STT 511-STT411: DESIGN OF EXPERIMENTS AND ANALYSIS OF VARIANCE Dr. Cuixian Chen Chapter 14: Nested and Split-Plot Designs Design & Analysis of Experiments 8E 2012 Montgomery 1 Chapter 14

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Chapter 14Design and Analysis of Experiments 8E 2012 Montgomery 3 Design of Engineering Experiments – Nested and Split-Plot Designs  Text reference, Chapter 14  These are multifactor experiments that have some important industrial applications  There are many variations of these designs – we consider only some basic situations

Chapter 14Design and Analysis of Experiments 8E 2012 Montgomery 4 Two-Stage Nested Design  In a nested design, the levels of one factor (B) is similar to but not identical to each other at different levels of another factor (A)  Consider a company that purchases material from three suppliers  The material comes in batches  Is the purity of the material uniform?  Experimental design  Select four batches at random from each supplier  Make three purity determinations from each batch In some two-factor experiments the level of one factor, say B, is not “cross” or “cross classified” with the other factor, say A, but is “NESTED” with it. The levels of B are different for different levels of A. For example: 2 Areas (Study vs Control) 4 sites per area, each with 5 replicates. There is no link from any sites on one area to any sites on another area.

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Chapter 14Design and Analysis of Experiments 8E 2012 Montgomery 6 Two-Stage Nested Design Statistical Model and ANOVA i indexes “A” (often called the “major factor”) (i)j indexes “B” within “A” (B is often called the “minor factor”) (ij)k indexes replication

Chapter 14Design and Analysis of Experiments 8E 2012 Montgomery 7 Two-Stage Nested Design Example 14.1 Three suppliers, four batches (selected randomly) from each supplier, three samples of material taken (at random) from each batch Experiment and data, Table 14.3 Data is coded JMP and Minitab balanced ANOVA will analyze nested designs Mixed model, assume restricted form

Chapter 14 Design and Analysis of Experiments 8E 2012 Montgomery 8 Questions to answer: 1. Are the suppliers different? 2. Are the batches within each supplier uniform? H 01 : τ i =0 for i=1,2,3 v.s. H a1 : τ i ≠0 for some i in{1,2,3} H 02 : β j(i) =0 for i=1,2,3 and j=1,2,3,4 v.s. H a2 : β j(i)≠0 for some i in{1,2,3}, and j={1,2,3,4}

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Chapter 14Design and Analysis of Experiments 8E 2012 Montgomery 11 Minitab Analysis

JMP Analysis (REML estimates of variance components) Chapter 14Design and Analysis of Experiments 8E 2012 Montgomery 12

Chapter 14Design and Analysis of Experiments 8E 2012 Montgomery 13 Practical Interpretation – Example 14.1  There is no difference in purity among suppliers, but significant difference in purity among batches (within suppliers)  What are the practical implications of this conclusion?  Examine residual plots – plot of residuals versus supplier is very important (why?)  What if we had incorrectly analyzed this experiment as a factorial? (see Table 14.5)  Estimation of variance components (ANOVA method)

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Nested Experiments  In some two-factor experiments the level of one factor, say B, is not “cross” or “cross classified” with the other factor, say A, but is “NESTED” with it.  The levels of B are different for different levels of A.  For example: 2 Areas (Study vs Control)  4 sites per area, each with 5 replicates.  There is no link from any sites on one area to any sites on another area.

 That is, there are 8 sites, not 2. Study Area (A) Control Area (B) S1(A) S2(A) S3(A) S4(A) S5(B) S6(B) S7(B) S8(B) X X X X X X X X X X X X X X X X X = replications Number of sites (S)/replications need not be equal with each sites. Analysis is carried out using a nested ANOVA not a two-way ANOVA.

 A Nested design is not the same as a two-way ANOVA which is represented by: A1 A2 A3 B1 X X X X X X X X X X X X X X X B2 X X X X X X X X X X X X X X X B3 X X X X X X X X X X X X X X X Nested, or hierarchical designs are very common in environmental effects monitoring studies. There are several “Study” and several “Control” Areas.

2 nd example on Nested design a=3, b=4, n=3; 3 Areas, 4 sites within each area, 3 replications per site, total of (M.m.n = 36) data points M 1 M 2 M 3 Areas Sites Repl

ANOVA Table for Example Nested ANOVA: Observations versus Area, Sites Source DF SS MS F P Area Sites (A)B ** Error Total What are the “proper” ratios? E(MS A ) =  2 + V B(A) + V A E(MS (A)B )=  2 + V B(A) E(MS error ) =  2 = MS A /MS B(A) = MS B(A) /MS error

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Chapter 14Design and Analysis of Experiments 8E 2012 Montgomery 21 Example 14.2 Nested and Factorial Factors

Chapter 14Design and Analysis of Experiments 8E 2012 Montgomery 22 Example 14.2 – Minitab Analysis

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Chapter 14Design and Analysis of Experiments 8E 2012 Montgomery 25 The Split-Plot Design  Text reference, Section 14.4 page 621  The split-plot is a multifactor experiment where it is not possible to completely randomize the order of the runs  Example – paper manufacturing  Three pulp preparation methods  Four different temperatures  Each replicate requires 12 runs  The experimenters want to use three replicates  How many batches of pulp are required?

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Chapter 14Design and Analysis of Experiments 8E 2012 Montgomery 27 The Split-Plot Design  Pulp preparation methods is a hard-to-change factor  Consider an alternate experimental design:  In replicate 1, select a pulp preparation method, prepare a batch  Divide the batch into four sections or samples, and assign one of the temperature levels to each  Repeat for each pulp preparation method  Conduct replicates 2 and 3 similarly

Chapter 14Design and Analysis of Experiments 8E 2012 Montgomery 28 The Split-Plot Design  Each replicate (sometimes called blocks) has been divided into three parts, called the whole plots  Pulp preparation methods is the whole plot treatment  Each whole plot has been divided into four subplots or split-plots  Temperature is the subplot treatment  Generally, the hard-to-change factor is assigned to the whole plots  This design requires only 9 batches of pulp (assuming three replicates)

Chapter 14Design and Analysis of Experiments 8E 2012 Montgomery 29 The Split-Plot Design Model and Statistical Analysis There are two error structures; the whole-plot error and the subplot error

Chapter 14Design and Analysis of Experiments 8E 2012 Montgomery 30 Split-Plot ANOVA Calculations follow a three-factor ANOVA with one replicate Note the two different error structures; whole plot and subplot

Chapter 14Design and Analysis of Experiments 8E 2012 Montgomery 31 Alternate Model for the Split-Plot

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“Inadvertent” Split-Plot and CRD Analysis Chapter 14Design and Analysis of Experiments 8E 2012 Montgomery 33

Chapter 14Design and Analysis of Experiments 8E 2012 Montgomery 34 Variations of the basic split-plot design More than two factors – see page 627 A & B (gas flow & temperature) are hard to change; C & D (time and wafer position) are easy to change.

Chapter 14Design and Analysis of Experiments 8E 2012 Montgomery 35 Unreplicated designs and fractional factorial design in a split-plot framework

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Chapter 14 Design and Analysis of Experiments 8E 2012 Montgomery 42  A split-split-plot design  Two randomization restrictions present within each replicate

Chapter 14Design and Analysis of Experiments 8E 2012 Montgomery 43 The strip-split-plot design The “strips” are just another set of whole plots