5.3 Three-Stage Nested Design Example

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1 5.3 Three-Stage Nested Design Example A researcher designs an experiment to study the of a metal alloy. A three-stage nested design was conducted that included Two alloy chemistry compositions. Three ovens for each alloy chemistry composition (6 ovens were used). Four ingot molds were used to produce alloy ingots for each of the six combinations of alloy chemistry composition and oven (24 molds were used). Three ingots were produced from each of the 24 molds. Molds can only be used once. The experimental data in the table below contains alloy measurements. Alloy Chemistry 1 Oven Mold Alloy Chemistry 2 Oven Mold Analyze this data assuming alloy chemistry and ovens are fixed effects and ingot mold is a random effect. Assume data collection was randomized. Based on the expected mean squares, the F -statistics are F alloy = MS alloy MS mold(alloy oven) F oven(alloy) = MS oven(alloy) MS mold(alloy oven) F mold(oven alloy) = MS mold(oven alloy) MS E The tests for oven(alloy) and mold(oven*alloy) are significant with p-values of <.1 and.4, respectively. The variance component estimates are σ 2 = and σ mold 2 =. Thus, the random replication variability is estimated to be about 27% larger than the variability due to molds. THREE-STAGE NESTED DESIGN: ALLOY AND oven FIXED, MOLD RANDOM Class Level Information Class Levels Values alloy oven mold Number of Observations Read Number of Observations Used 72

2 Dependent Variable: Source DF Sum of Squares Mean Square F Value Pr > F Model <.1 Error Corrected Total R-Square Coeff Var Root MSE Mean alloy THREE-STAGE oven(alloy) NESTED DESIGN: ALLOY AND oven FIXED , MOLD <.1 RANDOM mold(alloy*oven) Source Type III Expected Mean Square alloy Var(Error) + 3 Var(mold(alloy*oven)) + Q(alloy,oven(alloy)) oven(alloy) Var(Error) + 3 Var(mold(alloy*oven)) + Q(oven(alloy)) THREE-STAGE NESTED DESIGN: ALLOY AND oven FIXED, MOLD RANDOM mold(alloy*oven) Var(Error) + 3 Var(mold(alloy*oven)) Tests of Hypotheses for Mixed Model Analysis of Variance Dependent Variable: * alloy oven(alloy) <.1 Error Error: MS(mold(alloy*oven)) * This test assumes one or more other fixed effects are zero. mold(alloy*oven) Error: MS(Error)

3 THREE-STAGE NESTED DESIGN: ALLOY AND oven FIXED, MOLD RANDOM 5 6 Distribution of THREE-STAGE NESTED DESIGN: ALLOY AND oven FIXED, MOLD RANDOM alloy Distribution of alloy N Mean Std Dev alloy 2 N Mean Std Dev THREE-STAGE NESTED DESIGN: ALLOY AND oven FIXED, MOLD RANDOM 5 6 Distribution of oven oven oven(alloy) alloy N Mean Std Dev alloy N Mean Std Dev

4 Var(mold(alloy*oven)) Number of Observations Used THREE-STAGE NESTED DESIGN: ALLOY AND oven FIXED, MOLD RANDOM THREE-STAGE NESTED DESIGN: ALLOY AND oven FIXED, MOLD RANDOM Distribution of Distribution of 6 6 VARIANCE COMPONENTS ANALYSIS 5 5 Variance Components Estimation Procedure mold(alloy*oven) Class Level Information Class Levels Values alloy oven mold(alloy*oven) mold mold alloy oven N Mean Std Dev mold alloy oven N Mean Std Dev Number 2. of Observations Read Number 1 THREE-STAGE of 1Observations 3NESTED Used DESIGN: ALLOY AND oven FIXED, MOLD RANDOM mold 59.8alloy oven N Mean Std Dev Dependent Variable: REML 3 Iterations Iteration Objective 2.Var(mold(alloy*oven)) Var(Error) VARIANCE COMPONENTS ANALYSIS Convergence criteria met. Variance Components Estimation Procedure REML Class Estimates Level Information Variance Component Estimate Class Levels Values Var(mold(alloy*oven)) alloy Var(Error) oven mold Asymptotic Covariance Matrix of Estimates Number of Observations Var(mold(alloy*oven)) 28 Read 72 Var(Error)

