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Chapter 6 Factorial Treatment Designs 1 Factorial Treatment Designs Factorial Treatment Designs all us to look at how multiple treatment factors affect our response variable AND how the treatments interact to affect our response variable. This is all about interactions. Table of means for a 2 X 2 factorial. Factor A Factor B Level B1 Level B2 Level A1 Level A2

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Chapter 6 Factorial Treatment Designs 2 Tensile strength (psi) of asphalt specimens. Aggregate Type (A) Compaction Method (B) Aggregate Means Static (B1) Kneading (B2) Silicious (A1) 68 60 64.0 Basalt (A2) 65 97 81.0 Compaction Means 66.5 78.5 Grand Mean 72.5
Chapter 6 Factorial Treatment Designs 3 SIMPLE EFFECTS - contrasts between levels of one factor at a single level of another factor. Example 1. The effect of aggregate type [i.e. the difference in tensile strength between Basalt (A2) and Silicious (A1)] when the static compaction method (B1) is used. 1 21 11 21 11 l = μ = 0 - 0 = 65 - 68 = -3 Example 2. The effect of aggregate type [i.e. the difference in tensile strength between Basalt (A2) and Silicious (A1)] when the Kneading compaction method (B2) is used. 2 22 12 22 12 l = μ - μ = 0 - 0 = 97 - 60 = 37 MAIN EFFECTS - contrasts between levels of one factor averaged over all levels of another factor. Example 3. The main effect of aggregate type (i.e. A2-A1) averaged over all levels of compaction method B. 3 2. 1. 2. 1. l = μ - μ = 0 - 0 = 81 - 64 = 17 Note that this main effect of aggregate type is also the 1 2 average of the two simple effects l and l 3 2. 1. 21 22 11 12 l = μ - μ = 1/2( μ - μ ) - 1/2( μ - μ ) 22 12 21 11 =1/2( μ - μ ) - 1/2( μ -μ ) 1 2 =1/2(l + l ) =1/2(-3 + 37) = 34/2 = 17

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Chapter 6 Factorial Treatment Designs 4 INTERACTION EFFECTS - differences between simple effects of a factor at different levels of the other factor. Example 4. The interaction of A and B can be written as AxB or simply AB. It can be thought of as the differences between the effect of A (i.e. A2-A1) in the different levels of B (i.e. B1 and B2) or conversely the differences between the effect of B (i.e. B2-B1) in the different levels of A (i.e. A1 and A2). Let’s look at each of these. The differences between the effect of A (i.e. A2-A1) in the different levels of B (i.e. B1 and B2) : 4 22 12 21 11 2 1 l = (μ - μ ) - (μ ) = l - l 22 12 21 11 = ( 0 - 0 ) - ( 0 - 0 ) = 37 - (-3) =40 So our estimate of the AB term is 40. The differences between the effect of A (i.e. A2-A1) in the different levels of B (i.e. B1 and B2) : 5 22 21 12 11 l = (μ - μ ) - (μ - μ ) 22 21 12 11 = ( 0 - 0 ) - ( 0 - 0 ) = (97-65) - (60-68) = 32 - (-8) =40 So our estimate of the AB term is 40 this way also. The point here is that you get the same answer, but that is just mathematics. One interpretation may be preferable to the other because of the system or the science.
Chapter 6 Factorial Treatment Designs 5 No Interaction Lines are parallel. Magnitude Slopes have same sign, but differ in magnitude. Response Slopes have different sign. Note, that one could have a slope of zero.

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Chapter 6 Factorial Treatment Designs 6 The Statistical Model for Two Treatment Factors Section 6.3 pp 181-183 Additivity and Factor Effects pp 182-183 An interaction is often referred to as a lack of additivity.
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