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[SIX SIGMA GLOSSARY ALPHABETICAL INDEX] [SIX SIGMA GLOSSARY INDEX OF TOPICS]

Design of Experiments Terms

The Design of Experiments is covered in the MiC Quality basic Design of Experiments course and Advanced Design of Experiments course.

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Alias

In experimental design when two interactions, or a main effect and an interaction, share the same column, and so cannot be individually analyzed then their effects are aliased.

In the 23-1 design the factor C is aliased with the interaction AB:

Run
A
B
AB + C
1
-1
-1
+1
2
+1
-1
-1
3
-1
+1
-1
4
+1
+1
+1


Balanced Design

Also Balanced Experiment. A factorial design in which each factor is run the same number of times at the high and low levels. Most factorial designs are balanced, unbalanced designs are only used in exceptional circumstances.

Blocking

Blocking is a technique used to eliminate identified possible sources of variation that cannot be randomized. The experiment is organized into blocks, where the possible source of variation is held level in each block.

Suppose that there was insufficient material in one batch to carry out all the runs in an experiment. Half the runs could be done on one batch and half on another. The experiment would have to be carried out in such a way that the aggregate result was balanced between the two batches. At its simplest, if the experiment involved an even number of replications of each run then this would simply involve carrying out half the replications on one batch, and half on the other. Things are not usually so simple, and there are many ways of approaching more challenging situations.

Coded Factors

In experimental design it is usual to substitute 'coded factors' for the actual physical values. In a factorial design the coded factors are given the levels '-1' and '+1'. This assists the analysis in various ways.

Completely Randomized Design

A design where the experimental units are to the treatments in a completely random fashion. Suppose we wanted to compare four brands of washing powder. We decide to carry out four replications for each brand.

We would prepare 16 apparently identical pieces of soiled cloth and allocate them to the powders at random. To do this we could mix them up in a tub and pull them out one at a time. To be sure there is no bias, we might take 16 slips of paper and mark them Brand 1#1 up to Brand 4#4. We would then mix the pieces of paper up in a hat and match a slip of paper drawn at random with a piece of cloth drawn at random.

Confounding

Factors or interactions are confounded when the design array is configured so that the effect of one factor is combined with the other. The effect of the individual factors/interactions cannot be isolated by the analysis.

Confounding is very similar to aliasing, although aliasing is used to describe factors/interactions that are fully confounded, rather than partially confounded.

Defining Relationship

The relationship that determines the aliasing structure in a design.

Run
A
B
AB + C
1
-1
-1
+1
2
+1
-1
-1
3
-1
+1
-1
4
+1
+1
+1

The Defining Relationship is I=ABC

Dependent Variable

When an experiment is conducted the variables manipulated by the experimenter are called "independent variables" or factors and the response or output variables are the "dependent variables".

Design Generator

The relationship used to create a fractional factorial design, and which creates the defining relation:

Run
A
B
AB + C
1
-1
-1
+1
2
+1
-1
-1
3
-1
+1
-1
4
+1
+1
+1

The Design Generator is A=BC

Design of Experiments

Experimental design involves conducting a systematic series of tests to discover the relationship between the factors effecting a process (X) and the critical outputs (responses). The inputs are varied in a systematic fashion and the effects on the response(s) are observed.

The Design of Experiments (Experimental Design) is used when a process is affected by many separate factors. It is far more efficient than the 'One Factor at a Time' (OFAT) method in which each factor is varied in turn whilst the others remain constant.

The Design of Experiments will give more information for less testing. It will also allow 'interactions' between factors to be evaluated, and the effects of interactions are usually important.

The two main approaches to experimental design are the 'classical' and 'Taguchi' methods. Experts are divided on the merits of the Taguchi method, but the emphasis on variation and the methods it uses to address variation are important. The types of design commonly used included Full Factorial Designs, Fractional Factorial Designs and Plackett-Burman Designs.

A more powerful approach is to use Response Surface Methods; this group includes the Box Behnken and Central Composite Designs (CCD).

Designs that involve mixtures require a different method of analysis, see the topic on Mixture Designs.

