Design of Statistical Investigations

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SJS SDI_11 1 Design of Statistical Investigations Stephen Senn 11 Nested Factors

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Design of Statistical Investigations. 11 Nested Factors. Stephen Senn. Crossed Factors. So far the treatment and blocking factors we have considered have been “crossed”. In principle every level of one could be observed with every level of the other. Every treatment in each block - PowerPoint PPT Presentation

Transcript of Design of Statistical Investigations

Page 1: Design of Statistical Investigations

SJS SDI_11 1

Design of Statistical Investigations

Stephen Senn

11 Nested Factors

Page 2: Design of Statistical Investigations

SJS SDI_11 2

Crossed Factors

• So far the treatment and blocking factors we have considered have been “crossed”.

• In principle every level of one could be observed with every level of the other.– Every treatment in each block

• Or at least the same treatments in various blocks

– Each level of a factor in combination with each of another

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Nested Factors

• Sometimes some factors can only appear within other factors

• Blocks with sub-blocks– Example: Patients within given group allocated

a particular sequence• Episodes of treatment within patients

• Treatments with sub-treatments

• Such factors are “nested”

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Exp_15Nested “Treatments”

• Suppose that we wish to compare two beta-agonists in asthma, formoterol and salmeterol

• Formoterol has three formulations• solution, single-dose dry-powder inhaler, multi-

dose dry-powder inhaler

• Salmeterol has two• suspension, multi-dose dry powder inhaler

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Exp_15Treatment Structure

F orm otero lS o lu tion

F orm otero lP owd er

S in g le -d ose

F orm otero lP owd er

M u lti-d ose

F orm otero l P ow d er

F orm otero l

S a lm etero lS u sp en s ion

S a lm etero lP owd er

(m u lt i-d ose)

S a lm etero l

Trea tm en tsB eta-ag on is ts

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Exp_15Treatments

• From one point of view we have five treatments– defined by combination of molecule and

formulation

• We may have a hierarchy of interest– primarily to compare molecules

• then to compare formulations within molecules– possibly delivery type within formulations

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Exp_15

Possible factors (levels)

A: Treatments ( Formoterol, Salmeterol)

B: Formoterol formulation (Solution, Powder)

B*: Salmeterol formulation (Suspension, Powder)

C: Formoterol powder device (Single, Multi)

Note that B* is not really the same as B and each of the lower level factors only has meaning in the context of the higher level

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Wilkinson and Roger NotationWe encountered this in connection with factorial designs

Now we add an operator / for nested designs

A/B = A + A:B

Not that if B is a factor nested within A, it has no meaning on its own. Hence the main effect B does not exist on its own.

NB In their original papers Applied Statistics,1973,22,392-399, W&R used instead of : as used in S-PLUS

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Exp_13

• We encountered this example before

• We could regard this as an example of a nested design

• Treatments, placebo, ISF, MTA

• Doses within treatments

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Exp_13As nested design

P lacebo

6 12 24

IS F

6 12 24

M T A

F o rm u la tion

A c tiv e?

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Exp_13Nested Analysis

> #As before but treat as nested factorsfit2 <- aov(AUC ~ Patient + Period + Active/Formul/Dose, na.action = na.exclude)> summary(fit2, corr = F) Df Sum of Sq Mean Sq F Value Pr(F) Patient 157 80.29301 0.511420 70.5027 0.0000000 Period 4 0.02092 0.005230 0.7210 0.5777861 Active 1 1.63959 1.639591 226.0286 0.0000000 Formul %in% Active 1 0.66308 0.663078 91.4097 0.0000000Dose %in% (Active/Formul) 4 0.22666 0.056664 7.8115 0.0000038 Residuals 603 4.37411 0.007254

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Random Treatment Effects

• We now pick up a theme we alluded to in lecture 10

• Cases where our principle interest is in random effects– not random blocks– random treatments

• This example has nesting

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Exp_16Clarke and Kempson Example 13.1

1 2 3 4 5

A

1 2 3 4 5 6 7

B

1 2 3 4 5 6

C

1 2 3 4 5 6

D

Four labs, A,B,C,D. Six samples of uniform batch given to each. However a sample intended for A is sent to B by mistake

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Fixed or Random?

