Some Curiosities in Optimal Designs for Random Slopes Thomas Schmelter 12 , Norbert Benda 3 , Rainer Schwabe 1 1 Otto-von-Guericke-Universität, Magdeburg 2 Bayer Schering Pharma AG, Berlin 3 Novartis Pharma AG, Basel mODa 8 T. Schmelter, N. Benda, and R. Schwabe Optimal Designs for Random Slopes
Motivation Original motivation: 2.0 Population PK studies Konzentration 1.5 1.0 0.5 0.0 0 5 10 15 20 Zeit T. Schmelter, N. Benda, and R. Schwabe Optimal Designs for Random Slopes
Motivation Original motivation: 2.0 Population PK studies Konzentration 1.5 1.0 0.5 0.0 Linear Random Coefficient 0 5 10 15 20 Regression Model Zeit T. Schmelter, N. Benda, and R. Schwabe Optimal Designs for Random Slopes
Motivation Original motivation: 2.0 Population PK studies Konzentration 1.5 1.0 0.5 0.0 Linear Random Coefficient 0 5 10 15 20 Regression Model Zeit Random Slope Model T. Schmelter, N. Benda, and R. Schwabe Optimal Designs for Random Slopes
Linear Random Coefficient Regression Model j th observation of individual i given by Y ij = f ( x ij ) ⊤ b i + ε ij , i = 1 , . . . , n , j = 1 , . . . , m i T. Schmelter, N. Benda, and R. Schwabe Optimal Designs for Random Slopes
Linear Random Coefficient Regression Model j th observation of individual i given by Y ij = f ( x ij ) ⊤ b i + ε ij , i = 1 , . . . , n , j = 1 , . . . , m i ε ij ∼ iid N ( 0 , σ 2 ) T. Schmelter, N. Benda, and R. Schwabe Optimal Designs for Random Slopes
Linear Random Coefficient Regression Model j th observation of individual i given by Y ij = f ( x ij ) ⊤ b i + ε ij , i = 1 , . . . , n , j = 1 , . . . , m i � ε ij ∼ iid N ( 0 , σ 2 ) independent b i ∼ iid N ( β , σ 2 D ) T. Schmelter, N. Benda, and R. Schwabe Optimal Designs for Random Slopes
Linear Random Coefficient Regression Model j th observation of individual i given by Y ij = f ( x ij ) ⊤ b i + ε ij , i = 1 , . . . , n , j = 1 , . . . , m i � ε ij ∼ iid N ( 0 , σ 2 ) independent b i ∼ iid N ( β , σ 2 D ) Individual observation vector Y i = F i β + F i ( b i − β ) + ε i Cov ( Y i ) = σ 2 V i with V i = I m i + F i DF ⊤ i T. Schmelter, N. Benda, and R. Schwabe Optimal Designs for Random Slopes
Estimation of Population Parameters Weighted least squares: � n � − 1 n � � ˆ F ⊤ F ⊤ β = i V i F i i V i Y i i = 1 i = 1 � Covariance matrix � n � − 1 � Cov (ˆ β ) = σ 2 F ⊤ i V i F i i = 1 T. Schmelter, N. Benda, and R. Schwabe Optimal Designs for Random Slopes
Definition of Designs Elementary design � x i 1 � x ij exp. settings . . . x ik i ξ i = m ij replications m i 1 m ik i . . . T. Schmelter, N. Benda, and R. Schwabe Optimal Designs for Random Slopes
Definition of Designs Elementary design � x i 1 � x ij exp. settings . . . x ik i ξ i = m ij replications m i 1 m ik i . . . Population design � ξ 1 � ξ r elementary design . . . ξ l ζ = g 1 g l g r proportion of individuals . . . T. Schmelter, N. Benda, and R. Schwabe Optimal Designs for Random Slopes
Uniform Designs All individuals are observed uniformly using the same setting T. Schmelter, N. Benda, and R. Schwabe Optimal Designs for Random Slopes
Uniform Designs All individuals are observed uniformly using the same setting m i = m , x ij = x j , F i = F 1 , V i = V 1 � T. Schmelter, N. Benda, and R. Schwabe Optimal Designs for Random Slopes
Uniform Designs All individuals are observed uniformly using the same setting m i = m , x ij = x j , F i = F 1 , V i = V 1 � ◮ Estimation of β does not require knowledge of D (WLSE=OLSE). T. Schmelter, N. Benda, and R. Schwabe Optimal Designs for Random Slopes
Uniform Designs All individuals are observed uniformly using the same setting m i = m , x ij = x j , F i = F 1 , V i = V 1 � ◮ Estimation of β does not require knowledge of D (WLSE=OLSE). ◮ σ 2 � � − 1 Cov (ˆ F ⊤ i V − 1 β ) = F i i n σ 2 � � i F i ) − 1 + D ( F ⊤ = n T. Schmelter, N. Benda, and R. Schwabe Optimal Designs for Random Slopes
Uniform Designs All individuals are observed uniformly using the same setting m i = m , x ij = x j , F i = F 1 , V i = V 1 � ◮ Estimation of β does not require knowledge of D (WLSE=OLSE). ◮ σ 2 � � − 1 Cov (ˆ F ⊤ i V − 1 β ) = F i i n σ 2 � � i F i ) − 1 + D ( F ⊤ = n ◮ If non-integer replications are allowed for the observations, one cannot improve by going away from the uniform designs. T. Schmelter, N. Benda, and R. Schwabe Optimal Designs for Random Slopes
