The Quality Think Tank is dedicated to providing systematic professional knowledge, methodologies, and practical tools to quality management practitioners, helping enterprises continuously enhance their quality capabilities.