留学生辅导 Math 558 Lecture #20

Math 558 Lecture #20

Factorial Experiments
In a factorial experiment the effects of two or more factors are tested at two more levels each. These effects are tested simultaneously. The treatments consist of all possible combinations of the levels of different factors.

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Example; Text pg 7, 75 “An experiment was conducted to compare three different varieties of rye-grass in combination with four quantities of nitrogen fertilizer. The response measured was the total weight of dry-matter harvested from each plot. The three varieties of rye-grass were called Cropper, Melle and Melba. The four amounts of fertilizer were 0 kg/ha, 80 kg/ha, 160 kg/ha and 240 kg/ha.” The plan according to which the treatments are applied to the experimental units is given in the next slide.

Rye grass example (Text Pg 7, 75)
Mella Mella 0 160 240 160 80 0
160 80 80 0 10 80 80 0 160 240 0 240 240 240 0 80 240 160

Rye grass example (Text Pg 7, 75)
Here the twelve treatments are all combinations of the levels of the following two treatment factors with two factor levels.
Cultivar (C) Cropper, Melle, Melba Fertilizer (F) 0, 80, 160, 240 kg/ha
The treatments may be labelled 1, . . . , 12 according to the following table.
F C 0 80 160 240
Cropper 1 2 3 4 Melle 5 6 7 8
Melba 9 10 11 12

A factorial design
By a factorial design, we mean that in each complete trial or replicate of the experiment all possible combinations of the lev- els of the factors are investigated. For example, if there are a levels of factor A and b levels of factor B, each replicate con- tains all ab treatment combinations. When factors are arranged in a factorial design, they are often said to be crossed. (Mont- gomery, 2012 pg 184)

Example: Sugar Beet, Cox and Cochran pg 148
Consider an experiment on sugar beet with two factors. Factor 1. Amount of fertiliser
Factor 2. Depth of Winter ploughing
Fertilizer 0, cwt sulphate of ammonia Depth 7 inch, 11 inch
The treatments are the combinations of all factor levels given below.
Treatments 0, 7 inch 3cwt, 7 inch 0, 11 inch 3cwt, 11inch
yields of sugar(cwt/acre 40.9

Example: Sugar Beet, Cox and Cochran pg 148
11in Mean response to depth
fertilizer Mean 40.9 47.8 44.4 42.4 50.2 46.3
Response to the fertilizer (3cwt) +6.9
Results The effect of nitrogen increased the yield by 6.9 cwt for the depth of 7 inches and by 7.8 cwt for the depth of 11 inches. These effects are called simple effects of the fertilizer. The simple effects of the increasing the depth are +1.5 for 0 nitrogen and +2.4 for 3cwt of nitrogen.

Independent factors and their effects
Consider a two factor experiment. If the effect of a factor is same at all levels of the other factor, the factors are independent of each other. Considering the previous example we notice that the response to nitrogen for the two levels of depth are 6.9 and 7.8 cwt which are the estimates of two simple effects and also assume that they differ only by experimental errors. Similarly, the effects of two levels of depth are the same, in the presence and absence of the fertilizer (differing by the experimental error only). We can assume the two factors to be independent. Based on this assumption define the average
6.9 + 7.8 = 7.4 = Main Effect of the fertilizer 2
2.4 + 1.5 = 1.9 = Main effect of the depth 2

Definitions
Definition 1
A “main effect” is the effect of one of the factors on the response, ignoring the effects of all other factors.
Definition 2
A statistical interaction occurs when the effect of one factor on the response changes depending on the levels of the other factor.

Independent factors and their effects
Hence if two factors, the fertilizer and the dept are independent we can discuss our results as
” The application of 3cwt of nitrogen increased the yield by 7.4 cwt where as the increase in ploughing depth increased the yield by 1.9 cwt.” We do need to support our assumption of independence by a statistical test in addition to the contextual knowledge. We can test the difference of responses by a t-test. If the difference is significant the assumption of independence is rejected. This difference is the interaction between the fertilizer and the depth of ploughing.

Calculation of the main effects and interactions 2×2 Factorial Experiment
We will use the notation by Yates for these calculations. Let A and B denote the factors and a and b, denote the second levels of these factors. The first level will be indicated by the absence of the corresponding letter. The symbol (ab) will denote the mean of of all observations which receive the treatment combination ab. The letters A, B, and AB, when they refer to numbers, will present, respectively, the main effects of A and B and the interaction by AB.

Calculation of the main effects and interactions 2×2 Factorial Experiment
A = 12[(ab)−(b)+(a)−(1)]
B = 12[(ab)+(b)−(a)−(1)] AB = 12[(ab)−(b)−(a)+(1)]
The simple effects can be calculated as
(a) − (1) = simple effect of A when b is at the first level
(ab) − (b) = simple effect of A when b is at the second level

A simple example: Case 1
Factor A has two levels and Factor B has two levels. when Factor A is at level 1, Factor B changes by 3 units. When Factor A is at level 2, Factor B again changes by 3 units. Similarly, when Factor B is at level 1, Factor A changes by 2 units. When Factor B is at level 2, Factor A again changes by 2 units. There is no interaction. The change in the true average response when the level of either factor changes from 1 to 2 is the same for each level of the other factor. In this case, changes in levels of the two factors affect the true average response separately, or in an additive manner.

A simple example: Case 2
When Factor A is at level 1, Factor B changes by 3 units but when Factor A is at level 2, Factor B changes by 6 units. When Factor B is at level 1, Factor A changes by 2 units but when Factor B is at level 2, Factor A changes by 5 units. The change in the true average response when the levels of both factors change simultaneously from level 1 to level 2 is 8 units, which is much larger than the separate changes suggest. In this case, there is an interaction between the two factors, so the effect of simultaneous changes cannot be determined from the individual effects of the separate changes. Change in the true average response when the level of one factor changes depends on the level of the other factor. You cannot determine the separate effect of Factor A or Factor B on the response because of the interaction.

A simple example: Case 2

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