5 Analyze the data again but assume chemistry alloy is a fixed effect and oven and mold are random effects. That is, the 6 ovens were randomly selected from a larger population of ovens. Based on the expected mean squares, the F -statistics are F alloy = MS alloy MS oven(alloy) F oven(alloy) = MS oven(alloy) MS mold(alloy oven) F mold(oven alloy) = MS mold(oven alloy) MS E The tests for variance components for oven(alloy) and mold(oven*alloy) are significant with p-values of <.1 and.4, respectively. The variance component estimates are σ 2 =, σ 2 mold =, and σ2 oven = This implies there is very large variability due to changes in oven. It is estimated to be about 1.5 s larger than the random replication variability, and it is about 13.3 s larger than the variability due to molds. The engineers need to first focus on reducing the oven-to-oven variability. THREE-STAGE NESTED DESIGN: ALLOY FIXED, oven AND MOLD RANDOM Dependent Variable: Source DF Sum of Squares Mean Square F Value Pr > F Model <.1 Error Corrected Total R-Square Coeff Var Root MSE Mean alloy THREE-STAGE NESTED DESIGN: ALLOY FIXED, oven AND MOLD RANDOM oven(alloy) <.1 mold(alloy*oven) 18 The GLM Procedure Source alloy oven(alloy) Type III Expected Mean Square Var(Error) + 3 Var(mold(alloy*oven)) + 12 Var(oven(alloy)) + Q(alloy) Var(Error) + 3 Var(mold(alloy*oven)) + 12 Var(oven(alloy)) mold(alloy*oven) Var(Error) + 3 Var(mold(alloy*oven)) 29

6 THREE-STAGE NESTED DESIGN: ALLOY FIXED, oven AND MOLD RANDOM Tests of Hypotheses for Mixed Model Analysis of Variance Dependent Variable: Source VARIANCE DF Type COMPONENTS III SS Mean Square ANALYSIS F Value Pr > F alloy Variance Components Estimation Procedure.886 Error Class Level Error: MS(oven(alloy)) Information Class Levels Values Source DF alloy Type III SS2 Mean 1 2 Square F Value Pr > F oven(alloy) 4 oven <.1 Error 18 mold Error: MS(mold(alloy*oven)) Number of Observations Read 72 Source Number DFof Type Observations III SS Mean Used Square 72 F Value Pr > F mold(alloy*oven) Error: MS(Error) Dependent Variable: REML Iterations Iteration Objective Var(oven(alloy)) Var(mold(alloy*oven)) Var(Error) Convergence criteria met. REML Estimates Variance Component Estimate Var(oven(alloy)) Var(mold(alloy*oven)) Var(Error) Asymptotic Covariance Matrix of Estimates Var(oven(alloy)) Var(mold(alloy*oven)) Var(Error) Var(oven(alloy)) E-13 Var(mold(alloy*oven)) Var(Error) -6.2E

7 SAS Code for Three-Stage Nested Design Example DM LOG; CLEAR; OUT; CLEAR; ; ODS GRAPHICS ON; ODS PRINTER PDF file= C:\COURSES\ST541\NESTED3.PDF ; OPTIONS NODATE NONUMBER; **********************************; *** THREE-STAGE NESTED DESIGNS ***; **********************************; DATA in; DO alloy=1 TO 2; DO oven=1 TO 3; DO mold=1 TO 4; DO ingot = 1 to 3; OUTPUT; END; END; END; END; LINES; ; PROC GLM DATA=in; CLASS alloy oven mold; MODEL = alloy oven(alloy) mold(alloy oven) / SS3; RANDOM mold(alloy oven) / TEST; MEANS alloy oven(alloy) mold(alloy oven); TITLE THREE-STAGE NESTED DESIGN: ALLOY and OVEN FIXED, MOLD RANDOM ; PROC VARCOMP DATA=in METHOD=REML; CLASS alloy oven mold; MODEL = alloy oven(alloy) mold(alloy oven) / FIXED=2; TITLE VARIANCE COMPONENTS ANALYSIS ; PROC GLM DATA=in; CLASS alloy oven mold; MODEL = alloy oven(alloy) mold(alloy oven) / SS3; RANDOM oven(alloy) mold(alloy oven) / TEST; * MEANS alloy oven(alloy) mold(alloy oven); TITLE THREE-STAGE NESTED DESIGN: ALLOY FIXED, OVEN and MOLD RANDOM ; PROC VARCOMP DATA=in METHOD=REML; CLASS alloy oven mold; MODEL = alloy oven(alloy) mold(alloy oven) / FIXED=1; TITLE VARIANCE COMPONENTS ANALYSIS ; RUN; 211

