Design of Experiments: Key Concepts and Designs
Design of Experiments: Key Concepts
Design of Experiments (DOE) is the rational planning of experiments to obtain the maximum amount of information with the minimum number of trials. Its function is to analyze results and obtain evidence to test previously established hypotheses.
DOE is directly related to the problem statement and the hypothesis. The design aims to collect data from reality to solve the problem, and the hypothesis guides the type of experimentation needed to obtain, analyze, and compare information.
Key Terms in Design of Experiments
- Factor: The independent variable in an experiment. These are controllable variables that may or may not influence the response variable.
- Treatment: A specific combination of factor levels in an experiment.
- Level: A particular value of a factor.
- Response: The value of the dependent variable for each treatment.
- Block: A portion of experimental material with similar, but not identical, characteristics.
- Effect: The influence of factors, either combined or independently, on the response variable.
Common Experimental Designs
- Completely Randomized Design: Treatments are assigned to experimental units completely at random, often using a table of random numbers.
- Randomized Block Design: Treatments cannot be repeated within a block, unlike the completely randomized design. This accounts for potential block effects.
Yates Algorithm
The Yates algorithm is a practical method for calculating the contrasts of all effects studied in an experiment. From these contrasts, it’s possible to obtain the sum of squares, which provides evidence to test a hypothesis.
For a 23 design (Y = Y(A, B, C)), at least 8 experiments are required. In the Yates algorithm, all treatments are placed in a column, followed by the Y values (or the sum of Y values) for each treatment.
Calculating Contrasts:
- Column (1) is divided into two parts. Pairs of Y values are grouped.
- To get the top half of column (1), add the pairs of Y values.
- To get the bottom half, subtract the pairs of Y values.
- Repeat the same procedure for column (2) to obtain column (3).
- Column (3) represents the contrasts of each effect. These contrasts are used to determine the sum of squares of effects and perform an analysis of variance.
Example (Yates Algorithm Table):
| Treatment | (1) | (2) | (3) | SC |
|---|---|---|---|---|
| 1 | Y1 | y1 + y2 | C | ([col (3)]^2)/2^3 |
| a | Y2 | y3 + y4 | O | |
| b | Y3 | y5 + y6 | N | |
| ab | Y4 | y7 + y8 | T | |
| c | Y5 | y2 – y1 | A | |
| ac | Y6 | y4 – y3 | S | |
| bc | Y7 | y6 – y5 | T | |
| abc | Y8 | y8 – y7 | E |
23 Design
In a 23 design, there are 8 treatments, corresponding to the combinations of 2 levels for each of the 3 factors. The order of experiments is randomized using a random number table. Each treatment is assigned a number, and the conditions are determined before randomization. Data is then sorted in a standard way, and the sum of squares of treatments is obtained. The Yates algorithm for this design is as explained above.
An analysis of variance (ANOVA) is then performed. If there’s only one replicate, the ABC interaction is often considered the error term. The hypothesis of mean effects is then tested:
- If F > Fns, reject H0 (null hypothesis): n = ne. This indicates the factor has a significant influence.
- If F ≤ Fns, there is no significant effect.
Rotatable Design
Factorial designs (2k, 2(k-p)) involve varying the level of all factors simultaneously.
Composite Design
Additional points are added to the central part of the design. These points are obtained by adding +A and -A to one factor while keeping the others at their central value.
Factorial Designs at 2 Levels
Factorial designs are used for hypotheses of the type Y = (A, B, C), meaning more than one factor is involved. Examples include confounded, fractional factorial, and partially confounded designs.
- Partially Confounded: Replicates are used so that each replicate confounds the blocks in different effects.
- Factorial 2k: 2 = level, k = factor, 2k = n, where n indicates the minimum number of treatments to be performed. These are taken from the design matrix and sorted before being used with the Yates algorithm, culminating in an analysis of variance.