Design and Analysis of Experiments Lecture 2.1

Diploma in StatisticsDesign and Analysis of Experiments

Lecture 2.1 1

Design and Analysis of ExperimentsLecture 2.1

1. Review of Lecture 1.2

2. Randomised Block Design and Analysis– Illustration– Explaining ANOVA– Interaction?– Effect of Blocking– Matched pairs as Randomised blocks

3. Introduction to 2-level factorial designs– A 22 experiment– Set up– Analysis– Application

Lecture 2.1 2

Minute Test - How Much

How Much

ntChart of How Much

Lecture 2.1 3

Minute Test - How Fast

How Fast

ntChart of How Fast

Lecture 2.1 4

Was the blocking effective?

10987654321

Material AMaterial BDifference

Variable

Profile Plots of Material A, Material B, Difference

Lecture 2.1 5

Comparing several meansMembrane A: standardMembrane B: alternative using new materialMembrane C: other manufacturerMembrane D: other manufacturer

Burst strength (kPa) of 10 samplesof each of four filter membrane types

Lecture 2.1 6

Comparing several meansTukey 95% Simultaneous Confidence IntervalsAll Pairwise Comparisons among Levels of Membrane

Membrane = A subtracted from:Membrane Lower Center Upper ------+---------+---------+---------+-B -1.46 3.24 7.94 (---*----)C -12.91 -8.21 -3.51 (----*---)D -7.65 -2.95 1.75 (----*----) ------+---------+---------+---------+- -10 0 10 20Membrane = B subtracted from:Membrane Lower Center Upper ------+---------+---------+---------+---C -16.15 -11.45 -6.75 (----*---)D -10.89 -6.19 -1.49 (----*----) ------+---------+---------+---------+--- -10 0 10 20Membrane = C subtracted from:Membrane Lower Center Upper ------+---------+---------+---------+---D 0.560 5.260 9.960 (---*----) ------+---------+---------+---------+--- -10 0 10 20

Lecture 2.1 7

Comparing several means

• Membrane B mean is significantly bigger than Membranes C and D means and close to significantly bigger than Membrane A mean.

• Membrane C mean is significantly smaller than the other three means.

• Membranes A and D means are not significantly different.

Lecture 2.1 8

Comparing several means;Conclusions

• Membrane C can be eliminated from our inquiries.

• Membrane D shows no sign of being an improvement on the existing Membrane A and so need not be considered further.

• Membrane B shows some improvement on Membrane A but not enough to recommend a change.

• It may be worth while carrying out further comparisons between Membranes A and B.

Lecture 2.1 9

Characteristics of an experimentExperimental units:

entities on which observations are made

Experimental Factor:controllable input variable

Factor Levels / Treatments:values of the factor

Response:output variable measured on the units

Lecture 2.1 10

2 Randomised blocksIllustration

Manufacture of an organic chemical using a filtration process

• Three step process:

– input chemical blended from different stocks

– chemical reaction results in end product suspended in an intermediate liquid product

– liquid filtered to recover end product.

Lecture 2.1 11

Randomised blocksIllustration

• Problem: yield loss at filtration stage

• Proposal: adjust initial blend to reduce yield loss

• Plan:

– prepare five different blends

– use each blend in successive process runs, in random order

– repeat at later times (blocks)

Lecture 2.1 12

Results

Lecture 2.1 13

Exercise 2.1.1

What were the

experimental units

factor

factor levels

response

blocks

randomisation procedure

Lecture 2.1 14

Minitab AnalysisGeneral Linear Model ANOVA

General Linear Model: Loss, per cent versus Blend, Block

Analysis of Variance for Loss,%, using Adjusted SS for Tests

Source DF Seq SS Adj SS Adj MS F PBlend 4 11.5560 11.5560 2.8890 3.31 0.071Block 2 1.6480 1.6480 0.8240 0.94 0.429Error 8 6.9920 6.9920 0.8740Total 14 20.1960

S = 0.934880 R-Sq = 65.38% R-Sq(adj) = 39.41%

Unusual Observations for Loss, per cent

Loss, perObs cent Fit SE Fit Residual St Resid 12 17.1000 18.5267 0.6386 -1.4267 -2.09 R

