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Design Lecture: week3 HSTS212
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One Factor Analysis
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ANOVA Table
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Example
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Box plot
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ANOVA
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Randomized Block Design
HSTS312
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Randomized Block Design
1. Experimental Units (Subjects) Are Assigned Randomly within Blocks Blocks are Assumed Homogeneous 2. One Factor or Independent Variable of Interest 2 or More Treatment Levels or Classifications 3. One Blocking Factor HSTS312
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The Randomized Complete Block Design
Nuisance factor: a design factor that probably has an effect on the response, but we are not interested in that factor. If the nuisance variable is known and controllable, we use blocking If the nuisance factor is known and uncontrollable, sometimes we can use the analysis of covariance to remove the effect of the nuisance factor from the analysis
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If the nuisance factor is unknown and uncontrollable (a “lurking” variable), we hope that randomization balances out its impact across the experiment Sometimes several sources of variability are combined in a block, so the block becomes an aggregate variable
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We wish to determine whether 4 different tips produce different (mean) hardness reading on a Rockwell hardness tester Assignment of the tips to an experimental unit; that is, a test coupon Structure of a completely randomized experiment The test coupons are a source of nuisance variability Alternatively, the experimenter may want to test the tips across coupons of various hardness levels The need for blocking
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To conduct this experiment as a RCBD, assign all 4 tips to each coupon
Each coupon is called a “block”; that is, it’s a more homogenous experimental unit on which to test the tips Variability between blocks can be large, variability within a block should be relatively small In general, a block is a specific level of the nuisance factor A complete replicate of the basic experiment is conducted in each block A block represents a restriction on randomization All runs within a block are randomized
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Suppose that we use b = 4 blocks:
Once again, we are interested in testing the equality of treatment means, but now we have to remove the variability associated with the nuisance factor (the blocks)
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Statistical Analysis of the RCBD
Suppose that there are a treatments (factor levels) and b blocks A statistical model (effects model) for the RCBD is is an overall mean, i is the effect of the ith treatment, and j is the effect of the jth block ij ~ NID(0,2)
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Model
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Hypothesis
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Randomized Block Design
Factor Levels: (Treatments) A, B, C, D Experimental Units Treatments are randomly assigned within blocks Block 1 A C D B Block 2 Block 3 . Block b Are the mean training times the same for 3 different methods? 9 subjects 3 methods (factor levels) HSTS312 78
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Randomized Block F-Test
1. Tests the Equality of 2 or More (p) Population Means 2. Variables One Nominal Independent Variable One Nominal Blocking Variable One Continuous Dependent Variable Note: There is one dependent variable in the ANOVA model. MANOVA has more than one dependent variable. Ask, what are nominal & interval scales? HSTS312
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Randomized Block F-Test Assumptions
1. Normality Probability Distribution of each Block-Treatment combination is Normal 2. Homogeneity of Variance Probability Distributions of all Block-Treatment combinations have Equal Variances HSTS312
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Manual ANOVA
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Table
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Example
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Table
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Balance Incomplete Block Designs
May not run all the treatment combinations in each block. Randomized incomplete block design (BIBD) Any two treatments appear together an equal number of times. There are a treatments and each block can hold exactly k (k < a) treatments. For example: A chemical process is a function of the type of catalyst employed.
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Balanced Incomplete Block Design (BIBD)
Notation a-number of treatments k-number of treatments/block r-number of treatment reps b-number of blocks N=ar=bk r(k-1)=l(a-1)
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Statistical Analysis of the BIBD
a treatments and b blocks. Each block contains k treatments, and each treatment occurs r times. There are N = ar = bk total observations. The number of times each pair of treatments appears together in the same block is The statistical model for the BIBD is
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Example
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The sum of squares
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The testing statistic for testing equality of the treatment effects:
The degree of freedom: Treatments(adjusted): a – 1 Error: N – a – b – 1 The testing statistic for testing equality of the treatment effects: ANOVA table (see Table 4.23) Example 4.5
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Table
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THE LATIN SQUARE DESIGN
used to eliminate two nuisance sources of variability Latin square, is a square containing p rows and p columns contains one of the p letters that corresponds to the treatments each letter occurs once and only once in each row and column
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4 x 4 A B D C B C A D C D B A D A C B
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The model completely additive
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Example
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SPSS A 1 1 8 C B 1 3 4 D 1 4 6 E 1 5 4 B 2 1 7 E 2 2 2 A 2 3 9 C 2 4 8 D 2 5 2 D 3 1 1 A 3 2 7 C E 3 4 6 B 3 5 3 C 4 1 7 D 4 2 3 E 4 3 1 B 4 4 6 A 4 5 8 E 5 1 3 B 5 2 8 D 5 3 5 A C 5 5 8
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ANOVA TABLE
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