Optimal designs for comparing curves

Transcription

1 Optimal designs for comparing curves Holger Dette, Ruhr-Universität Bochum Maria Konstantinou, Ruhr-Universität Bochum Kirsten Schorning, Ruhr-Universität Bochum FP7 HEALTH

2 Outline 1 Motivation 2 Testing for similarity 3 Optimal designs for comparing curves 4 Robust designs 5 Conclusions

3 Holger Dette Optimal designs for comparing curves 1 / 36 Motivation: four examples Populations of different geographic regions may bear differences in efficacy (or safety) dose response Objective: Ability to extrapolate study results Demonstrating equivalence of curves becomes an issue Comparison of dose response relationships for two regimens For example, demonstrate that once-daily and twice-daily applications of a drug are similar over a given dose range

4 Holger Dette Optimal designs for comparing curves 1 / 36 Motivation: four examples Populations of different geographic regions may bear differences in efficacy (or safety) dose response Objective: Ability to extrapolate study results Demonstrating equivalence of curves becomes an issue Comparison of dose response relationships for two regimens For example, demonstrate that once-daily and twice-daily applications of a drug are similar over a given dose range

5 Holger Dette Optimal designs for comparing curves 2 / 36 Motivation: four examples Comparison of different drugs containing the same active substance in order to claim bioequivalence. Traditional approaches based on AUC or Cmax may be misleading. Example: Both curves have the same AUC and Cmax It might be better to compare the curves directly

6 Holger Dette Optimal designs for comparing curves 2 / 36 Motivation: four examples Comparison of different drugs containing the same active substance in order to claim bioequivalence. Traditional approaches based on AUC or Cmax may be misleading. Example: Both curves have the same AUC and Cmax It might be better to compare the curves directly

7 Holger Dette Optimal designs for comparing curves 3 / 36 Motivation: four examples Comparison of dissolution profiles (correlated observations) In vitro dissolution profile comparison of two formulations (test vs. reference product) in order to demonstrate bioequivalence Figure: twelve tablets per product, each measured at six time points % Dissolved Time (min)

8 Comparing curves Two dose response curves from different populations (e.g. European, Japanese) Are the two curves similar? If they are: extrapolating information from one population to the other might be possible Dose Response Dose Response Holger Dette Optimal designs for comparing curves 4 / 36

9 Holger Dette Optimal designs for comparing curves 5 / 36 Comparing curves Two dose response models (from two samples) Y ijk = m i (x ij, ϑ i ) + ε ijk ; i = 1, 2; j = 1,..., l i ; k = 1,..., n ij, ε ijk independent N (0, σi 2 ) (i = 1, 2) x ij X (dose levels) X : design space (dose range) ϑ i R d i (parameter in model m i )

10 Holger Dette Optimal designs for comparing curves 6 / 36 Problem of similarity: Problem: Are the dose response curves m 1 and m 2 similar? If they are: Information from one population can be transferred to the other! We are particularly interested in optimal designs for a test for similarity (in order to minimize the sample size) IDEAL - Integrated DEsign and AnaLysis of small population group trials (EU-Project 7th Framework Program)

11 Holger Dette Optimal designs for comparing curves 7 / 36 Measures of similarity d: a metric measuring the distance between m 1 and m 2. Hypothesis of similarity: H 0 : d = d(m 1, m 2 ) versus H 1 : d = d(m 1, m 2 ) < (here is a pre-specified constant). In this talk: maximum difference d = d (m 1, m 2 ) = max x X m 1(x, ϑ 1 ) m 2 (x, ϑ 2 )

12 Holger Dette Optimal designs for comparing curves 7 / 36 Measures of similarity d: a metric measuring the distance between m 1 and m 2. Hypothesis of similarity: H 0 : d = d(m 1, m 2 ) versus H 1 : d = d(m 1, m 2 ) < (here is a pre-specified constant). In this talk: maximum difference d = d (m 1, m 2 ) = max x X m 1(x, ϑ 1 ) m 2 (x, ϑ 2 )