Lecture 2.1 15

5% critical values for the F distribution

1 1 2 3 4 5 6 7 8 10 12 24 ∞ 2 1 161 200 216 225 230 234 237 239 242 244 249 254 2 18.5 19.0 19.2 19.2 19.3 19.3 19.4 19.4 19.4 19.4 19.5 19.5 3 10.1 9.6 9.3 9.1 9.0 8.9 8.9 8.8 8.8 8.7 8.6 8.5 4 7.7 6.9 6.6 6.4 6.3 6.2 6.1 6.0 6.0 5.9 5.8 5.6 5 6.6 5.8 5.4 5.2 5.1 5.0 4.9 4.8 4.7 4.7 4.5 4.4 6 6.0 5.1 4.8 4.5 4.4 4.3 4.2 4.1 4.1 4.0 3.8 3.7 7 5.6 4.7 4.3 4.1 4.0 3.9 3.8 3.7 3.6 3.6 3.4 3.2 8 5.3 4.5 4.1 3.8 3.7 3.6 3.5 3.4 3.3 3.3 3.1 2.9 9 5.1 4.3 3.9 3.6 3.5 3.4 3.3 3.2 3.1 3.1 2.9 2.7 10 5.0 4.1 3.7 3.5 3.3 3.2 3.1 3.1 3.0 2.9 2.7 2.5 12 4.7 3.9 3.5 3.3 3.1 3.0 2.9 2.8 2.8 2.7 2.5 2.3 15 4.5 3.7 3.3 3.1 2.9 2.8 2.7 2.6 2.5 2.5 2.3 2.1 20 4.4 3.5 3.1 2.9 2.7 2.6 2.5 2.4 2.3 2.3 2.1 1.8 30 4.2 3.3 2.9 2.7 2.5 2.4 2.3 2.3 2.2 2.1 1.9 1.6 40 4.1 3.2 2.8 2.6 2.4 2.3 2.2 2.2 2.1 2.0 1.8 1.5 120 3.9 3.1 2.7 2.4 2.3 2.2 2.1 2.0 1.9 1.8 1.6 1.3 ∞ 3.8 3.0 2.6 2.4 2.2 2.1 2.0 1.9 1.8 1.8 1.5 1.0

Lecture 2.1 16

Conclusions (prelim.)

F(Blends) is almost statistically significant, p = 0.07

F(Blocks) is not statistically significant, p = 0.4

Prediction standard deviation: S = 0.93

Lecture 2.1 17

Deleted diagnostics

19181716

Fitted Value

210-1-2

lScore

N 15AD 0.245P-Value 0.712

Versus Fits(response is Loss)

Normal Probability Plot(response is Loss)

Lecture 2.1 18

Iterated analysis:delete Case 12

General Linear Model: Loss versus Blend, Block

Analysis of Variance for Loss

Source DF Seq SS Adj SS Adj MS F P

Blend 4 13.0552 14.5723 3.6431 8.03 0.009Block 2 3.7577 3.7577 1.8788 4.14 0.065Error 7 3.1757 3.1757 0.4537

Total 13 19.9886

S = 0.673548

Lecture 2.1 19

Deleted diagnostics

2019181716

Fitted Value

-3210-1-2

lScore

N 14AD 0.189P-Value 0.881

Versus Fits(response is Loss)

Normal Probability Plot(response is Loss)

Lecture 2.1 20

Conclusions (prelim.)

