📘 Chapter 13 Summary – Design and Analysis of Single-Factor Experiments (FYI)

🔗 Slides for Chapter 13

📂 Download Chapter 13 PowerPoint (ch13.pptx)

🎯 Learning Objectives

• Design single-factor experiments with multiple levels
• Apply ANOVA to test for treatment effects
• Use residual analysis for model validation
• Conduct multiple comparisons (e.g., Fisher's LSD)
• Differentiate between fixed vs. random factors
• Estimate variance components
• Design randomized complete block experiments (RCBD)

🔧 13.1 – Designing Engineering Experiments

• Used for process improvement, product optimization, material evaluation
• Helps reduce cost, variability, development time
• Sequential design: screening → refinement → optimization

🧪 13.2 – Completely Randomized Single-Factor Experiment

Model:

Yij = μ + τi + εij

• μ: overall mean
• τi: treatment effect
• εij: random error ~ NID(0, σ²)

ANOVA Hypotheses:

• H₀: All τi = 0 (no treatment effect)
• H₁: At least one τi ≠ 0

F-statistic:

F₀ = MSTreatments / MSError
Reject H₀ if F₀ > Fα, a−1, a(n−1)

ANOVA Table:

Source        | SS         | df        | MS         | F
-------------|------------|-----------|------------|-----------
Treatments    | SSTreat    | a - 1     | MSTreat    | MSTreat / MSE
Error         | SSE        | a(n - 1)  | MSE        |           
Total         | SST        | an - 1    |            |           
  

Confidence Intervals:

• For mean: ȳi ± tα/2 √(MSE/n)
• For mean difference: (ȳi - ȳj) ± tα/2 √(2MSE/n)

📊 13.2.3 – Multiple Comparisons (Fisher's LSD)

LSD = tα/2 × √(2 × MSE / n)

• Compare all pairs: if |ȳ_i - ȳ_j| > LSD → significant difference
• Graphical method: underline means that are not different

📈 13.2.4 – Residual Analysis

• Check assumptions: normality, equal variance, independence
• Plots: residuals vs treatment, residuals vs fitted values, normal plot
• Transformations (e.g., log, √y) if variance changes with mean

🔢 13.2.5 – Sample Size & Power

• Use OC curves or power formula: Φ² = (n Στ²ᵢ) / (aσ²)
• Determine how many replicates (n) are needed for target power

🎲 13.3 – Random Effects Model

• Used when treatment levels are randomly selected
• Same model as fixed, but τ_i ~ N(0, σ²_τ)

Variance Components:

• σ̂² = MSE
• σ̂²τ = (MSTreat - MSE) / n

🧱 13.4 – Randomized Complete Block Design (RCBD)

Model:

Yij = μ + τi + βj + εij

• β_j: block effect (e.g., material batch, operator)
• Block design helps reduce error from nuisance variables

ANOVA Partition:

SST = SSTreatments + SSBlocks + SSE
  

ANOVA Table:

Source        | SS         | df           | MS         | F
-------------|------------|--------------|------------|-----------
Treatments    | SSTreat    | a - 1        | MSTreat    | MSTreat / MSE
Blocks        | SSBlocks   | b - 1        | MSBlocks   |           
Error         | SSE        | (a−1)(b−1)   | MSE        |           
Total         | SST        | ab - 1       |            |           
  

Fisher's LSD (adjusted):

LSD = tα/2, (a−1)(b−1) × √(2 × MSE / b)

📝 Important Terms

• ANOVA (Analysis of Variance)
• Fixed vs. Random Effects
• F-statistic, MSTreatments, MSError
• Residuals, Normal Probability Plot
• Multiple Comparisons (LSD)
• Blocking, RCBD
• OC Curves and Power
• Variance Components