Issue 85, March 2008

Introduction to Design of Experiments (DOE) - DOE Types

This article continues the discussion of Design of Experiments (DOE) that started in last month's issue of the Reliability HotWire. This article gives a summary of the various types of DOE. Future articles will cover more DOE fundamentals in addition to applications and discussion of DOE analyses accomplished with the soon-to-be-introduced DOE++ software!

The design and analysis of experiments revolves around the understanding of the effects of different variables on other variable(s). In mathematical jargon, the objective is to establish a cause-and-effect relationship between a number of independent variables and a dependent variable of interest. The dependent variable, in the context of DOE, is called the response, and the independent variables are called factors. Experiments are run at different factor values, called levels. Each run of an experiment involves a combination of the levels of the investigated factors. Each of the combinations is referred to as a treatment. In a single factor experiment, each level of the factor is referred to as a treatment. In experiments with many factors, each combination of the levels of the factors is referred to as a treatment. When the same number of response observations are taken for each of the treatments of an experiment, the design of the experiment is said to be balanced. Repeated observations at a given treatment are called replicates. The number of treatments of an experiment is determined on the basis of the number of factor levels being investigated in the experiment. For example, if an experiment involving two factors is to be performed, with the first factor having x levels and the second factor having z levels, then x × z treatment combinations can possibly be run, and the experiment is an x × z factorial design. If all x × z combinations are run, then the experiment is a full factorial. If only some of the x × z treatment combinations are run, then the experiment is a fractional factorial. In full factorial experiments, all of the factors and their interactions are investigated, whereas in fractional factorial experiments, all interactions are not considered because not all treatment combinations are run.

It can be seen that the size of an experiment escalates rapidly as the number of factors, or the number of the levels of the factors, increases. For example, if two factors at three levels each are to be used, nine different treatments are required for a full factorial experiment (3 × 3 = 9). If a third factor with three levels is added, 27 treatments are required (3×3×3 = 27) and 81 treatments are required if a fourth factor with three levels is added (3×3×3×3 = 81). If only two levels are used for each factor, then in the four factor case, 16 treatments are required (2 × 2 × 2 × 2 = 16). For this reason, many experiments are restricted to two levels. Fractional factorial experiments further reduce the number of treatments to be executed in an experiment.

DOE Types

The following is a summary of some of the most common DOE types.

1 One Factor Designs
These are the designs where only one factor is under investigation, and the objective is to determine whether the response is significantly different at different factor levels. The factor can be qualitative or quantitative. In the case of qualitative factors (e.g. different suppliers, different materials, etc.), no extrapolations (i.e. predictions) can be performed outside the tested levels, and only the effect of the factor on the response can be determined. On the other hand, data from tests where the factor is quantitative (e.g. temperature, voltage, load, etc.) can be used for both effect investigation and prediction, provided that sufficient data are available.

2 Factorial Designs
In factorial designs, multiple factors are investigated simultaneously during the test. As in one factor designs, qualitative and/or quantitative factors can be considered. The objective of these designs is to identify the factors that have a significant effect on the response, as well as investigate the effect of interactions (depending on the experiment design used). Predictions can also be performed when quantitative factors are present, but care must be taken since certain designs are very limited in the choice of the predictive model. For example, in two level designs only a linear relationship between the response and the factors can be used, which may not be realistic.

• General Full Factorial Designs
In general full factorial designs, each factor can have a different number of levels, and the factors can be quantitative, qualitative or both.

• Two Level Full Factorial Designs
These are factorial designs where the number of levels for each factor is restricted to two. Restricting the levels to two and running a full factorial experiment reduces the number of treatments (compared to a general full factorial experiment) and allows for the investigation of all the factors and all their interactions. If all factors are quantitative, then the data from such experiments can be used for predictive purposes, provided a linear model is appropriate for modeling the response (since only two levels are used, curvature cannot be modeled).

• Two Level Fractional Factorial Designs
This is a special category of two level designs where not all factor level combinations are considered and the experimenter can choose which combinations are to be excluded. Based on the excluded combinations, certain interactions cannot be determined.

• Plackett-Burman Designs
This is a special category of two level fractional factorial designs, proposed by R. L. Plackett and J. P. Burman, where only a few specifically chosen runs are performed to investigate just the main effects (i.e. no interactions).

• Taguchi’s Orthogonal Arrays
Taguchi’s orthogonal arrays are highly fractional designs, used to estimate main effects using only a few experimental runs. These designs are not only applicable to two level factorial experiments, but also can investigate main effects when factors have more than two levels. Designs are also available to investigate main effects for certain mixed level experiments where the factors included do not have the same number of levels.

3 Response Surface Method Designs
These are special designs that are used to determine the settings of the factors to achieve an optimum value of the response.

4 Reliability DOE
This is a special category of DOE where traditional designs, such as the two level designs, are combined with reliability methods to investigate effects of different factors on the life of a unit. In Reliability DOE, the response is a life metric (e.g. age, miles, cycles, etc.), and the data may contain censored observations (suspensions, interval data). One factor designs and two level factorial designs (full, fractional, and Plackett-Burman) are available in DOE++ to conduct a Reliability DOE analysis.

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