13 Holger Dette Optimal designs for comparing curves 8 / 36 Example Emax and Loglinear model (X = [0, 1]) m 1 (x, ϑ 1 ) = x x, m 2(x, ϑ 2 ) = log(x+0.2) dose effect EMAX model loglinear model maximum distance dose range Maximum deviation between Emax and loglinear model

14 Holger Dette Optimal designs for comparing curves 9 / 36 Testing for similarity (Gsteiger, Bretz, and Liu, 2011) Use the maximum distance d = d (m 1, m 2 ) = max x X m 1(x, ϑ 1 ) m 2 (x, ϑ 2 ) Construct a confidence band for the difference m 1 (, ϑ 1 ) m 2 (, ϑ 2 ) Reject and decide for similarity H 0 : d H 1 : d < if the confidence band is contained in the rectangle X [, ]

15 Holger Dette Optimal designs for comparing curves 9 / 36 Testing for similarity (Gsteiger, Bretz, and Liu, 2011) Use the maximum distance d = d (m 1, m 2 ) = max x X m 1(x, ϑ 1 ) m 2 (x, ϑ 2 ) Construct a confidence band for the difference m 1 (, ϑ 1 ) m 2 (, ϑ 2 ) Reject and decide for similarity H 0 : d H 1 : d < if the confidence band is contained in the rectangle X [, ]

16 Holger Dette Optimal designs for comparing curves 9 / 36 Testing for similarity (Gsteiger, Bretz, and Liu, 2011) Use the maximum distance d = d (m 1, m 2 ) = max x X m 1(x, ϑ 1 ) m 2 (x, ϑ 2 ) Construct a confidence band for the difference m 1 (, ϑ 1 ) m 2 (, ϑ 2 ) Reject and decide for similarity H 0 : d H 1 : d < if the confidence band is contained in the rectangle X [, ]

17 Holger Dette Optimal designs for comparing curves 10 / 36 Testing for similarity - example H 0 : d = 1 versus H 1 : d < = 1 Difference = 1 Difference = Design Space Design Space H 0 : is rejected! H 0 : is not rejected!

18 Holger Dette Optimal designs for comparing curves 11 / 36 Improvements Optimal designs In this talk we will improve this test: Efficient designs for the construction of confidence bands Dette, H., Schorning, K. Optimal designs for comparing curves (Annals of Statistics, 2016) Idea: Minimize the width of the confidence band by the construction of good designs (increase in power) This is a non standard optimal design problem!

19 Holger Dette Optimal designs for comparing curves 11 / 36 Improvements Optimal designs In this talk we will improve this test: Efficient designs for the construction of confidence bands Dette, H., Schorning, K. Optimal designs for comparing curves (Annals of Statistics, 2016) Idea: Minimize the width of the confidence band by the construction of good designs (increase in power) This is a non standard optimal design problem!

20 Holger Dette Optimal designs for comparing curves 12 / 36 Testing for similarity - more details More details for Gsteiger, Bretz, Liu (2011): Two designs for two samples (of sizes n 1 and n 2 ) ( x11 x 1l1 ξ 1 = n 11 n 1 n 1l1 n 1 ) ( x21 x 2l2 ξ 2 = n 21 n 2 n 2l2 n 2 ) Note: The design for the ith sample is called ξ i (i = 1, 2) n i is the sample size of sample i (i = 1, 2) l i is the number of different dose levels in sample i x ij is the jth dose level in the ith sample (j = 1,..., l i ) n ij is the number of patients treated assigned to dose level x ij n ij is the relative proportion of patients assigned to dose level n i x ij