8 5.4 Examples with Crossed and Nested Effects Three Factor Example An industrial engineer is studying the manual assembly of electronic components on circuit boards. The goal is to improve the speed of the assembly operation. The engineer has designed three assembly fixtures and two workplace layouts that seem promising. Operators are required to manually perform the assembly, and it is was decided to randomly select four operators for each fixture and layout combination. Because the workplaces are in different locations within the plant, it is difficult to use the same four operators for each layout. Therefore, the four operators chosen for layout 1 are different individuals than the four operators chosen for layout 2. Because the eight operators were also chosen at random, we have a mixed model containing both crossed (factorial) and nested effects. The data collection was completely randomized and two replicates were obtained. The assembly s measured in seconds are given in the table below. Layout 1 Layout 2 Operator y i Fixture Fixture Fixture y jk y j y = 1252 What is the model equatioin associated with this design? Class Level Information Class Levels Values fixture layout operator Number of Observations Read 48 Number of Observations Used

9 Dependent Variable: Source DF Sum of Squares Mean Square F Value Pr > F Model Error Corrected Total R-Square Coeff Var Root MSE Mean fixture <.1 layout fixture*layout operator(layout) fixtu*operat(layout) Fit Diagnostics for Residual RStudent RStudent Predicted Value Predicted Value Leverage Percent Residual Quantile Residual Predicted Value Fit Mean Residual Proportion Less Cook's D Observation Observations 48 Parameters 24 Error DF 24 MSE R-Square.818 Adj R-Square

10 Source fixture layout fixture*layout operator(layout) fixtu*operat(layout) ependent Variable: Type III Expected Mean Square Var(Error) + 2 Var(fixtu*operat(layout)) + Q(fixture,fixture*layout) Var(Error) + 2 Var(fixtu*operat(layout)) + 6 Var(operator(layout)) + Q(layout,fixture*layout) Var(Error) + 2 Var(fixtu*operat(layout)) + Q(fixture*layout) Var(Error) + 2 Var(fixtu*operat(layout)) + 6 Var(operator(layout)) Var(Error) + 2 Var(fixtu*operat(layout)) Tests of Hypotheses for Mixed Model Analysis of Variance * fixture fixture*layout operator(layout) Error Error: MS(fixtu*operat(layout)) * This test assumes one or more other fixed effects are zero. * layout Error Error: MS(operator(layout)) * This test assumes one or more other fixed effects are zero. fixtu*operat(layout) Error: MS(Error)

11 Distribution of 32.5 Distribution of fixture fixture 32.5 fixture N Mean Std Dev Distribution of fixture N Mean Std Dev Distribution of layout 2 Distribution of layout layout N Mean Std Dev layout N Mean Std Dev Distribution of fixture*layout fixture layout N Mean Std Dev fixture fixture*layout layout N Mean Std Dev operator operator(layout) operator layout N Mean Std Dev layout N Mean Std Dev

12 Distribution DESIGN of WITH NESTED AND CROSSED FACTORS 32.5 Distribution of The UNIVARIATE Procedure Variable: resid Moments N 48 Sum Weights fixture operator Mean Sum Observations fixtu*operat(layout) fixture operator fixtu*operat(layout) Std Deviation Variance Skewness Kurtosis Uncorrected SS 56 Corrected SS 56 Coeff Variation. Std Error Mean layout N Mean Std Dev layout N 1 Mean1Basic Std Statistical Dev 1 2 Measures Location Mean. Std Deviation Median Variance Mode Range Level 31. of Level of fixture operator 1Interquartile layout N Mean Std Dev Range Tests for Location: Mu= Test 1 1 Statistic p Value Student's 24.5t t 3 Pr > 1 t Variability Note: The mode displayed is the smallest of 2 modes with a count of 1. Sign M Pr >= M 1. Signed Rank S -67 Pr >= S Tests for Normality Test Statistic p Value Shapiro-Wilk W.9695 Pr < W.11 Kolmogorov-Smirnov D Pr > D <.1 Cramer-von Mises W-Sq Pr > W-Sq <.5 Anderson-Darling A-Sq Pr > A-Sq <.5 216

13 SAS Code for Three Factor Example DM LOG; CLEAR; OUT; CLEAR; ; ODS GRAPHICS ON; ODS PRINTER PDF file= C:\COURSES\ST541\NESTWF.PDF ; OPTIONS NODATE NONUMBER; **********************************; *** NESTED AND CROSSED FACTORS ***; **********************************; DATA in; DO fixture=1 TO 3; DO layout=1 TO 2; DO operator=1 TO 4; DO rep=1 TO 2; OUTPUT; END; END; END; END; LINES; ; PROC GLM DATA=in PLOTS=(ALL); CLASS fixture layout operator; MODEL = fixture layout fixture operator(layout) / SS3; RANDOM operator(layout) fixture*operator(layout) / TEST; MEANS fixture layout fixture operator(layout) ; OUTPUT OUT=diag R=resid; TITLE ; PROC UNIVARIATE DATA= diag NORMAL; VAR resid; RUN; Partitioning the SS total 217

