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Lecture 2.13 © 2016 Michael Stuart Minute Test: How Fast Postgraduate Certificate in Statistics Design and Analysis of Experiments
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Lecture 2.1 1© 2016 Michael Stuart
Design and Analysis of ExperimentsLecture 2.1
1. Review– Minute tests 1.2– Homework– Experimental factors with several levels
2. Analysis of Variance3. Randomised blocks design:
illustrations4. Randomised blocks design and analysis:
case study
Postgraduate Certificate in Statistics Design and Analysis of Experiments
Lecture 2.1 2© 2016 Michael Stuart
Minute Test: How Much
Postgraduate Certificate in Statistics Design and Analysis of Experiments
Lecture 2.1 3© 2016 Michael Stuart
Minute Test: How Fast
Postgraduate Certificate in Statistics Design and Analysis of Experiments
Lecture 2.1 4© 2016 Michael Stuart
Exercise 1.2.1Process Development Study
Process: pellet makingRequirement: specification limits for pellet sizeProblem: proportion meeting specification
too lowProposal: change machine speed from A to B
Postgraduate Certificate in Statistics Design and Analysis of Experiments
Lecture 2.1 5© 2016 Michael Stuart
Postgraduate Certificate in Statistics Design and Analysis of Experiments
Replication
Naive experiment:run process once at speed A,run process once at speed B,calculate response difference
Q: is response difference due tochange
orchance?
Lecture 2.1 6© 2016 Michael Stuart
Postgraduate Certificate in Statistics Design and Analysis of Experiments
Replication
Improved experiment:run process twice at speed A,run process twice at speed B,calculate mean response at each speed,
difference in mean responsesmeasures change effect
calculate response difference at each speed,mean of response differences measures chance effect
Lecture 2.1 7© 2016 Michael Stuart
Process Development Study
Postgraduate Certificate in Statistics Design and Analysis of Experiments
Mean 78.75 75.25
Difference 5.1 3.5
Speed B Speed A
76.2 73.581.3 77.0
Lecture 2.1 8© 2016 Michael Stuart
Exercise 1.2.1
Formal test:
Numerator measures change effect,
Denominator measures chance effect.
Carry out the test using the results from the first two runs at each speed. Compare with test using complete data
n/s2yyt
2AB
Postgraduate Certificate in Statistics Design and Analysis of Experiments
Lecture 2.1 9© 2016 Michael Stuart
Process Development Study
Variable N Mean StDev
Speed B 2 78.75 3.61Speed A 2 75.25 2.47
Speed B Speed A
76.2 73.581.3 77.0
Postgraduate Certificate in Statistics Design and Analysis of Experiments
Lecture 2.1 10© 2016 Michael Stuart
Replication
Two measurements per sample provides a valid test– but not a powerful test
More measurements per sample provides – more precision in estimating within-sample variation,
i.e., estimating sand, therefore,
– more power in testing between-sample variation.Recall the discussion in Base Module Chapter 4:
– 11 replications needed to detect a 5% improvement
Postgraduate Certificate in Statistics Design and Analysis of Experiments
Lecture 2.1 11© 2016 Michael Stuart
Design and Analysis of ExperimentsLecture 2.1
1. Review– Minute tests 1.2– Homework– Experimental factors with several levels
2. Analysis of Variance3. Randomised blocks design:
illustrations4. Randomised blocks design and analysis:
case study
Postgraduate Certificate in Statistics Design and Analysis of Experiments
Lecture 2.1 12© 2016 Michael Stuart
A multi-level experimental factorFilter membrane improvement project
Four membrane types:
A: current standardB: newly developed alternativeC: OEM 1D: OEM 2
Criterion: failure pressure level (kPa)
Objectives: (i) is Type B better than Type A?
(ii) are OEM membranes better?
Postgraduate Certificate in Statistics Design and Analysis of Experiments
Lecture 2.1 13© 2016 Michael Stuart
Filter membrane improvement project
Procedure: from each of 10 production batchesof each membrane type,sample 5 membranes,
for each sample of 5, run the filtering process using each membrane, increasing pressure until membrane failure,
calculate sample mean failure pressure reading.
