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ISE 426-526 - Lecture 2 - Single Factor Experiments

# ISE 426-526 - Lecture 2 - Single Factor Experiments - ISE...

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1 ISE 426/526 Lecture 2 - 2007 Single Factor Experiments Models, Equations, & Theory What’s a CRD*? * Completely Randomized Design An experiment in which there is no restriction on randomization of the trials that are run Completely Randomized Design Single Factor Multiple Factors Fixed versus Random Factors • Fixed Effects Model – Treatments specifically chosen by experimenter – Conclusions cannot be extended to similar treatments • Random Effects Model – Treatments are a random sample from a larger population of treatments – Conclusions extend to all treatments in population

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2 Building a Model Building a Model (cont.) Analyzing the Fixed Effects Model
3 Partitioning SS Total Partitioning SS Total (cont.) Expected Mean Squares

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4 Example 3-1 • Plasma Etching Experiment – Looks at impact of RF power on Etch Rate Power 12345 T o t a l A v e r a g e s 160 575 542 530 539 570 2756 551.2 180 565 593 590 579 610 2937 587.4 200 600 651 610 637 629 3127 625.4 220 725 700 715 685 710 3535 707.0 Treatments Review of Example 3-1 Impact of RF Power on Observed Etch Rate RF Power Etch Rate 220 200 180 160 750 700 650 600 550 Boxplot of Etch Rate vs RF Power ANOVA Calculations
5 ANOVA Calculations (cont.) Reference Distribution F(3,16) Frequency 14 12 10 8 6 4 2 0 900 800 700 600 500 400 300 200 100 0 Why Does the ANOVA Work? 22 10 ( 1 ) 0 We are sampling from normal populations, so if is true, and Cochran's theorem gives the independence of these two chi-square random variables /( So Treamtents E aa n Treatments SS SS H SS F χχ σσ −− = ∼∼ 2 1 1, ( 1) 2 (1 ) 2 1 1) /( /[ ( 1)] /[ ( 1)] Finally, ( ) and ( ) 1 Therefore an upper-tail test is appropriate. a n Ea n n i i Treatments E F SS an n EMS a F χ τ σ = =+ =

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6 Estimating Model Parameters Calculating Residuals Example 3-1 Residuals Residual Percent 50 25 0 -25 -50 99 90 50 10 1 Fitted Value Residual 700 650 600 550 20 10 0 -10 -20 Residual Frequency 30 20 10 0 -10 -20 -30 4 3 2 1 0 Observation Order 20 18 16 14 12 10 8 6 4 2 20 10 0 -10 -20 Normal Probability Plot of the Residuals Residuals Versus the Fitted Values Histogram of the Residuals Residuals Versus the Order of the Data Residual Plots for Etch Rate
7

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ISE 426-526 - Lecture 2 - Single Factor Experiments - ISE...

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