Chapter 13 Analysis of Multifactor Experiment

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Chapter 13 Analysis of Multifactor Experiment

• Dec 7th, 2006

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?Our Group?
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Group Members I

• Shengnan Cai Why we work on this topic? What can
  it be used?
• Tingting He Who developed related technology?
• Weixin Guo Theory about Two-factor experiments
• Yinghua Li Theory about 2k experiment

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Group Members IIData Examples

• Yi Su Parameter Estimates
• Siuying 22 factors experiment
• Sandy 23 Experiment
• Yi Zhang 2k factors examples

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Group Members III

• Tianyi Zhang Model Diagnostics and SAS
  Programming
• Ling Leng Regression Approach Conclusion

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The Goal of Analysis of Multifactor Experiment

• Shengnan Cai
• Why we work on this topic?
• What can it be used?

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About factors

• A Factor is a linked set of experimental
• conditions we may wish to compare.
• e.g. Levels of temperature
• different methods to teach
• group from different academic
  background
• Factors are also sometimes called independent
  variables.

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Two-factor experiments

• We have seen how to use one-way ANOVA related
  to samples designs to compare responses for a
  factor in the previous chapter.
• We need not to restrict ourselves to just one
  factor. Several different factors can be studied
  in a single experiment.
• We combine their levels to provide treatment
  combinations which can be compared in either
  related or unrelated samples.

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• The methods developed for two factors can be
  generalized to three or more factors

Two-factor experiments
2k factors experiments
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Why the multifactor analysis is important?

• Multifactor world
• Multifactor problems
• Example
• Meteorologists wants to know what influences
  the amount of the snowfall in Long Island.
• The influences could be temperature, moisture
  capacity, wind speed etc. These factors could
  form a multifactor system.

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Three sections

1. Two-factor Experiments with Fixed Crossed Factors
2. 23 Factorial Experiments
3. 2k Factorial Experiments

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History of this Techonology

• Tingting He
• Introduction to ANOVA

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ANOVA---Analysis of variance

• A collection of statistical models and their
  associated procedures which compare means by
  splitting the overall observed variance into
  different parts.
• The initial techniques of the analysis of
  variance
• were pioneered by the
• statistician and geneticist
• R.A.Fisher in the 1920s
• and 1930s.

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Two-Way ANOVAfor Balanced Design

• Assumptions
• The populations from which the samples were
  obtained must be normally or approximately
  normally distributed.
• The samples must be independent.
• The variances of the populations must be equal
• The groups must have the same sample size

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• Hypotheses
• The null hypotheses for each of the sets are
  given below.
• The population means of the first factor are
  equal.
• The population means of the second factor are
  equal.
• There is no interaction between the two factors.

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Breakdown of variability
TOTAL SS
Residual SS
Between Treatments SS
Between Subjects SS
Interaction
Factor 2
Factor 1
Extent to which factors influence each other
important information
Main effects
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• F-test
• There is an F-test for each of the hypotheses,
  and the F-test is the mean square for each main
  effect and the interaction effect divided by the
  within variance. The numerator degrees of
  freedom come from each effect, and the
  denominator degrees of freedom is the degrees of
  freedom for the within variance in each case.

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• Two-Way ANOVA table ( for an ab factorial
  experiment)

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Test Supplementing ANOVA

• Necessary condition for pairwise comparisons
• When the interactions are nonsignificant (
  H0AB is not rejected) pairwise comparisons
  between the row main effects and/or between the
  column main effects are generally of interest.

Method apply Tukey method, recommended by Tukey,
is highly valued in statistics. We find better
and accurate confidence intervals by this method.
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Things I havent told you

• What happens if you have unequal sample sizes.
• Answer is that the method of calculation is
  modified
• 2 What happens if sample is not normal?
• Dont worry too much. ANOVA is robust and
  can endure violations of assumptions. However,
  you might consider transforming

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• What happens if samples do not have same
  variance?
• Again, ANOVA is robust and can deal with
  this (to some extent). If homogeneity of
  variance is seriously violated, then Howell
  advises using Welchs Procedure.

