Full Factorial Design Lecture Notes

Overview

• Full factorial design: A systematic method to examine effects and interactions of several factors on a response variable.
• Purpose: Analyze the relationship between input variables (factors) and an output variable (response variable).

Key Concepts

• Factors: Variables believed to affect the response variable.
• Levels: Specific values that a factor can take.
• Response variable: Output variable affected by factors.

Example: Bike Bearings

• Response variable: Frictional torque of bearings.
• Factors:
  • Lubrication (oiled, greased)
  • Temperature (low, medium, high)
• Objective: Determine the influence of different levels on frictional torque.

Importance of Full Factorial Design

• Useful for multiple factors (3, 4, 5 variables).
• Helps in reducing experimental costs and time by minimizing the number of runs.
• Can analyze complex systems (processes, machines, products, human studies).

Estimating Number of Experiments

1. Start with a random sample: Initial trial with 10 bearings (5 oiled, 5 greased).
2. Calculate mean values: Difference in frictional torque between samples.
3. Sample size influence: Larger sample sizes reduce variability and provide more precise mean estimates.
4. Formula for runs: n = (σ^2/Δ^2) * (Zα/2 + Zβ)^2
   • n: Number of runs
   • σ: Standard deviation
   • Δ: Effect size
   • Example: With σ=3 and Δ=5, 22 runs needed.*

Reducing Number of Experiments with Full Factorial Design

• Combining factors: Example with lubrication and temperature, reducing from 24 to 16 runs.
• Interaction analysis: Identifying interactions (e.g., lubrication affected by temperature).
• Extension to three factors: More significant reduction in required runs (e.g., from 32 to 16).

Practical Application

• Creating a full factorial design: Use online tools like data.net.

1. Define factors and levels: Example factors (Temperature: low, high; Lubrication: oil, grease; Material: steel, ceramic).
2. Generate test plan: Perform experiments in random order to avoid bias.
3. Analyze results: Import data back, perform variance analysis.

Analyzing Results

• Significant factors & interactions: Use P-value (<0.05) to identify significant influences.
• Regression coefficients: Differences between coded and uncoded coefficients.
• Interpretation: Remove non-significant factors/interactions to refine the model.

Conclusion

• Optimal parameters: Identifying best input parameters to maximize/minimize target values.
• Applications: Valuable in various fields, especially when dealing with multiple factors.

Final Note: Full factorial designs are best suited for up to 6-7 factors due to exponential growth in required runs.