SJS SDI_11 1

Design of Statistical Investigations

Stephen Senn

11 Nested Factors

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

SJS SDI_11 3

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”

SJS SDI_11 4

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

SJS SDI_11 5

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

SJS SDI_11 6

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

SJS SDI_11 7

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

SJS SDI_11 8

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

SJS SDI_11 9

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

SJS SDI_11 10

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?

SJS SDI_11 11

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

SJS SDI_11 12

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

SJS SDI_11 13

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

SJS SDI_11 14

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

SJS SDI_11 15

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

SJS SDI_11 16

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

SJS SDI_11 17

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

SJS SDI_11 18

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

SJS SDI_11 19

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

SJS SDI_11 20

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

SJS SDI_11 21

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

SJS SDI_11 22

ANOVA Exp_16

Source d.f. Sum ofSquares

Meansquare

BetweenLaboratories

3 4.1750 1.3917

WithinLaboratories

20 3.8546 0.1927

Total 8.0296

SJS SDI_11 23

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

SJS SDI_11 24

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

SJS SDI_11 25

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