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2.830J / 6.780J / ESD.63J Control of Manufacturing Processes (SMA 6303)
Spring 2008
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M
anufacturing
Control of
Manufacturing Processes
Subject 2.830/6.780/ESD.63
Spring 2008
Lecture #17
Nested Variance Components
April 15, 2008
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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M
anufacturing
Agenda
• Standard ANOVA
– Looking for fixed effect vs. chance/sampling
• Nested variance structures
– More than one zeromean variance at work
– Want to estimate these variances
•E
x
a
m
p
l
e
s
– Based on simple ANOVA
– Twolevel example (from Drain)
– Threelevel example (from Drain)
• Implications for design of sampling and
experimental plans
VS.
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 withingroup variance)
– Estimate the betweengroup 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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M
anufacturing
ANOVA Example
Group 1
Group 2
withingroup
variation
betweengroup
variation
withingroup
variation
• Groups are different
levels of some
treatment
• Goal – determine if
there is a nonzero
fixedeffect or not
3
5
7
9
6
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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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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.
 Spring '08
 DavidHardt

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