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1 Recall: One-Way ANOVA 1.Calculate and check residuals, e ij = O i - E i –plot residuals vs treatments –normal probability plot 2.Perform ANOVA and determine if there is a difference in the means 3.Identify which means are different using Tukey’s procedure: 4.Model: y ij = μ + α i + ε ij

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2 Blocking Creating a group of one or more people, machines, processes, etc. in such a manner that the entities within the block are more similar to each other than to entities outside the block. Balanced design: each treatment appears in each block. Model: y ij = μ + α i + β j + ε ij

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3 Example: Robins Air Force Base uses CO 2 to strip paint from F-15’s. You have been asked to design a test to determine the optimal pressure for spraying the CO 2. You realize that there are five machines that are being used in the paint stripping operation. Therefore, you have designed an experiment that uses the machines as blocking variables. You emphasized the importance of balanced design and a random order of testing. The test has been run with these results (values are minutes to strip one fighter):

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4 ANOVA: One-Way with Blocking 1.Construct the ANOVA table Where,

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5 Blocking Example Your turn: fill in the blanks in the following ANOVA table (from Excel): 2.Make decision and draw conclusions: ANOVA Source of VariationSSdfMSFP-valueF crit Rows89.733244.8678.4920.01054.458968 Columns77.733____________0.0553_______ Error42.26785.2833 Total209.73___

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6 Two-Way ANOVA Blocking is used to keep extraneous factors from masking the effects of the treatments you are interested in studying. A two-way ANOVA is used when you are interested in determining the effect of two treatments. Model: y ijk = μ + α i + β j + ( α β) ijk + ε ij

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7 Two-Way ANOVA w/ Replication Your fame as an experimental design expert grows. You have been called in as a consultant to help the Pratt and Whitney plant in Columbus determine the best method of applying the reflective stripe that is used to guide the Automated Guided Vehicles (AGVs) along their path. There are two ways of applying the stripe (paint and coated adhesive tape) and three types of flooring (linoleum and two types of concrete) in the facilities using the AGVs. You have set up two identical “test tracks” on each type of flooring and applied the stripe using the two methods under study. You run 3 replications in random order and count the number of tracking errors per 1000 ft of track. The results are as follows:

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8 Two-Way ANOVA Example Analysis is the same as with blocking, except we are now concerned with interaction effects

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9 Two-Way ANOVA

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10 Your Turn Fill in the blanks … What does this mean? ANOVA Source of VariationSSdfMSFP-valueF crit Sample0.43561_____2.39760.147484.74722 Columns4.4822.2412.330.001233.88529 Interaction0.9644___0.4822_____0.111043.88529 Within2.18___0.1817 Total8.0617

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11 What if Interaction Effects are Significant? For example, suppose a new test was run using different types of paint and adhesive, with the following results: ANOVA Source of VariationSSdfMSFP-valueF crit Sample0.10910.10891.0710.32114.7472 Columns1.9620.989.6390.00323.8853 Interaction2.83121.415613.920.00073.8853 Within1.22120.1017 Total6.1217

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12 Understanding Interaction Effects Graphical methods: –graph means vs factors –identify where the effect will change the result for one factor based on the value of the other.

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