F(Blends) is highly statistically significant, p = 0.01

F(Blocks) is not statistically significant, p = 0.65

Prediction standard deviation: S = 0.67

Lecture 2.1 21

Explaining ANOVA

ANOVA depends on a decompostion of "Total variation" into components:

Total Variation = Blend effect + Block effect

+ chance variation;

)YYYY(

)YY(k)YY(n)YY(

Lecture 2.1 22

Decomposition of results Blocks

I II III Mean A 16.9 16.5 17.5 17.0 B 18.2 19.2 17.1 18.2 Blends C 17.0 18.1 17.3 17.5

D 15.1 16.0 17.8 16.3 E 18.3 18.3 19.8 18.8

Mean 17.1 17.6 17.9 17.5

Lecture 2.1 23

Decomposition of results

Overall Deviations Blend Deviations Block Deviations Residuals

YYrc = YYr + YYc + YYYY crrc

I II III I II III I II III I II III A -0.6 -1.0 0.0 -0.6 -0.6 -0.6 -0.4 0.1 0.4 0.4 -0.5 0.2 B 0.7 1.7 -0.4 0.6 0.6 0.6 -0.4 0.1 0.4 0.5 1.0 -1.4 C -0.5 0.6 -0.2 = -0.1 -0.1 -0.1 + -0.4 0.1 0.4 + 0.0 0.6 -0.5 D -2.4 -1.5 0.3 -1.2 -1.2 -1.2 -0.4 0.1 0.4 -0.8 -0.4 1.1 E 0.8 0.8 2.3 1.3 1.3 1.3 -0.4 0.1 0.4 -0.1 -0.6 0.6

SSTO = 20.20 SS(Blends) = 11.56 SS(Blocks) = 1.65 SSE = 6.99

dfTO = 14 df(Blends) = 4 df(Blocks) = 2 dfE = 8

Blocks

D 15.1 16.0 17.8 16.3 E 18.3 18.3 19.8 18.8

Mean 17.1 17.6 17.9 17.5

Lecture 2.1 24

Blocks

D 15.1 16.0 17.8 16.3 E 18.3 18.3 19.8 18.8

Mean 17.1 17.6 17.9 17.5

Lecture 2.1 25

Blocks

D 15.1 16.0 17.8 16.3 E 18.3 18.3 19.8 18.8

Mean 17.1 17.6 17.9 17.5

Lecture 2.1 26

Blocks

D 15.1 16.0 17.8 16.3 E 18.3 18.3 19.8 18.8

Mean 17.1 17.6 17.9 17.5

Lecture 2.1 27

Blocks

D 15.1 16.0 17.8 16.3 E 18.3 18.3 19.8 18.8

Mean 17.1 17.6 17.9 17.5

Lecture 2.1 28

Blocks

D 15.1 16.0 17.8 16.3 E 18.3 18.3 19.8 18.8

Mean 17.1 17.6 17.9 17.5

Lecture 2.1 29

Interaction?

Blend profiles

Blend x Block interaction?

Lecture 2.1 30

Interaction?

Blend x Block interaction?

Block 1Block 2Block 3

Block Profiles

Lecture 2.1 31

Exercise 2.1.2Calculate fitted values:

Overall Mean + Blend Deviation + Block deviation

17.5 +

Lecture 2.1 32

Exercise 2.1.2 (cont'd)

Make a Block profile plot

Lecture 2.1 33

Fitted values; NO INTERACTION

Line Plot of Fitted Values

Lecture 2.1 34

Actual plot: Interaction?

tBlock 1Block 2Block 3

Block Profiles

Blend effects (the contributions of each blend to Loss) are similar for Blocks 1 and 2 but quite different for Block 3.

Lecture 2.1 35

Effect of BlockingAnalysis of Variance for Loss (one run deleted)

Blend 4 13.0552 14.5723 3.6431 8.03 0.009Block 2 3.7577 3.7577 1.8788 4.14 0.065Error 7 3.1757 3.1757 0.4537

Total 13 19.9886

Analysis of Variance for Loss (one run deleted) unblocked

Blend 4 13.0552 13.0552 3.2638 4.24 0.034Error 9 6.9333 6.9333 0.7704

Total 13 19.9886

Lecture 2.1 36

Matched pairs as Randomised blocks

Wear of shoe solesmade of two materials, A and B,

worn on opposite feet by each of 10 boys

Boy Material A Material B Difference 1 13.2 14.0 0.8 2 8.2 8.8 0.6 3 10.9 11.2 0.3 4 14.3 14.2 -0.1 5 10.7 11.8 1.1 6 6.6 6.4 -0.2 7 9.5 9.8 0.3 8 10.8 11.3 0.5 9 8.8 9.3 0.5