21 Holger Dette Optimal designs for comparing curves 13 / 36 Testing for similarity - more details ˆϑ i : maximum likelihood estimate of ϑ i (i = 1, 2) where (i = 1, 2) ˆϑ i M i (ξ i, ϑ i ) = l i n ij j=1 a N (ϑi, σ2 i n i M 1 i (ξ i, ϑ i )) n i ( )( ϑ i m i (x ij, ϑ i ) ϑ i m i (x ij, ϑ i ) is the information matrix and depends on the design ξ i for the ith sample Note: Other estimates could be considered as well ) T

22 Holger Dette Optimal designs for comparing curves 13 / 36 Testing for similarity - more details ˆϑ i : maximum likelihood estimate of ϑ i (i = 1, 2) where (i = 1, 2) ˆϑ i M i (ξ i, ϑ i ) = l i n ij j=1 a N (ϑi, σ2 i n i M 1 i (ξ i, ϑ i )) n i ( )( ϑ i m i (x ij, ϑ i ) ϑ i m i (x ij, ϑ i ) is the information matrix and depends on the design ξ i for the ith sample Note: Other estimates could be considered as well ) T

23 Holger Dette Optimal designs for comparing curves 14 / 36 Testing for similarity - more details Confidence band { x X where m 1(x, ˆϑ 1 ) m 2 (x, ˆϑ 2 ) (m 1 (x, ϑ 1 ) m 2 (x, ϑ 2 )) } D ˆγ(x, ξ 1, ξ 2 ) ˆγ 2 (x, ξ 1, ξ 2 ) = ˆσ2 1 n 1 x m 1(x, ˆϑ 1 )M 1 1 (ξ 1, ˆϑ 1 ) ( x m 1(x, ˆϑ 1 ) ) T + ˆσ2 2 n 2 x m 2(x, ˆϑ 2 )M 1 2 (ξ 2, ˆϑ 2 ) ( x m 2(x, ˆϑ 2 ) ) T is an estimate of the variance of the predicted difference Critical value D is determined by parametric bootstrap Note: the width of the band at the point x depends on the designs ξ 1 and ξ 2 and is approximately given by 2Dˆγ(x, ξ 1, ξ 2 )

24 Holger Dette Optimal designs for comparing curves 15 / 36 Example Emax and loglinear model (X = [0, 1]) m 1(x, ϑ 1) = x, m2(x, ϑ2) = log(x + 0.2) x Hypotheses d 2.5 versus H 1 : d < 2.5 dose effect EMAX model loglinear model maximum distance dose range Emax and loglinear model.

25 Holger Dette Optimal designs for comparing curves 16 / 36 Example Simulated confidence bands (n 1 = n 2 = 100, σ 1 = σ 2 = 1.478) using standard designs ( ) ξ 1 = ξ 2 = Difference Design Space Confidence bands for the difference of the Emax and loglinear model. The test is very conservative! Can we do better by choosing a nice design?

26 Holger Dette Optimal designs for comparing curves 17 / 36 Approximate design theory Two designs ( x11 x 1l1 ξ 1 = n 11 n 1 n 1l1 n 1 ) ( x21 x 2l2 ξ 2 = n 21 n 2 n 2l2 n 2 ) Approximate design theory: replace n ij n i by w ij (0, 1), Approximate designs ξ 1 = ( x11 x 1l1 w 11 w 1l1 ) n 1 n 1 + n 2 by λ (0, 1) ξ 2 = ( x21 x 2l2 w 21 w 2l2 )

