In Six Sigma’s Design of Experiments, curvature describes a condition where the relationship between a process input (X) and a process output (Y) is not linear. In a two-level factorial design, the model assumes a straight-line relationship between each X and Y.
Curvature occurs when that assumption fails — when the output at the center of the factor range does not lie on the straight line connecting the low and high settings. Detecting curvature before building a predictive model is critical, because a linear model fitted to curved data will produce biased predictions and mislead the Improve phase team.
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Curvature in Six Sigma DOE
Curvature in Design of Experiments (DOE) refers to a statistically detectable non-linear relationship between an X factor and a Y response in a factorial design.
A standard two-level factorial design can only model linear (straight-line) relationships. If the true X-Y relationship is curved, the center point of the experimental range will produce a response that does not lie on the line between the low and high outputs. Center points are added to the design specifically to test for this condition.
When the ANOVA curvature p-value is below 0.05, the test confirms that significant curvature exists and the team must switch to a Response Surface Design — typically a Central Composite Design — to model and optimize the non-linear relationship.
Key Takeaways
- Curvature in DOE describes a non-linear (curved) relationship between an X factor and Y response. Standard two-level factorial designs assume linearity. They cannot model curvature.
- Center points — experimental runs where all factors are set halfway between their low and high levels — are added to a factorial design specifically to test whether curvature is present.
- The curvature test examines whether the observed response at the center point differs significantly from the predicted response under a linear model. A significant difference signals curvature.
- According to Minitab’s official documentation, when a design has center points, the sum of squares for residual error is partitioned into: pure error, curvature, and lack-of-fit. Curvature contributes 1 degree of freedom.
- A curvature p-value below 0.05 confirms significant curvature. The linear model is insufficient. The team must move to a Response Surface Methodology (RSM) design, typically a Central Composite Design (CCD).
- A curvature p-value above 0.05 means no significant curvature was detected. The linear model is acceptable for the tested range. The team can proceed with the factorial model.
- In Minitab, the curvature test appears in the ANOVA table as a separate row labeled “Curvature” or “CT Pt” when center points are included in a factorial design analysis.
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What Is Curvature in Six Sigma DOE?
When a Six Sigma team runs a factorial experiment, the goal is to understand how X factors affect the Y response. A two-level factorial design tests each factor at two settings: a low level and a high level. The model the design produces is linear — it assumes that the relationship between each X and Y follows a straight line across the tested range.
This linearity assumption is often valid within a narrow, well-chosen experimental range. But it fails when the X-Y relationship is curved. In a curved relationship, the response at the midpoint of the factor range does not fall on the straight line connecting the low-level and high-level responses. It falls either above or below that line.
Why Factorial Designs Cannot Model Curvature
A two-level factorial design uses only two data points per factor. Two points define a straight line — and nothing more. With only low and high levels, the design has no information about what happens in between. It cannot detect whether the relationship curves upward or downward in the middle of the range.
A straight-line model fitted to curved data produces biased predictions. If the true optimum lies near the center of the factor range — which is often where optima occur in physical processes — a linear model will miss it. The team may optimize in the wrong direction, implementing settings that perform worse than the actual best conditions.
According to the Minitab blog, “factorial designs assume there’s a linear relationship between each X and Y. Therefore, if the relationship between any X and Y exhibits curvature, you shouldn’t use a factorial design because the results may mislead you.”
This is why testing for curvature before interpreting a factorial design’s results is a sound experimental practice, not an optional step.
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Center Points: The Tool for Detecting Curvature
Center points are the standard tool for detecting curvature in factorial designs. A center point is an experimental run where every factor is set to the midpoint between its low and high level.
For a temperature factor ranging from 150°C (low) to 170°C (high), the center point temperature is 160°C. For a pressure factor ranging from 100 PSI (low) to 140 PSI (high), the center point pressure is 120 PSI. When multiple factors are in the design, all factors are set to their center simultaneously in a center point run.
Center points serve two purposes in factorial DOE.
Purpose 1: Detect curvature. The response observed at the center point is compared against the response the linear model predicts for that point. If the observed value matches the linear model’s prediction, the linear model is adequate. If the observed value differs significantly from the linear prediction, curvature is present.
Purpose 2: Estimate pure error. Multiple center point replicates allow estimation of experimental error that is independent of any model assumptions. This pure error estimate supports the lack-of-fit test and gives the team a baseline variability estimate for the experimental conditions.
The Minitab blog confirms that by adding center points, practitioners “can increase the probability of detecting significant X’s, and estimate the variability (or pure error, statistically speaking).”
