Fractional Factorials (Statistics). Design Identification

Modulo 2 means that the interaction is considered significant if the number of positive signs in the defining interaction is even. In this case, there are 6 positive signs, which is even, so the interaction is significant.
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Northbysouth
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Homework Statement


Anand, Bhadkamkar, and Moghe (1995) used a fractional factorial design to determine which of the six possible factors influenced the determination of manganese in cast iron. The six factors and their levels follow:

A-Titration speed ; Medium ; Fast
B-Dissolution time ; 20 ; 30
C-AgNO3 addition ; 20 ; 10
D-Persulfate addition ; 2 ; 3
E-Volume HMnO4 ; 100 ; 150
F-Sodium arsenite ; 0.10 ; 0.15

I have attached a data table

X1 is Titration speed
X2 is Dissolution time
etca) Identify this design

b) Give the defining interaction or interactions

Homework Equations


The Attempt at a Solution



I've read over this material in my textbook and viewed the lecture video that came with it and I'm still quite confused.

a) is it just asking for:

26-3

The 2 refers to the levels (-1 and +1+

The 6 refers to the number of factors

The -3 to how there are 8 rows of data,

2-3=1/8

64/8 = 8

b) I have no idea what it is asking for with the defining interaction or interactions

My professor sent this email out:

I = BCDE = -ADE = -ABC = -BDF = -CEF = ABEF = ACDF. Since I am giving you the defining relation, please state the alias structure for main effect A. You need to go through and add A to each part of the defining interaction to determine the answer modulo 2.

But I'm still confused.

Any clarification would be appreciated
 

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a) The design used in this study is a 2^6-3 fractional factorial design. This means that there are 6 factors with 2 levels each, and the design is constructed to be a fraction of the full factorial design. The -3 indicates that there are 8 rows of data in the design, which is 1/8 of the full factorial design.

b) The defining interaction in this design is I = BCDE = -ADE = -ABC = -BDF = -CEF = ABEF = ACDF. This means that the interaction between factors B, C, D, and E is the same as the interactions between factors A and B, A and D, A and C, B and D, C and E, and A, B, E, and F. This defining interaction can also be written as A*B*C*D*E*F, which means that all 6 factors are involved in this interaction. This is known as a full factorial interaction.
 

1. What are fractional factorial designs in statistics?

Fractional factorial designs in statistics are experimental designs that are used to study the effects of multiple factors on a response variable. They are a type of design of experiments that aim to reduce the number of experimental runs needed while still allowing for the identification of important factors and their interactions.

2. How are fractional factorial designs identified?

Fractional factorial designs are identified through the use of a specific notation, also known as the defining relation, which describes the structure of the design. This notation includes the number of factors, the number of levels for each factor, and the alias structure, which identifies which factors are confounded with each other.

3. What are the advantages of using fractional factorial designs?

One of the main advantages of using fractional factorial designs is that they allow for the study of multiple factors simultaneously, which can save time and resources compared to studying each factor individually. They also provide information on the interactions between factors, which can be useful in understanding the underlying relationships between variables.

4. What are the limitations of using fractional factorial designs?

One limitation of using fractional factorial designs is that they may not be able to capture all possible interactions between factors. This is because some interactions may be confounded with each other and cannot be separated out in the design. Additionally, these designs may not be suitable for studying factors with nonlinear relationships.

5. How are fractional factorial designs different from full factorial designs?

Fractional factorial designs differ from full factorial designs in that they use fewer experimental runs to study multiple factors. Full factorial designs require all possible combinations of factor levels to be tested, while fractional factorial designs only test a subset of these combinations, leading to a reduction in the number of runs needed. However, this also means that fractional factorial designs may not be able to provide as much information as full factorial designs.

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