Postgraduate Certificate in Statistics Design and Analysis of Experiments
Lecture 2.1 14© 2016 Michael Stuart
Filter membrane improvement projectthe response
the experimental factor
the factor levels
the treatments
an experimental unit
an observational unit
unit structure
treatment assignment
replication
burst strength
membrane type
A, B, C, D
A, B, C, D
5 process test runs
process test run
simple
no information
10Postgraduate Certificate in Statistics Design and Analysis of Experiments
Lecture 2.1 15© 2016 Michael Stuart
ResultsMean burst strengths (failure pressure level, kPa) of
10 samples from each of 4 filter membrane types
Classwork 1.2.3 Make dotplots of the breaking strengths
Membrane A Membrane B Membrane C Membrane D 95.5 90.5 86.3 89.5 103.2 98.1 84.0 93.4 93.1 97.8 86.2 87.5 89.3 97.0 80.2 89.4 90.4 98.0 83.7 87.9 92.1 95.2 93.4 86.2 93.1 95.3 77.1 89.9 91.9 97.1 86.8 89.5 95.3 90.5 83.7 90.0 84.5 101.3 84.9 95.6
Ref: Membrane strength.xls
Postgraduate Certificate in Statistics Design and Analysis of Experiments
Lecture 2.1 16© 2016 Michael Stuart
Initial data analysisBurst strength (kPa) of 10 samples
of each of four filter membrane types
Variable Membrane N Mean StDev Minimum Maximum RangeStrength A 10 93 4.8 85 103 19 B 10 96 3.4 91 101 11 C 10 85 4.3 77 93 16 D 10 90 2.8 86 96 9
1051009590858075
A
BC
D
Strength
Membran
e
Postgraduate Certificate in Statistics Design and Analysis of Experiments
Lecture 2.1 17© 2016 Michael Stuart
Design and Analysis of ExperimentsLecture 2.1
1. Review– Minute tests 1.2– Homework– Experimental factors with several levels
2. Analysis of Variance3. Randomised blocks design:
illustrations4. Randomised blocks design and analysis:
case study
Postgraduate Certificate in Statistics Design and Analysis of Experiments
Lecture 2.1 18© 2016 Michael Stuart
One-way ANOVA: Strength versus Membrane
Source DF SS MS F PMembrane 3 709.2 236.4 15.54 0.000Error 36 547.8 15.2Total 39 1257.0
S = 3.901
F3,36;0.05 ≈ 2.85
Conclusion:
Differences between means are highly statistically significant:
process variation has standard deviation of almost 4.
Comparing several means
Postgraduate Certificate in Statistics Design and Analysis of Experiments
Lecture 2.1 19© 2016 Michael Stuart
One-way ANOVA: Strength versus Membrane
Source DF SS MS F PMembrane 3 709.2 236.4 15.54 0.000Error 36 547.8 15.2Total 39 1257.0
S = 3.901
F3,36;0.05 ≈ 2.85
Conclusion:
Differences between means are highly statistically significant:
process variation has standard deviation of almost 4.
Comparing several means
Postgraduate Certificate in Statistics Design and Analysis of Experiments
Lecture 2.1 20© 2016 Michael Stuart
Analysis of Variance Explained
Decomposing Total Variation
Expected Mean Squares
Ref: Base Module Chapter 5, §5.1
Postgraduate Certificate in Statistics Design and Analysis of Experiments
Lecture 2.1 21© 2016 Michael Stuart
Decomposing Total VariationElementary decomposition:
Analysis of Variance decomposition:
SSTO = SSM + SSE DFTO = DFM + DFE
)yy()yy(yy iijiij
totaldeviation
membranedeviation
errordeviation= +
dataall
2iij
dataall
2i
dataall
2ij )yy()yy()yy(
Postgraduate Certificate in Statistics Design and Analysis of Experiments
Lecture 2.1 22© 2016 Michael Stuart
Decomposing Total Variation
Classwork 1.2.4
Confirm the degrees of freedom (DF) and sum of squares (SS) decompostion and confirm the calculation of the mean squares and the F-ratio in the membrane analysis of variance table.
