1 Chihiro HIROTSU Meisei （明星） University Estimating the dose response pattern via multiple...

1

Chihiro HIROTSU

Meisei （明星） University

Estimating the dose response pattern via multiple decision processes

2

Phase Clinical Trial (Binomial or Normal Model)Ⅱ

,: 21 KH

1. Proving the monotone dose-response relationship,

2. Estimating the recommended dose for the ordinary clinical treatments, which shall be confirmed by a Phase Ⅲ trial.

with at least oneinequality strong.

The mcp for the interested dose-response patterns should be preferable to fitting a particular parametric model such as logistic distribution.

(1)

3

Table 1. Monotone Dose-Response Patterns of Interest (K=4)

Model Coefficient of

ContrastLiner Contrast

Statistic

-1 -1 -1 3

-1 -1 1 1

-3 1 1 1

-1 -1 0 2

-2 0 1 1

-3 -1 1 3

1M 1t

2t

3t

4t

4321

4321

4321

4321

4321

4321 5t

6t

2M

3M

4M

5M

6M

4321Exclude ( Non-Sigmoidal )

4

Maximal Contrast Type Tests

max acc. t method (Hirotsu, Kuriki & Hayter, 1992; Hirotsu & Srivastava, 2000)

,,

,,

,ˆ11

),,max(.max

11

11

21

2

11

KkkkKkk

kkkkk

kkkk

k

K

nnNNyyY

nnNNyyY

YYNN

t

ttt

acc

sizesampleTotal:

modelBinomial,1

modelNormalvarianceUnbiased:ˆ 2

K

K

NN

Yyyy

(Changepoint soon after the level k)

5

Merits of max acc. t method

b. The K-1 components of max acc. t are the projections of the observation vector on to the corner vectors of the convex cone defined by the monotone hypothesis H and every monotone contrast can be expressed by a unique positive liner combination of those basic contrasts (Hirotsu & Marumo, Scand. J. Statist, 2002).

a. Immediate correspondence to the complete class lemma for the tests of monotone hypothesis (Hirotsu, Biometrika 1982).

c. The simultaneous confidence intervals for the basic contrasts of max acc. t can be extended to all the monotone contrasts uniquely whose significance can therefore be evaluated also (Hirotsu & Srivastava, Statistics and Probability Letters, 2000) .

d. A very efficient and exact algorithm for calculating the distribution function is available based on the Markov property of those components (Hawkins, 1977 ; Worsley,1986 ; Hirotsu, Kuriki & Hayter, Biometrika, 1992).

e. High power against wide range of the monotone hypothesis H as compared with other tests such as lrt or William’s (Hirotsu, Kuriki & Hayter, 1992).

6

Estimating the Dose-Response Patterns 1

1. Apply closed testing procedure based on max acc. t.

① Test H0 : μ1= μ2= μ3= μ4 and if it is not significant stop here (0-stopping), otherwise

② test H0 : μ1= μ2= μ3 and if it is not significant stop here (1-stopping), otherwise

③ test H0 : μ1= μ2 and if it is not significant we call it (2-stopping), otherwise we call it 3-stopping.

(model selection by the maximal contrast)

7

Estimating the Dose-Response Patterns 1 (continued)

2. Model selection based on maximal contrast

0-stopping : Accept the null model

1-stopping : Uniquely select Model 1 iff the corresponding contrast is significant.

2-stopping : Select either Model 2 or 4 corresponding to the largest contrast of Models 2 and 4 iff it is significant.

3-stopping : Select either Model 3, 5 or 6 corresponding to the largest contrast of Models 3, 5 and 6 iff it is significant.

For evaluating significance of those contrasts that are not included in the basic contrasts of max acc. t an extension to the simultaneous lower bounds by Hirotsu & Srivastava (2000) is applied. Especially this time we need a lemma for evaluating a linear trend.

