Optimal Crossover Designs for Comparing Test Treatments to a Control - PowerPoint PPT Presentation

Optimal Crossover Designs for Comparing Test Treatments to a Control Treatment When Subject Effects are Random A.S. Hedayat and Wei Zheng Department of Mathematics, Statistics, and Computer Science University of Illinois at Chicago

Outline A Taste of Optimal Designs Motivation from Statistics Background Results Construction of Optimal Designs Characteristics of Optimal Designs Guidelines for Construction Methods of Constructions Further Problems

A Taste of Optimal Designs Motivation from Statistics Construction of Optimal Designs Further Problems A design as a mapping 1 2 3 0 0 2 0 3 2 1 0 3 0 1 4 0 1 2 3 4 2 0 3 3 0 4 4 2 4 0 d : ( k , u ) �→ i where 1 ≤ k ≤ p , 1 ≤ u ≤ n and 0 ≤ i ≤ t In this example, n = 10 , p = 3 , t = 4. A.S. Hedayat and Wei Zheng Optimal Crossover Designs

A Taste of Optimal Designs Motivation from Statistics Construction of Optimal Designs Further Problems What’s special about these two designs? 0 0 0 2 3 1 2 3 1 1 2 3 0 0 0 1 2 3 2 3 1 1 2 3 0 0 0 1 3 2 3 2 0 1 3 0 2 1 0 2 1 3 2 0 3 3 0 1 1 0 2 0 0 0 1 1 1 2 2 2 3 3 3 A.S. Hedayat and Wei Zheng Optimal Crossover Designs

A Taste of Optimal Designs Motivation from Statistics Construction of Optimal Designs Further Problems Notations n diu = � p k =1 I [ d ( k , u )= i ] . n diu = � p − 1 ˜ k =1 I [ d ( k , u )= i ] . l dik = � n u =1 I [ d ( k , u )= i ] . � p − 1 m dij = � n k =1 I [ d ( k , u )= i , d ( k +1 , u )= j ] . u =1 � p r di = � n k =1 I [ d ( k , u )= i ] . u =1 r di = � n � p − 1 ˜ k =1 I [ d ( k , u )= i ] . u =1 A.S. Hedayat and Wei Zheng Optimal Crossover Designs

A Taste of Optimal Designs Motivation from Statistics Construction of Optimal Designs Further Problems In general, we need... A design d is saided to be a totally balanced test-control incomplete crossover design (TBTCI) if: 1 Each element from { 1 , 2 , ..., t } show up in each column at most once. 2 Each element from { 0 , 1 , ..., t } is equally replicated in each row. 3 | n d 0 u − n d 0 v | ≤ 1 and | ˜ n d 0 u − ˜ n d 0 v | ≤ 1 for all 1 ≤ u , v ≤ n . 4 m d 0 i , m di 0 and m dij are constants across all 1 ≤ i � = j ≤ t and m dii = 0 for all 0 ≤ i ≤ t . 5 r di and ˜ r di are constants across all 1 ≤ i ≤ t . 6 � n u =1 n d 0 u n diu , � n u =1 n diu n dju , � n n diu , � n u =1 ˜ n d 0 u ˜ u =1 ˜ n diu ˜ n dju , � n n diu , � n n d 0 u n diu , and � n u =1 n d 0 u ˜ u =1 ˜ u =1 n diu ˜ n dju , are constants across all 1 ≤ i � = j ≤ t . A.S. Hedayat and Wei Zheng Optimal Crossover Designs

A Taste of Optimal Designs Motivation from Statistics Construction of Optimal Designs Further Problems Let N d = ( n diu ) and ˜ N d = (˜ n diu ) when 0 ≤ i ≤ t and 1 ≤ u ≤ n . Conditions 5 and 6 are equivalent to b 1 1 ′ � � a 1 N d N ′ t d = (1) b 1 1 t ( e 1 − f 1 ) I t + f 1 J t b 2 1 ′ � � a 2 N d ˜ N ′ t d = (2) c 2 1 t ( e 2 − f 2 ) I t + f 2 J t � b 3 1 ′ � a 3 N d ˜ ˜ N ′ t d = (3) b 3 1 t ( e 3 − f 3 ) I t + f 3 J t a 2 = � n u =1 n d 0 u ˜ n d 0 u a 1 = � n a 3 = � n u =1 n 2 n 2 u =1 ˜ b 2 = � n d 0 u d 0 u u =1 n d 0 u ˜ n d 1 u b 1 = � n b 3 = � n u =1 ˜ n d 0 u ˜ u =1 n d 0 u n d 1 u n d 1 u c 2 = � n u =1 ˜ n d 0 u n d 1 u e 1 = � n e 3 = � n u =1 n 2 n 2 u =1 ˜ e 2 = � n d 1 u d 1 u u =1 n d 1 u ˜ n d 1 u f 1 = � n f 3 = � n u =1 ˜ n d 1 u ˜ u =1 n d 1 u n d 2 u n d 2 u f 2 = � n u =1 n d 1 u ˜ n d 2 u A.S. Hedayat and Wei Zheng Optimal Crossover Designs