27 Holger Dette Optimal designs for comparing curves 18 / 36 Approximate design theory Note: two designs ξ 1, ξ 2 have to be determined (λ is fixed) Recall: the width of the band at the point x is proportional to γ(x, ξ 1, ξ 2 ) = γ(x, ϑ 1, ϑ 2, ξ 1, ξ 2 ) An optimal pair of designs minimizes the maximum width of the confidence band for the difference m 1 (x, ϑ 1 ) m 2 (x, ϑ 2 )! More mathematically: an optimal pair of designs (ξ 1, ξ 2 ) minimizes where γ 2 (ξ 1, ξ 2 ) = max x X γ2 (x, ξ 1, ξ 2 ) γ 2 (x, ξ 1, ξ 2 ) = σ2 1 λ 1 ϑ 1 m 1 (x, ϑ 1 )M 1 1 (ξ 1, ϑ 1 ) ( ϑ 2 m 1 (x, ϑ 1 ) ) T + σ2 2 λ 2 ϑ 2 m 2 (x, ϑ 2 )M 1 2 (ξ 2, ϑ 2 ) ( ϑ 2 m 2 (x, ϑ 2 ) ) T

28 Holger Dette Optimal designs for comparing curves 18 / 36 Approximate design theory Note: two designs ξ 1, ξ 2 have to be determined (λ is fixed) Recall: the width of the band at the point x is proportional to γ(x, ξ 1, ξ 2 ) = γ(x, ϑ 1, ϑ 2, ξ 1, ξ 2 ) An optimal pair of designs minimizes the maximum width of the confidence band for the difference m 1 (x, ϑ 1 ) m 2 (x, ϑ 2 )! More mathematically: an optimal pair of designs (ξ 1, ξ 2 ) minimizes where γ 2 (ξ 1, ξ 2 ) = max x X γ2 (x, ξ 1, ξ 2 ) γ 2 (x, ξ 1, ξ 2 ) = σ2 1 λ 1 ϑ 1 m 1 (x, ϑ 1 )M 1 1 (ξ 1, ϑ 1 ) ( ϑ 2 m 1 (x, ϑ 1 ) ) T + σ2 2 λ 2 ϑ 2 m 2 (x, ϑ 2 )M 1 2 (ξ 2, ϑ 2 ) ( ϑ 2 m 2 (x, ϑ 2 ) ) T

29 Holger Dette Optimal designs for comparing curves 19 / 36 Approximate design theory Note: we have to find l 1, l 2 (number of different dose levels) x ij (dose levels) w ij (proportions of patients allocated to each dose level) Optimal designs depend on parameters ϑ 1, ϑ 2 In this talk: we mainly assume some prior knowledge for the parameters and determine (locally optimal designs) Alternative designs can be constructed by standard methodology Bayesian optimal designs (later) (standardized) minimax designs Adaptive designs

30 Holger Dette Optimal designs for comparing curves 19 / 36 Approximate design theory Note: we have to find l 1, l 2 (number of different dose levels) x ij (dose levels) w ij (proportions of patients allocated to each dose level) Optimal designs depend on parameters ϑ 1, ϑ 2 In this talk: we mainly assume some prior knowledge for the parameters and determine (locally optimal designs) Alternative designs can be constructed by standard methodology Bayesian optimal designs (later) (standardized) minimax designs Adaptive designs

31 Holger Dette Optimal designs for comparing curves 19 / 36 Approximate design theory Note: we have to find l 1, l 2 (number of different dose levels) x ij (dose levels) w ij (proportions of patients allocated to each dose level) Optimal designs depend on parameters ϑ 1, ϑ 2 In this talk: we mainly assume some prior knowledge for the parameters and determine (locally optimal designs) Alternative designs can be constructed by standard methodology Bayesian optimal designs (later) (standardized) minimax designs Adaptive designs

32 Holger Dette Optimal designs for comparing curves 20 / 36 Approximate design theory Locally optimal designs have to be found numerically - Particle Swarm Optimization (PSO - metaheuristic) Check the optimality of the calculated designs Theorem (equivalence theorem) (ξ1, ξ 2 ) is optimal if and only if there exists a measure µ supported on the set {z X γ 2 (z, ξ 1, ξ 2) = max x X γ2 (x, ξ 1, ξ 2)} such that for all x 1, x 2 X L(x 1, x 2 ) := h(x 1, x 2, ξ1, ξ2, z)µ (dz) 0 X

