# 2K Factorial DOE

## Summary

Note: When you insert this tool into the Roadmap, you can use this form to record the data analysis from your experiment. Use the DOE Planning Worksheet form to help you design the experiment.

Provides a cost-effective methodology for conducting controlled experiments (DOEs) where all of the factors (process inputs) are held at one of two levels (settings) during each run of the experiment (plus optional center points).

The two types of 2K factorial DOEs are the 2K full-factorial and the 2K fractional-factorial:
• 2K full-factorial DOE - The experiment uses all possible combinations of factor settings with 8 runs for 3 factors, 16 runs for 4 factors, 32 runs for 5 factors, and so on. The goals of this type of experiment are usually focused on developing a full predictive model (Y = f(X)) describing how the process inputs jointly affect the process output and determining the optimal settings of the inputs.
• 2K fractional-factorial DOE - The experiment uses a fraction (one-half, one-fourth, and so on) of all possible combinations of factor settings, with a smaller number of runs than the 2K full-factorial DOE. The goals of this type of experiment can vary, from eliminating factors to developing a full predictive model (Y = f(X)) describing how the process inputs jointly affect the process output and determining the optimal settings of the inputs.
• Which process inputs (factors) have the largest effects on the process output (which inputs are the key inputs)?
• Do any important interactions between factors exist?
• Is the current testing space near an optimal condition for the process output?
• If no, what direction do you need to move to get closer to the area where the optimal condition can be found?
• If yes, what settings of the key inputs will result in the optimal process output?
• What is the equation (Y = f(X)) relating the process output to the levels of the factors?
• If I change a factor from its low setting to its high setting, how much will the process output change?
• How much of the variation in the process output can be explained by varying the process inputs?
When to Use Purpose
Mid-project Low resolution (III or IV) 2K fractional-factorial DOEs can be used as an early screening tool to perform a first-pass elimination of noncritical inputs, especially when you have many inputs (for example, more than five) and cost or time is a significant issue.
Mid-project You can use 2K full-factorial DOEs (especially for 3 or 4 factors) and resolution V or higher 2K factorial DOEs (for 5 or more factors) to model 2-way interactions and determine the settings for the key variables that result in the optimal process output.
Mid-project If all factors are numeric and no significant curvature is present, these designs can be used to determine the direction in which to continue experimenting (to locate an area closer to the optimal solution).
Mid-project If all factors are continuous and significant curvature is present, you can expand the 2K full-factorial DOE and resolution V or higher 2K fractional-factorial DOEs to allow the fitting of a quadratic model (3-dimensional modeling using central-composite designs) to find optimal settings.

### Data

Continuous Y, categorical X's or numeric X's tested at two discrete levels.

## How-To

1. State your factors (typically less than eight factors) and their levels of interest (only two levels plus an optional center point allowed).
2. If you are using a 2K fractional-factorial DOE, determine your fraction (one-half, one-fourth, and so on) based on your budget and desired resolution.
3. Verify that the measurement systems for the Y data and the inputs (factors) are adequate.
4. Develop a data-collection strategy (who should collect the data, as well as where and when; how many data values are needed; the preciseness of the data; how to record the data, and so on).
5. Run the experiment and reduce to a final model by eliminating terms with high p-values (typically greater than 0.05). Note: Eliminate terms in order with the more complex terms evaluated and eliminated first. For example, eliminate all nonsignificant 3-factor interactions before evaluating 2-factor interactions.
6. Use either the response optimizer or the main effects and interactions plots to determine optimal settings of significant factors.
7. Generate the prediction equation.

## Guidelines

• First, you should decide whether you want to run a full-factorial or fractional-factorial DOE.
• If the number of factors is less than 5, run the 2K full-factorial DOEs because they allow for modeling all 2-factor interactions with only 8 (3 factors) or 16 (4 factors) runs.
• If the number of factors is 5 or more, run the resolution V or higher 2K fractional-factorial DOEs because they reduce the number of runs while still allowing you to model all 2-factor interactions.
• Second, develop a sound data-collection strategy to ensure that your conclusions are based on truly representative data.
• Whenever possible, do the runs in the experiment in random order to prevent confusing a factor effect with the effect of an untested factor (sometimes called a lurking variable).
• All 2K factorial DOEs (full and fractional) rely on the assumption that the effects of the factors on the response are reasonably linear (can be modeled adequately with a straight line) in the inference space. You should include center points in your 2K factorial DOE whenever you doubt the linearity of the effects. The center points produce a test for curvature; in other words, they test the assumption of linearity. If the curvature is statistically significant, you must still decide, from a practical standpoint, whether the amount of curvature present is of concern.
• When adding center points to the DOE, the following procedures are often recommended:
• Use the current process factor settings as the center point to give the operators running the experiment a comfort level with familiar factor settings.
• Do not fully randomize the center points in the DOE. Instead, put one or two center points at the start of the experiment, one or two in the middle, and one or two at the end. This placement provides a check for trends during the experiment.
• The residuals of the final model must be independent, reasonably normal, and have reasonably equal variance. The residuals are usually analyzed by a histogram, normal probability plot, and plots of residuals versus fits and residuals versus order. You can create these plots simultaneously using the “four-in-one” option. Note: Due to the small size of many DOEs you may find it difficult to check these assumptions.
• If you must evaluate any factor at more than two levels, you must use the general full-factorial DOE.
• Do not extrapolate beyond your inference space.
• You can expand the 2K full-factorial DOE and the resolution V or higher 2K fractional-factorial DOE easily and use it as the basis for a 3-dimensional DOE using central composite designs.
• Check for possible outliers in the table of unusual observations in Minitab's Session window.
• While this discussion focuses on designed experiments created by Minitab, you can use the "Analyze" portion of the factorial DOE to analyze any numeric experimental data (for example, 2 factors at each of 10 levels). To do this analysis, enter the Y and X data in Minitab and then using Stat > DOE > Factorial > Define Custom Factorial Design to define the factors. You can then analyze this newly defined custom design in the usual manner.
• If you have discrete numeric data from which you can obtain every equally spaced value, and you have measured at least 10 possible values, your data often are evaluated as though they are continuous.
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