One-way ANOVA: Strength versus Membrane
Source DF SS MS F PMembrane 3 709.2 236.4 15.54 0.000Error 36 547.8 15.2Total 39 1257.0
Postgraduate Certificate in Statistics Design and Analysis of Experiments
Lecture 2.1 23© 2016 Michael Stuart
Expected Mean Squares
All mi = m ↔ E(MSM) = E(MSE)
All mi ≠ m ↔ E(MSM) > E(MSE)
Hence, MSM ≈ MSE suggests all mi ≈ m, and
MSM >> MSE suggests all mi ≠ m.
F = measures by how much MSM exceeds MSE
1I)(
J)MSM(E2
i2
mms
2)MSE(E s
MSEMSM
Postgraduate Certificate in Statistics Design and Analysis of Experiments
Lecture 2.1 24© 2016 Michael Stuart
Multiple comparisons
Postgraduate Certificate in Statistics Design and Analysis of Experiments
Lecture 2.1 25© 2016 Michael Stuart
Multiple comparisons
Confidence interval for difference between means:
If 0 is not within the interval,then
0 is more than 2SE fromso
is more than 2SE from 0,that is
means are statististically significantly different
Postgraduate Certificate in Statistics Design and Analysis of Experiments
SE2YX
YX
YX
Lecture 2.1 26© 2016 Michael Stuart
Interpreting multiple comparisons
• Membrane B mean is significantly stronger than Membranes C and D means and close to significantly stronger than Membrane A mean.
• Membrane C mean is significantly less strong than the other three means.
• Membranes A and D means are not significantly different.
Membrane Type
Mean Strength
B 96.08 A 92.84 D 89.89 C 84.63
Postgraduate Certificate in Statistics Design and Analysis of Experiments
Lecture 2.1 27© 2016 Michael Stuart
Multiple comparisons explained
• Simultaneous confidence intervals slightly wider than individual confidence intervals.– level of confidence in
• several intervals simultaneouslyversus– level of confidence in
• a single interval; – more opportunities for being wrong
• Widening intervals increases confidence.– extent of widening chosen to compensate for
reduction in confidence involved.Postgraduate Certificate in Statistics Design and Analysis of Experiments
Lecture 2.1 28© 2016 Michael Stuart
Diagnostic analysis
95.092.590.087.585.0
10
5
0
-5
-10
Fitted Value
Res
idua
l
10
5
0
-5
-10210-1-2
Res
idua
l
Score
N 40AD 0.736
P-Value 0.051
Versus Fits(response is Strength)
Normal Probability Plot(response is Strength)
Postgraduate Certificate in Statistics Design and Analysis of Experiments
Lecture 2.1 29© 2016 Michael Stuart
Report
Postgraduate Certificate in Statistics Design and Analysis of Experiments
Reminder of objectives of the experiment:(i) is the company's newly developed membrane Type
B better than the standard Type A?(ii) is there any advantage in introducing other
companies' membranes?
Answer (ii): NOAnswer (i): "Some evidence" that B is better than A,
but other factors may be more important.Possibly make further comparisons.
Lecture 2.1 30© 2016 Michael Stuart
Brief management report
Membrane Type C can be eliminated from our inquiries.
Membrane Type D shows no sign of being an improvement on the existing Membrane Type A and so need not be considered further.
Membrane Type B shows some improvement on Membrane Type A but not enough to recommend a change.
It may be worth while carrying out further comparisons between Membranes Types A and B.
Postgraduate Certificate in Statistics Design and Analysis of Experiments
Lecture 2.1 31© 2016 Michael Stuart
Fisher on Analysis of Variance Table
"a convenient method of arranging the arithmetic"
(so don't show it in management reports!)
Postgraduate Certificate in Statistics Design and Analysis of Experiments
Lecture 2.1 32© 2016 Michael Stuart
Design and Analysis of ExperimentsLecture 2.1
1. Review– Minute tests 1.2– Homework– Experimental factors with several levels
2. Analysis of Variance3. Randomised blocks design:
illustrations4. Randomised blocks design and analysis:
case study
Postgraduate Certificate in Statistics Design and Analysis of Experiments
Lecture 2.1 33© 2016 Michael Stuart
Part 3Randomized complete blocks design
Example 1: treating crops with fertiliser to improve yield.
