A two-level factorial experiment efficiently compares the low and high corners. However, if the actual response curves upward or downward in the middle, a plane fit only to the corners will not capture that curvature.

The center point sets all factors to coded 0

If you ran a two-factor experiment with factors A and B at −1 and +1, the center point is (A, B)=(0, 0). This is not simply the average of the four corners, but a separate run where the actual factors are set to the middle of each range.

If the two-level plane model is correct, the average response at the center should not be much different from the midpoint predicted by the corners. If the average of several center points is systematically different, that is a signal of curvature.

Curvature contrast and pure error are different numbers

Consider the following educational synthetic function:

Y = 70 + 4A + 3B − 4(A² + B²)

With no noise, the average of the four corners is 62, and the average of the center point is 70. Therefore, center mean − corner mean = 8. This difference shows that the straight-line plane is missing the curvature in the middle.

If you repeat the center point five times independently and get slightly different values, you can estimate the pure error as the variation within the same condition. The curvature contrast is the systematic difference in location, and the pure error is the variation within the same condition.

Lack of fit looks at the remaining error relative to the pure error.

The sum of squares of the residuals can be divided into the pure error within the repeated points and the structure missed by the model.

SS_error = SS_pure error + SS_lack of fit

If there are no repeated identical conditions, it is difficult to separate these two based on the data alone. A single center point adds a location in the center but does not create degrees of freedom for pure error. It must be independent repeats, and technical repeats of the same sample should not be counted as experimental error.

In Actual vs. Predicted, the center point is considered separately.

In a synthetic response where the linear model is adequate, the center points will be near the diagonal of the Actual vs. Predicted plot. In a response with curvature, the repeated center points may cluster and fall on the same side of the diagonal. Even with a high R², this systematic deviation will not disappear.

In residual plots, we distinguish the following:

• Do only the center points fall in the same direction?
• Does a curved pattern remain at the corner points?
• Is there a drift depending on the execution order?
• Is the scatter of the repeated points larger under specific conditions?

If curvature signals are detected, reinforce the measurement locations rather than arbitrarily adding terms to the same two-level model.

In-Silico Lab: Separately control curvature and pure error

1. Set the curvature coefficient to 0 and observe the center-corner contrast.
2. Set the noise to 0 and increase only the curvature to see how the contrast changes.
3. Fix the curvature and increase the noise to distinguish the standard deviation of the center points from the contrast.
4. Explain why the pure error estimate weakens when the number of center point repetitions is reduced to 1.

The four rows of corner points and the row of center point repetitions in the Lab are synthetic independent runs. The same seed, version, and input will produce the same values, and in actual experiments, execution order, block, and experimental unit are additionally required.

Do not choose only one table from the JMP output

The Lack of Fit table is read together with the degrees of freedom for pure error. Actual by Predicted shows the direction of the center-point cluster, and residual plots are used to check for patterns by predicted value, order, and factors. Curvature in Surface or Profiler is an expression derived from the current model, not an identification of its cause.

Choices after the center point

• If the curvature signal is small and the residuals are random, the first-order model within the current range can be used with caution.
• If the curvature signal is large, move to a design that estimates the second-order terms, such as the axial point of CCD or Box-Behnken.
• If the pure error is large, do not pursue the optimal point without first reducing the process, measurement, or block causes.
• If the variance of the center point and the corner point differs significantly, check the assumption of homoscedasticity and the transformation/model structure.

Example of Result Statement

We compared four corner points and five independent center points. The average of the center points was 3.1 units higher than the predicted value at the center of the two-level plane, and the standard deviation of the repeated center points was 0.42. This result supports the possibility of average curvature within the current range, but it does not identify individual second-order terms, so we planned a subsequent response surface design.

Concluding the Section

• The center point is an actual run where all factors are set at the center of the range.
• The center-corner difference is a signal of average curvature.
• Repeated center points allow for the estimation of pure error.
• Lack of fit looks at the model discrepancy remaining beyond the pure error.
• Individual second-order terms or the optimal point cannot be determined using only center points.

In the next section, we will examine where CCD and Box-Behnken place new measurement points after detecting curvature.

Official Supplementary Material

• NIST/SEMATECH · Response surface model example
• NIST/SEMATECH · Lack of fit
• JMP Help · Lack of Fit

This article and the Lab are educational synthetic materials and should not be used as evidence for actual research, process, quality, or regulatory decisions.