• If we are interested in the performance of these four labs, we can consider them as fixed

• However we may be interested in using them to tell us how measurements vary in general from lab to lab

• If they are a sample of such labs, we could consider the effects as random

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Exp_16The Data

Lab Sample Result 1 A 1 16.0 2 A 2 17.1 3 A 3 16.9 4 A 4 17.2 5 A 5 17.0 6 B 1 17.0 7 B 2 17.3 8 B 3 16.2 9 B 4 17.110 B 5 16.011 B 6 17.212 B 7 17.0

Lab Sample Result 13 C 1 16.914 C 2 16.115 C 3 16.416 C 4 16.117 C 5 16.618 C 6 16.319 D 1 15.020 D 2 15.921 D 3 16.022 D 4 15.923 D 5 16.224 D 6 15.9

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A

B

C

D

15.0 15.5 16.0 16.5 17.0

Result

lab

ora

tory

Data from Exp_13Data from Exp_13Data from Exp_13Data from Exp_13Data from Exp_13Data from Exp_13Data from Exp_13Data from Exp_13Data from Exp_13Data from Exp_13Data from Exp_13Data from Exp_13Data from Exp_13Data from Exp_13Data from Exp_13Data from Exp_13

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Model

1

2 2

.1

2 2 2 2.

22 2 2 2 2 2 2.

( 1 , 1 )

, 0, 0

,

,i

ij i ij i

v

i i iji

i ij

r

i i i i ij i ij

i i i

i i i i i i

y i v j r

r N E E

Var Var

T r r E T r

Var T r r

E T Var T E T r r r

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Sums of Squares & Expectations

2 2

1

2 2

1

2

2. .

1 1 1 1

2.

1 1

2.

1 1

/

( ) ( ) ( )

( )

( ) (

i i

i

i

v

between i ii

vi

betweeni i

r rv v

within ij i i ij i ii j i j

rv

within ij ii j

rv

within ij i iji j

SS T r G N

E T E GE SS

r N

SS y y

SS

E SS E E

2.

1 1

2 2

1

)

( 1) ( )

irv

ii j

v

ii

r N r

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1 1 1 1 1

22 2 2 2 2

1

2 2

1

2 2

2 2 2 2 21

1 1

2

2 21 ( 1)

i ir rv v v

ij i i iji j i i j

v

ii

vi

betweeni i

v

iv vi

i ii i

v

ii

G y N r

E G N

E G Var G E G r N N

E T E GE SS

r N

rr v r N

N

rN v

N

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ANOVAExpected value ofmean square

Source ofVariation

d.f. GeneralCase

Equalreplication

BetweenGroups

1v 2 2 2 2r

W ithinGroups

N v 2 2

Total 1N 2

1

1 1

1

v

ii

N rv N

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Calculations Exp_16

20

2 2 2 2

84.1; 117.8; 98.4; 94.9

395.3; 24; 395.3 / 24 6510.9204;

6518.95;

84.2 117.8 98.4 94.96515.0954

5 7 6 6

A B C D

labs

y y y y

G N S

S

S

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ANOVA Exp_16

Source d.f. Sum ofSquares

Meansquare

BetweenLaboratories

3 4.1750 1.3917

WithinLaboratories

20 3.8546 0.1927

Total 8.0296

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Exp_16Components of Variance

2 2 2 2

2 2

1 124 5 7 6 6 5.972

3 24

1ˆ ˆ0.1927, 1.3917 0.1927 0.2008

5.972

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Exp_16S-PLUS Analysis

> is.random(one.frame) <- T> varcomp.1 <- varcomp(Result ~ Lab, data = one.frame, method = "reml")> summary(varcomp.1)Call:varcomp(formula = Result ~ Lab, data = one.frame, method = "reml")Variance Estimates: Variance Lab 0.2000226Residuals 0.1927181Method: reml Approximate Covariance Matrix of Variance Estimates: Lab Residuals Lab 0.03612192 -0.00063555Residuals -0.00063555 0.00379463

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Exp_14 Revisited

> #Variance components analysisSubject.ran <- data.frame(Subject)> is.random(Subject.ran) <- T> varcomp(lAUC ~ Subject + Formulation, data = Subject.ran)Variances: Subject Residuals 0.0766226 0.003424223> varcomp(lAUC ~ Subject * Formulation, data = Subject.ran)Variances: Subject Subject:Formulation Residuals 0.07679968 -0.0005244036 0.003764744