Chapter 7: Design of Experiment (DOE) – Exam Revision Notes

1. Introduction

• Design of Experiment (DOE): A structured method for planning experiments, collecting data, and analyzing results.
  
• Used to study the effect of one or more factors on a response variable.
  
• Provides valid and statistically efficient conclusions.
  

2. Concept of Analysis of Variance (ANOVA)

• ANOVA: Statistical technique to compare means of two or more groups.
  
• Determines whether observed differences are statistically significant or due to random variation.
  
• Key components:
  
  • F-statistic: Ratio of variance between groups to variance within groups.
    
  • Null Hypothesis (H₀): All group means are equal.
    
  • Alternative Hypothesis (H₁): At least one group mean is different.
    

3. Linear Model in ANOVA

• General form:
  Yij=μ+τi+ϵijY_{ij} = \mu + \tau_i + \epsilon_{ij}
  1. YijY_{ij} = Observation from i-th treatment and j-th replicate
     
  2. μ\mu = Overall mean
     
  3. τi\tau_i = Effect of i-th treatment
     
  4. ϵij\epsilon_{ij} = Random error
     
• Assumptions:
  
  1. Errors are independent and normally distributed.
     
  2. Homogeneity of variance (equal variance across groups).
     

4. Types of ANOVA

4.1 One-Way ANOVA

• Tests differences among means of one factor with multiple levels.
  
• Steps:
  
  1. Compute group means and overall mean
     
  2. Calculate Sum of Squares Between (SSB) and Sum of Squares Within (SSW)
     
  3. Compute F-statistic:
     F=MSBMSWF = \frac{MSB}{MSW}
     where MSB = Mean Square Between, MSW = Mean Square Within
     
  4. Compare F with critical value from F-table.
     

4.2 Two-Way ANOVA

• Analyzes two factors simultaneously.
  
• Can include interactions between factors.
  
• Example: Study effect of fertilizer type and irrigation level on crop yield.
  

5. Observations per Cell

• Balanced Design: Same number of observations in each group.
  
• Unbalanced Design: Unequal number of observations; requires special computation methods.
  

6. Fixed Effect Model

• Factor levels are fixed and pre-determined.
  
• Interest lies in comparing specific treatments rather than generalizing to all possible levels.
  

7. Key Takeaways

• DOE increases efficiency, precision, and reliability of experimental results.
  
• ANOVA is the core tool in DOE for comparing group means.
  
• One-way ANOVA → Single factor, multiple levels.
  
• Two-way ANOVA → Two factors, can assess interactions.
  
• Balanced design simplifies calculations; fixed effect model focuses on specific treatments.