How Minitab Handles Curvature in ANOVA
When a factorial design includes center points and the data is analyzed in Minitab, the software partitions the residual error in the ANOVA table into distinct components. According to Minitab’s official documentation on error partitioning:
“In a DOE analysis, the sum of squares (and degrees of freedom) for residual error can be partitioned in up to three parts: pure error, curvature, and lack of fit.”
The curvature row in the ANOVA table specifically tests whether the center point response differs significantly from the response predicted by the linear factorial model. Minitab states: “If the design has any center points, you can choose to include a center point term as a parameter in the model or treat the curvature as a component of the error. In both cases, there is 1 degree of freedom associated with curvature.
The curvature sum of squares is the reduction in the sum of squares of the residual error you obtain when you add the center point term to the model.”
This means the curvature test in Minitab’s ANOVA table directly quantifies how much the center point response deviates from the linear model’s prediction. The F-statistic and p-value for that row test whether this deviation is statistically significant.
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Interpreting the Curvature P-Value
The curvature p-value is the primary decision criterion for the curvature test. It follows the same interpretation rule as all other p-values in Six Sigma hypothesis testing.
Curvature p-value < 0.05 (significant curvature detected):
The observed center point response differs significantly from the linear model’s prediction. A statistically significant non-linear component exists in the X-Y relationship. The linear factorial model is not adequate. The team must move to a Response Surface Design to model the curvature before optimizing the process.
As the Stat-Ease blog states: “If the curvature test is significant, this is considered evidence that a quadratic or higher order model is required to model the relationship between the factors and the response.”
Curvature p-value ≥ 0.05 (no significant curvature detected):
There is insufficient statistical evidence to conclude that the X-Y relationship deviates significantly from linearity within the tested range. The linear factorial model is adequate for the current design space. The team can proceed with interpreting the factorial results and making improvement decisions based on the linear model.
As Stat-Ease confirms: “If the curvature test is not significant, then it is okay to assume that the linear model fits in the middle of the design space.”
An important caution: a non-significant curvature test does not prove the relationship is perfectly linear. It means the experiment did not detect significant curvature with the power available from the current design. Small curvature effects may remain undetected if the center point sample size is too small or the experimental range is too narrow.
What to Do When Curvature Is Significant
A significant curvature p-value tells the team that the linear factorial model cannot adequately describe the X-Y relationship. The next step is to transition to a Response Surface Methodology (RSM) design, specifically one that can estimate quadratic (second-order) terms.
Central Composite Design (CCD)
The Central Composite Design is the most widely used RSM design for modeling curvature. A CCD extends a two-level factorial design by adding star points (also called axial points) at a specified distance from the center, plus center point replicates.
The star points provide the additional information needed to estimate quadratic terms. Where a two-level factorial design produces a flat plane (linear model), a CCD produces a curved surface (quadratic model) that can capture the curved relationship and identify the true optimum within the experimental region.
Lean Outside the Box confirms this directly: “Central Composite Designs are a type of DOE used to model curvature. They extend a two-level factorial or fractional factorial design. Instead of stopping at linear effects, they add points that estimate quadratic terms.”
Box-Behnken Design
The Box-Behnken Design is an alternative RSM design that avoids extreme corner combinations. It is particularly useful when corner points represent factor combinations that are physically impossible or unsafe to test. Like the CCD, it estimates quadratic effects and can model curvature.
The Decision Sequence
A good experimental strategy follows a sequential logic:
- Start with a two-level factorial design with center points to screen main effects and test for curvature.
- If curvature is not significant, interpret the factorial results and optimize using the linear model.
- If curvature is significant, augment the factorial design with star points (creating a CCD) or transition to a Box-Behnken design.
- Analyze the RSM design to build a quadratic model and find the optimal factor settings.
The JMP community documentation describes this transition clearly: “The center points are added as a hedge to tell the experimenter if they need to consider quadratic terms in later rounds of experimentation.”
Curvature and the Six Sigma Improve Phase
Curvature testing belongs to the Improve phase of DMAIC. After the Analyze phase has confirmed which X factors significantly drive the Y response, the Improve phase uses DOE to find the optimal settings for those factors.
In a typical Improve phase DOE sequence:
- The team selects the confirmed X factors and chooses the experimental range based on Analyze phase findings.
- They build a two-level factorial design with center points.
- They run the experiment, collect Y responses, and analyze the design in Minitab.
- They examine the curvature p-value in the ANOVA table.
- If curvature is not significant, they optimize using the linear factorial model.
- If curvature is significant, they transition to a Central Composite Design to model the curvature and find the true optimum.
The curvature test prevents a costly mistake: optimizing a process based on a linear model when the true relationship is curved. Without center points, the team would never discover that the optimal setting lies between their high and low levels — precisely where the linear model provides no information.