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Theory Derivation I

• Weixin Guo
• Theory derivation about Two-factor experiments

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The parameter of interest
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The Sets of testing hypothesis
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The first Chi Square Variable
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The second Kai Square Variable
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The pivotal quantity
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F-test
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Theory Derivation II

• Yinghua Li
• Theory derivation about 2k experiment

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Data Example I

• Yi Su
• An Example on Parameter Estimates

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Example 13.1 Parameter Estimates ANOVA

• Bonding Strength of Capacitors
• Capacitors are bonded to a circuit board used
  in high voltage electronic equipment. Engineers
  designed and carried out an experiment to study
  how the mechanical bonding strength of capacitors
  depends on the type of substrate (factor A) and
  the bonding material (factor B). There were 3
  types of substrates aluminum oxide (Al2O3) with
  bracket, Al2O3 without bracket, and beryllium
  oxide (BeO) without bracket. Four types of
  bonding materials were used Epoxy I, Epoxy II,
  Solder I and Solder II. Four capacitors
  were tested at each factor level combination.
• Calculate the estimates of the parameters for
  these data.

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Bonding Strength of Capacitors
Substrate Bonding Material Bonding Material Bonding Material Bonding Material
Substrate Epoxy I Epoxy II Solder I Solder II
Al2O3 1.51, 1.96 1.83, 1.98 2.62, 2.82 2.69, 2.93 2.96, 2.82 3.11, 3.11 3.67, 3.40 3.25, 2.90
(Al2O3) 1.63, 1.80 1.92, 1.71 3.12, 2.94 3.23, 2.99 2.91, 2.93 3.01, 2.93 3.48, 3.51 3.24, 3.45
BeO 3.04, 3.16 3.09, 3.50 1.91, 2.11 1.78, 2.25 3.04, 2.91 2.48, 2.83 3.47, 3.42 3.31, 3.76
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Parameters Estimates

• yijk the kth observation on the (i, j)th
  treatment combination
• the mean of cell (i, j)
• i.i.d random error, normal
  distribution
• i th row main effect j th
  column main effect
• (i, j)th row-column interaction

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Parameters Estimates
sample mean of the (i, j)th cell least
square estimate of
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Parameters Estimates

• Sample variance for the (i, j)th cell is

The pooled estimate of is
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Parameters estimates Bonding Strength of
Capacitors
Sample Means
Substrate Bonding Material Bonding Material Bonding Material Bonding Material Row mean
Substrate Epoxy I Epoxy II Solder I Solder II Row mean
Al2O3 1.820 2.765 3.000 3.305 2.723
(Al2O3) 1.765 3.070 2.945 3.420 2.800
BeO 3.198 2.013 2.815 3.490 2.879
Column mean 2.261 2.616 2.920 3.405 2.800
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Parameters estimates Bonding Strength of
Capacitors
The cell sample SDs are s110.217 s120.138
s130.139 s140.321 s210.124 s220.131
s230.044 s240.122 s310.208 s320.209
s330.240 s340.192 The pooled sample SD0.187
with 36 d.f.
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The Estimates of Model Parameters for Capacitor
Bonding Strength Data
Parameters estimates Bonding Strength of
Capacitors
Substrate Bonding Material Bonding Material Bonding Material Bonding Material Row effects
Substrate Epoxy I Epoxy II Solder I Solder II Row effects
Al2O3
(Al2O3)
BeO
Column effects
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Analysis of Variance
ANOVA Table for Crossed Two-Way Layout
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Bonding Strength of Capacitors ANOVA
Analysis of Variance for Bonding Strength Data
Conclusion The main effect of bonding material
and the interaction between the bonding material
and substrate are both highly significant, but
the main effect of substrate is NOT significant
at the .05 level.
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Data Example II

• Siuying
• An example on 22 factorial experiment

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Calculate the estimated main effects A and B, and
the interaction AB.

• Factor B
• Low High
• Low y1110 y1215
  ?1.12.5
• Factor A
• High y2120 y2235
  ?2.27.5
• ?.115
  ?.225 ?..40

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• The estimated main effects are
• A (y22-y12)(y21-y11)/2
• (35-15)(20-10)/2 15
• B (y22-y21)(y12-y11)/2
• (35-20)(15-10)/2 10
• The estimated interaction effect is
• AB (y22-y12)-(y21-y11)/2
• (35-15)-(20-10)/2 5

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• ANOVA Table (Two-Way Layout with Fixed Factors)
• Source d.f. SS
  MS F
• A
• B
• AB
• ___________________________________________
• Total

MSA
MSAB
MSB
MSAB
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• Analysis of Variance
• (Two-Way Layout with Fixed Factors)
• Source d.f. SS MS
  F
• A 1 225
  225 9
• B 1 100
  100 4
• AB 1 25
  25
• Total 3 350

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Data Example III

• Sandy
• An example on 23 factors experiment

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Table 1. 3-Factor Full-Factorial Experiment
Design Computing Table
RUN Comb. I A B AB C AC BC ABC
1 (1) - - - -
2 a - - - -
3 b - - - -
4 ab - - - -
5 c - - - -
6 ac - - - -
7 bc - - - -
8 23 abc
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Goal

• To determine how the yield of an adhesive
  application process can be improved by adjusting
  three (3) process parameters 1. mixture
  ratio2. curing temperature3. curing time

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• ? The output response monitored is process
  yield.
• ? Assume further that the data were gathered by
  performing just a single replicate (n1) per
  combination treatment.