33 Holger Dette Optimal designs for comparing curves 20 / 36 Approximate design theory Locally optimal designs have to be found numerically - Particle Swarm Optimization (PSO - metaheuristic) Check the optimality of the calculated designs Theorem (equivalence theorem) (ξ1, ξ 2 ) is optimal if and only if there exists a measure µ supported on the set {z X γ 2 (z, ξ 1, ξ 2) = max x X γ2 (x, ξ 1, ξ 2)} such that for all x 1, x 2 X L(x 1, x 2 ) := h(x 1, x 2, ξ1, ξ2, z)µ (dz) 0 X

34 Holger Dette Optimal designs for comparing curves 21 / 36 Approximate design theory The function h is complicated, but known: h(x 1, x 2, ξ1, ξ2 (, z) = m 1 (x 1, ϑ 1 )M 1 1 (ξ1, ϑ 1 ) m 1 (z, ϑ 2 ) ϑ 1 ϑ 1 ( + m 2 (x 2, ϑ 2 )M 1 2 (ξ2, ϑ 2 ) m 2 (z, ϑ 2 ) ϑ 2 ϑ 2 γ(ξ1, ξ2 ) µ is not known (the optimality criterion is not differentiable - solution of a minimax problem) ) )

35 Holger Dette Optimal designs for comparing curves 22 / 36 Example Emax and Log-linear model (X = [0, 1]) m 1 (x, ϑ 1 ) = x x, m 2(x, ϑ 2 ) = log(x+0.2) dose effect EMAX model loglinear model maximum distance dose range Emax and loglinear model.

36 Holger Dette Optimal designs for comparing curves 23 / 36 Optimal designs for Emax- and Log-linear model Emax- and Log-linear model m 1 (x, ϑ 1 ) = x, x [0, 1] x m 2 (x, ϑ 2 ) = log(x + 0.2), x [0, 1] d (m 1, m 2 ) = Numerically calculated optimal designs ( ) ( ξ = ξ = )

37 Holger Dette Optimal designs for comparing curves 24 / 36 Equivalence theorem Checking condition: L(x 1, x 2 ) = X h(x 1, x 2, ξ 1, ξ 2, z)µ (dz) Design space Design space The designs ξ 1 and ξ 2 are optimal

38 Holger Dette Optimal designs for comparing curves 25 / 36 Example (continued) Confidence bands for the difference of the Emax and loglinear model using a standard design (left panel) and the optimal design (right panel).

39 Holger Dette Optimal designs for comparing curves 26 / 36 Power of the test of Gsteiger et al (2011) Simulated power of (n 1 = n 2 = 100; 100 runs) of the test for the hypothesis H 0 : d versus H 1 : d < optimal design standard design

40 Holger Dette Optimal designs for comparing curves 27 / 36 Sensitivity Emax and Exponential model with assumed parameters Emax (m 1 ) ϑ 1 + ϑ2t t+ϑ 3 (0.2, 0.7, 0.2) exponential (m 2 ) ϑ 1 + ϑ 2 exp(t/ϑ 3 ) (0.183, 0.017, 0.28) Simulated confidence bands obtained from the standard design (dashed lines) and optimal design Difference Difference Design space Design space ϑ Emax = (0.1, 0.35, 0.1) ϑ Emax = (0.4, 0.7, 0.4) ϑ exp = (0.1, 0.05, 0.167) ϑ exp = (0.4, 0.2, 0.66)

41 Holger Dette Optimal designs for comparing curves 28 / 36 Sensitivity Simulated confidence bands obtained from the standard design (dashed lines) and optimal design Difference Difference Design space Design space ϑ Emax = (0.1, 0.35, 0.1) ϑ Emax = (0.4, 0.7, 0.4) ϑ exp = (0.1, 0.05, 0.167) ϑ exp = (0.4, 0.2, 0.66) An improvement can be obtained, even if the parameters are misspecified Can we do even better, incorporating the uncertainty about the parameters in the construction of the optimal designs?

