Chapter 3 - Chapter 3 Factorial Treatment Structure and...

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Chapter 3 Factorial Treatment Structure and Interaction STAT 332 Spring 2017
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Motivating example: Darts A student wants to find out how accurately he can throw a dart. He suspects that it depends on which hand he uses ( Hand ) and on how far from the target he is standing ( Distance ). He used two levels of the factor Distance : Near and Far , and two levels of the factor Hand : Left and Right . In random order, he threw six darts at a dartboard under each of the four treatments, and measured the distance each dart landed from the bull’s eye in inches. STAT 332 Spring 2017
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Unlike the experiments we have seen so far, this experiment has two factors of interest. This type of experiment is sometimes referred to as a two-factor experiment There are two hypothesis tests. Each of those hypotheses asks whether the mean of the response variable is the same for each of the treatment levels. In the previous chapter, we know that any excess variation gets in the way. We used blocking to remove or avoid extra variation in designing experiments. In a two-factor experiment, both factors have an effect on the responses. By removing the effects of one factor, we should be able to see and assess the effects of the other more easily. STAT 332 Spring 2017
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One-way ANOVA Consider a complete randomized design from the previous chapter. This design has one factor of interest and the model is written as: 𝑌 ?? = 𝜇 + 𝜏 ? + ? ?? , ? ?? ~ ?(0, 𝜎) where ? = 1, … , ? and ? = 1, … ,? ?=1 𝑡 𝜏 ? = 0 STAT 332 Spring 2017
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Two-way ANOVA Now that we have two factors of interest, the CRD model can be extended to reflect the effects of both models: 𝑌 ??? = 𝜇 + 𝜏 ? + 𝛾 ? + ? ??? , ? ??? ~ ?(0, 𝜎) where ? = 1,… ,? , ? = 1,… , 𝑔 and ? = 1, … , ? ?=1 𝑡 𝜏 ? = 0 ?=1 𝑔 𝛾 ? = 0 Why the k subscript? The student recorded six trials at each treatment, so the subscript k denotes the different observations at each combination of the two factors. The error has three subscripts because we associate a possibly different error with each observation. STAT 332 Spring 2017
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𝑌 ??? = 𝜇 + 𝜏 ? + 𝛾 ? + ? ??? 𝜇 is the overall mean, 𝜏 ? is the i -th (main) effect of Factor 1 used to represent the increase (or decrease) from the overall average response, 𝛾 ? is the j -th (main) effect of Factor 2 used to represent the increase (or decrease) from the overall average response, and ? ??? is the independent random errors from a Normal distribution with zero mean and constant variance STAT 332 Spring 2017
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The model appears very similar to the CRBD model. In fact the estimates of the parameters are 𝜇 = 𝑦 +++ , 𝜏 ? = 𝑦 ?++ 𝑦 +++ , 𝛾 ? = 𝑦 +?+ 𝑦 +++ 𝜎 2 = ?=1 𝑡 ?=1 𝑔 ?=1 𝑟 𝑦 ??? 𝑦 ?++ 𝑦 +?+ + 𝑦 +++ 2 ?𝑔? − ? − 𝑔 + 1 STAT 332 Spring 2017
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Two hypotheses In general, we would like to know whether factors (both) have any effect on the response The two hypotheses of interest are: ? 0 :𝜏 1 = 𝜏 2 = … = 𝜏 𝑡 = 0 ?
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