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Table 2. Results of the Example 23 Factorial
Experiment
RUN Comb. Factors Factors Factors Yield
RUN Comb. Mix Ratio Temp Time Yield
1 (1) 45 (-) 100C (-) 30m (-) 8
2 a 55 () 100C (-) 30m (-) 9
3 b 45 (-) 150C () 30m (-) 34
4 ab 55 () 150C () 30m (-) 52
5 c 45 (-) 100C (-) 90m () 16
6 ac 55 () 100C (-) 90m () 22
7 bc 45 (-) 150C () 90m () 45
8 abc 55 () 150C () 90m () 56
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A 1/(4n) x -(1)a-bab-cac-bcabc 1/4
x-89-3452-1622-4556 9 B 1/4 x
-8-93452-16-224556 33AB 1/4 x
8-9-345216-22-4556 5.5C 1/4 x
-8-9-34-5216224556 9AC 1/4 x
8-934-52-1622-4556 -0.5BC 1/4 x
89-34-52-16-224556 -1.5ABC 1/4 x
-8934-5216-22-4556 -3
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Conclusion

• ? The main effect of temperature (B33) has the
  greatest influence on the process yield.
• ? The main effects of mixture ratio (A9) and
  time (C9) are also significant.
• ? The interaction between mixture ratio and
  temperature also produces a positive effect on
  yield (AB5.5).
• ? But the rest of the factorial interactions
  affect the yield in the negative direction.

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Example of 2k Design

• Yi Zhang

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Problem Statement

• A soft drink bottler is interested in obtaining
  more uniform fill heights in the bottles produced
  by his manufacturing process. (Response variable
  is fill heights)
• The process engineer can control three variables
  during the filling process
• The percent carbonation (A)
• The operating pressure in the filler (B)
• The bottles produced per minute or the line speed
  (C)

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The Fill Height Experiment
Coded Factors Coded Factors Coded Factors Fill Height Deviation Fill Height Deviation Fill Height Deviation Fill Height Deviation Factor Levels Factor Levels
Run A B C   Replica1 Replica2     Low (-1) High(1)
1 -1 -1 -1 -3 -1 A() 10 12
2 1 -1 -1 0 1 B(psi) 25 30
3 -1 1 -1 -1 0 C(b/min) 200 250
4 1 1 -1 2 3
5 -1 -1 1 -1 0
6 1 -1 1 2 1
7 -1 1 1 1 1
8 1 1 1   6 5        
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The Geometric View of 23 Design
Run A B C   Treatment
1 - - -   (1)
2 - - a
3 - - b
4 - ab
5 - - c
6 - ac
7 - bc
8   abc
Speed (C)
Pressure (B)
Carbonation (A)
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Contrasts Calculations
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Factor Effects and Sum of Squares
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Analysis of Variance for the Fill Height Data
Source of   Sum of Degrees of Mean    
Variance   Squares Freedom Square F0 P-Value
Percent carbonation(A) 36.00 1 36.00 57.60 lt0.0001
Pressure(B) 20.25 1 20.25 32.40 0.0005
Line Speed(C ) 12.25 1 12.25 19.60 0.0022
AB 2.25 1 2.25 3.60 0.0943
AC 0.25 1 0.25 0.40 0.5447
BC 1.00 1 1.00 1.60 0.2415
ABC 1.00 1 1.00 1.60 0.2415
Error 5.00 8 0.625
Total   78.00 15      
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Conclusions

• Main effects are highly significant (all have
  very small P-values).
• The AB interaction is significant at about the 10
  percent level (Between carbonation and pressure).
• Other interactions are not significant at even
  the 20 percent level. (This is referred to as
  the effect sparsity principle)

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Model Diagnostics

• Tianyi Zhang
• Model Diagnostics using residual plots
• SAS code

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Three Assumptions

• Constant Variance Assumption
• Normality Assumption
• Independence Assumption

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1. Check constant variance assumption

• Residuals
• The plot of residuals against the fitted values.
• Whether fairly constant dispersed

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2. Check normality assumption

• Normal scores of residuals
• If the normal plot is linear, the assumption is
  valid.