14 5.4.2 Four Factor Example For the following quality control example, Suppliers and Operators are fixed effects. The two production runs for operator D are different than the two production runs for operator N. From each supplier, two random batches of raw materials were used in this study. Hence, Production Run and Batches are random effects. Supplier A B C Batch Batch Batch Run Operator D Run Run Operator N Run What is the model equation associated with this design? There are exact F -tests for Run(Operator), Batch(Supplier), Supplier*Run(Operator), Operator*Batch(Supplier), and Run(Operator)*Batch(Supplier). There are approximate F -tests for Operator, Supplier, and Operator*Supplier. The REML (restricted maximum likelihood) estimates of the variance components were generated. REML estimates do not allow for negative variance components estimates. That is why the one of the estimates is set to. The Supplier*Operator interaction plots indicate parallel lines which is consistent with the p-value of QUALITY CONTROL EXAMPLE-- 2 CROSSED AND 2 NESTED EFFECTS Class Level Information Class Levels Values operator 2 D N supplier 3 A B C run batch Number of Observations Read 72 Number of Observations 218 Used 72

15 Dependent Variable: Source DF Sum of Squares Mean Square F Value Pr > F Model <.1 Error Corrected Total R-Square Coeff Var Root MSE Mean operator <.1 supplier <.1 operator*supplier run(operator) <.1 batch(supplier) <.1 supplie*run(operato) operat*batch(suppli) QUALITY CONTROL EXAMPLE CROSSED AND NESTED EFFECTS <.1 run*batc(oper*suppl) 6The GLM Procedure <.1 Fit Diagnostics for Residual RStudent 2 RStudent Predicted Value Predicted Value Leverage Residual Cook's D Quantile Predicted Value Observation 3 Fit Mean Residual 1 Percent Observations 72 Parameters 24 Error DF 48 MSE.2587 R-Square.9946 Adj R-Square Residual Proportion Less 219

16 QUALITY CONTROL EXAMPLE-- 2 CROSSED AND 2 NESTED EFFECTS Distribution of Distribution of A B C supplier A B C supplier 35 QUALITY CONTROL EXAMPLE-- 2 CROSSED AND 2 NESTED EFFECTS supplier N Mean Std Dev A 24 Distribution of B C QUALITY CONTROL supplier EXAMPLE-- N 2 CROSSED Mean AND Std Dev 2 NESTED EFFEC A The 24GLM Procedure B Distribution of C D N operator operator N Mean Std Dev D D operator QUALITY CONTROL EXAMPLE-- 2 CROSSED AND 2 NESTED EFFECTS operator N Mean Std Dev 35 D Distribution of N N N D A D B D C N A N B N C operator*supplier operator supplier N Mean Std Dev D A D B D C N A D A D B D C N A N B N C operator*supplier N B N C operator supplier N Mean Std Dev D A D B D C N A N B N C

17 QUALITY CONTROL EXAMPLE-- 2 CROSSED AND 2 NESTED EFFECTS Tests of Hypotheses for Mixed Model Analysis of Variance Dependent Variable: * operator Error Error: MS(run(operator)) + MS(operat*batch(suppli)) - MS(run*batc(oper*suppl)) + 12E-16*MS(Error) * This test assumes one or more other fixed effects are zero. * supplier Error Error: MS(batch(supplier)) + MS(supplie*run(operato)) - MS(run*batc(oper*suppl)) + 89E-17*MS(Error) * This test assumes one or more other fixed effects are zero. operator*supplier Error Error: MS(supplie*run(operato)) + MS(operat*batch(suppli)) - MS(run*batc(oper*suppl)) - 89E-17*MS(Error) run(operator) Error Error: MS(supplie*run(operato)) QUALITY Source CONTROL EXAMPLE-- DF Type III 2 SS CROSSED Mean Square AND F 2 Value NESTED Pr > EFFECTS F batch(supplier) Error Tests of Hypotheses 3 for Mixed Model Analysis of Variance Error: MS(operat*batch(suppli)) Dependent Variable: supplie*run(operato) operat*batch(suppli) Error Error: MS(run*batc(oper*suppl)) run*batc(oper*suppl) <.1 Error: MS(Error)

