Stat 470-11

• Today: More Chapter 3

Analysis of Location and Dispersion Effects

• The epitaxial growth layer experiment is a 24 factorial design

• Have looked at ways to analyze response of a factorial experiment

– Plotting effects on a normal probability plot

– Regression

• May wish to model mean and also the variance

Analysis of Location and Dispersion Effects

• Recall, from Section 3.2, the quadratic loss function

• The expected loss E(y,t)=cVar(y)+c(E(y)-t)2 suggested1. Selecting levels of some factors to minimize V(y)

2. Selecting levels of other factors to adjust the mean as close as possible to the target, t.

• Need a model for the variance (dispersion)

2)(),( tyctyL

Analysis of Location and Dispersion Effects

• Let be the sample mean of observations taken at the ith treatment of the experiment

• Let si2 be the sample variance of observations taken at the ith treatment

of the experiment

• That is,

• Can model both the mean and variance using regression

in

jij

ii yn

y1

1

2

1

2 )()1(

1i

n

jij

ii yy

ns

i

Analysis of Location and Dispersion Effects

• Would like to model the variance as a function of the factors

• Regression assumes that quantities measured at each treatment be normally distributed

• Is it likely that is normally distributed?

•

2

1

2 )()1(

1i

n

jij

ii yy

ns

i

Example: Original Growth Layer Experiment

A B C D Thickness -1 -1 -1 +1 14.812 14.774 14.772 14.794 14.860 14.914 -1 -1 -1 -1 13.768 13.778 13.870 13.896 13.932 13.914 -1 -1 +1 +1 14.722 14.736 14.774 14.778 14.682 14.850 -1 -1 +1 -1 13.860 13.876 13.932 13.846 13.896 13.870 -1 +1 -1 +1 14.886 14.810 14.868 14.876 14.958 14.932 -1 +1 -1 -1 14.182 14.172 14.126 14.274 14.154 14.082 -1 +1 +1 +1 14.758 14.784 15.054 15.058 14.938 14.936 -1 +1 +1 -1 13.996 13.988 14.044 14.028 14.108 14.060 +1 -1 -1 +1 15.272 14.656 14.258 14.718 15.198 15.490 +1 -1 -1 -1 14.324 14.092 13.536 13.588 13.964 14.328 +1 -1 +1 +1 13.918 14.044 14.926 14.962 14.504 14.136 +1 -1 +1 -1 13.614 13.202 13.704 14.264 14.432 14.228 +1 +1 -1 +1 14.648 14.350 14.682 15.034 15.384 15.170 +1 +1 -1 -1 13.970 14.448 14.326 13.970 13.738 13.738 +1 +1 +1 +1 14.184 14.402 15.544 15.424 15.036 14.470 +1 +1 +1 -1 13.866 14.130 14.256 14.000 13.640 13.592

Example: Original Growth Layer Experiment

A B C D y s2 ln(s2)

-1 -1 -1 +1 14.82 .0031 -5.771 -1 -1 -1 -1 13.86 .0049 -5.311 -1 -1 +1 +1 14.76 .0033 -5.704 -1 -1 +1 -1 13.88 .0009 -6.984 -1 +1 -1 +1 14.89 .0027 -5.917 -1 +1 -1 -1 14.17 .0041 -5.485 -1 +1 +1 +1 14.92 .0165 -4.107 -1 +1 +1 -1 14.04 .0020 -6.237 +1 -1 -1 +1 14.93 .2148 -1.538 +1 -1 -1 -1 13.97 .1205 -2.116 +1 -1 +1 +1 14.42 .2061 -1.579 +1 -1 +1 -1 13.91 .2260 -1.487 +1 +1 -1 +1 14.88 .1471 -1.916 +1 +1 -1 -1 14.03 .0880 -2.430 +1 +1 +1 +1 14.84 .3268 -1.118 +1 +1 +1 -1 13.91 .0704 -2.653

Example: Original Growth Layer Experiment

• Model Matrix for a single replicate:

A B C D AB AC AD BC BD CD ABC ABD ACD BCD ABCD-1 -1 -1 +1 +1 +1 -1 +1 -1 -1 -1 +1 +1 +1 -1-1 -1 -1 -1 +1 +1 +1 +1 +1 +1 -1 -1 -1 -1 +1-1 -1 +1 +1 +1 -1 -1 -1 -1 +1 +1 +1 -1 -1 +1-1 -1 +1 -1 +1 -1 +1 -1 +1 -1 +1 -1 +1 +1 -1-1 +1 -1 +1 -1 +1 -1 -1 +1 -1 +1 -1 +1 -1 +1-1 +1 -1 -1 -1 +1 +1 -1 -1 +1 +1 +1 -1 +1 -1-1 +1 +1 +1 -1 -1 -1 +1 +1 +1 -1 -1 -1 +1 -1-1 +1 +1 -1 -1 -1 +1 +1 -1 -1 -1 +1 +1 -1 +1+1 -1 -1 +1 -1 -1 +1 +1 -1 -1 +1 -1 -1 +1 +1+1 -1 -1 -1 -1 -1 -1 +1 +1 +1 +1 +1 +1 -1 -1+1 -1 +1 +1 -1 +1 +1 -1 -1 +1 -1 -1 +1 -1 -1+1 -1 +1 -1 -1 +1 -1 -1 +1 -1 -1 +1 -1 +1 +1+1 +1 -1 +1 +1 -1 +1 -1 +1 -1 -1 +1 -1 -1 -1+1 +1 -1 -1 +1 -1 -1 -1 -1 +1 -1 -1 +1 +1 +1+1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1+1 +1 +1 -1 +1 +1 -1 +1 -1 -1 +1 -1 -1 -1 -1

