Design and Analysis of Experiments Lecture 6.2 …€“Mixed Models –Restricted Maximum...

Lecture 6.2 1

© 2016 Michael Stuart

Design and Analysis of Experiments

Lecture 6.2 Review topics

1. Repeated measures

2. Complex block structures

3. Missing / unbalanced data

4. Analysis of Covariance

5. Clinical trials

6. Response surface methodology

7. Transformations

8. Strategies for Experimentation

Postgraduate Certificare in Statistics


Lecture 6.2 2


Repeated Measures

Example:

Different calves fed different diet supplements to

improve growth, starting at 9 weeks old.

Initial weight recorded, Y0

blocks formed based on initial weight,

weights recorded at

4 weeks Y1

8 weeks Y2

12 weeks Y3

16 weeks Y4

Postgraduate Certificate in Statistics


Lecture 6.2 3


Split plots analysis?

• Calves are whole units

– treatment factor = Supplement Type

• Time periods are sub units

– treatment factor = Supplement Yes / No

• Problems:

− correlation structure

– varying standard deviation

• Solutions:

– DF adjustments

– Multivariate analysis



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Crossover designs

• Repeated measures designs compares different

diets on different calves,

• reduce variation by comparing diets on same

calves,

• e.g. diet A for weeks 1 to 4

diet B for weeks 5 to 8

diet C for weeks 9 to 12

diet D for weeks 13 to 16

• requires attention to order of diets



Lecture 6.2 5


Crossover design

• Every diet occurs

– once for each calf,

– once in each time

period

– Latin square

• Problems:

? correlation structure

? carry over effect

? experimental set up

versus actual use

Calf Time Period

1 - 4 5 – 8 9 – 12 13 - 16 1 A B C D 2 B D A C 3 C A D B 4 D C B A



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5. Clinical trials


7. Transformations




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Complex blocking

• 2 blocking factors

– calves and time periods

– Latin square

– Latin rectangle

• Incomplete blocks

– more treatments than plots in a block

– balanced incomplete blocks



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5. Clinical trials


7. Transformations




Lecture 6.2 9


Missing / Unbalanced Data

• Balanced data allows unambiguous interpretation

of effect estimates

• Unbalanced data does not

• e.g. multiple regression:

Y = b0 + b1X1 + b2X2 + e

• Solution: regression like calculation, interpret

with care

• Split plots analysis:

– Mixed Models

– Restricted Maximum Likelihood Estimation



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5. Clinical trials


7. Transformations




Lecture 6.2 11


Analysis of Covariance

Objective: take account of variation in uncontrolled

environmental variables.

Solution: measure the environmental variables at

each design point and incorporate in the

analysis through regression methods

(Analysis of Covariance)

Effects: reduces "error" variation, makes factor

effects more significant

adjusts factor effect estimates to take

account of extra variation source.



Lecture 6.2 12


Analysis of Covariance; Illustration

Breaking strength of monofilament fibre (Y)

produced by three different machines (1, 2, 3)

allowing for variation in fibre thickness (X)

Machine 1 Machine 2 Machine 3

Y X Y X Y X

36 20 40 22 35 21 41 25 48 28 37 23 39 24 39 22 42 26 42 25 45 30 34 21 49 32 44 28 32 15



Lecture 6.2 13


Analysis of Covariance; Minitab

General Linear Model: Y versus Machine

Source DF SS MS F P

Machine 2 140.40 70.20 4.09 0.044

Error 12 206.00 17.17

Total 14 346.40

S = 4.14

General Linear Model: Y versus Machine, X

Source DF SS MS F P

X 1 178.01 178.01 69.97 0.000

Machine 2 13.28 6.64 2.61 0.118

Error 11 27.99 2.54

Total 14 346.40

S = 1.60



Lecture 6.2 14


Covariance vs Blocking

Chance causes and assignable causes of variation

(W. Shewhart, 1931)

Chance causes of variation are the

many individually negligible and unpredictable

but

collectively influential

factors that affect a process or system.

Assignable causes of variation are the

few individually influential and predictable effect

factors that affect a process or system.