8

Table 1. Monotone Dose-Response Patterns of Interest (K=4)

Model Coefficient of

ContrastLiner Contrast

StatisticPhase of dosed test

-1 -1 -1 3 1-stopping

-1 -1 1 1 2-stopping

-3 1 1 1 3-stopping

-1 -1 0 2 2-stopping

-2 0 1 1 3-stopping

-3 -1 1 3 3-stopping

1M 1t

2t

3t

4t

4321

4321

4321

4321

4321 4321 5t

6t

2M

3M

4M

5M

6M

4321Exclude

9

Simultaneous Lower Bounds by max acc. t

)1,1(11

:

)2,2(11

:

)3,3(11

:

21

*11

1*

11*13

21

*22

2*

22*22

21

*33

3*

33*31

SLBTNN

YYM

SLBTNN

YYM

SLBTNN

YYM

Basic contrasts

(Each interpreted as estimating under the respective assumed model)1 K

)1,1()2,2()2,1(:

)2,2()3,3()3,2(:

121

*25

232

*34

SLBN

NSLB

N

NSLBM

SLBN

NSLB

N

NSLBM

10

General formula 1

Corresponding to the model with changepoint soon after level and saturating at level The basic contrasts correspond to the case

i

iii

Kj

kKjjj

ijij

nn

nn

nn

nn

iiSLBN

NjjSLB

N

NjiSLB

1

11

1

1*

*

,

),(),(),(

i .1j

.ji

Simultaneous Lower Bounds by max acc. t (continued)

11

Simultaneous Lower Bounds by max acc. t (continued)

Kkk

K

kkk

K

k kk

nkKnN

nnkNN

C

kkSLBNN

CSLB

)(

11

),(11

)(

1

1

111

*

1

1

1

*1

Corresponding to the linear regression model : 1,,1,1 Kk kk

Estimating the difference under the assumed linear regression model like other monotone contrasts.

)()1()( SLBKlinearSLB

1 K

Lemma

General formula 2

12

Proof of Lemma

Deriving SLB for 　　 as the best linear combination

of the basic contrasts :

under the assumption :

kkkk NYNY

kk

kkkKKkkk

k

k

k

k

NN

nnNnnN

N

Y

N

YE

)()()( 1111

kk

kkKkkkkkk

NN

nnkNkKnnnkNnnnkNn 11112111 )1()()1(2)1(

)

)1()(2( 1121

k

k

k

Kkk

N

nnk

N

nkKnn

.1,,1, Kkck 　 （ Inhomogeneous and complicated structure )

.11),(),(

KkvNN

N

N

Y

N

Y

N

Y

N

YCov kl

lkl

l

l

l

k

k

k

k 　 　

d

d （ Markov structure )

1,1,1 Kkkk

13

By Markov structure we have

.N

NNdiag1,,1,1V

*ii1

-c

N

NNdiag

nnn-0

n-

n-nnn-

0n-nn

N

NNdiagV

*ii

111

1

1

13

13

13

12

12

12

12

11

*ii1

KKK

K

-

n

1

1*

1

*1

1

1*

1

*1

N

Y

N

Y,,

N

Y

N

Y

K

K

K

KZ

klK vcc ZccZZ V;,,,E 1-1

⇒ Zccc 11 VVˆ

d

Proof of Lemma (continue)

and

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Final result （ simple and explicit form ）

・ The weights are proportional to the reciprocal of the respective variances.

・ This is the formula for independent components with equal expectations.

・ The inhomogeneity of expectations and the correlation are nicely cancelling out.

This increases the usefulness of max acc t.

k

k*

k

*k

1

1-*

1-

1

*11

1-

N

Y

N

Y

N

1

N

1,...,

N

1

N

1V

KK

Zc

Proof of Lemma (continue)

15

Comparing SLB(3,3), SLB(2,2), SLB(1,1), SLB(2,3), SLB(1,2) and 3×SLB(linear) for patterns M1, M2, M3, M4, M5 and M6, respectively, will make sense.

1-stopping : Uniquely select M1 iff SLB(3,3)>0.

2-stopping : Select either M2 or M4 corresponding to the largest of SLB(2,2) and SLB(2,3) iff it is above 0.

3-stopping : Select either M3, M5 or M6 corresponding to the largest of SLB(1,1), SLB(1,2) and 3×SLB(linear) iff it is above 0.

