1 Statistical Analysis Professor Lynne Stokes Department of Statistical Science Lecture 20 Nested Designs and Analyses

2 Torque Experiment Crossed Factors Levels have the same physical meaning regardless of the other levels with which they are combined in the experiment Factors Shaft Alloy Sleeve Porosity Lubricant Levels Steel, Aluminum Porous, Nonporous Lub 1, Lub 2, Lub 3, Lub 4 Can be fixed or random effects Can be fixed or random effects MGH Exhibit 11.1

3 Torque Experiment : Crossed Factors Alloy Porosity Lubricant Steel Aluminum Porous Nonporous 1234

4 Torque Experiment : Crossed Factors Alloy Porosity Lubricant Steel Aluminum Porous Nonporous 1234 MGH, Fig. 11.1

5 Lawnmower Cutoff Times Experiment Nested Factors Unique levels of a factor for each level or combination of levels of one or more other factors Factors Manufacturer Lawnmower Speed Levels A, B 1, 2, 3, 4, 5, 6 High, Low Crossed Nested Crossed Usually random effects Usually random effects MGH Exhibit 11.2

6 High Low AB Lawnmower Cutoff Times : Nested Factors Speed Manufacturer Lawnmower MGH, Fig. 11.3

7 Polyethylene Density Experiment Polyethylene pellets heated, melted, pressed into a plaque Disks cut from plaques, density of each polyethylene disk then measured Two shifts of technicians Each shift prepares two plaques 4 disks cut from each plaque Factors ShiftPlaque Disk 1, 2 1, 21, 2, 3, 4 Crossed or Nested ? Fixed or Random ? CrossedNested Nested in Shift in Plaque

8 Polyethylene Density Experiment Shift Plaque Disk MGH, Fig. 11.2

9 Polyethylene Density Experiment Polyethylene pellets heated, melted, pressed into a plaque Disks cut from plaques, density of Each polyethylene disk then measured Two shifts of technicians Each shift prepares two plaques 4 disks cut from each plaque Factors ShiftPlaque Disk 1, 2 1, 21, 2, 3, 4 FixedRandom Random

10 Hierarchically Nested Designs Each factor is progressively nested within the levels of the preceding factor Design is usually balanced Equal number of levels of one factor within levels of the preceding factor More information is obtained on factors lower in the hierarchy than those higher Source df A a - 1 B(A) a(b - 1) C(AB)ab(c - 1)

11 Polyethylene Density Experiment Shift 12 MGH, 11.2 df (Shift) = a-1 = 1

12 Polyethylene Density Experiment Shift Plaque df (Plaques) = a(b-1) = 2(1) = 2

13 Polyethylene Density Experiment Shift Plaque Disk df (Disks) = ab(c-1) = 2 x 2 x (3) =

14 Hierarchically Nested Designs Establish the hierarchy of factors Select, randomly if appropriate, the levels to be nested with each level of the preceding factor Randomize the test sequence or the assignment of combinations to experimental units Each factor is progressively nested within the preceding factors

15 Fertilizers can be applied to individual fields; Insecticides must be applied to an entire farm from an airplane Fertilizers can be applied to individual fields; Insecticides must be applied to an entire farm from an airplane Agricultural Field Trial Investigate the yield of a new variety of crop Factors Insecticides Fertilizers Experimental Units Farms Fields within farms Experimental Design ?

16 Agricultural Field Trial Insecticides applied to farms Insecticides applied to farms One-factor ANOVA One-factor ANOVA Main effect: Insecticides Main effect: Insecticides MSE: Farm-to-farm variability MSE: Farm-to-farm variability Farms

17 Agricultural Field Trial Fertilizers applied to fields Fertilizers applied to fields One-factor ANOVA One-factor ANOVA Main Effect: Fertilizers Main Effect: Fertilizers MSE: Field-to-field variability MSE: Field-to-field variability Fields

18 Agricultural Field Trial Insecticides applied to farms, fertilizers to fields Insecticides applied to farms, fertilizers to fields Two sources of variability Two sources of variability Insecticides subject to farm-to-farm variability Insecticides subject to farm-to-farm variability Fertilizers and insecticides x fertilizers subject to field-to-field variability Fertilizers and insecticides x fertilizers subject to field-to-field variability Farms Fields

19 Split-Plot Design Two hierarchically nested factors, with additional crossed factors occurring within levels of the nested factor Two sizes of experimental units, one nested within the other, with crossed factors applied to the smaller units An Experiment Can Have Either of these Features

20 Split-Plot Design Whole-Plot Experiment : Whole-Plot Factor = A Level A 1 Level A 2 Level A 1

21 Split Plot Designs Analysis of Variance Table

22 Split-Plot Design Split-Plot Experiment : Split-Plot Factor = B Level A 1 Level A 2 Level A 1 B2B2 B1B1 B2B2 B1B1 B1B1 B1B1 B2B2 B2B2 B1B1 B2B2 B2B2 B1B1 B2B2 B1B1 B1B1 B2B2

