# Full Factorial Design

A full factorial design is a systematic method for investigating the effects and possible interactions of several factors on a response variable. Unlike a fractional factorial design, a full factorial design includes all possible combinations of factor levels.

The goal is to study how various input variables affect an output variable. The input variables in the system are called factors, and the output variable is referred to as the response or target variable.

Factors are the variables believed to influence the target variable. Each factor has at least two levels or settings. These levels are the specific values a factor can take. The aim is to determine if changing the factor levels has a significant impact on the target variable.

### Advantages of Full Factorial Design

**Comprehensive Analysis:** By testing all possible combinations, you get a
complete picture of how the factors influence the outcome.

**Interactions:** A full factorial design allows for the identification of
interactions between factors. This means you can not only measure the isolated effect
of each factor but also study how factors work together to influence the outcome.

### Disadvantages of Full Factorial Design

**Resource Intensive:** Since all possible combinations must be tested, the design
can become very time-consuming and expensive when a large number of factors and levels
are involved. The number of experiments and associated costs increase exponentially
with the number of factors.

## Full Factorial Design Example

Using a full factorial design, we want to determine what influences the friction torque of a bearing. In this case, friction torque is our response variable. Potential factors could be lubrication and temperature. For lubrication, the factor levels could be "oiled" and "greased," and for temperature, the factor levels could be "low," "medium," and "high."

Now, we want to understand the effect of the different levels on the response variable. This is where a full factorial design can help!

## Why Do You Need a Full Factorial Design?

Experiments take time and money! Therefore, it’s essential to keep the number of experiments as low as possible. But be careful: if the sample size is too small, there’s a high risk of missing significant differences.

Let’s look at an example. We want to investigate which factors influence the frictional torque of a bearing.

Let’s start with a single factor: lubrication. We want to determine whether the type of lubrication, either oil or grease, affects the frictional torque.

To investigate this, we take a sample of 10 bearings, lubricating half with oil and the other half with grease.

We can then measure the frictional torque of the 5 oiled bearings and the 5 greased bearings. But why exactly 10 bearings?

Each experiment costs money, so it’s worth considering if we can achieve our objective with fewer trials.

How large does the sample need to be to determine whether the lubricant affects the frictional torque?

Let's start with 10 bearings and calculate the average frictional torque for both the oiled and greased bearings. We can then compute the difference between these two averages.

This sample shows a difference between oiled and greased bearings. However, there is noticeable variation in frictional torque within each group. If we take another sample of 5 bearings for each condition, the difference might be larger, smaller, or even reversed.

This variation indicates that the frictional torque of the bearings has a certain spread. The greater this spread, the harder it becomes to detect a specific difference or effect.

Fortunately, the spread of the mean can be reduced by increasing the sample size. The larger the sample, the more confidently we can determine the location of the mean.

Therefore, the smaller the effect we are interested in and the greater the variability of the target variable, the larger the required sample size.

## Calculate number of runs

To calculate the required number of experiments, you can use this approximation formula. Here, N represents the sample size, Sigma the standard deviation, and Delta the detectable effect size.

For instance, if we have a standard deviation of 3 Nmm and a relevant difference of 5 Nmm, we need 22 trials. With a standard deviation of 2 Nmm, we need only 10 trials, and with a standard deviation of 1 Nmm, only 4 trials (2 with oiled and 2 with greased bearings) would suffice.

But how can a full factorial design help reduce the number of runs? Let’s assume we add a second factor, such as temperature, with levels for low and high temperatures.

To account for this additional factor, we would need 8 more trials.

This means we would need a total of 24 trials to check for the effects of both lubrication and temperature.

Is it possible to reduce the number of trials? This is where the full factorial design becomes beneficial.

Why test each factor independently? We can create an experimental design that also considers the fourth combination, grease and high temperature.

This way, we need a total of 16 trials by performing 4 repetitions for each of the four combinations. We now have 8 trials with oil, 8 with grease, 8 at low temperature, and 8 at high temperature.

In total, we now have 16 trials. Previously, we needed 24 trials, but now we need fewer trials and even gain additional information.

Why additional information? We can now also determine if there’s an interaction between temperature and lubrication. For instance, there might be a temperature effect with oil but not with grease—this valuable insight would have been lost otherwise.

## Savings with 3 Factors

If we consider not just 2 factors but 3, the potential savings increase even more. Testing each factor individually with three factors would require 32 trials. However, by using a full factorial design with two trials for each combination, we still only need 16 trials.

For each factor level, we still perform 8 trials per level. For instance, for the lubrication factor, we have 8 trials with oil and 8 trials with grease.

### Full Factorial Designs with 3 Levels

Full factorial designs can also include more than two levels per factor. For example, the temperature factor could have three levels: low, medium, and high.

### Number of runs Required

The number of required trials in a full factorial design with two levels per factor increases rapidly as the number of factors grows.

In a full factorial design, the total number of trials, *n*, can be calculated as
*n = 2 ^{k}*, where

*k*is the number of factors. Here is a quick reference: with three factors, we need a minimum of 8 trials; with seven factors, at least 128 trials; and with ten factors, a minimum of 1,024 trials. Therefore, full factorial designs are typically limited to a maximum of 6 factors in practice.

Note that this table applies to designs where each factor has only two levels; if additional levels are used, the required number of trials increases significantly.

Number of Factors | Number of Runs |
---|---|

2 | 4 |

3 | 8 |

4 | 16 |

5 | 32 |

6 | 64 |

7 | 128 |

8 | 256 |

9 | 512 |

10 | 1,024 |

… | … |

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