42 Holger Dette Optimal designs for comparing curves 29 / 36 Bayesian optimal designs Recall: the width of the band at the point x is proportional to γ(x, ϑ 1, ϑ 2, ξ 1, ξ 2 ), where γ 2 (x, ϑ 1, ϑ 2, ξ 1, ξ 2 ) = σ2 1 n 1 x m 1(x, ϑ 1 )M 1 1 (ξ 1, ϑ 1 ) ( x m 1(x, ϑ 1 ) ) T + σ2 2 n 2 x m 2(x, ϑ 2 )M 1 2 (ξ 2, ϑ 2 ) ( x m 2(x, ϑ 2 ) ) T Local optimal designs: fix ϑ 1 and ϑ 2 and minimize max x X γ2 (x, ϑ 1, ϑ 2, ξ 1, ξ 2 ) Bayesian optimal designs: put prior distributions π 1 and π 2 on ϑ 1 and ϑ 2, respectively, and minimize max x X γ2 (x, ϑ 1, ϑ 2, ξ 1, ξ 2 )π 1 (dϑ 1 )π 2 (dϑ 2 )

43 Holger Dette Optimal designs for comparing curves 30 / 36 Bayesian optimal designs Usually the designs are sensitive with respect to the model assumption, e.g. An optimal design for comparing two Emax models might be inefficient, if the true models are an exponential and a loglinear model Model uncertainty can be addressed as well assume that m 1,..., m p are competing models assign to each pair (m i, m j ) a nonnegative weight α ij, which reflects the experimenters belief about the pair (m i, m j ) for group 1 and 2

44 Holger Dette Optimal designs for comparing curves 31 / 36 Bayesian optimal designs Model robust Bayesian optimal designs minimize p α ij i,j=1 max x X γ2 ij(x, ϑ i, ϑ j, ξ 1, ξ 2 )π i (dϑ i )π j (dϑ j ) where π 1,..., π p are prior distributions for the parameters ϑ 1,..., ϑ p in the models m 1,..., m p, respectively. Similar results are available An algorithm to calculate the model robust Bayesian optimal designs An equivalence theorem, to check the optimality of the numerical results

45 Holger Dette Optimal designs for comparing curves 32 / 36 Model robust Bayesian optimal designs - an example Commonly used dose response models model m(t, ϑ) parameters Emax (m 1 ) ϑ 1 + ϑ2t t+ϑ 3 (0.2, 0.7, 0.2) exponential (m 2 ) ϑ 1 + ϑ 2 exp(t/ϑ 3 ) (0.183, 0.017, 0.28) loginear (m 3 ) ϑ 1 + ϑ 2 log(t + ϑ 3 ) (0.74, 0.33, 0.2) We are interested in a comparison of two Emax or two exponential or two loglinear models, i.e α 11 = α 22 = α 33 = 1/3 and α ij = 0 else We fix parameters for each model (prior distribution concentrated in one point)

46 Holger Dette Optimal designs for comparing curves 33 / 36 Model robust Bayesian optimal designs - an example Model robust Bayesian optimal designs First sample: allocate 30.27%, 30.72% 19.56%, 19.44% of the patients to dose levels 0.00, 0.17, 0.77, 1.00 Second sample: allocate 29.96%, 30.18%, 19.38%, 20.48% at the points 0.00, 0.17, 0.76, 1.00 how good are the designs? model 1 / model 2 Emax / Emax exp / exp loglin / loglin standard design 67.98% 42.87% 74.29% robust optimal design 90.40% 78.45% 90.79%

