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• Purpose to compare average response among levels of the factors Chapters 2-4

• Purpose to compare average response among levels of the factors Chapters 2-4 -predict future response at specific factor level -recommend best factor level for future use. • Purpose to Study Sources of Variability Chapter 5

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• Purpose to compare average response among levels of the factors Chapters 2-4

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  1. •Purpose to compare average response among levels of the factors Chapters 2-4 -predict future response at specific factor level -recommend best factor level for future use •Purpose to Study Sources of Variability Chapter 5 - Medical diagnostic tests (procedures, equipment, reagents etc. - Educational testing (student, testing instrument, repeat tests) - Quality Control Measurements (operator, equipment, environment)

  2. Random Sampling Experiment or RSE

  3. Model for CRD Determine if changes in factor levels cause difference in mean response Estimate variance in response across levels of this factor Model for RSE

  4. samples preparations variance components

  5. From Chapter 2

  6. Notice interval estimate for 2t much wider than 2 What if MST < MSE ?

  7. Variability in dry soup mix “intermix” (vegetable oil, salt flavorings etc.) - too little not enough flavor - too much too strong Make batch of soup and dry it on a rotary dryer Place dry soup in a mixer where intermix is injected through ports Package Ingredients Cook temperature Dryer temperature Dryer RPM, etc. number of mixer ports for Vegetable oil temperature of mixer jacket Mixing time Batch weight delay time between mixing and packaging, etc.

  8. If you would like the width of the confidence interval to be  ½ (2), search for t and r such that the multiplier of  above is  1.0

  9. Degrees of freedom for error is t(r-1)

  10. Rule of thumb for determining t and r • When 2t is expected to be larger than 2choose t = 2, r = 2 • Another way is to determine the power of the F-test of H0: 2t = 0

  11. Repeat measurement part operator gage reproducibility

  12. Iteration Objective Var(part) Var(oper) Var(part*oper) Var(Error) 0 -305.4599806158 0.0220118090 0 0.0128684142 0.000740262090 1 -305.4737662997 0.0225616136 0 0.0124709323 0.000751340496 2 -305.4737720510 0.0225514859 0 0.0124650026 0.000751666525 3 -305.4737720510 0.0225514859 0 0.0124650026 0.000751666525 Convergence criteria met. REML Estimates Variance Component Estimate Var(part) 0.02255 Var(oper) 0 Var(part*oper) 0.01247 Var(Error) 0.0007517 Asymptotic Covariance Matrix of Estimates Var(part) Var(oper) Var(part*oper) Var(Error) Var(part) 0.0001618 0 -5.4962E-6 -2.201E-13 Var(oper) 0 0 0 0 Var(part*oper) -5.4962E-6 0 0.00001650 -1.8834E-8 Var(Error) -2.201E-13 0 -1.8834E-8 3.76668E-8

  13. • Replace t(r-1) by ab(r-1) in formula 5.8 in order to get the desired width for a confidence interval on 2 • When 2a , 2b, and 2abare expected to be larger than 2choose r = 2, partition ab according to your belief in the relative size of 2a and 2b

  14. • Method of Moments Estimators are not unique, because they depend on whether type I or type III sums of squares are used • Maximum Likelihood and REML estimates are unique

  15. procglm; class lab sol; model conc=lab sol lab*sol/e1; random lab sol lab*sol; run; Source DF Type I SS Mean Square F Value Pr > F lab 2 2919.97407 1459.98704 1.05 0.3731 sol 3 34868.60688 11622.86896 8.39 0.0016 lab*sol 6 1384.32645 230.72108 0.17 0.9820 Source DF Type III SS Mean Square F Value Pr > F lab 2 1665.07016 832.53508 0.60 0.5611 sol 3 35185.75293 11728.58431 8.46 0.0016 lab*sol 6 1384.32645 230.721080.17 0.9820

  16. Source Type III Expected Mean Square lab Var(Error) + 1.8498 Var(lab*sol) + 7.3992 Var(lab) sol Var(Error) + 2.1077 Var(lab*sol) + 6.3232 Var(sol lab*sol Var(Error) + 2.1335 Var(lab*sol) Source Type I Expected Mean Square lab Var(Error)+2.5250Var(lab*sol)+0.0806Var(sol)+8.963Var(lab) sol Var(Error)+2.1979Var(lab*sol)+6.4648Var(sol) lab*sol Var(Error)+2.1335Var(lab*sol)

  17. • Maximum Likelihood and REML estimates are unique proc varcomp method=reml; class lab sol; model conc=lab sol lab*sol; run;

  18. Approximate Confidence Intervals

  19. crossed factors – levels are uniquely defined (example operators and levels in Gage RR study) nested factors – levels are physically different depending on the level of the factor they are nested in.

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