The purpose of a design of experiments is to provide the most efficient and economical methods of reaching valid and relevant conclusions from the experiment being conducted. As stated in previous blog posts, there are many experimental designs to chose from and in my opinion this is part of the problem. The novice experimenter when faced with so many choices can become confused and overwhelmed. Here are some of the designs available:

**One-factor experiments:**

- One-way ANOVA completely randomized
- Randomized complete block design (RCBD) where each block contains all treatments and randomization is restricted within blocks
- Latin square design is generally used to eliminate two block effects by balancing out their contributions.

**Screening experiments:**

These experiments are used to study a large number of variables for the purpose of identifying the most important ones. Some, such as Plackett-Burman designs can handle seven variables in eight experimental runs or eleven variables in twelve runs. Screening designs use only two levels of each design factor and cannot resolve interactions between factors.

**Two-level fractional factorial experiments**

These designs are used to reduce the size of the experiments. Lets say that after you conducted a Placket-Burman screening experiment, you were left with five significant factors you want to study further. If you were to run a full factorial, two-level experiment where every possible combination is ran, the design would be 2^{5 }or 32 experimental runs without any replicates. If the experiment was conducted at three levels, the design would be 3^{5} or 243 experimental runs. This could be very tedious requiring a lot of time and resources. In order to overcome this problem and reduce the experiment to a more manageable size, a fraction of the full factorial experiment can be ran. For instance, if a 1/2 fractional factorial experiment is ran for the 2^{5 }experiment only 16 experimental runs would be required, for a 1/4 fraction, only 8 experimental runs, etc.

The problem with fractional factorial experiments is that because not all combinations are ran some of the information is lost and confounded with other effects, especially interactions. Interactions exist between variables when the effect of one variable on the response depends on the level of another variable. We normally overcome this problem by assuming that higher order interactions are not statistically significant and in most cases we are safe in this assumption. My only experience where this higher-order interactions must be considered is in processes where a chemical reaction takes place.

Here is an example of the confounding that occurs. Let say we’re looking at a 3-factor, 2-level experiment where A, B and C are the three factors being studied. In a full-factorial experiment there would be eight experimental runs as shown below. Run 1 would have all three factors set at the low level, run 2 would have factor A set at the high level, and B and C factors set at the low level and so on. In addition to the three main factors, the experimental array shows the interactions. The two-way interactions are AB, AC and BC. There is also the three-way interaction, ABC.

It’s important to note that the interactions AB, AC, BC and ABC are the result of the analysis. They are the result of setting the three-factors (low or high level) in each experimental run. Looking at run 1, both A and B are low and indicated by -1. The resultant interaction AB is the product of A (-1) and B (-1) or A times B = +1. The three-way interaction is the product of all three factors and is A (-1) X B (-1) X C (-1) = (-1).

In a 1/2 fraction design four of the eight combinations would be run. These four runs represent half of the total combinations that exist for three-factors at two-levels.

Showing the interactions for this half-fractions results in the following

Take a close look at column A and the BC interaction column, the signs are exactly the same. The effects of those two columns are confounded which means we cannot separate the effect of A from that of the interaction BC. The same goes for factor B and the AC interaction. This is the drawback or problem that exists in using fractional-factorial experiments.

**Full factorial experiments**

Full factorial experiments measure the response at all combinations of factor levels and are broken down into two-level full factorial designs and general full factorial designs. In a two-level design, each experimental factor is study at two levels, a low level and high level. In a general full factorial design, the experimental factors can have any number of levels. As an example, one factor may have two levels, another factor may have three levels and another factor may have five levels. The experimental runs would include all combinations of these factor levels.

**Choosing a design**

When choosing a design there are basically two things to consider. The first is how many experimental factors have you identified and at what levels will they need to be studied? The second is how many experimental runs can you afford to perform? This will include the cost of scrap incurred, the time it takes to conduct the experiment and the lost productivity, and the resources required to do the experiments.