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Factorial Design

Full Factorial Design

A full factorial design combines the levels for each factor with all the levels of every other factor. It covers all combinations and provides the best data. However, it consumes time and resources.

Fractional Factorial Design

A fractional factorial design, does not take into account each and every factor. If a full-factorial design uses too many resources, or if a slightly non-orthogonal array is acceptable, a fractional factorial design is used. To analyze a data from a DOE, the team must first evaluate the statistical significance by computing the one-way ANOVA, or for more than one factor, the N-Way ANOVA.

The practical significance can be evaluated through the study of sum of squares, pie charts, Pareto diagrams, main effects plots and normal probability plots. The factorials are also known as 2-k factorials.

2 = no. of levels, k = no. of factors.

Full factorial = 2k,

Fractional factorial = 2k-1

Main Effects versus Interaction Effects:

Traditional one-factor-at-a-time experimentation tests only single factors, known as main effects. DOE tests for interactions between or among factors, are known as interaction effects.

Two-Level Fractional Factorial:

The following seven step procedure is followed:

  • Select a process

  • Identify the output factors of concern

  • Identify the input factors and levels to be investigated

  • Select a design (from a catalogue, Taguchi, self-created, etc.)

  • Conduct the experiment under the predetermined conditions

  • Collect the data (relative to the identified outputs)

  • Analyse the data and draw conclusions

Linear and Quadratic Mathematical Models


A level is a value or a setting for a factor, an assignment of high or low for the different states of the X’ s (inputs) being used in the DOE.


Runs are calculated through the use of design matrix (also known as array), the table of treatment combinations that will be used to set the as for the levels that are defined and collect the Yes (outputs) that will be analyzed.

Repetition / Replication:

When an experiment needs to run for more than once, the subsequent run might be repetition or replication. For repetition, the factors are not reset; for replication the factors are reset. Replications give a better estimate of experimental error, but cost more.


When the runs are ordered randomly, and there are concerns about the possibility of unknown external factors affecting variation, it is called randomization.

Balanced and Orthogonal Designs


The term orthogonally is a mathematical property of a matrix, indicating that the DOE is of very good design. Generally, orthogonally describes independence among factors. An orthogonal design matrix, is balanced both vertically and horizontally. For each factor, there are equal numbers of high and low values (vertical and horizontal balance).


Fractional factorial designs reduce the number of runs by screening factors. A disadvantage of that can be that some of the effects of those factors might be confounded or mixed together, so that it cannot be estimated separately. This is called aliasing of effects.

Plackett-Burman Designs

Plackett-Burman (1946) designs are used for screening experiments. These designs are very economical. The run number is a multiple of four rather than a power of 2. Plackett-Burman geometric designs are two-level designs with 4, 8, 16, 32, 64, and 128 runs and work best as screening designs.

Each interaction effect is confounded with exactly one main effect. All other two-level Plackett-Burman designs (12, 20, 24, 28, etc.) are non-geometric designs. In these designs a two-factor interaction will be partially confounded with each of the other main effects in the study. Thus, the non-geometric designs are essentially (“main effect designs,” when there is reason to believe that any interactions are of little significance) a Three-Factor, Three-Level Experiment. Often, a Three-Factor experiment is required after screening a large number of variables. These experiments may be full or fractional factorial.

Generally the (-) and (+) levels in two-level designs are expressed as 0 and 1 in most design catalogues. Here-level designs are often represented as 0,1, and 2.