10 13.3 13.6 0.3 Mean 10.63 11.04 0.41 St Dev 2.451 2.518 0.387

Lecture 2.1 37

Pairing equals Blocking

Paired T for Material B - Material A

T-Test of mean difference = 0 (vs not = 0): T-Value = 3.35 P-Value = 0.009

Two-way ANOVA: Wear versus Material, Boy

Source DF SS MS F PMaterial 1 0.841 0.8405 11.21 0.009Boy 9 110.491 12.2767 163.81 0.000Error 9 0.675 0.0749Total 19 112.006

Lecture 2.1 38

Selected critical values for the t-distribution .25 .10 .05 .02 .01 .002 .001

= 1 2.41 6.31 12.71 31.82 63.66 318.32 636.61 2 1.60 2.92 4.30 6.96 9.92 22.33 31.60 3 1.42 2.35 3.18 4.54 5.84 10.22 12.92 4 1.34 2.13 2.78 3.75 4.60 7.17 8.61 5 1.30 2.02 2.57 3.36 4.03 5.89 6.87 6 1.27 1.94 2.45 3.14 3.71 5.21 5.96 7 1.25 1.89 2.36 3.00 3.50 4.79 5.41 8 1.24 1.86 2.31 2.90 3.36 4.50 5.04 9 1.23 1.83 2.26 2.82 3.25 4.30 4.78 10 1.22 1.81 2.23 2.76 3.17 4.14 4.59 12 1.21 1.78 2.18 2.68 3.05 3.93 4.32 15 1.20 1.75 2.13 2.60 2.95 3.73 4.07 20 1.18 1.72 2.09 2.53 2.85 3.55 3.85 24 1.18 1.71 2.06 2.49 2.80 3.47 3.75 30 1.17 1.70 2.04 2.46 2.75 3.39 3.65 40 1.17 1.68 2.02 2.42 2.70 3.31 3.55 60 1.16 1.67 2.00 2.39 2.66 3.23 3.46 120 1.16 1.66 1.98 2.36 2.62 3.16 3.37 ∞ 1.15 1.64 1.96 2.33 2.58 3.09 3.29

t and F

Lecture 2.1 39

1 1 2 3 4 5 6 7 8 10 12 24 ∞ 2 1 161 200 216 225 230 234 237 239 242 244 249 254 2 18.5 19.0 19.2 19.2 19.3 19.3 19.4 19.4 19.4 19.4 19.5 19.5 3 10.1 9.6 9.3 9.1 9.0 8.9 8.9 8.8 8.8 8.7 8.6 8.5 4 7.7 6.9 6.6 6.4 6.3 6.2 6.1 6.0 6.0 5.9 5.8 5.6 5 6.6 5.8 5.4 5.2 5.1 5.0 4.9 4.8 4.7 4.7 4.5 4.4 6 6.0 5.1 4.8 4.5 4.4 4.3 4.2 4.1 4.1 4.0 3.8 3.7 7 5.6 4.7 4.3 4.1 4.0 3.9 3.8 3.7 3.6 3.6 3.4 3.2 8 5.3 4.5 4.1 3.8 3.7 3.6 3.5 3.4 3.3 3.3 3.1 2.9 9 5.1 4.3 3.9 3.6 3.5 3.4 3.3 3.2 3.1 3.1 2.9 2.7 10 5.0 4.1 3.7 3.5 3.3 3.2 3.1 3.1 3.0 2.9 2.7 2.5 12 4.7 3.9 3.5 3.3 3.1 3.0 2.9 2.8 2.8 2.7 2.5 2.3 15 4.5 3.7 3.3 3.1 2.9 2.8 2.7 2.6 2.5 2.5 2.3 2.1 20 4.4 3.5 3.1 2.9 2.7 2.6 2.5 2.4 2.3 2.3 2.1 1.8 30 4.2 3.3 2.9 2.7 2.5 2.4 2.3 2.3 2.2 2.1 1.9 1.6 40 4.1 3.2 2.8 2.6 2.4 2.3 2.2 2.2 2.1 2.0 1.8 1.5 120 3.9 3.1 2.7 2.4 2.3 2.2 2.1 2.0 1.9 1.8 1.6 1.3 ∞ 3.8 3.0 2.6 2.4 2.2 2.1 2.0 1.9 1.8 1.8 1.5 1.0