Curvature vs. Lack of Fit
These two ANOVA terms often appear together in Minitab’s factorial design output. They are related but distinct.
| Term | What It Tests | When It Appears |
| Curvature | Whether the center point response deviates significantly from the linear model | When center points are included in the design |
| Lack of Fit | Whether the model overall fails to describe the data adequately | When the design has replicates and unsaturated degrees of freedom |
Minitab’s documentation states: “The lack-of-fit degrees of freedom is found by subtracting the degrees of freedom for pure error and curvature (if appropriate) from the residual-error degrees of freedom.”
A significant curvature test and a significant lack-of-fit test both point toward model inadequacy, but for different reasons. Significant curvature specifically indicates that a quadratic term is needed. Significant lack of fit may indicate missing interaction terms, higher-order terms, or other model specification issues.
Frequently Asked Questions: Curvature in DOE
Q: What does curvature mean in Six Sigma DOE?
A: In Six Sigma Design of Experiments, curvature refers to a statistically detectable non-linear relationship between an X factor and a Y response. A two-level factorial design assumes that the X-Y relationship is linear — a straight line connecting the low-level and high-level responses. Curvature occurs when the actual response at the midpoint of the factor range differs significantly from what the linear model predicts for that point, indicating that the true relationship bends rather than follows a straight line.
Q: How do you test for curvature in a factorial DOE?
A: Curvature is tested by adding center points to the factorial design. Center points are experimental runs where all factors are set to their midpoint between low and high levels. The ANOVA table produced by Minitab includes a curvature row that tests whether the observed center point response differs significantly from the linear model’s prediction. A p-value below 0.05 in that row confirms significant curvature.
Q: What does a significant curvature p-value mean?
A: A significant curvature p-value (below 0.05) means the observed response at the center point differs significantly from what the linear factorial model predicts for that point. This confirms that the X-Y relationship is non-linear within the tested range. The linear factorial model is not adequate. The team must transition to a Response Surface Design — typically a Central Composite Design — to model the curvature and find the true process optimum.
Q: What should you do when curvature is detected?
A: When the curvature test is significant, the team should transition from the two-level factorial design to a Response Surface Methodology (RSM) design. The most common choice is a Central Composite Design (CCD), which extends the factorial design with star points to estimate quadratic terms. The CCD produces a curved model surface that can accurately describe the non-linear X-Y relationship and locate the true optimum.
Q: What is the difference between curvature and lack of fit in Minitab DOE ANOVA?
A: Both terms indicate model inadequacy, but they address different issues. Curvature specifically tests whether the center point response deviates from the linear model, indicating that a quadratic term is needed. Lack of fit tests whether the overall model fails to describe the data, which may indicate missing interactions, higher-order terms, or other specification problems. Minitab partitions the residual error into pure error, curvature, and lack of fit as separate components.
Q: Can a two-level factorial design model curvature?
A: No. A two-level factorial design has only two data points per factor (low and high). Two points define a straight line and cannot capture any curvature. The design can detect the presence of curvature through the center point test, but it cannot model or quantify the curved relationship. Modeling curvature requires a Response Surface Design such as a Central Composite Design or Box-Behnken Design, which include additional factor levels (star points) that provide the information needed to estimate quadratic terms.
Q: What is the connection between curvature and Response Surface Methodology?
A: Response Surface Methodology (RSM) is specifically designed to model curved X-Y relationships. When the curvature test in a factorial design is significant, RSM provides the next step. RSM designs — particularly the Central Composite Design — add star points to the factorial design that enable estimation of quadratic (second-order) terms. This produces a curved model surface rather than a flat plane, allowing the team to identify the true process optimum even when it lies in the middle of the experimental range rather than at one of the factor extremes.
Curvature Training in Six Sigma
Curvature testing, center point design, and the transition to Response Surface Methodology are advanced DOE topics taught at the Black Belt level. These concepts require practitioners to understand when the linear factorial model is adequate, when it fails, and what design to use when it does.
At Six Sigma Development Solutions Inc, DOE including center points, curvature testing, and Response Surface Methodology is covered as part of our Black Belt training curriculum. Practitioners learn to design experiments that detect curvature efficiently, interpret ANOVA curvature p-values correctly, and decide when to transition from a factorial design to a Central Composite Design.
We offer training in three formats:
- Onsite training — delivered at your facility, with real process DOE examples from your industry.
- Live virtual training — instructor-led sessions online with guided Minitab exercises covering factorial design, center points, and curvature analysis.
- Online training — self-paced Black Belt certification programs covering all IASSC-testable DOE content.
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