Definition of Full Factorial DOE:

DOE, or Design of Experiments, is a method of designed experimentation where you manipulate the controllable factors (independent variables or inputs) in your process at different levels to see their effect on some response variable (dependent variable or output).

This article will explore the different approaches to DOE with a specific focus on the full factorial design. We will discuss the benefits of using the full factorial design and offer some best practices for a successful experiment.

## Overview: What is a full factorial DOE?

As stated above, a full factorial DOE design is one of several approaches to designing and carrying out an experiment to determine the effect that various levels of your inputs will have on your outputs. The purpose of the DOE is to determine at what levels of the inputs will you optimize your outputs. For example, if your output is a thickness of coating to be applied to a metal sheet, and your primary process variables are speed, temperature, and viscosity of the coating, then what combination of speed, temperature, and viscosity should you use to get an optimal and consistent thickness on the metal sheet?

With three variables, speed, temperature, and viscosity, how many different unique combinations would we have to try to fully examine all the possibilities? Which combination of speed, temperature and viscosity will give us the best coating thickness? The experimentation using all possible factor combinations is called a full factorial design, and the minimum number of experiments you would have to do is called *Runs*.

We can calculate the total number of unique factor combinations with the simple formula of *# Runs=2^k, *where *k* is the number of variables and *2 *is the number of levels, such as (high/low) or (400 rpm/800 rpm). In our coating example, we would call this design a *2 level, 3 factor full factorial DOE*. The number of runs would then be calculated as 2^3, or 2x2x2, which equals 8 total runs.

There are other designs that you can use such as a fractional factorial, which uses only a fraction of the total runs. That fraction can be one-half, one-quarter, one-eighth, and so forth depending on the number of factors or variables.

## 3 benefits of doing a full factorial DOE

Doing a full factorial as opposed to a fractional factorial or other screening design has a number of benefits.

### 1. You can determine main effects

Main effects describe the impact of each individual factor on the output or response variable. In our example, one of the main effects would be the impact or change in the coating thickness that would be attributable to speed alone if we changed from a run speed of 400 rpm to a speed of 800 rpm.

### 2. You can determine the effects of interactions on the response variable

An interaction occurs when the effect of one factor on the response depends on the setting of another factor. For our example, if we ran at a speed of 800 rpm, what temperature should we run at to optimize our coating thickness? On the other hand, what temperature should we run at if the speed is 400 rpm?

### 3. The optimal settings for the independent variables can be estimated

The final full factorial analysis will tell us what setting or levels of our speed, temperature, and viscosity we should use to optimize our coating thickness.

Our example answer might look like this: Run the machine at 400 rpm, a temperature of 350 degrees using a coating viscosity of 6,000 cps.

## Why is a full factorial DOE important to understand?

Using intuition to set the optimal settings of your process variables or factors is insufficient if the goal is to understand the impact of the factors on your output. Using trial and error may miss the important combinations, or optimal combination, and you might end up with a less-than-optimized process or product.

### You need to know the full effect of your variables on the process

Your process variables have different impacts on your output. You need the whole picture.

### The world is not just made up of main effects

As explained earlier, main effects are the individual impacts of each factor on the output. But the world is more complex than that, and most outcomes are a function of interactions, not just main effects.

### State your conclusions with statistical certainty

You can determine main effects, interactions, and other outcomes of a full factorial DOE using statistics, so decisions are based on statistical significance rather than hunches or “seat of the pants” conclusions.

## An industry example of a full factorial DOE

A beverage manufacturer wanted to reduce the amount of overfilled bottles on its manufacturing line. Company leadership felt that the major factors in the process were the run speed of the machine, size of bottle, type of product, and degree of carbonation. A 2-level, 4-factor full factorial experiment was selected. This would require 16 runs.

The company’s Master Black Belt designed the DOE and ran it on a preselected machine. The experiment was restricted to a single machine to block out any impact that might be attributable to machine differences.

Each run consisted of 100 bottles, and they took the average of the fill level of those 100 to use in the calculations. Running a single bottle would have been impractical. They determined that, after doing the calculations and analysis all four factors were statistically significant — and that there were interactions between speed and carbonation. The optimal settings suggested by the experiment were used in a confirmatory run to see if the changes actually improved fill level.

It did, and they replicated the new settings on the rest of the machines.

## 3 best practices when thinking about a full factorial DOE

Running a DOE needs planning and discipline. The results of your experiment can become contaminated and untrustworthy if you don’t take care.

### 1. Clearly define your factors and desired outcomes

Know what factors are likely to be the most important and how you are going to measure them.

### 2. Minimize the “noise”

Since you’re only interested in the impact of your chosen factors on the response, remove any other factors than might contaminate your experiments. Control and minimize any other factors around you so that they don’t inadvertently affect your experiment.

### 3. Use screening experiments if appropriate

If you have a large number of possible factors, you will be doing a large number of runs that can get costly and time-consuming. To help screen out the factors that are not really important, use appropriate screening experiments such as a fractional factorial.

## Frequently Asked Questions (FAQ) about full factorial DOE

** How do you calculate the number of runs or experiments to do for a full factorial DOE?**

Use the simple formula *# Runs=X^K**, *where X is the number of levels or settings, and K is the number of variables for factors.

** What are the main effects of a DOE?**

They are the specific impact of a single factor on the response variable. They are determined by calculating the difference in response when a factor is run at different levels or settings.

** Are Interactions in a Full Factorial DOE important? **

Yes, they are very important. If interactions exist between the factors or variables in your process, you’ll want to understand them so you can optimize your settings based on the crossed impact of your factors.

## In summary: Full factorial DOE

Designed experiments in general and a full factorial DOE design in particular, are powerful statistical tools to understand your process and optimize your output. You must take care to do the DOE with planning and discipline so the results are meaningful.

In a full factorial DOE, you will identify the appropriate output that you want to improve and the factors or variables that you believe impact that output. Once you’ve identified the factors, determine the levels or settings you’d like to explore and the number of unique combinations of the factors and levels.

After running your experiment, you’ll usually use a statistical software package to analyze your results. From there, you will be able to statistically determine the main effects of your factors, the interactions between the factors, and the optimal levels or settings.

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