DOE 61a ANOVA

1
DOE 6-1a ANOVA Residuals
2
DOE 6-1a ANOVA Residuals

• The Analysis of Variance (ANOVA) depends on
  certain assumptions
• Observations (treatment i, observation j) may be
  described by

Random error
Overall mean of very large sample
Treatment influence
3
DOE 6-1 ANOVA Residuals

• The Analysis of Variance (ANOVA) depends on
  certain assumptions
• Observations (treatment i, observation j) may be
  described by
• The experimental errors eij are normally and
  independently distributed with mean 0 and a
  constant (but unknown) variance.

Random error
Overall mean of very large sample
Treatment influence
4
In practice, these assumptions are not always
obeyed exactlyat least in a first attempt at an
experiment design. It is always wise to check
these conditions before carrying through an ANOVA
calculation.
5
In practice, these assumptions are not always
obeyed exactlyat least in a first attempt at an
experiment design. It is always wise to check
these conditions before carrying through an ANOVA
calculation. We can do this by looking at
residuals, eij yij - yi?/n where yi?/n is the
average of the ith treatment.
6
In practice, these assumptions are not always
obeyed exactlyat least in a first attempt at an
experiment design. It is always wise to check
these conditions before carrying through an ANOVA
calculation. We can do this by looking at
residuals, eij yij - yi?/n where yi?/n is the
average of the ith treatment. If ANOVA can be
applied then the residuals should be normally
distributedthey should be structureless and
show no obvious patterns. If the residuals follow
a pattern then the supposedly random experimental
errors (eij) are not fully random!
7
In practice, these assumptions are not always
obeyed exactlyat least in a first attempt at an
experiment design. It is always wise to check
these conditions before carrying through an ANOVA
calculation. We can do this by looking at
residuals, eij yij - yi?/n where yi?/n is the
average of the ith treatment. If ANOVA can be
applied then the residuals should be normally
distributedthey should be structureless and
show no obvious patterns. If the residuals follow
a pattern then the supposedly random experimental
errors (eij) are not fully random! Checking
whether a sample is normally distributed takes us
back to Normal Probability plots.
8
Normal Probability plot of the residuals for the
plasma etch rate example of DOE 5-1 2 ANOVA.
Percent (cumulative normal probability ? 100)
Probability paper at left provided by
www.weibull.com/GPaper/
Residuals for plasma etch rate (Å/min)
-25.4
-12.65
0.10
12.85
25.60
38.35
9
Its clear that the plasma etch rate data set
(Manufacturing Integrated Circuits) was a good
candidate for ANOVA.
10
Its clear that the plasma etch rate data set
(Manufacturing Integrated Circuits) was a good
candidate for ANOVA. What if the normality test
of the residuals had shown a non-normal
distribution (i.e., non-linear on the plot)?
11
Its clear that the plasma etch rate data set
(Manufacturing Integrated Circuits) was a good
candidate for ANOVA. What if the normality test
of the residuals had shown a non-normal
distribution (i.e., non-linear on the plot)? Wed
have to go back to the experimental setup to look
for hidden bias. A redesign of the experiment
might be necessary. (We should likely check our
math first, just to make sure its not an
artifact of the calculation.)