Linear Criteria ◮ minimize � 1 F 1 ) − 1 + D ) L ⊤ � L (( F ⊤ tr � 1 F 1 ) − 1 ) L ⊤ � � LDL ⊤ � L (( F ⊤ = tr + tr � �� � constant T. Schmelter, N. Benda, and R. Schwabe Optimal Designs for Random Slopes
Linear Criteria ◮ minimize � 1 F 1 ) − 1 + D ) L ⊤ � L (( F ⊤ tr � 1 F 1 ) − 1 ) L ⊤ � � LDL ⊤ � L (( F ⊤ = tr + tr � �� � constant T. Schmelter, N. Benda, and R. Schwabe Optimal Designs for Random Slopes
Linear Criteria ◮ minimize � 1 F 1 ) − 1 + D ) L ⊤ � L (( F ⊤ tr � 1 F 1 ) − 1 ) L ⊤ � � LDL ⊤ � L (( F ⊤ = tr + tr � �� � constant ( x 1 , . . . , x m ) optimal in reduced model ⇒ ( x 1 , . . . , x m ) optimal in RCR model Luoma (2000), Liski et al. (2002) T. Schmelter, N. Benda, and R. Schwabe Optimal Designs for Random Slopes
Random Slope Model Y ij = µ + b i x ij + ε ij , T. Schmelter, N. Benda, and R. Schwabe Optimal Designs for Random Slopes
Random Slope Model � 0 � 0 Y ij = µ + b i x ij + ε ij , D = 0 d T. Schmelter, N. Benda, and R. Schwabe Optimal Designs for Random Slopes
Random Slope Model � 0 � 0 Y ij = µ + b i x ij + ε ij , D = 0 d 1.0 ● 0.8 ◮ standard interval 0.6 0 ≤ x ij ≤ 1 0.4 ◮ σ 2 = 1 0.2 0.0 ● 0.0 0.2 0.4 0.6 0.8 1.0 x T. Schmelter, N. Benda, and R. Schwabe Optimal Designs for Random Slopes
◮ Optimal designs are supported only on { 0 , 1 } T. Schmelter, N. Benda, and R. Schwabe Optimal Designs for Random Slopes
◮ Optimal designs are supported only on { 0 , 1 } ◮ Candidates for optimal designs can be characterized by number of observations m 1 to be taken at x = 1 or by w = m 1 m T. Schmelter, N. Benda, and R. Schwabe Optimal Designs for Random Slopes
◮ Optimal designs are supported only on { 0 , 1 } ◮ Candidates for optimal designs can be characterized by number of observations m 1 to be taken at x = 1 or by w = m 1 m allow wm to be non-integer T. Schmelter, N. Benda, and R. Schwabe Optimal Designs for Random Slopes
◮ Optimal designs are supported only on { 0 , 1 } ◮ Candidates for optimal designs can be characterized by number of observations m 1 to be taken at x = 1 or by w = m 1 m allow wm to be non-integer ◮ Then � w � ˆ � � µ 1 1 − w Cov = ˆ w ( 1 − w ) − w 1 + mdw ( 1 − w ) β nm T. Schmelter, N. Benda, and R. Schwabe Optimal Designs for Random Slopes
D- vs. G-optimality The D-optimal proportion at x = 1 is √ w ∗ md + 1 ) − 1 D = ( 1 + T. Schmelter, N. Benda, and R. Schwabe Optimal Designs for Random Slopes
D- vs. G-optimality The D-optimal proportion at x = 1 is √ w ∗ md + 1 ) − 1 D = ( 1 + The G-optimal proportion at x = 1 is � 1 � 2 ( 1 − 2 ( md ) − 1 + 1 + 4 ( md ) − 2 ) , d > 0 w ∗ G = 1 d = 0 2 , T. Schmelter, N. Benda, and R. Schwabe Optimal Designs for Random Slopes
D- vs. G-optimality Optimal proportions at x = 1 1.0 0.8 0.6 IMSE-optimal proportion w 0.4 0.2 0.0 0 20 40 60 80 100 m*d T. Schmelter, N. Benda, and R. Schwabe Optimal Designs for Random Slopes
D- vs. G-optimality Optimal proportions at x = 1 1.0 0.8 0.6 IMSE-optimal proportion w D-optimal proportion 0.4 0.2 0.0 0 20 40 60 80 100 m*d T. Schmelter, N. Benda, and R. Schwabe Optimal Designs for Random Slopes
D- vs. G-optimality Optimal proportions at x = 1 1.0 0.8 0.6 IMSE-optimal proportion w D-optimal proportion 0.4 G-optimal proportion 0.2 0.0 0 20 40 60 80 100 m*d T. Schmelter, N. Benda, and R. Schwabe Optimal Designs for Random Slopes
Efficiency of w = 1 2 1.00 0.90 D-efficiency of w = 1 eff 2 0.80 0.70 0 20 40 60 80 100 m*d T. Schmelter, N. Benda, and R. Schwabe Optimal Designs for Random Slopes
Efficiency of w = 1 2 1.00 0.90 D-efficiency of w = 1 eff 2 G-efficiency of w = 1 0.80 2 0.70 0 20 40 60 80 100 m*d T. Schmelter, N. Benda, and R. Schwabe Optimal Designs for Random Slopes
Discretization Uniform designs: T. Schmelter, N. Benda, and R. Schwabe Optimal Designs for Random Slopes
Discretization Uniform designs: m = 2: T. Schmelter, N. Benda, and R. Schwabe Optimal Designs for Random Slopes
Discretization Uniform designs: m = 2: m ∗ 1 = 1 (number of obs. at x = 1) D- and G-optimal T. Schmelter, N. Benda, and R. Schwabe Optimal Designs for Random Slopes
Discretization Uniform designs: m = 2: m ∗ 1 = 1 (number of obs. at x = 1) D- and G-optimal m = 4: T. Schmelter, N. Benda, and R. Schwabe Optimal Designs for Random Slopes
Recommend
More recommend