23 Split Plot Designs Analysis of Variance Table

24 Agricultural Field Trial

25 Agricultural Field Trial Insecticide 2 Insecticide 1

26 Agricultural Field Trial Fert B Fert A Fert B Insecticide 2 Insecticide 1

27 Agricultural Field Trial Whole Plots = Farms Split Plots = Fields Large Experimental Units Small Experimental Units

28 Agricultural Field Trial Whole Plots = Farms Split Plots = Fields Large Experimental Units Small Experimental Units Whole-Plot Factor = Insecticide Whole-Plot Error = Whole-Plot Replicates Split-Plot Factor = Fertilizer Split-Plot Error = Split-Plot Replicates

29 Connector Pin Experiment FactorLevels Vendor Wire1, 2 EnvironmentAmbient, High Humidity Heat TreatmentYes, No

30 Connector Pin Experiment FactorLevels Vendor Wire1, 2 EnvironmentAmbient, High Humidity Heat TreatmentYes, No Whole Plot : Reel of Wire Split Plot : 6” Samples of Wire

31 Connector Pin Split-Plot Design Vendor #1Vendor #2 Vendor #1 Amb, Yes High, Yes Amb, No High, No Amb, Yes High, Yes Amb, No High, No Amb, Yes High, Yes Amb, No High, No Amb, Yes High, Yes Amb, No High, No Vendor Error: Reel-to-Reel VariationEnv., Heat Error: Sample-to-Sample Variation Reel #1Reel #2Reel #3Reel #4

32 Constructing Split-Plot Designs Identify the whole plots and the factors to be assigned to them Randomly select a whole plot Randomly assign a whole-plot factor combination to the whole plot Continue until all combinations have been assigned Identify the split plots and the factors to be assigned to them Randomly select a split plot Randomly assign a split-plot factor combination to the split plot Continue until all combinations have been assigned

33 Cylinder Wear Study Cylinder Wall Piston Piston Ring Air-Fuel Mixture Air-Fuel InletExhaust Spark Plug MGH, Fig. 11.7

34 Cylinder Wear Study Factors Levels Piston Rings (A)Type 1, Type 2 Engine Oil (B)Oil A, Oil B Engine Speed (C)1500, 3000 rpm Intake Temperature (D)30, 90 o F Air-Fuel Mixture (E)Lean, Rich Fuel Oxygen Content (F)2.5, 7.5 %

35 Cylinder Wear Study MGH Table 11.6

36 Restricted Randomization : Split-Plot Design MGH Table 11.7

37 Restricted Randomization Sometimes Randomization is Restricted Cost Difficulty of experimentation Inability to (or failure to) reset operating conditions, machine settings Split plot experiments are an example Whole plot factors must be assigned to the whole plots Split plot factors can be randomized over any of the split plots – Useful to have replicate or repeat runs

38 Restricted Randomization : Split-Plot Design MGH Table 11.7 Whole Plot Factor Changes Split Plot Factor Changes

Mixed Models Two-Factor Model y ijk =  +  i + b j + (ab) ij + e ijk i = 1,..., a j = 1,..., b k = 1,..., r One or more factors fixed, One or more factors random One or more factors fixed, One or more factors random Convention: Interaction of a fixed and a random factor is a random effect

Implications of Alternative Assumptions Interaction Random (Preferred Assumption) (ab) ij ~ NID(0,  ab 2 ) No “constraints” required Both main effects tested against the interaction effect Interaction both Fixed and Random (Alternative Assumption) For each i (ab) ij ~ NID(0,  ab 2 ) For each j

Nested Effects Notation Enclose subscripts of the factor(s) in which an effects is nested in parentheses  j (i) c k (ij)  jk (i) Nested Factor Levels Three-Factor, Hierarchically Nested Design Model y ijkl =  + a i + b j(i) + c k(ij) + e ijkl e ijkl = e l(ijk)

Sums of Squares Nested Effects, Balanced Designs B Nested Within A y ijk =  + a i + b j(i) + c k(ij) + e ijkl Compare Averages for Nested Levels with the Overall Average for Levels having the Same Nesting Compare Averages for Nested Levels with the Overall Average for Levels having the Same Nesting

Sums of Squares Nested Effects, Balanced Designs B Nested Within A y ijk =  + a i + b j(i) + e ijk df: a(b-1) = b-1 + (a-1)(b-1)

Analysis of Variance Table

Split Plot Design Model y ijkl =  +  i + b j(i) +  k + (  ) ik + e ijkl Whole Plots Design Factor: A(fixed) Whole Plot Error: B(A)(random) Split Plots Design Factor: C(fixed)

Split Plot Design Model y ijkl =  +  i + b j(i) +  k + (  ) ik + e ijkl Whole Plot Analysis Main Effects & Interactions of Factors Applied to the Whole Plots; Uses Whole Plot Error MS Split Plot Analysis Main Effects & Interactions of Factors Applied to the Split Plot, Interactions of Whole Plot & Split Plot Factors; Uses Error MS Whole Plots Design Factor: A(fixed) Whole Plot Error: B(A)(random) Split Plots Design Factor: C(fixed)

Automatic Cutoff Times: Split-Plot Design MGH Table 13.6 Factors Manufacturers: A, B Lawnmowers: 3 for Each Manufacturer Speeds: High, Low Whole Plot W.P. Error Split Plot Whole Plot Analysis Split Plot Analysis

Split-Plot ANOVA Table