t and F

Lecture 2.1 40

Selected critical values for the t-distribution .25 .10 .05 .02 .01 .002 .001

= 1 2.41 6.31 12.71 31.82 63.66 318.32 636.61 2 1.60 2.92 4.30 6.96 9.92 22.33 31.60 3 1.42 2.35 3.18 4.54 5.84 10.22 12.92 4 1.34 2.13 2.78 3.75 4.60 7.17 8.61 5 1.30 2.02 2.57 3.36 4.03 5.89 6.87 6 1.27 1.94 2.45 3.14 3.71 5.21 5.96 7 1.25 1.89 2.36 3.00 3.50 4.79 5.41 8 1.24 1.86 2.31 2.90 3.36 4.50 5.04 9 1.23 1.83 2.26 2.82 3.25 4.30 4.78 10 1.22 1.81 2.23 2.76 3.17 4.14 4.59 12 1.21 1.78 2.18 2.68 3.05 3.93 4.32 15 1.20 1.75 2.13 2.60 2.95 3.73 4.07 20 1.18 1.72 2.09 2.53 2.85 3.55 3.85 24 1.18 1.71 2.06 2.49 2.80 3.47 3.75 30 1.17 1.70 2.04 2.46 2.75 3.39 3.65 40 1.17 1.68 2.02 2.42 2.70 3.31 3.55 60 1.16 1.67 2.00 2.39 2.66 3.23 3.46 120 1.16 1.66 1.98 2.36 2.62 3.16 3.37 ∞ 1.15 1.64 1.96 2.33 2.58 3.09 3.29

More on F

Lecture 2.1 42

Paired Comparison:Effect of Pairing / Blocking

Paired T for Material B - Material A

T-Test of mean difference = 0 (vs not = 0): T-Value = 3.35 P-Value = 0.009

Two-sample T for Material B vs Material A

T-Value = 0.37 P-Value = 0.716

Lecture 2.1 43

Paired Comparison:Effect of Pairing / Blocking

Two-way ANOVA: Wear versus Material, Boy

Source DF SS MS F PMaterial 1 0.841 0.8405 11.21 0.009Boy 9 110.491 12.2767 163.81 0.000Error 9 0.675 0.0749Total 19 112.006

One-way ANOVA: Wear versus Material

Source DF SS MS F PMaterial 1 0.84 0.84 0.14 0.716Error 18 111.17 6.18Total 19 112.01

Lecture 2.1 44

3 Introduction to 2-levelfactorial designs

A 22 experiment

Project:

optimisation of a chemical process yield

Factors (with levels):

operating temperature (Low, High)

catalyst (C1, C2)

Design:

Process run at all four possible combinations of factor levels, in duplicate, in random order.

Lecture 2.1 45

Exercise 2.1.3

What were the

experimental units

factors

factor levels

response

blocks

randomisation procedure

Lecture 2.1 46

Standard Order Temperature Catalyst

1 Low 1 2 High 1 3 Low 2 4 High 2 5 Low 1 6 High 1 7 Low 2 8 High 2

Set up

Lecture 2.1 47

Standard Order Temperature Catalyst Run

Order 1 Low 1 6 2 High 1 8 3 Low 2 1 4 High 2 4 5 Low 1 3 6 High 1 7 7 Low 2 2 8 High 2 5

Set up:Randomisation

Lecture 2.1 48

Set up:Run order

Standard Order Temperature Catalyst Run

Order 3 Low 2 1 7 Low 2 2 5 Low 1 3 4 High 2 4 8 High 2 5 1 Low 1 6 6 High 1 7 2 High 1 8

Lecture 2.1 49

Results (run order)