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SAS programThe example is the 23 experiment for
fill height deviation.

• data sasuser.gp
• do c-1 to 1 by 2
• do b-1 to 1 by 2
• do a-1 to 1 by 2
• do r1 to 2
• Input height _at__at_
• abab acac bcbc abcabc
• output
• end
• end
• end
• end
• cards
• -3 -1 0 1 -1 0 2 3 -1 0 2 1 1 1 6 5
• run

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• proc anova datasasuser.gp
• class a b c ab ac bc abc
• model height a b c ab ac bc abc
• run
• The ANOVA Procedure
• Class Level
  Information
• Class
  Levels Values
• a
  2 -1 1
• b
  2 -1 1
• c
  2 -1 1

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• The ANOVA Procedure
• Dependent Variable height
• 
  Sum of
• Source DF
  Squares Mean Square F Value Pr gt F
• Model 7
  73.00000000 10.42857143 16.69 0.0003
• Error 8
  5.00000000 0.62500000
• Corrected Total 15 78.00000000
• R-Square Coeff Var
  Root MSE height Mean
• 0.935897 79.05694
  0.790569 1.000000

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• The ANOVA Procedure
• Dependent Variable height
• 
  Sum of
• Source DF
  Squares Mean Square F Value Pr gt F
• Model 3
  68.50000000 22.83333333 28.84 lt.0001
• Error 12
  9.50000000 0.79166667
• Corrected Total 15 78.00000000
• R-Square Coeff Var
  Root MSE height Mean
• 0.878205 88.97565
  0.889757 1.000000

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Regression Approach and Summary

• Ling Leng
• Regression Approach.
• Summary on this topic.

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Regression Approach

• Why we need regression approach?
• A unified approach to the analysis of balanced
  or unbalanced designs is provided by multiple
  regression.

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Regression Approach

• Define indicator variables x1 and x2 to represent
  the levels of A and B
• if A is low
  if B is low
• if A is high
  if B is high

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• We can use SAS programs to analyze the data and
  get the result.
• When put the same data in Yizhangs example
  about the 23 experiment, we got the same result.
• SAS programming as follows
• proc reg datasasuser.project
• //project is the name of
  set we put the data in
• var y x1 x2 x3 x12 x23 x13 x123
• model yx1 x2 x3 x12 x23 x13 x123
• run

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SAS result in regression

• Parameter Estimates
• Parameter
  Standard
• Variable DF Estimate
  Error t Value Pr gt t
• Intercept 1 1.00000
  0.19764 5.06 0.0010
• a 1 1.50000
  0.19764 7.59 lt.0001
• b 1 1.12500
  0.19764 5.69 0.0005
• c 1 0.87500
  0.19764 4.43 0.0022
• ab 1 0.37500
  0.19764 1.90 0.0943
• ac 1 0.12500
  0.19764 0.63 0.5447
• bc 1 0.25000
  0.19764 1.26 0.2415
• abc 1 0.25000
  0.19764 1.26 0.2415

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Regression Approach

• Will the model of regression affect the
  parameter?
• For orthogonal design, the parameter will remain
  the same.
• For nonorthogonal design, the parameter will be
  different.

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Summary

• This chapter gives us the solution of
  experimental design and the method of analyzing
  data we collected.
• 1?The design of the experiment is called complete
  factorial design compared to fractional
  factorial design.
• 2?We just need to give the entire set of possible
  level
• of factors.
• If we have k factors and each have 2
  levels( thats the situation in many cases),
  then we can get the 2k experiment.
• 3?When we get the data of our experiment, we just
  need to find the parameter to our linear model
• SSTSS( all single factor
  effects)SS(all interactive effects)e(noise)
• 4?We can use regression to get the parameters of
  the model above.

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• Why is 2k experiment better than
  One-Factor-a-time?
• 1) The factors may not be additively.
• 2) Can detect interaction.
• Although sometimes we can have better ways than
  genuine replication, replicated trials can give
  us the estimation of the variance.
• ANOVA table gives us a great way to analyze the
  data to get and give us the significance of each
  factor.

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• Thank you for your attention
• Do you have any questions to ask us?