Lecture 6.2 15


Covariance vs Blocking

Blocking Chance causes

Covariance Assignable causes



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5. Clinical trials


7. Transformations




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5 Clinical trials

• Simple treatment structure

• Elaborate covariate structure

– blocking / matching

– covariate analysis

• Randomised allocation of treatments to subjects

– randomised controlled trial = "Gold Standard"

• Placebo effect?



Lecture 6.2 18


Ethical issues

• withholding medical treatment?

• blinding,

• double blinding,

• inadequate budget

– puts patients at risk

for non-informative results



Lecture 6.2 19








5. Clinical trials


7. Transformations




Lecture 6.2 20


Initial experiment used to suggest

second experiment with improved results.

Sequence of experiments leads to

best results,

sequential assembly

6 Response surface methodology:

Optimisation using Factorials Designs



Lecture 6.2 21


Optimising performance; hill climbing

50 52 54 56 58 60 62 64

Temperature

65

66

67

68

69

70

Yield



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Hill climbing



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Hill climbing



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Hill climbing



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Hill climbing



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Use 2-level factors to locate optimum;

use multi-level factors to explore the response

surface in more detail.

Analysis may involve use of regression methods

and transformation of variables.

NB: Avoid the rush into multi-level factors;

Sequential assembly (Box)



Lecture 6.2 27








5. Clinical trials


7. Transformations




Lecture 6.2 28


7 Transformations

• log transformation

• other transformations

• generalised linear models



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Why transform?



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Why transform?

Corresponding skew distribution



Lecture 6.2 31


Why transform?



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Changing spread with log



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Why the log transform works

High spread at high X

transformed to

low spread at high Y

Low spread at low X

transformed to

high spread at low Y



Lecture 6.2 42


Multiplicative models

Instead of simple linear regression:

Y = a + bX + e

suppose the model is multiplicative:

Y = gXw d

Log transform:

log(Y) = log(g) + w log(X) + log(d)

simple linear regression!



Lecture 6.2 44


Generalised linear model etc.

Linear model: Y = lm(X) + e

Generalised linear model: f (Y) = lm(X) + g(e)

Generalised additive model: f (Y) = f1(X1) + f2(X2) + …

+ g(e)

Multilevel model



Lecture 6.2 45








5. Clinical trials


7. Transformations




Lecture 6.2 46


8 Strategies for Experimenting

• A list:

– Consultation

– Design

– Planning

– Resources

– Ethical issues

– Implementation (of design)

– Application (of results)



Lecture 6.2 47


Strategy for Experimenting

Shewhart's PDCA Cycle

Check

Act

Plan

Do



Lecture 6.2 48


Strategy for Experimenting

Shewhart's PDCA Cycle

• Plan: Plan a change to the process, predict its effect, plan to measure the effect

• Do: Implement the change as an experiment and measure the results

• Check: Analyse the results to learn what effect the change had, if any

• Act: If successful, make the change permanent, proceed to plan the next improvement

or

if not, proceed to plan an alternative change



Lecture 6.2 49


Robinson's outline

Ref: GKR p.6, see also p.7



Lecture 6.2 50


Strategy

When you see the credits roll at the end of a

successful movie you realize there are many more

things that must be attended to in addition to

choosing a good script.

Similarly in running a successful experiment there

are many more things that must be attended to in

addition to choosing a good experimental design.

George Box



Lecture 6.2 51


Readings from DCM (and BHH)

and others

Time related issues

repeated measures

Kuehl, Robert O., Design of experiments : statistical

principles of research design and analysis, Ch. 15

cross-over designs

Complex block structures

Analysis of Covariance

Robustness studies

Clinical trials

Response surface designs

Transformations

Strategies for Experimentation

x§15.4

§4.2

§4.4 (§§4.4, 4.5)

§§15.3, 15.3.2, 15.3.4

§12.1 (§13.1)

Pocock, SJ

§11.1 (Ch. 12)

§3.4.3, §15.1 (Ch. 8)

§1.1 Postgraduate Certificare in Statistics


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Design and Analysis of Experiments Lecture 6.2 …€“Mixed Models –Restricted Maximum...

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