Standard Order

Run Order Temperature Catalyst Yield

3 1 Low 2 52 7 2 Low 2 45 5 3 Low 1 54 4 4 High 2 83 8 5 High 2 80 1 6 Low 1 60 6 7 High 1 68 2 8 High 1 72

Lecture 2.1 50

Results (standard order)

Standard Order

1 6 Low 1 60 2 8 High 1 72 3 1 Low 2 52 4 4 High 2 83 5 3 Low 1 54 6 7 High 1 68 7 2 Low 2 45 8 5 High 2 80

Lecture 2.1 51

Analysis (Minitab)

• Main effects and Interaction plots

• Pareto plot of effects

• ANOVA results

– with diagnostics

• Calculation of t-statistic

Lecture 2.1 52

HighLow

Temperature

Catalyst

CatalystM

LowHigh

Temperature

Main Effects Plot for YieldData Means

Interaction Plot for YieldData Means

Main Effects and Interactions

Lecture 2.1 53

Pareto plot of effects

Bar height = t value (see slide 31)

Reference line is at critical t value (4 df)

9876543210

Standardized Effect

A TemperatureB Cataly st

Factor Name

Pareto Chart of the Standardized Effects(response is Yield, Alpha = 0.05)

Lecture 2.1 54

Minitab DOEAnalyze Factorial Design

Estimated Effects and Coefficients for Yield (coded units)

Term Effect Coef SE Coef T PConstant 64.2500 1.311 49.01 0.000Temperature 23.0000 11.5000 1.311 8.77 0.001Catalyst 1.5000 0.7500 1.311 0.57 0.598Temperature*Catalyst 10.0000 5.0000 1.311 3.81 0.019

S = 3.70810 R-Sq = 95.83% R-Sq(adj) = 92.69%

Analysis of Variance for Yield (coded units)

Source DF Seq SS Adj SS Adj MS F PMain Effects 2 1062.50 1062.50 531.25 38.64 0.0022-Way Interactions 1 200.00 200.00 200.00 14.55 0.019Residual Error 4 55.00 55.00 13.75 Pure Error 4 55.00 55.00 13.75Total 7 1317.50

Lecture 2.1 55

1 1 2 3 4 5 6 7 8 10 12 24 ∞ 2 1 161 200 216 225 230 234 237 239 242 244 249 254 2 18.5 19.0 19.2 19.2 19.3 19.3 19.4 19.4 19.4 19.4 19.5 19.5 3 10.1 9.6 9.3 9.1 9.0 8.9 8.9 8.8 8.8 8.7 8.6 8.5 4 7.7 6.9 6.6 6.4 6.3 6.2 6.1 6.0 6.0 5.9 5.8 5.6 5 6.6 5.8 5.4 5.2 5.1 5.0 4.9 4.8 4.7 4.7 4.5 4.4 6 6.0 5.1 4.8 4.5 4.4 4.3 4.2 4.1 4.1 4.0 3.8 3.7 7 5.6 4.7 4.3 4.1 4.0 3.9 3.8 3.7 3.6 3.6 3.4 3.2 8 5.3 4.5 4.1 3.8 3.7 3.6 3.5 3.4 3.3 3.3 3.1 2.9 9 5.1 4.3 3.9 3.6 3.5 3.4 3.3 3.2 3.1 3.1 2.9 2.7 10 5.0 4.1 3.7 3.5 3.3 3.2 3.1 3.1 3.0 2.9 2.7 2.5 12 4.7 3.9 3.5 3.3 3.1 3.0 2.9 2.8 2.8 2.7 2.5 2.3 15 4.5 3.7 3.3 3.1 2.9 2.8 2.7 2.6 2.5 2.5 2.3 2.1 20 4.4 3.5 3.1 2.9 2.7 2.6 2.5 2.4 2.3 2.3 2.1 1.8 30 4.2 3.3 2.9 2.7 2.5 2.4 2.3 2.3 2.2 2.1 1.9 1.6 40 4.1 3.2 2.8 2.6 2.4 2.3 2.2 2.2 2.1 2.0 1.8 1.5 120 3.9 3.1 2.7 2.4 2.3 2.2 2.1 2.0 1.9 1.8 1.6 1.3 ∞ 3.8 3.0 2.6 2.4 2.2 2.1 2.0 1.9 1.8 1.8 1.5 1.0

