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Lecture_2010_09_15

Lecture_2010_09_15 - Stat 430/830 Analysis of Variance...

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1 Stat Stat 430/830: Analysis of Variance Lecture September 15, 2010 Motivation Motivation Methods for comparing two populations, or two formulations or two levels of a factor are studied in introductory courses of statistics (e.g., comparison of two means). However, in many scientific, engineering, clinical situations it is necessary to compare more than two populations. The analysis of variance (ANOVA) is a statistical method to compare more than two populations. Example - Wafer Plasma Etching Tool Wafer Plasma Etching Tool Factorial Experiments A factor is a set of treatments which can be applied to experimental units A level of a factor is a particular treatment of the factor A factorial experiment investigates the effect of one or more factors An experimental treatment is the description of the way in which a particular unit is treated and comprises one level from each factor The replication is the number of units per experimental treatment ( n )
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2 A Single Factor Experiment A Single Factor Experiment Assumes that only one factor (although with different number of levels) can affect the outcome. Example (textbook). An engineer is interested in investigating the effect of Radio Frequency (RF) power setting on the etch rate (A/min) of a wafer plasma etching tool in an integrated circuit manufacturing process. She is interested in investigating the effect of the power (160,180, 200 and 220 Watts) when the gas is hexafluroethane (C 2 F 6 ) and the gap is set at 0.80 cm. Example (Cont) Example (Cont) • This is a single factor experiment with 4 levels . • The engineer can run 5 wafers at each power level. We say that the experiment has 5 replicates . • If the order in which the 20 runs are made is random, we say that the experiment is randomized . Example (Cont) Example (Cont) How can you formally test the assumption that all the means are equal? Analysis of Variance Analysis of Variance Subdivision of the total variation between the experimental units into separate components, each representing a different source of variation, so that the relative importance of the different sources can be assessed. To do this it is useful to have a model for the observations from the experiment
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3 Models for Observations Models for Observations Assume have a model with a levels and n replications Can use the means model y ij = μ i + e ij (mean of treatment + error) i = 1,…, a j = 1,…,n Can use the effects model y il = μ + τ i + e il (overall mean, treatment effect & error) i = 1,…, a j = 1,…,n where . This implies that 0 1 = = a i i τ a a i i = = 1 μ μ Fixed and Random Effects Models Fixed and Random Effects Models If the a levels of the factor are specifically chosen, then the model is called a fixed effects model . The parameters of interest are ( μ , τ i , σ 2 ).
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