Four fertilisers being tested:
divide a single field into four plots (experimental units) to form one block,
assign treatments at random to the four plots,
repeat with several other fields to form several blocks,
choose blocks in varying locations, for generalising.
Postgraduate Certificate in Statistics Design and Analysis of Experiments
Lecture 2.1 34© 2016 Michael Stuart
Blocks of experimental plots at Rothamstead
© Rothamsted ResearchPostgraduate Certificate in Statistics Design and Analysis of Experiments
Lecture 2.1 35© 2016 Michael Stuart
Randomized blocks design
Example 2: treating long spools of rubber to improve abrasion resistance.
Four treatments being tested:
cut a single piece into four experimental units to form one block,
assign treatments at random to the four units,
repeat with several other pieces to form several blocks.
Postgraduate Certificate in Statistics Design and Analysis of Experiments
Lecture 2.1 36© 2016 Michael Stuart
Randomized blocks design
Piece 1
B
Piece 2
A
Piece 3
C
Piece 4
D
Block 1Postgraduate Certificate in Statistics Design and Analysis of Experiments
Lecture 2.1 37© 2016 Michael Stuart
Randomized blocks design
Piece 1
A
Piece 2
D
Piece 3
C
Piece 4
B
Block 2Postgraduate Certificate in Statistics Design and Analysis of Experiments
Lecture 2.1 38© 2016 Michael Stuart
Randomized blocks design
Block 3
Piece 1
B
Piece 2
A
Piece 3
D
Piece 4
C
Postgraduate Certificate in Statistics Design and Analysis of Experiments
Lecture 2.1 39© 2016 Michael Stuart
Randomized blocks design
Block 1 Block 2 Block 3 Block 4 etc.
Blocking accounts for anticipated variation patterns along the length of the spool of rubber
Randomization allows for unanticipated sources of variation within blocks,
e.g., side to side, diagonal, any other
B A C A A D D B
D C B D C B C A
Postgraduate Certificate in Statistics Design and Analysis of Experiments
Lecture 2.1 40© 2016 Michael Stuart
Randomized blocks design
Example 3: assessing process changes.
Five versions of the process being assessed:
assess the five versions on five successive days in a working week,
Randomize the time order in which the versions are used,
repeat over several weeks to form several blocks.
NB: Pairing = Blocking with two units per blockPostgraduate Certificate in Statistics Design and Analysis of Experiments
Lecture 2.1 41© 2016 Michael Stuart
Randomized block design
Where replication entails increased variation, replicate the full experiment in several blocks so that• non-experimental variation within blocks is as
small as possible,
– comparison of experimental effects subject to minimal chance variation,
• variation between blocks may be substantial,
– comparison of experimental effects not affected
Postgraduate Certificate in Statistics Design and Analysis of Experiments
Lecture 2.1 42© 2016 Michael Stuart
Illustrations of blocking variables
Agriculture:
fertility levels in a field or farm,
moisture levels in a field or farm,
genetic similarity in animals, litters as blocks,
etc.
Postgraduate Certificate in Statistics Design and Analysis of Experiments
Lecture 2.1 43© 2016 Michael Stuart
Illustrations of blocking variables
Clinical trials (stratification)
age,
sex,
height, weight,
social class,
medical history
etc.
Postgraduate Certificate in Statistics Design and Analysis of Experiments
Lecture 2.1 44© 2016 Michael Stuart
Illustrations of blocking variables
Clinical trials
body parts as blocks,
hands, feet, eyes, ears,
different treatments applied to the same individual at different times,
cross-over, carry-over, correlation,
etc.
Postgraduate Certificate in Statistics Design and Analysis of Experiments
Lecture 2.1 45© 2016 Michael Stuart
Illustrations of blocking variables
Industrial trials
multiple machines,
multiple test laboratories,
time based blocks,
time of day, day of week, shift
etc.