Lecture 2.1 56

Minitab DOEAnalyze Factorial Design

Estimated Effects and Coefficients for Yield (coded units)

Term Effect Coef SE Coef T PConstant 64.2500 1.311 49.01 0.000Temperature 23.0000 11.5000 1.311 8.77 0.001Catalyst 1.5000 0.7500 1.311 0.57 0.598Temperature*Catalyst 10.0000 5.0000 1.311 3.81 0.019

S = 3.70810 R-Sq = 95.83% R-Sq(adj) = 92.69%

Analysis of Variance for Yield (coded units)

Source DF Seq SS Adj SS Adj MS F PMain Effects 2 1062.50 1062.50 531.25 38.64 0.0022-Way Interactions 1 200.00 200.00 200.00 14.55 0.019Residual Error 4 55.00 55.00 13.75 Pure Error 4 55.00 55.00 13.75Total 7 1317.50

Lecture 2.1 57

ANOVA results

ANOVA superfluous for 2k experiments

"There is nothing to justify this complexity other than a misplaced belief in the universal value of an ANOVA table".

BHH (2nd ed.), Section 5.10

"a convenient method of arranging the arithmetic" R.A. Fisher

Lecture 2.1 58

Diagnostic Plots

80706050

Fitted Value

-3210-1-2

N 8AD 0.261P-Value 0.600

Versus Fits(response is Yield)

Normal Probability Plot(response is Yield)

Lecture 2.1 59

Calculation of t-statistic

Standard Order

3 1 Low 2 52 7 2 Low 2 45 5 3 Low 1 54 1 6 Low 1 60 4 4 High 2 83 8 5 High 2 80 6 7 High 1 68 2 8 High 1 72

4s)YY(SE

LowHigh

LowHighHighLow

Results (Temperature order)

Lecture 2.1 60

Exercise 2.1.4

Calculate a confidence interval for the Temperature effect.

All effects may be estimated and tested in this way.

Homework 2.1.2

Test the statistical significance of and calculate confidence intervals for the Catalyst effect and the Temperature × Catalyst interaction.

Lecture 2.1 61

ApplicationFinding the optimum

More Minitab results

Least Squares Means for Yield

Mean SE MeanTemperature Low 52.75 1.854 High 75.75 1.854

Catalyst 1 63.50 1.854 2 65.00 1.854

Temperature*Catalyst Low 1 57.00 2.622 High 1 70.00 2.622 Low 2 48.50 2.622 High 2 81.50 2.622

Lecture 2.1 62

HighLow

Catalyst

Temperature

70.057.0

Cube Plot (data means) for Yield

8.5 11.5

Lecture 2.1 63

Optimum operating conditions

Highest yield achieved

with Catalyst 2

at High temperature.

Estimated yield: 81.5%

95% confidence interval:

81.5 ± 2.78 × 2.622,

i.e., 81.5 ± 7.3,

i.e., ( 74.2 , 88.8 )

Lecture 2.1 64

Homework 2.1.3As part of a project to develop a GC method for analysing trace compounds in wine without the need for prior extraction of the compounds, a synthetic mixture of aroma compounds in ethanol-water was prepared. The effects of two factors, Injection volume and Solvent flow rate, on GC measured peak areas given by the mixture were assessed using a 22 factorial design with 3 replicate measurements at each design point. The results are shown in the table that follows.

What conclusions can be drawn from these data? Display results numerically and graphically. Check model assumptions by using appropriate residual plots.

Lecture 2.1 65

Peak areas for GC study

Injection volume, LSolvent flow rate,

mL/min 100 200

13.1 126.5 400 15.3 118.5

17.7 122.1 48.8 134.5

200 42.1 135.4 39.2 128.6

(EM, Exercise 5.2)

Lecture 2.1 66

Reading

EM §5.3, §7.4.2

DCM §§4-1, 5-1, 5-2, 6-1, 6-2

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