Lecture17 - http/ocw.mit.edu MIT OpenCourseWare 2.830J 6.780J ESD.63J Control of Manufacturing Processes(SMA 6303 Spring 2008 For information about

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MIT OpenCourseWare ____________ http://ocw.mit.edu 2.830J / 6.780J / ESD.63J Control of Manufacturing Processes (SMA 6303) Spring 2008 For information about citing these materials or our Terms of Use, visit: ________________ http://ocw.mit.edu/terms .
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1 M anufacturing Control of Manufacturing Processes Subject 2.830/6.780/ESD.63 Spring 2008 Lecture #17 Nested Variance Components April 15, 2008
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2 M anufacturing Readings/References • D. Drain, Statistical Methods for Industrial Process Control , Chapter 3: Variance Components and Process Sampling Design, Chapman & Hall, New York, 1997.
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3 M anufacturing Agenda • Standard ANOVA – Looking for fixed effect vs. chance/sampling • Nested variance structures – More than one zero-mean variance at work – Want to estimate these variances •E x a m p l e s – Based on simple ANOVA – Two-level example (from Drain) – Three-level example (from Drain) • Implications for design of sampling and experimental plans VS.
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4 M anufacturing Standard Analysis of Variance (ANOVA) • Question in single variable ANOVA: – Are we seeing anything other than random sampling from a single (Normal) distribution? • Approach: – Estimate variance of the natural variation from observed replication for each treatment level (i.e., estimate the within-group variance) – Estimate the between-group variance • Could be due to a fixed effect • Could be due to chance (random sampling) – Consider probability of a ratio of these two variances as large as what was observed, if only a single (Normal) distribution is at work
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5 M anufacturing ANOVA Example Group 1 Group 2 within-group variation between-group variation within-group variation • Groups are different levels of some treatment • Goal – determine if there is a non-zero fixed-effect or not 3 5 7 9
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6 M anufacturing Hypotheses in ANOVA • Null Hypothesis: Random Sampling from Single Distribution – E.g. we draw multiple samples of some size – What range of variance ratios among these samples would we expect to see purely by chance? – Assumed model: • Fixed Effects Model – The alternative hypothesis is that there is a fixed effect between the treatment groups (where i indicates group, and j indicates replicate within group) – Assumed model:
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7 M anufacturing Some Definitions (for ANOVA Calculations) Deviations from grand mean – Individual data point from grand mean: – Squared deviation of point from grand mean: – Sum of squared deviations from grand mean: Deviations of group mean from grand mean – Deviation of group i mean from grand mean: – Squared dev of group mean from grand mean: – Sum of squared deviations of group means: Deviations from local group mean – Deviation of individual point j (within group i ) from the group mean: – Squared deviation from group mean: – Sum of squared deviations from group mean:
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This note was uploaded on 09/24/2010 for the course MECHE 2.830J taught by Professor Davidhardt during the Spring '08 term at MIT.

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Lecture17 - http/ocw.mit.edu MIT OpenCourseWare 2.830J 6.780J ESD.63J Control of Manufacturing Processes(SMA 6303 Spring 2008 For information about

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