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

2 Phase Ⅱ Clinical Trial (Binomial or Normal Model) 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 one inequality 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 Contrast Liner Contrast Statistic ( Non-Sigmoidal )

4 Maximal Contrast Type Tests max acc. t method (Hirotsu, Kuriki & Hayter, 1992; Hirotsu & Srivastava, 2000) (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 H 0 : μ 1 = μ 2 = μ 3 = μ 4 and if it is not significant stop here (0-stopping), otherwise ② test H 0 : μ 1 = μ 2 = μ 3 and if it is not significant stop here (1-stopping), otherwise ③ test H 0 : μ 1 = μ 2 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 Contrast Liner Contrast Statistic Phase of dosed test stopping stopping stopping stopping stopping stopping

9 Simultaneous Lower Bounds by max acc. t Basic contrasts (Each interpreted as estimating under the respective assumed model)

10 General formula 1 Corresponding to the model with changepoint soon after level and saturating at level The basic contrasts correspond to the case Simultaneous Lower Bounds by max acc. t (continued)

11 Simultaneous Lower Bounds by max acc. t (continued) Corresponding to the linear regression model : Estimating the difference under the assumed linear regression model like other monotone contrasts. Lemma General formula 2

12 Proof of Lemma Deriving SLB for as the best linear combination of the basic contrasts : under the assumption : （ Inhomogeneous and complicated structure ) （ Markov structure )

13 By Markov structure we have ⇒ Proof of Lemma (continue) and

14 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. 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 M 1, M 2, M 3, M 4, M 5 and M 6, respectively, will make sense. 1-stopping : Uniquely select M 1 iff SLB(3,3)>0. 2-stopping : Select either M 2 or M 4 corresponding to the largest of SLB(2,2) and SLB(2,3) iff it is above 0. 3-stopping : Select either M 3, M 5 or M 6 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))

16 Estimating the Dose-Response Patterns 3 (1) Step-down procedure for ⇒ and or ⇒ (Model selection by SLB due to multiple decision processes) Acceptance sets :

17 Confidence sets : with inequality strict if the limit is 0.

18 Model selection

19 (2) Model selection by step-down procedure for

20 Simulation result 1 Comparing 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 modelSelected pattern acc. tHMLOrthogonal type ◎ ○ ◎ or ○ ◎ ◎ or ○ ○ ◎ ◎ or ○

21 Simulation result 2 Effects of adding monotone contrasts,, to max acc. t Table 3. Probability of selecting a model ( ) Method : statistic corresponding to Remarkably small effects of adding, and / or True modelSelected pattern ◎ 85.5 ◎ or ○ ◎ ◎ or ○ ◎ ◎ or ○ ◎ ◎ or ○ ◎ 27.7 ◎ or ○ ◎ ◎ or ○

22 Simulation result 3 Comparing maximal contrast method and SLB method based on max acc. t Table 4. Probability of selecting a model ( ) Method True modelSelected pattern contrastSLB (closed test)SLB (mult. dec.) ◎ ○ ◎ ◎ or ○ ◎ ◎ ◎ or ○ ○ ◎ ◎ ◎ or ○ Total ◎ ◎ or ○

23 Simulation result 4 Comparing maximal contrast method and SLB method based on max acc. t Table 5. Probability of selecting a model ( ) Method True ModelSelected pattern. contrastSLB (closed test)SLB (mult. Dec.) ◎ ◎ or ○ ◎ ◎ or ○ ◎ ◎ or ○ ◎ ◎ or ○ ◎ ◎ or ○ ◎ ◎ or ○ Total ◎ ◎ or○

24 Adding Contrasts t 4, t 5 and/or t 6 to the Basic Contrasts (t 1, t 2, t 3 ) of max acc. t Intending the Detection of Patterns M 4, M 5 and M 6 (Japanese Practice) Method 1 : Method 2 : Method 3 :

25 Calculating the Critical Point (Normal Theory) easy to evaluate

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.

27 References 1. Hirotsu,C.(1982). Use of cumulative efficient scores for testing ordered alternatives in discrete models. Biometrika 69, Hirotsu,C., Kuriki, S. & Hayter,A.J.(1992). Multiple comparison procedures based on the maximal component of the cumulative chisquared statistic. Biometrika 79, Hirotsu,C. & Srivastava, M. S.(2000). Simultaneous confidence intervals based on one-sided max t test. Statistics & Probability Letters 49, Hirotsu,C. & Marumo, K.(2002). Changepoint analysis as a method for isotonic inference. Scandinavian J. Statist. 29, 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,