Postgraduate Certificate in Statistics Design and Analysis of Experiments
Lecture 2.1 46© 2016 Michael Stuart
Design and Analysis of ExperimentsLecture 2.1
1. Review– Minute tests 1.2– Homework– Experimental factors with several levels
2. Analysis of Variance3. Randomised blocks design:
illustrations4. Randomised blocks design and analysis:
case study
Postgraduate Certificate in Statistics Design and Analysis of Experiments
Lecture 2.1 47© 2016 Michael Stuart
Case StudyReducing yield loss in a chemical process• Process: chemicals blended, filtered and dried• 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)
Postgraduate Certificate in Statistics Design and Analysis of Experiments
Lecture 2.1 48© 2016 Michael Stuart
Classwork 1.2.5: What were the
response:experimental factor(s):factor levels:treatments:experimental units:observational units:unit structure:treatment allocation:replication:
Postgraduate Certificate in Statistics Design and Analysis of Experiments
Lecture 2.1 49© 2016 Michael Stuart
Classwork 1.2.5: What were the
response:experimental factor(s):factor levels:treatments:experimental units:observational units:unit structure:treatment allocation:
replication:
yield lossBlendA, B, C, D, EA, B, C, D, Eprocess runsprocess runs3 blocks of 5 unitsrandom order of blends within blocks3
Postgraduate Certificate in Statistics Design and Analysis of Experiments
Lecture 2.1 50© 2016 Michael Stuart
Unit Structure
Block 1 Block 2 Block 3
Postgraduate Certificate in Statistics Design and Analysis of Experiments
Unit 1Unit 2Unit 3Unit 4Unit 5
Unit 1Unit 2Unit 3Unit 4Unit 5
Unit 1Unit 2Unit 3Unit 4Unit 5
Lecture 2.1 51© 2016 Michael Stuart
Unit Structure
Block 1 Block 2 Block 3
Postgraduate Certificate in Statistics Design and Analysis of Experiments
Unit 1_1Unit 1_2Unit 1_3Unit 1_4Unit 1_5
Unit 2_1Unit 2_2Unit 2_3Unit 2_4Unit 2_5
Unit 3_1Unit 3_2Unit 3_3Unit 3_4Unit 3_5
Blocks
Units
Units nested in Blocks
Lecture 2.1 52© 2016 Michael Stuart
Randomization procedure1. enter numbers 1 to 5 in Column A of a spreadsheet,
headed Run,
2. enter letters A-E in Column B, headed Blend,
3. generate 5 random numbers into Column C, headed Random
4. sort Blend by Random,
5. allocate Treatments as sorted to Runs in Block I,
6. repeat Steps 3 - 5 for Blocks II and III.
Go to ExcelPostgraduate Certificate in Statistics Design and Analysis of Experiments
Lecture 2.1 53© 2016 Michael Stuart
Part 2 Randomised blocks analysis
• Exploratory analysis• Analysis of Variance• Block or not?• Diagnostic analysis
– deleted residuals• Analysis of variance explained
Postgraduate Certificate in Statistics Design and Analysis of Experiments
Lecture 2.1 54© 2016 Michael Stuart
Results
Ref: BlendLoss.xls
Block Run Blend Loss, per cent I 1 B 18.2 2 A 16.9 3 C 17.0 4 E 18.3 5 D 15.1 II 6 A 16.5 7 E 18.3 8 B 19.2 9 C 18.1 10 D 16.0 III 11 B 17.1 12 D 17.8 13 C 17.3 14 E 19.8 15 A 17.5
Postgraduate Certificate in Statistics Design and Analysis of Experiments
Lecture 2.1 55© 2016 Michael Stuart
Initial data analysis
• Little variation between blocks• More variation between blends• Disturbing interaction pattern; see laterPostgraduate Certificate in Statistics Design and Analysis of Experiments
Lecture 2.1 56© 2016 Michael Stuart
Formal Analysis
Analysis of Variance: Loss vs Block, Blend
Source DF SS MS F PBlock 2 1.648 0.824 0.94 Blend 4 11.556 2.889 3.31 0.071Error 8 6.992 0.874Total 14 20.196
Classwork 1.2.6: Confirm the calculation of• Total DF, • Total SS, • MS(Block), MS(Blend), MS(Error) • F(Block), F(Blend)
