Design of Experiments

Design of Experiments (DOE) is a systematic method used to determine the relationship between factors affecting a process and the output of that process. It is widely used in scientific research and industrial applications to optimize processes, improve product quality, and reduce costs.

Key Concepts in Design of Experiments

1. Factors and Levels:
   • Factors: Independent variables that are manipulated in an experiment e.g., number of cylinders (cyl)
   • Levels: Different values or settings of the factors e.g., 4, 6, 8
2. Response Variable:
   • The dependent variable that is measured in the experiment to assess the effect of the factors e.g. mpg.
3. Treatment:
   • A specific combination of factor levels.
4. Experimental Unit:
   • The smallest division of the experimental material to which a treatment is applied.
5. Randomization:
   • The process of randomly assigning treatments to experimental units to avoid bias.
6. Replication:
   • Repeating the experiment to estimate variability and improve the reliability of the results.
7. Blocking:
   • Grouping experimental units that are similar to reduce the variability caused by extraneous factors.

Understanding types of Experimental Designs with mtcars as an example

1. Completely Randomized Design (CRD)

Objective: Investigate the effect of the number of cylinders (factor) on miles per gallon (mpg) (response variable).

Factors and Levels:

• Factor: Number of Cylinders (cyl)
• Levels: 4, 6, 8

Example: Randomly select cars with 4, 6, and 8 cylinders and measure their mpg.

2. Randomized Block Design (RBD)

Objective: Investigate the effect of the number of cylinders (cyl) on mpg, blocking by transmission type (am).

Factors and Levels:

• Factor: Number of Cylinders (cyl)
• Levels: 4, 6, 8
• Block: Transmission Type (am)
  • Levels: 0 (automatic), 1 (manual)

Example: Group cars into two blocks based on transmission type and randomly assign cars with 4, 6, and 8 cylinders within each block to measure mpg.

3. Factorial Design

Objective: Investigate the combined effect of the number of cylinders (cyl) and weight (wt) on mpg.

Factors and Levels:

• Factor 1: Number of Cylinders (cyl)
  • Levels: 4, 6, 8
• Factor 2: Weight (wt)
  • Levels: Low, Medium, High (e.g., categorize weight into three groups)

Example: Randomly select cars with different combinations of cylinder counts and weight categories to measure mpg.

4. Fractional Factorial Design

Objective: Investigate the effect of multiple factors on mpg using a fraction of the full factorial design.

Factors and Levels:

• Factor 1: Number of Cylinders (cyl)
  • Levels: 4, 6, 8
• Factor 2: Weight (wt)
  • Levels: Low, Medium, High
• Factor 3: Horsepower (hp)
  • Levels: Low, Medium, High

Example: Select a subset of the possible combinations of cyl, wt, and hp levels to measure mpg, reducing the number of experimental runs.

5. Response Surface Methodology (RSM)

Objective: Optimize mpg by exploring the relationships between factors such as weight (wt) and horsepower (hp).

Factors and Levels:

• Factor 1: Weight (wt)
• Factor 2: Horsepower (hp)

Example: Conduct experiments at various levels of weight and horsepower, fit a quadratic model to the data, and use response surface plots to identify the optimal combination of weight and horsepower for maximum mpg.

Example of RSM with mtcars

1. Experimental Design:
   • Central Composite Design (CCD) with factors wt and hp.
2. Modeling:
   • Fit a quadratic model to the data: mpg ~ wt + hp + I(wt^2) + I(hp^2) + wt:hp.
3. Optimization:
   • Use the model to find the optimal levels of wt and hp for maximum mpg.
4. Visualization:
   • Create contour plots and response surface plots to visualize the relationships and identify optimal conditions.

Design of Experiments (DOE) is a powerful tool for understanding the relationships between factors and response variables. Using the mtcars dataset, we can illustrate various experimental designs and their applications, such as CRD, RBD, factorial designs, fractional factorial designs, and RSM. By carefully designing and analyzing experiments, we can optimize processes and make informed decisions based on data. Remember, the dataset is just an example to illustrate how experiments can be designed.