Full Factorial Design.
Full Factorial Design. Photo by Arkangel

Full Factorial Design leads to experiments where at least one trial is included for all possible combinations of factors and levels. This exhaustive approach makes it impossible to miss any interactions, as all factor interactions are accounted for. However, the thoroughness of this approach makes it quite expensive and time-consuming for experiments with multiple factors – this increases exponentially with the number of factors and levels.

 Factor 1Factor 2Factor 3
TrialsL1L2L1L2L1L2
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Sample factorial design table for a three-factor experiment with two levels per factor

Calculating the Number of Trials

The number of trials required for a full factorial experimental run is the product of the levels of each factor:

No. of trials = F1 level count x F2 level count x … x Fn level count

How Many trials in a Full Factorial Design?

Found by taking the number of levels as the base and the number of factors as the exponent:

Ex1. a design of 4 factors with 3 levels each would be: 3 x 3 x 3 x 3 = 3^4 = 81

Ex 2. 4 factors (A = 3, B = 2, C = 5, D = 4 levels).   3 x 2 x 5 x 4 = 120 observations.

Example

Let’s look at an experiment with four factors:

  • The first factor has two possible levels.
  • The second factor has five possible levels.
  • The third factor has three possible levels.
  • The fourth factor has six possible levels.

To cover all of the potential combinations, the experiment will need:

No. of trials = 2 x 5 x 3 x 6 = 180 trials

Full Factorial Design Notes

  • Full factorial designs include all possible combinations of every level of every factor.
  • Full factorial designs can require a lot of trials, which can take a lot of time.
  • Full factorial designs can require a lot of trials, which can cost a lot of money.
  • Requires at least one observation for every combination of factors and levels.
  • Allows for the measurement of all possible interactions.
  • Expensive and time-consuming.

Analyzing Full Factorial Designs

Factorial ANOVA

You can use an Analysis of Variation – ANOVA to determine the results of full factorial design experiments.

Yates Analysis

Yates analysis is used in experiments with multiple factors, all having two levels. In some circumstances, the two levels can be ‘high’ and ‘low’ data points. It can be used in both full and fractional factorial design experiments. Read more about Yates analysis.

Why You Would Use Partial or Fractional Factorial Design Instead

One of the big drawbacks of fractional factorial design is the potential to miss important interactions.

Fractional factorials (like Latin and Graeco-Latin Squares) will not allow the analysis of interactions. The interactions are confounded with other effects.

Moving from Full Factorial to Partial Factorial

  • There will be fewer trials
  • There will be confounding
  • Resolution will decrease

Design of Experiments Factorial Design Video

Author

Comments (14)

Hi Ted,

Thank you for the detailed explanation.

Can you please include some details about Linear & Quadratic Mathematical Models. Would be grateful for it.

Thanks
VJ

How many experimental runs exist in a Full Factorial and fully randomized design for 4 factors with 2 replicates for the Corner Points and no Center Points? The factors in the experiment are only at 2-levels *

To calculate the number of experimental runs in a Full Factorial and fully randomized design with 4 factors at 2 levels, and with 2 replicates for the corner points and no center points, follow this logic:

Step-by-Step Calculation

1. Determine the number of runs for a 2-level factorial design:

For 4 factors at 2 levels: 2^4 = 16 corner points (unique treatment combinations).

2. Account for the number of replicates:

Each corner point is replicated 2 times: 16 × 2 = 32 total runs.

3. No center points:

Since there are no center points, we do not add any additional runs.

Answer:

Total experimental runs = 32

Additionally, if you’re interested in preparing for your certification, consider our comprehensive courses:

When the search space is large, genetic algorithms can be tailored to find the optimal parameters. Better than a Partial design.

Hi,
i want to ask a question,i have been working on minitab lately.
i saw a question and i cant this solve question.I see error=0 in my anova table
Would you help me please?
0 (Low)1 (High)
Factors Repeats
A B C 1 2 3 4
0 0 0 46 50 50 58
1 0 0 70 59 52 61
0 1 0 51 39 59 53
1 1 0 51 55 52 40
0 0 1 48 48 44 66
1 0 1 36 40 37 33
0 1 1 48 47 65 47
1 1 1 39 66 62 47

Thank you for your question! When you encounter an error = 0 in your ANOVA table in Minitab, it typically indicates that your model is saturated. This means that all the degrees of freedom (DF) have been used to estimate the model terms, leaving none to estimate the error term. As a result, Minitab cannot compute the Mean Square Error (MSE), F-values, or p-values, and these may appear as asterisks (*) in the output.

Understanding the Saturated Model

In your case, you have a 2-level, 3-factor full factorial design with 4 replicates per treatment combination, totaling 32 observations. If you include all main effects and interactions (A, B, C, AB, AC, BC, ABC) in your model, the degrees of freedom are allocated as follows:

  • Main effects: 3 factors × (2 levels – 1) = 3 DF
  • Two-way interactions: 3 combinations × (2 levels – 1) × (2 levels – 1) = 3 DF
  • Three-way interaction: 1 combination × (2 levels – 1) × (2 levels – 1) × (2 levels – 1) = 1 DF
  • Total model DF: 3 + 3 + 1 = 7 DF

With 32 observations, the total DF is 31 (n – 1). Subtracting the model DF (7), you should have 24 DF for error. However, if your model includes all possible terms and interactions, and the data perfectly fits the model (i.e., no variability left unexplained), the residual error becomes zero, leading to an error DF of zero.

How to Address This Issue

To resolve this, consider the following steps:

  1. Reduce the Model Complexity: Remove higher-order interactions (e.g., the three-way interaction ABC) from the model. This will free up degrees of freedom to estimate the error term.
  2. Use Hierarchical Modeling: Include only significant main effects and lower-order interactions, based on prior knowledge or preliminary analysis.
  3. Increase Replication: If feasible, collect more data to increase the total degrees of freedom, allowing for a more complex model without exhausting the error DF.

By simplifying the model or increasing the data, you can obtain a more reliable estimate of the error term, enabling Minitab to compute the necessary statistics for your ANOVA.

Additionally, if you’re interested in preparing for your certification, consider our comprehensive courses:

Full factorial designs do not adequately account for any factor interactions that may exist. Pls explain about this?

Hi Tharanga,

I’m unsure where you are seeing that. My text above states that full factor design is exhaustive. “This exhaustive approach makes it impossible for any interactions to be missed as all factor interactions are accounted for.”

Best, Ted.

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