Postgraduate Certificate in Statistics Design and Analysis of Experiments
Lecture 2.1 57© 2016 Michael Stuart
Formal Analysis
Analysis of Variance: Loss vs Block, Blend
Source DF SS MS F PBlock 2 1.648 0.824 0.94 Blend 4 11.556 2.889 3.31 0.071Error 8 6.992 0.874Total 14 20.196
Classwork 1.2.6: Confirm the calculation of• Total DF, • Total SS, • MS(Block), MS(Blend), MS(Error) • F(Block), F(Blend)
Postgraduate Certificate in Statistics Design and Analysis of Experiments
Lecture 2.1 58© 2016 Michael Stuart
Assessing variation between blendsF(Blends) = 3.3F4,8;0.1 = 2.8
F4,8;0.05 = 3.8
p = 0.07F(Blends) is "almost statistically significant"
Multiple comparisons:All intervals cover 0;Blends B and E difference "almost significant"
Ref: Lecture Note 1.2, p. 20.Postgraduate Certificate in Statistics Design and Analysis of Experiments
Lecture 2.1 59© 2016 Michael Stuart
Assessing variation between blocks
F(Blocks) = 0.94 < 1; MS(Blocks) < (MS(Error)
differences between blocks consistent with chance variation;
Source DF SS MS F PBlock 2 1.648 0.824 0.94 0.429Blend 4 11.556 2.889 3.31 0.071Error 8 6.992 0.874Total 14 20.196
Source DF SS MS F PBlend 4 11.556 2.889 3.34 0.055Error 10 8.640 0.864Total 14 20.196Postgraduate Certificate in Statistics Design and Analysis of Experiments
Lecture 2.1 60© 2016 Michael Stuart
Was the blocking effective?
Source DF SS MS F PBlock 2 1.648 0.824 0.94Blend 4 11.556 2.889 3.31 0.071Error 8 6.992 0.874Total 14 20.196
S = 0.9349
Source DF SS MS F PBlend 4 11.556 2.889 3.34 0.055Error 10 8.640 0.864Total 14 20.196
S = 0.9295
Postgraduate Certificate in Statistics Design and Analysis of Experiments
Lecture 2.1 61© 2016 Michael Stuart
Was the blocking effective?• F(Blocks) < 1• Blocks MS smaller than Error MS
• When blocks deleted from analysis– Residual standard deviation almost unchanged
and– F(Blends) almost unchanged
• Blocking NOT effective.
Postgraduate Certificate in Statistics Design and Analysis of Experiments
Lecture 2.1 62© 2016 Michael Stuart
Block or not?
Source DF SS MS F PBlock 2 1.648 0.824 0.94 0.429Blend 4 11.556 2.889 3.31 0.071Error 8 6.992 0.874Total 14 20.196
Source DF SS MS F PBlend 4 11.556 2.889 3.34 0.055Error 10 8.640 0.864Total 14 20.196
Postgraduate Certificate in Statistics Design and Analysis of Experiments
Lecture 2.1 63© 2016 Michael Stuart
Block or not?
Source DF SS MS F PBlock 2 1.648 0.824 0.94 0.429Blend 4 11.556 2.889 3.31 0.071Error 8 6.992 0.874Total 14 20.196
Source DF SS MS F PBlend 4 11.556 2.889 3.34 0.055Error 10 8.640 0.864Total 14 20.196
Postgraduate Certificate in Statistics Design and Analysis of Experiments
Lecture 2.1 64© 2016 Michael Stuart
Block or not?
Omitting blocks
increases DF(Error),therefore
increases precision of estimate of s,and
increases power of F(Blends)
F4,8:0.10 = 2.8; F4,8:0.05 = 3.8
F4,10:0.10 = 2.6 F4,10:0.05 = 3.5
Smaller critical value easier to exceed, more power.Postgraduate Certificate in Statistics Design and Analysis of Experiments
Lecture 2.1 65© 2016 Michael Stuart
Block or not?
Statistical theory suggests no blocking.
Practical knowledge may suggest otherwise.
Quote from Davies et al (1956):
"Although the apparent variation among the blocks is not confirmed (i.e. it might well be ascribed to experimental error), future experiments should still be carried out in the same way.
There is no clear evidence of a trend in this set of trials, but it might well appear in another set, and no complication in experimental arrangement is involved".
