Full and Fractional Factorial Designs
Full factorial designs are a foundational approach in experimental design when you want to study the effects of multiple factors simultaneously. In a full factorial design, you run an experiment at every possible combination of factor levels, ensuring that you can estimate all main effects and interactions. Consider a simple example with two factors, each at two levels (often called a 2x2 design). Suppose you want to test the effect of fertilizer type (A: organic or synthetic) and watering frequency (B: daily or every other day) on plant growth. You would run experiments for all possible combinations:
- Fertilizer: organic, Watering: daily;
- Fertilizer: organic, Watering: every other day;
- Fertilizer: synthetic, Watering: daily;
- Fertilizer: synthetic, Watering: every other day.
This gives you four unique treatment groups, allowing you to measure not only the individual effects of fertilizer and watering, but also whether their combination produces a different outcome than expected from their separate effects.
As the number of factors in an experiment increases, the number of experimental runs required for a full factorial design grows exponentially. The following table shows how quickly the number of runs increases for full versus fractional factorial designs as you add more factors (each at two levels):
Fractional factorial designs allow you to run only a subset of all possible combinations, dramatically reducing the number of experiments needed as the number of factors grows.
The core idea behind fractional factorial designs is to trade some information for efficiency. By running only a carefully chosen subset of all possible combinations, you can still estimate the main effects of each factor and some interactions, while ignoring higher-order interactions that are assumed to be negligible. This approach is especially useful when resource constraints make full factorial designs impractical. Mathematically, fractional factorials use the structure of the full factorial to select runs that maximize the information gained per experiment, but you must be aware that some effects may be confounded or indistinguishable from one another as a result.
1. Which of the following is a main advantage of fractional factorial designs?
2. In a 2^3 full factorial design (three factors, each at two levels), how many runs are required?
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Full factorial designs are a foundational approach in experimental design when you want to study the effects of multiple factors simultaneously. In a full factorial design, you run an experiment at every possible combination of factor levels, ensuring that you can estimate all main effects and interactions. Consider a simple example with two factors, each at two levels (often called a 2x2 design). Suppose you want to test the effect of fertilizer type (A: organic or synthetic) and watering frequency (B: daily or every other day) on plant growth. You would run experiments for all possible combinations:
- Fertilizer: organic, Watering: daily;
- Fertilizer: organic, Watering: every other day;
- Fertilizer: synthetic, Watering: daily;
- Fertilizer: synthetic, Watering: every other day.
This gives you four unique treatment groups, allowing you to measure not only the individual effects of fertilizer and watering, but also whether their combination produces a different outcome than expected from their separate effects.
As the number of factors in an experiment increases, the number of experimental runs required for a full factorial design grows exponentially. The following table shows how quickly the number of runs increases for full versus fractional factorial designs as you add more factors (each at two levels):
Fractional factorial designs allow you to run only a subset of all possible combinations, dramatically reducing the number of experiments needed as the number of factors grows.
The core idea behind fractional factorial designs is to trade some information for efficiency. By running only a carefully chosen subset of all possible combinations, you can still estimate the main effects of each factor and some interactions, while ignoring higher-order interactions that are assumed to be negligible. This approach is especially useful when resource constraints make full factorial designs impractical. Mathematically, fractional factorials use the structure of the full factorial to select runs that maximize the information gained per experiment, but you must be aware that some effects may be confounded or indistinguishable from one another as a result.
1. Which of the following is a main advantage of fractional factorial designs?
2. In a 2^3 full factorial design (three factors, each at two levels), how many runs are required?
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