Estimating the Dose-Response Patterns 2

(Model selection by the simultaneous lower bounds (SLB))

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Estimating the Dose-Response Patterns 3

(1) Step-down procedure for 3K

1*11 2

*22 d d

0, 21 0, 21 ⇒

and

or

3,ˆ11

,11

max

22

2*2

*2

21

2*2

11

1*1

*1

21

1*1

NnTN

Y

N

Y

NN

N

Y

N

Y

NN

0,0 ji ⇒ 3ˆ11

*

*21

*

NtN

Y

N

Y

NN jj

j

j

j

jj

(Model selection by SLB due to multiple decision processes)

Acceptance sets :

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Confidence sets :

KNt

KNnnTiiSLBiiSLB

byreplaced

,with),(),(*

0)2,2(*),1,1(*min0)2,2(),1,1(max SLBSLBSLBSLB

d

with inequality strict if the limit is 0.

),(,0max iiSLBi

,2,1),,( iiiSLBi

),,(*,0 jjSLBji

0)2,2(,0)1,1( SLBSLB

0),(*,0),( jjSLBiiSLB

18

Model selection

.modelnulltheAccept

2,1,0),( iiiSLB

);(2and)2,2(),1,1(among

largestthetongconespondipatternaSelect

linearSLBSLBSLB

0),(,0),( jjSLBiiSLB

2,1,0),( iiiSLB

;tongconespondipatterntheSelect i

19

(2) Model selection by step-down procedure for

3,2,1,0),( iiiSLB

4K

Otherwise

;),,(and),(),,(among

largestthetoingcorrespondpatternaSelect

jijiSLBjjSLBiiSLB

);(3and

)2,1(),3,2(,3,2,1),,(among

largestthetoingcorrespondpatternaSelect

linearSLB

SLBSLBiiiSLB

0),(,0),(,0),( kkSLBjjSLBiiSLB

;toingcorrespondpatternaSelect i 0),(,0),(,0),( kkSLBjjSLBiiSLB

model.nulltheAccept

20

Simulation result 1Comparing with other maximal contrasts methods.

Table 2. Probability of selecting a model( )

Method

◎: Correct selection; ○: Correct optimal dose

HML: by Liu, Miwa & Hayter (2000)

Orthogonal :

True model Selected pattern acc. t HML Orthogonal type

◎ 85.6 76.6 86.4

0.7 0.5 0.3

○ 4.3 3.1 0.2

◎ or ○ 89.8 79.7 86.6

6.9 1.3 7.6

◎ 71.4 69.4 50.7

11.7 8.7 1.7

◎ or ○ 71.4 69.4 50.7

○ 44.2 26.2 50.0

17.9 16.0 7.8

◎ 28.4 21.8 0.9

◎ or ○ 72.6 48.0 50.9

3K

1M

1M

1M

)2,1,1(1

M 2M

2M

3M

2M

3M

3M)1,0,1(3

M

)1,1,2(2

M

2312 ,max yyyy 32,max 21312 yyyyy

max

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Simulation result 2Effects of adding monotone contrasts 　　 , 　　 , 　　 to max

acc. tTable 3. Probability of selecting a model ( )

Method

: statistic corresponding to : statistic corresponding to : statistic corresponding to Remarkably small effects of adding , and / or

True model Selected pattern

◎ 85.5 85.5 85.5

◎ or ○ 88.7 88.7 88.3

◎ 68.4 68.3 68.1

◎ or ○ 71.6 71.5 71.4

◎ 63.0 62.8 62.6

◎ or ○ 63.0 62.8 62.6

◎ 9.0 9.2 9.2

◎ or ○ 78.0 78.3 78.8

◎ 27.7 27.7 27.7

◎ or ○ 50.6 50.9 50.7

◎ 10.0 9.9 9.9

◎ or ○ 49.7 50.2 50.4

4K

641 ,, MMM

)(6

linear

M

)1,1,0,2(5

M

641 ,, MMM

641 ,, MMM

6M

5M

)2,0,1,1(4

M 4M

)1,1,1,3(3

M 3M

)1,1,1,1(2

M 2M

3M

)3,1,1,1(1

M 1M

52 , MM

52 , MM

6321 ,,,max tttt 654321 ,,,,,max tttttttacc.max

4M

5M

6M

4t5t

6t

ttt

4t 5t .6t

4t 5t 6t

22

Simulation result 3Comparing maximal contrast method and SLB method based on max acc. t

Table 4. Probability of selecting a model ( ) Method

True model Selected pattern contrast SLB (closed test) SLB (mult. dec.)