Design Resolution

Used in experimental design to help select a fractional factorial design:

Resolution III RIII two factor interactions are aliased with main effects
Resolution IV RIV two factor interactions are aliased with other two factor interactions, but not with main effects. Main effects are aliased with three factor interactions.
Resolution 5 RV two factor interactions are not aliased with each other, but are aliased with three factor interaction

An RIII design would usually be used for screening experiments. There would rarely be any point in going beyond an RV design.

In general the resolution is the length of the shortest word in the defining relationship.

Dispersion

The degree to which data tend to spread about the mean. It is measured by the standard deviation, or variance.

EVOP

Evolutionary Operation. An experimental design technique that involves making small changes to a process progressively, and during normal operation, to find the optimum operating conditions.

Experimental Design

See Design of Experiments

Factor

A factor is a parameter whose potential effect on the response is being evaluated in the experiment.

The experiment involves varying the levels of 'factors' to determine their effect on the 'response'.

Fixed Effects Model

An experimental design where the factor levels are specifically selected by the experimenter (cf. Random Effects Model)

Independent Variables

See Dependent Variable for an explanation

Interaction

In many processes the factors interact, the combined effect is not the sum of the individual effects. The figure below uses the well known danger of combining alcohol with some medications to illustrate the idea:

Interactions are an important consideration in experimental design.

Level

During the experiment the factors are set to different values or 'levels'. In a factorial experiment two levels, or sometimes three, are selected for each factor.

Main Effects

The effect of a factor, as opposed to an interaction effect.

Orthogonal Design

A design is orthogonal if each factor can be evaluated independently of all other factors. In a two level factorial design, this is achieved by matching each level of each factor with an equal number of each level of the other factors.

For example, in the array the '+1' level of Factor A (runs 2 and 4) is matched with one instance of Factor B at '-1' and one at '+1'. If any two columns are compared, the same thing will be found for both factor levels.

Run
A
B
AB + C
1
-1
-1
+1
2
+1
-1
-1
3
-1
+1
-1
4
+1
+1
+1

The term 'orthogonal array' is often used in the context of Taguchi designs; 'Taguchi Orthogonal Arrays'.

Random Effects Model

An experimental design where the factor levels are selected at random from a large number of possible levels. The analysis is based on estimating the variance.

This compares to a fixed effects model where a restricted number of levels are selected and set by the experimenter.

Randomization

Experiments should be run in random order to randomize the effects of any variables that are varying during the experiment, and might impose a pattern on the results.

Randomizing destroys any systematic variation, which could not be detected by the analysis, and converts it to either 'common cause' variation, or variation that will be detected in the residual analysis.

Randomized Block

Suppose we wanted to compare four brands of washing powder. We decide to carry out four replications for each brand. To make the test realistic we give four volunteers four clean handkerchiefs each; they are to use each for one day.

In a Completely Randomized design we would allocate the handkerchiefs to powder at random, so that one powder might get one, two, three or even all four handkerchiefs from one person.

In a Randomized Block design we would block on the handkerchiefs. We would allocate one, and only one, handkerchief from each volunteer to each powder. The four handkerchiefs from each volunteer would be allocated at random to the powders, using a similar scheme to the one described in the Completely Randomized design.

Repetition

See Replication

Replication

Carrying out several tests at each treatment. If each treatment is tested four times then there are four replications of the design.

It is important that the order of the runs is randomized to remove systematic error. If this is not done, but each treatment is repeated several times before moving on to the next, then that is 'repetition'. The ANOVA analysis should not be carried out using the repeats as individual readings. Instead the mean of the repeats should be treated as a single reading

Response Variable

An experimental design explores the effect of input factors on the process response. For example, an experiment might investigate the effects of altering the amounts of flour, water, yeast, oven temperature etc. on the consistency of bread. The consistency quality of the bread would be the response variable.

Rotatable

A desirable property of Response Surface Designs. It refers to designs where the variance is the same at all points that are the same distance from the design center.

Screening Design

A type of experimental design used when a large number of factors that may affect the process have been identified. The screening design identifies the factors that do not affect the process so those remaining can be studied more closely. Screening designs are Resolution III designs.

Transformations

Applying a mathematical operation to response data to make it conform to a normal distribution. Often necessary with experiments that involve rates because the results typically do not conform to a normal distribution.

Common transforms include natural log, reciprocal and squares.

Treatment

A certain combination of factor levels whose effect on the response variable is of interest. Often replaced by the term 'run'.

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