Postgraduate Certificate in Statistics Design and Analysis of Experiments
Lecture 2.1 66© 2016 Michael Stuart
Diagnostic plots
• The diagnostic plot, residuals vs fitted values– checking the homogeneity of chance variation
• The Normal residual plot,– checking the Normality of chance variation
Postgraduate Certificate in Statistics Design and Analysis of Experiments
Lecture 2.1 67© 2016 Michael Stuart
Diagnostic analysis
Postgraduate Certificate in Statistics Design and Analysis of Experiments
• One exceptional case– likely to be related to interaction pattern.
see Slide 55− resist deletion and refitting!
Lecture 2.1 68© 2016 Michael Stuart
Initial data analysis
• Little variation between blocks• More variation between blends• Disturbing interaction pattern; see laterPostgraduate Certificate in Statistics Design and Analysis of Experiments
Lecture 2.1 69© 2016 Michael Stuart
Analysis of Variance Explained
Decomposing Total Variation
Expected Mean Squares
Postgraduate Certificate in Statistics Design and Analysis of Experiments
Lecture 2.1 70© 2016 Michael Stuart
Decomposing Total Variation
Analysis of Variance: Loss vs Block, Blend
Source DF SS MS F PBlock 2 1.648 0.824 0.94 Blend 4 11.556 2.889 3.31 0.071Error 8 6.992 0.874Total 14 20.196
SS(TO) = SS(Block) + SS(Blend) + SS(Error)
Postgraduate Certificate in Statistics Design and Analysis of Experiments
Lecture 2.1 71© 2016 Michael Stuart
Model for analysis
Yield loss includes
– a contribution from each blend
plus
– a contribution from each block
plus
– a contribution due to chance variation.
Postgraduate Certificate in Statistics Design and Analysis of Experiments
Lecture 2.1 72© 2016 Michael Stuart
Model for analysis
Y = m + a + b + ewhere
m is the overall mean,a is the blend effect, above or below the mean,
depending on which blend is used,b is the block effect, above or below the mean,
depending on which block is involvede represents chance variation
Postgraduate Certificate in Statistics Design and Analysis of Experiments
Lecture 2.1 73© 2016 Michael Stuart
Estimating the model
Postgraduate Certificate in Statistics Design and Analysis of Experiments
Lecture 2.1 74© 2016 Michael Stuart
Estimating the model
Postgraduate Certificate in Statistics Design and Analysis of Experiments
Lecture 2.1 75© 2016 Michael Stuart
Estimating the model
Postgraduate Certificate in Statistics Design and Analysis of Experiments
Lecture 2.1 76© 2016 Michael Stuart
Decomposing Total Variation
statistical residual format
mathematically simplified format
)]yy()yy()yy[()yy()yy(
yy
jiij
j
i
ij
)yyyy( jiij
SSTO = SS(Blocks) + SS(Blends) + SS(Error)
dataall
2jiij
dataall
2j
dataall
2i
dataall
2ij )yyyy()yy()yy()yy(
Postgraduate Certificate in Statistics Design and Analysis of Experiments
Lecture 2.1 77© 2016 Michael Stuart
Expected Mean Squares
F(Blends) = tests equality of blend means
F(Blocks) = assesses effectiveness of blocking
1I
)(J= )EMS(Blends i
2i
2
mms
1J
)(I)Blocks(EMS j
2j
2
mm
s
2)Error(EMS s
)Error(MS)Blends(MS
)Error(MS)Blocks(MS
Postgraduate Certificate in Statistics Design and Analysis of Experiments
Lecture 2.1 78© 2016 Michael Stuart
Minute test
– How much did you get out of today's class?– How did you find the pace of today's class?– What single point caused you the most
difficulty?– What single change by the lecturer would have
most improved this class?
Postgraduate Certificate in Statistics Design and Analysis of Experiments
Lecture 2.1 79© 2016 Michael Stuart
Reading
EM Ch. 4, §7.2
Base Module, §2.2, §2.4, §§4.1- 4.3, §5.1
MGM §2.1, §§3.1,3,2
DCM §2-4.1 to §2-4.3, §2.5, §§3.1 to 3.4, §3.5.7, §4.1
DV §3.5, §§4.2.1-4.2.3, §4.3.2, §4.4.1, §§4.4.4-4.4.6, §10.3, §10.4 (with back references)
Postgraduate Certificate in Statistics Design and Analysis of Experiments