📘 Chapter 14 Summary – Multifactor Experiments (FYI)

🔗 Slides for Chapter 14

📂 Download Chapter 14 PowerPoint (ch14.pptx)

🎯 Learning Objectives

• Understand and analyze experiments with more than one factor
• Interpret interaction effects between factors
• Construct ANOVA models for two-factor and multi-factor experiments
• Evaluate factorial treatment structures
• Perform residual analysis to validate model assumptions
• Use graphical tools (interaction plots, cube plots) to visualize effects
• Apply factorial experiments in randomized block and split-plot designs

🔧 14.1 – Introduction to Factorial Experiments

• Involve two or more factors studied simultaneously
• Include all possible combinations of factor levels
• Efficient compared to one-factor-at-a-time studies
• Can detect interaction effects in addition to main effects

🧪 14.2 – Two-Factor Factorial Designs

Assume factors A (a levels) and B (b levels), each with n replicates.

Model: Yijk = μ + αi + βj + (αβ)ij + εijk

• μ: Overall mean
• α_i: Effect of factor A
• β_j: Effect of factor B
• (αβ)_ij: Interaction effect between A and B
• ε_ijk: Random error ~ N(0, σ²)

ANOVA Table:

Source         | df         | SS       | MS        | F
---------------|------------|----------|-----------|-------------------
Factor A       | a − 1      | SSA      | MSA       | MSA / MSE
Factor B       | b − 1      | SSB      | MSB       | MSB / MSE
Interaction AB | (a−1)(b−1) | SSAB     | MSAB      | MSAB / MSE
Error          | ab(n−1)    | SSE      | MSE       |
Total          | abn − 1    | SST      |           |
  

🔀 14.3 – Interaction Effects

• Interaction occurs when the effect of one factor depends on the level of another
• Plot means to detect interactions
• Significant interaction affects interpretation of main effects

📦 14.4 – Three-Factor and Higher-Order Designs

• Same principles apply for 3+ factors
• Model includes all main effects, two-way and three-way interactions
• Visualization: cube plots, slice plots
• Interpretation becomes more complex with added factors

📈 14.5 – Model Checking and Residual Analysis

• Check assumptions with residual plots:
• Normality: normal probability plot
• Equal variance: residuals vs fitted values
• Independence: residuals vs run order
• Transform data (log, √) if assumptions are violated

🧪 14.6 – Guidelines and Applications

• Factorial designs reduce required experimental runs
• Use blocks to control nuisance variables
• Use cube and interaction plots for insight
• Apply split-plot designs when some factors are hard to change

🧱 14.7 – Randomized Block and Split-Plot Designs

• Randomized Block Design: block one factor, use factorial on others
• Split-Plot Design: useful when one factor is hard to change
• Model has different error terms for whole plot vs subplot
• Less precision in whole plot effects; more complex analysis

📝 Important Terms

• Factorial Design
• Main Effect
• Interaction Effect
• Cube Plot, Slice Plot
• Split-Plot Design
• Randomized Block Design
• Residual Analysis
• ANOVA Table with 2+ factors