47 Holger Dette Optimal designs for comparing curves 34 / 36 Model robust Bayesian optimal designs - an example Model robust Bayesian optimal designs First sample: allocate 30.27%, 30.72% 19.56%, 19.44% of the patients to dose levels 0.00, 0.17, 0.77, 1.00 Second sample: allocate 29.96%, 30.18%, 19.38%, 20.48% at the points 0.00, 0.17, 0.76, 1.00 how good are the designs if we use other models? model 1 / model 2 loglin/exp loglin/emax exp/emax standard design 58.85% 72.83% 59.00% D-optimal designs (Emax) 2.21% 93.81% 2.24% D-optimal designs (loglinear) 7.31% 92.44% 7.40% D-optimal designs (exponential) 15.08% 3.72% 4.29% robust optimal design 67.30% 89.65% 69.57%

48 Holger Dette Optimal designs for comparing curves 35 / 36 Conclusions The problem of testing for the similarity of two (parametric) curves is fundamental interest in biostatistics The power of the test of Gsteiger et al. (2011) can be increased by optimal designs which produce narrower confidence bands The designs can be constructed such that they are robust with respect to model misspecification with respect to misspecification of model parameters Standard designs are not efficient in this context, e.g. uniform design individual D-optimal designs The resulting procedure is substantially more powerful, but it is still not a good test!

49 Holger Dette Optimal designs for comparing curves 35 / 36 Conclusions The problem of testing for the similarity of two (parametric) curves is fundamental interest in biostatistics The power of the test of Gsteiger et al. (2011) can be increased by optimal designs which produce narrower confidence bands The designs can be constructed such that they are robust with respect to model misspecification with respect to misspecification of model parameters Standard designs are not efficient in this context, e.g. uniform design individual D-optimal designs The resulting procedure is substantially more powerful, but it is still not a good test!

50 Holger Dette Optimal designs for comparing curves 35 / 36 Conclusions The problem of testing for the similarity of two (parametric) curves is fundamental interest in biostatistics The power of the test of Gsteiger et al. (2011) can be increased by optimal designs which produce narrower confidence bands The designs can be constructed such that they are robust with respect to model misspecification with respect to misspecification of model parameters Standard designs are not efficient in this context, e.g. uniform design individual D-optimal designs The resulting procedure is substantially more powerful, but it is still not a good test!

51 Holger Dette Optimal designs for comparing curves 35 / 36 Conclusions The problem of testing for the similarity of two (parametric) curves is fundamental interest in biostatistics The power of the test of Gsteiger et al. (2011) can be increased by optimal designs which produce narrower confidence bands The designs can be constructed such that they are robust with respect to model misspecification with respect to misspecification of model parameters Standard designs are not efficient in this context, e.g. uniform design individual D-optimal designs The resulting procedure is substantially more powerful, but it is still not a good test!

52 Holger Dette Optimal designs for comparing curves 35 / 36 Conclusions The problem of testing for the similarity of two (parametric) curves is fundamental interest in biostatistics The power of the test of Gsteiger et al. (2011) can be increased by optimal designs which produce narrower confidence bands The designs can be constructed such that they are robust with respect to model misspecification with respect to misspecification of model parameters Standard designs are not efficient in this context, e.g. uniform design individual D-optimal designs The resulting procedure is substantially more powerful, but it is still not a good test!

53 Holger Dette Optimal designs for comparing curves 36 / 36 Power of the test of Gsteiger et al (2011) Simulated power of (n 1 = n 2 = 100; 100 runs) of the test for the hypothesis H 0 : d versus H 1 : d < optimal design standard design Note: d (m 1, m 2 ) = (rather extreme alternatives!) In the next talk a more efficient test will be constructed Optimal designs for this test can be constructed as well

54 Holger Dette Optimal designs for comparing curves 36 / 36 Power of the test of Gsteiger et al (2011) Simulated power of (n 1 = n 2 = 100; 100 runs) of the test for the hypothesis H 0 : d versus H 1 : d < optimal design standard design Note: d (m 1, m 2 ) = (rather extreme alternatives!) In the next talk a more efficient test will be constructed Optimal designs for this test can be constructed as well