Example: Original Growth Layer Experiment

• Effect Estimates and QQ-Plot:

-Effect Estimate

A 0.055B 0.142C -0.109D 0.836

AB -0.032AC -0.074AD -0.025BC 0.047BD 0.010CD -0.037

ABC 0.060ABD 0.067ACD -0.056BCD 0.098

ABCD 0.036

Quantiles of Standard Normal

y

-1 0 1

0.0

0.2

0.4

0.6

0.8

Example: Original Growth Layer Experiment

• Regression equation for the mean response:

Example: Original Growth Layer Experiment

• Dispersion analysis:

-Effect Estimate

A 3.834B 0.078C 0.077D 0.632

AB -0.428AC 0.214AD 0.002BC 0.331BD 0.305CD 0.582

ABC -0.335ABD 0.086ACD -0.494BCD 0.314

ABCD 0.109

Quantiles of Standard Normal

ln(s

2)

-1 0 1

01

23

4

Example: Original Growth Layer Experiment

• Regression equation for the ln(s2) response:

Example: Original Growth Layer Experiment

• Suggested settings for the process:

Example: Original Growth Layer Experiment

• Suggested settings for the process in the original units of the factors:

Location-Dispersion Modeling

• Steps:

Example

• An experiment was conducted to improve a heat treatment process on truck leaf springs

• The heat treatment process, which forms the curvature of the leaf spring, consists of 1. Heating in a furnace

2. Processing by machine forming

3. Quenching in an oil bath

• The height of an unloaded spring, known as the free height, is the quality characteristic of interest and has a target of 8 inches

Example

• The experiment goals are to1. Minimize the variability about the target

2. Keep the process mean as close to the target of 8 inches as possible

• A 24 factorial experiment was conducted with factors:• A. Furnace Temperature

• B. Heating Time

• C. Transfer Time

• Q. Quench Oil Temperature

• There were 3 replicates of the experiment

Example

• DataA B C Q Free Height (inches) -1 +1 +1 -1 7.78 7.78 7.81 +1 +1 +1 -1 8.15 8.18 7.88 -1 -1 +1 -1 7.50 7.56 7.50 +1 -1 +1 -1 7.59 7.56 7.75 -1 +1 -1 -1 7.94 8.00 7.88 +1 +1 -1 -1 7.69 8.09 8.06 -1 -1 -1 -1 7.56 7.62 7.44 +1 -1 -1 -1 7.56 7.81 7.69 -1 +1 +1 +1 7.50 7.25 7.12 +1 +1 +1 +1 7.88 7.88 7.44 -1 -1 +1 +1 7.50 7.56 7.50 +1 -1 +1 +1 7.63 7.75 7.56 -1 +1 -1 +1 7.32 7.44 7.44 +1 +1 -1 +1 7.56 7.69 7.62 -1 -1 -1 +1 7.18 7.18 7.25 +1 -1 -1 +1 7.81 7.50 7.59

Example

• DataA B C Q y s2 ln(s2) -1 +1 +1 -1 7.7900 .0003 -8.1117 +1 +1 +1 -1 8.0700 .0273 -3.6009 -1 -1 +1 -1 7.5200 .0012 -6.7254 +1 -1 +1 -1 7.6333 .0104 -4.5627 -1 +1 -1 -1 7.9400 .0036 -5.6268 +1 +1 -1 -1 7.9467 .0496 -3.0031 -1 -1 -1 -1 7.5400 .0084 -4.7795 +1 -1 -1 -1 7.6867 .0156 -4.1583 -1 +1 +1 +1 7.2900 .0373 -3.2888 +1 +1 +1 +1 7.7333 .0645 -2.7406 -1 -1 +1 +1 7.5200 .0012 -6.7254 +1 -1 +1 +1 7.6467 .0092 -4.6849 -1 +1 -1 +1 7.4000 .0048 -5.3391 +1 +1 -1 +1 7.6233 .0042 -5.4648 -1 -1 -1 +1 7.2033 .0016 -6.4171 +1 -1 -1 +1 7.6333 .0254 -3.6717

Example: Location Model

Term Estimated Regression Coefficient

Effect Estimates

A .111 .222 B .088 .176 C .014 .029 Q -.130 -.260

AB .009 .017 AC .010 .020 AQ .042 .085 BC -.018 -.035 BQ -.083 -.165 CQ .027 .054

ABC .052 .104 ABQ .005 .010 ACQ -.020 -.040 BCQ -.024 -.047

ABCQ .014 .027

Example: Location Model

Quantiles of Standard Normal

x

-1 0 1

-0.1

0-0

.05

0.0

0.0

50

.10

Example

• Regression equation for the mean response:

Example: Dispersion Model

Term Estimated Regression Coefficient

Effect Estimates

A .945 1.890 B .284 .568 C -.124 -.248 Q .140 .280

AB -.001 -.002 AC .212 .424 AQ -.294 -.588 BC .335 .670 BQ .299 .598 CQ .555 1.110

ABC .108 .216 ABQ -.545 -1.090 ACQ -.216 -.432 BCQ .427 .854

ABCQ .065 .129

Example: Dispersion Model

Quantiles of Standard Normal

x

-1 0 1

-1.0

-0.5

0.0

0.5

1.0

Example

• Regression equation for the dispersion responses: