Transcript Lecture 10 - Statistics

Stat 470-10
•
Today: More Chapter 3
Full Factorial Designs at 2 Levels
• Notation/terminology: 2k experiment, where
– k is the number of factors
– each factor has two levels: low, high (denoted by -1, +1)
• Each replicate has a run-size of 2k
• Table 3.1 shows a 24 experiment with 6 replicates
• Experiment is performed as a completely randomized design
Notes
• Reasons for use:
• Two-level factors enable the largest no. of factors to be included in
experiment, for a given no. of runs
• Responses often assumed to be monotonic functions of the factors. So,
testing at low and high values will detect a factor’s effect
• 2-level factorial experiments can be used in a screening mode -- find
factors with large effects; eliminate those with small effects. Then, follow
up these experiments with subsequent experiments
• Experimental results are (relatively) easy to analyze and interpret
Computing Factorial Effects
• Experiment is run to see which factors are significant
• Suppose for the moment, there was only 1 factor with 2 levels
• Could compare average at the high level to the average at the low level
• Suppose there are several factors
• Variance of an effect estimate:
Example: Epitaxial Layer Growth
• Data from Table 3.1
Effect
A
B
C
D
AB
AC
AD
BC
BD
CD
Effect Estimate
-0.077
0.173
-0.77
0.490
0.07
-0.093
-0.050
0.058
-0.030
-.346
m14.05 eanofy 14.20
Example: Epitaxial Layer Growth
B
1
- 1
- 1
1
A
m14.05 eanofy 14.5
C
- 1
1
- 1
1
A
m
e
a
n
o
f
y
13.9 14.2
D
1
- 1
- 1
1
A
Example: Epitaxial Layer Growth
14.0 meanofy 14.5
C
- 1
1
- 1
1
B
13.8 meanofy 14.2
D
1
- 1
- 1
1
13.8 meanofy14.2 14.6
B
D
1
- 1
- 1
1
C
Common Assumptions
• Can address relative importance of effects and their relationships
• Hierarchical Ordering Principle: Lower order terms are more likely
to be important than higher order terms
• Effect Sparsity: Number of relatively important effects is small
• Effect Heredity: For an interaction to be significant, at least one of its
parents should be significant
Using Regression for the Analysis
• Can compute factorial effects directly as before
• Or can use linear regression to estimate effects
• Model:
• Regression Estimates:
Using Regression for the Analysis
• Consider an unreplicated 24 factorial design
• Effect estimate of factor A
• Regression estimate of factor A
One-at-a-Time Experiments
• Have discussed the factorial layout for experimentation
– all level combinations
– m replicates
– performed in random order
• Another obvious strategy is call the “one-factor-at-a-time approach”
1 identify the most important factor,
2 investigate this factor by itself, keeping other factors fixed,
3 decide on optimal level for this factor, and fix it at this level, and
4 move on to the next most important factor and repeat 2-3
Example
• Suppose have 2 factors A and B, each with 2-levels (-1,+1)
-1
-1
+1
+1
Comments
• Disadvantages of the on-factor-at-a-time approach
–
–
–
–
less efficient than factorial experiments
interactions may cause misleading conclusions
conclusions are less general
may miss optimal settings
Assessing Effect Significance
• For replicated experiments, can use regression to determine important
effects
• Can also use a graphical procedure
• The graphical procedure can be used for replicated and replicated
factorial experiments
Normal and Half-Normal Probability Plots
• Graphical method for assessing which effects are important are based
on normal probability plots (a.k.a normal qq-plots)
• Let ˆ(1)  ˆ( 2 )  ...  ˆ( I ) be the sorted (from smallest to largest) effect
estimates
 i  1
• Plot ˆ(i ) versus   I  , where  represents the cumulative


distribution function of the standard normal (N(0,1)) distribution
1
Normal and Half-Normal Probability Plots
• That is, we plot the quantiles of our sample of effects versus the
corresponding quantiles of the standard normal
• If no effect is important then the sample of effects appear to be a
random sample from a normal distribution…we observe:
• Otherwise:
Normal and Half-Normal Probability Plots
• Why does this work?
Normal and Half-Normal Probability Plots
• Half-Normal Plots
E
f
e
c
t
s
i
m
a
-0.2 0. 0.2 0.4
Example: Epitaxial Growth Layer Experiment
-1 .5
1.0
0 .5
0 .0
0 .5
1 .0
1 .5
Qu a n ti l e s
of