Design and Analysis of experiments | Factorial design 2*2 & 2*3 | Central composite design | Unit 5

Key Concepts:

• Design and Analysis of Experiment (DAE): Statistical methodology for designing, conducting, and analyzing experiments.
• Factorial Design: Research method to study how multiple factors affect a dependent variable.
• Response Surface Methodology (RSM): Explores the relationship between input and output variables.
• Central Composite Design (CCD): A type of RSM design combining center points and star points.
• Historical Design (Retrospective Design): Examines past events and data to draw conclusions.
• Optimization Technique: Mathematical method to find the best solution to a biostatistical problem.
• Extraneous Variable (Confounding Variable): Variables that introduce bias or error in the final result.
• 2^2 Factorial Design: Factorial design with two factors, each at two levels, resulting in four experimental runs.
• 2^3 Factorial Design: Factorial design with three factors, each at two levels, resulting in eight experimental runs.

1. Design and Analysis of Experiment (DAE)

• Definition: A statistical methodology used to design, conduct, and analyze experiments to draw meaningful conclusions and make informed decisions.
• Process: Involves planning, executing, and interpreting experiments to determine the relationship between variables.
• Application: Used in engineering, product development, quality control, agriculture (crop yield improvement), medicine (clinical trials), social sciences (psychology, education), and business (marketing, operations research).
• Design Phases:
  • Defining the research question or hypothesis.
  • Identifying independent variables.
  • Selecting dependent variables.
  • Choosing an experimental design (randomized, block, factorial).
  • Determining the sample size.
  • Controlling extraneous variables through blocking and randomization.
• Experimental Designs:
  • Completely Randomized Design (CRD)
  • Randomized Block Design (RBD)
  • Factorial Design
  • Response Surface Methodology

2. Factorial Design

• Definition: A research method that studies how multiple factors (independent variables) affect a dependent variable.
  • "Factorial design is a research method that studies how multiple factors affect the dependent variable."
• Example: Studying the effect of diet and exercise on weight loss.
• Purpose: Helps researchers understand how multiple factors interact and affect the outcome.
• Key Point: The factorial design helps the researcher to understand how multiple factors interact and affect the outcome.

3. 2^2 Factorial Design

• Structure: Two factors, each at two levels, resulting in 2 * 2 = 4 experimental runs (orders).
• Table Setup:
  • Columns: Order (1-4), Factor A, Factor B, A*B (interaction), Block.
  • Factor A: -, +, -, + (negative, positive, negative, positive)
  • Factor B: +, +, -, - (positive, positive, negative, negative)
  • A*B: Calculated by multiplying the signs of A and B.
  • Block: 2 for negative (-), 1 for positive (+).
• Interpretation: The table helps analyze the effects of each factor and their interaction on the dependent variable.

4. 2^3 Factorial Design

• Structure: Three factors, each at two levels, resulting in 2 * 2 * 2 = 8 experimental runs (orders).
• Table Setup:
  • Columns: Order (1-8), Factor A, Factor B, Factor C, ABC (interaction), Block.
  • Factor A: ++++---- (four positives, four negatives)
  • Factor B: +-+-+-+- (alternating positive and negative)
  • Factor C: ++--++-- (two positives, two negatives, repeated)
  • ABC: Calculated by multiplying the signs of A, B, and C.
  • Block: 1 for positive (+), 2 for negative (-).
• Interpretation: The table helps analyze the effects of each factor and their interactions on the dependent variable.

5. Advantages of Factorial Design

• Studies multiple factors simultaneously.
• Examines interactions between factors.
• Identifies the best combination of factors for maximum or minimum effect.
• Minimizes experimental error by controlling extraneous variables.
• Reduces costs by minimizing the number of experiments.
• Saves time by studying multiple factors in one experiment.

6. Response Surface Methodology (RSM)

• Definition: A method to explore the relationship between input and output variables.
  • "Response Surface Method explore the relationship between input variable and output variable."
• Origin: Introduced by George E.P. Box and K.B. Wilson in 1951.
• Main Idea: Uses a sequence of designed experiments to obtain an optimal response.
• Components:
  • Central Composite Design (CCD)
  • Historical Design
  • Optimization Technique

7. Central Composite Design (CCD)

• Definition: A type of RSM design that combines center points and star points.
  • "It is a type of response surface methodology design that combines center point and star point."
• Structure: Includes factorial points, center points, and axial (star) points.
• Diagram:
  • X and Y axes representing two factors.
  • Center point at (0,0).
  • Factorial points at the corners of a square.
  • Star points along the axes at a distance α from the center.
• Star Points: Located at (α,0), (-α,0), (0,α), (0,-α).
• Factorial Points: Determined by the combination of positive and negative levels of the factors.

8. Historical Design (Retrospective Design)

• Definition: A research methodology that examines past events and data to draw conclusions about cause-and-effect relationships.
  • "Historical design also known as retrospective design it is a research methodology that examines past event past data analyze current record current document and conclusion to draw what cause and effect relationship."
• Types: Retrospective cohort study, case-control study, historical comparative study.
• Characteristics: Non-experimental, uses existing data, no manipulation of variables.
• Advantages: Cost-effective, time-efficient, provides insights into past trends.
• Limitations: Data quality issues, potential for confounding variables.
• Applications: Epidemiology, business, economic research, social sciences, medical research, policy evaluation.
• Usefulness: When experimental design is impractical or impossible, and when historical data is available.

9. Optimization Technique

• Definition: A mathematical method used to find the best solution among possible solutions to a biostatistical problem.
  • "Optimization technique a mathematical method used to find the best solution among possible solution to a biostatistical problem."
• Objective: To estimate parameters, select models, test hypotheses, make predictions, and analyze data.
• Example: Determining the optimal dose of a new medication to maximize efficacy and minimize side effects.
• Goal: Minimize side effects and maximize therapeutic efficacy.
• Applications: Identifying genetic markers, optimizing treatment regimens, improving patient outcomes.
• Software: R, Python, SAS, STATA.

10. Synthesis/Conclusion

The lecture provides a detailed overview of experimental designs, focusing on factorial designs and response surface methodology. It covers the theoretical foundations, practical applications, and step-by-step procedures for implementing these techniques. The key takeaway is that these methods are essential for researchers seeking to understand complex relationships between multiple variables and optimize outcomes in various fields, including medicine, engineering, and business. The lecture emphasizes the importance of controlling extraneous variables and using appropriate statistical tools to ensure the validity and reliability of experimental results.