18 VARIANCE COMPONENTS ANALYSIS Variance Components The GLM Estimation Procedure Procedure Source Type III Expected Mean Class Square Level Information operator Var(Error) + 3 Var(run*batc(oper*suppl)) Class Levels+ 6 Values Var(operat*batch(suppli)) + 6 Var(supplie*run(operato)) + 18 Var(run(operator)) + Q(operator,operator*supplier) operator 2 D N supplier Var(Error) + 3 Var(run*batc(oper*suppl)) supplier 3+ 6 A Var(operat*batch(suppli)) B C + 6 Var(supplie*run(operato)) + 12 Var(batch(supplier)) + Q(supplier,operator*supplier) operator*supplier run Var(Error) + 3 Var(run*batc(oper*suppl)) + 6 Var(operat*batch(suppli)) + 6 Var(supplie*run(operato)) batch + Q(operator*supplier) run(operator) Var(Error) + 3 Var(run*batc(oper*suppl)) + 6 Var(supplie*run(operato)) + 18 Var(run(operator)) batch(supplier) Var(Error) + 3 Var(run*batc(oper*suppl)) Number of Observations + 6 Var(operat*batch(suppli)) Read Var(batch(supplier)) supplie*run(operato) Number of Observations Used 72 Var(Error) + 3 Var(run*batc(oper*suppl)) + 6 Var(supplie*run(operato)) operat*batch(suppli) Var(Error) + 3 Var(run*batc(oper*suppl)) + 6 Var(operat*batch(suppli)) run*batc(oper*suppl) Dependent Variable: Var(Error) + 3 Var(run*batc(oper*suppl)) REML Iterations Iteration Objective Var(run(operator)) Var(batch(supplier)) Var(supplie*run(operato)) REML Iterations Iteration Var(operat*batch(suppli)) Var(run*batc(oper*suppl)) Var(Error) Convergence criteria met. VARIANCE COMPONENTS ANALYSIS Variance Components Estimation Procedure REML Estimates Variance Component Estimate Var(run(operator)) Var(batch(supplier)) Var(supplie*run(operato)) Var(operat*batch(suppli)) Var(run*batc(oper*suppl)) Var(Error) Asymptotic Covariance Matrix of Estimates Var(run(operator)) Var(batch(supplier)) Var(supplie*run(operato)) 222 Var(run(operator)) E-16

19 SAS Code for Four Factor Example DM LOG; CLEAR; OUT; CLEAR; ; ODS GRAPHICS ON; ODS PRINTER PDF file= C:\COURSES\ST541\NEST4.PDF ; OPTIONS NODATE NONUMBER; DATA comp; INPUT operator $ run supplier $ LINES; D 1 A D 1 A D 1 A D 1 A D 1 A D 1 A D 1 B D 1 B D 1 B D 1 B D 1 B D 1 B D 1 C D 1 C D 1 C D 1 C D 1 C D 1 C D 2 A D 2 A D 2 A D 2 A D 2 A D 2 A 6 2 D 2 B D 2 B D 2 B D 2 B D 2 B D 2 B D 2 C D 2 C D 2 C 2 8 D 2 C D 2 C D 2 C N 3 A N 3 A N 3 A N 3 A N 3 A N 3 A N 3 B N 3 B 1 1 N 3 B N 3 B N 3 B N 3 B N 3 C N 3 C N 3 C N 3 C N 3 C N 3 C N 4 A N 4 A N 4 A N 4 A N 4 A N 4 A N 4 B N 4 B N 4 B N 4 B N 4 B N 4 B N 4 C N 4 C N 4 C N 4 C N 4 C N 4 C ; PROC GLM DATA=comp PLOTS=(ALL); CLASS operator supplier run batch; MODEL =operator supplier run(operator) batch(supplier) supplier*run(operator) operator*batch(supplier) run*batch(operator supplier) / SS3; MEANS supplier operator; RANDOM run(operator) batch(supplier) supplier*run(operator) operator*batch(supplier) run*batch(operator supplier) / TEST; TITLE QUALITY CONTROL EXAMPLE-- 2 CROSSED AND 2 NESTED EFFECTS ; PROC VARCOMP DATA=comp METHOD=REML; CLASS operator supplier run batch; MODEL =operator supplier operator*supplier run(operator) batch(supplier) supplier*run(operator) operator*batch(supplier) run*batch(operator supplier) / FIXED=3; TITLE VARIANCE COMPONENTS ANALYSIS ; RUN; 223

unadjusted model for baseline cholesterol 22:31 Monday, April 19,

unadjusted model for baseline cholesterol 22:31 Monday, April 19, 2004 1 Class Level Information Class Levels Values TRETGRP 3 3 4 5 SEX 2 0 1 Number of observations 916 unadjusted model for baseline cholesterol