◎ 85.6 85.6 77.7

0.7 1.2 1.8

○ 4.2 3.8 11.0

◎ 85.6 85.6 77.7

◎ or ○ 89.7 89.3 88.7

6.9 6.9 2.0

◎ 71.4 74.7 77.0

11.6 8.3 10.9

◎ 71.4 74.7 77.0

◎ or ○ 71.4 74.7 77.0

○ 44.0 44.0 25.4

17.9 22.3 25.5

◎ 28.5 24.1 39.5

◎ 28.5 24.1 39.5

◎ or ○ 72.4 68.1 64.9

Total

◎ 185.5 184.4 194.2

◎ or ○ 233.8 232.2 230.6

2M1M

3M

2M2M

1M

1M

3M

)(3 linearM

1M

2M

3M

.max

3K

23

Simulation result 4

Comparing maximal contrast method and SLB method based on max acc. t

Table 5. Probability of selecting a model ( )

Method

4K

True Model Selected pattern . contrast SLB (closed test) SLB (mult. Dec.)

◎ 85.5 85.5 69.7

◎ or ○ 88.7 88.4 87.6

◎ 68.4 66.6 61.4

◎ or ○ 71.6 69.2 71.2

◎ 63.0 72.2 69.0

◎ or ○ 63.0 72.2 69.0

◎ 9.0 10.1 36.5

◎ or ○ 78.0 78.4 75.8

◎ 27.7 18.0 37.0

◎ or ○ 50.6 40.9 48.1

◎ 10.0 7.1 20.6

◎ or ○ 49.7 48.2 47.3

Total

◎ 263.6 259.5 294.2

◎or○ 401.6 397.3 399

1M

2M

3M

4M

5M

)(6 linearM

max

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Adding Contrasts t4, t5 and/or t6 to the Basic Contrasts (t1, t2, t3) of max acc. t

Intending the Detection of Patterns M4, M5 and M6 (Japanese Practice)

Method 1 :

Method 2 :

Method 3 :

),,max(.accmax 321 tttt ),,,max( 6321 tttt

),,,,,max( 654321 tttttt

.32352

1,

6

1

2

1

,2

1

6

1,

33

4ˆ

,22ˆ

,33

4ˆ

3216215

3243214

21

3

21432

1432

21

1

ttttttt

tttn

YYY

n

Ynt

n

YY

n

YYnt

n

Y

n

YYYnt

25

Calculating the Critical Point (Normal Theory)

a a uuu

dtdudtutt

atatatatatat),min( ),,min(

3232

654321

2 321

)(),(

),,,,,Pr(

easy to evaluate

.1,3

12

,1,36

,1,3

2

3

52

3

212

211323

3

211

211232

3

212111321

t

tρρρuta

t

tρρρuta

t

tρaρρutta

26

Concluding Remarks

1. The SLB based on the basic contrasts of max. acc. t can be extended to any monotone contrasts including the linear trend.

2. The effects of adding , and to the basic contrasts of max acc. t are remarkably small.

3. The selection of the monotone contrasts of interest is almost good but the power is not homogeneous for those patterns. The linear trend is difficult to be detected, for example. This is the problem of early stopping due to the step down procedure and the consideration of the overall power is insufficient.

4. The simultaneous confidence internals based on the multiple decision processes behave better for the linear trend.

4t 5t 6t

27

References1. Hirotsu,C.(1982). Use of cumulative efficient scores for testing ordered altern

atives in discrete models. Biometrika 69, 567-577.

2. Hirotsu,C., Kuriki, S. & Hayter,A.J.(1992). Multiple comparison procedures based on the maximal component of the cumulative chisquared statistic. Biometrika 79, 381-392.

3. Hirotsu,C. & Srivastava, M. S.(2000). Simultaneous confidence intervals based on one-sided max t test. Statistics & Probability Letters 49, 25-37.

4. Hirotsu,C. & Marumo, K.(2002). Changepoint analysis as a method for isotonic inference. Scandinavian J. Statist. 29, 125-138.

5. Hothorn,L. A., Vaeth, M., & Hothorn, T.(2003). Trend tests for the evaluation of dose-response relationships in epidemiological exposure studies. Research Reports from the Department of Biostatistics, University of Aarhus.

6. Liu, W., Miwa, T. & Hayter, A. J.(2000). Simultaneous confidence interval estimation for successive comparisons of ordered treatment effects. JSPI 88, 75-86.