A Taste of Optimal Designs Motivation from Statistics Background Construction of Optimal Designs Results Further Problems Definition A p × n array with symbols from { 0 , 1 , 2 , ..., t } is said to be a crossover design if columns represent subjects, rows represent periods and symbols represent treatments. Our goal Compare the test treatments, { 1 , 2 , ..., t } , with the control treatment { 0 } . Important notations n : number of subjects/units/patients p : number of periods t : number of test treatments r d 0 : replications of the control treatment in design d . A.S. Hedayat and Wei Zheng Optimal Crossover Designs

A Taste of Optimal Designs Motivation from Statistics Background Construction of Optimal Designs Results Further Problems An Example 1 2 3 0 0 2 0 3 2 1 d: 0 3 0 1 4 0 1 2 3 4 2 0 3 3 0 4 4 2 4 0 A.S. Hedayat and Wei Zheng Optimal Crossover Designs

A Taste of Optimal Designs Motivation from Statistics Background Construction of Optimal Designs Results Further Problems An Example 1 2 3 0 0 2 0 3 2 1 d: 0 3 0 1 4 0 1 2 3 4 2 0 3 3 0 4 4 2 4 0 n = 10 , p = 3 , t = 4 r d 0 = 9 A.S. Hedayat and Wei Zheng Optimal Crossover Designs

A Taste of Optimal Designs Motivation from Statistics Background Construction of Optimal Designs Results Further Problems An Example 1 2 3 0 0 2 0 3 2 1 d: 0 3 0 1 4 0 1 2 3 4 2 0 3 3 0 4 4 2 4 0 n = 10 , p = 3 , t = 4 r d 0 = 9 n could be hundreds or thousands depending on the study. p is usually not large due to ethic or other issues. t is not large either; we will inverstigate t + 1 ≥ p ≥ 3. A.S. Hedayat and Wei Zheng Optimal Crossover Designs

A Taste of Optimal Designs Motivation from Statistics Background Construction of Optimal Designs Results Further Problems An Example 1 2 3 0 0 2 0 3 2 1 d: 0 3 0 1 4 0 1 2 3 4 2 0 3 3 0 4 4 2 4 0 n = 10 , p = 3 , t = 4 r d 0 = 9 n could be hundreds or thousands depending on the study. p is usually not large due to ethic or other issues. t is not large either; we will inverstigate t + 1 ≥ p ≥ 3. # of designs (identical up to an isomorphism): ( N + n − 1)! t !( N − 1)! n N − 1 , N = ( t + 1) p . 1 t ! n !( N − 1)! ≥ Isomorphism: in the sense of relabling the subjects and test treatments. A.S. Hedayat and Wei Zheng Optimal Crossover Designs

A Taste of Optimal Designs Motivation from Statistics Background Construction of Optimal Designs Results Further Problems Model Y dku = µ + α k + β u + τ d ( k , u ) + γ d ( k − 1 , u ) + ǫ ku (4) β u iid N (0 , σ 2 ǫ ku iid N (0 , σ 2 ) , β ) , β u ⊥ ⊥ ǫ ku Y dku : Response from unit (subject) u in period k in design d . α k : Effect of period k . β u : Effect of subject u . d ( k , u ): Treatment specified by the design d for unit u in period k . (Control { 0 } ; Test { 1 , 2 , ... t } ) τ i : Direct effect of treatment i γ i : Carryover effect of treatment i (by convention γ d (0 , u ) = 0) A.S. Hedayat and Wei Zheng Optimal Crossover Designs

A Taste of Optimal Designs Motivation from Statistics Background Construction of Optimal Designs Results Further Problems Model (In Matrix Form) E ( Y d ) = 1 np µ + P α + T d τ + F d γ (5) σ 2 ( I n ⊗ ( I p + θ J p )) var ( Y d ) = Where θ = σ 2 β /σ 2 . α = ( α 1 , ..., α p ) ′ , τ = ( τ 0 , ...τ t ) ′ , γ = ( γ 0 , ..., γ t ) ′ . P = 1 n ⊗ I p . T d and F d denote the treatment and carryover incidence matrices. ⊗ denote the Kronecker product. A.S. Hedayat and Wei Zheng Optimal Crossover Designs

A Taste of Optimal Designs Motivation from Statistics Background Construction of Optimal Designs Results Further Problems The information matrix C d for τ is C d = T ′ d V − 1 / 2 pr ⊥ ( V − 1 / 2 [1 np | P | F d ]) V − 1 / 2 T d (6) where V = I n ⊗ ( I p + θ J p ) which depends on θ only, and pr ⊥ A = I − A ( A ′ A ) − A ′ is a projection. If θ = ∞ (Hedayat and Yang (2005)) C d becomes the information matrix for the model with fixed subject effects ( β u is nonrandom.) If θ = 0 C d becomes the information matrix for the model without subject effects ( β u ≡ 0) A.S. Hedayat and Wei Zheng Optimal Crossover Designs

A Taste of Optimal Designs Motivation from Statistics Background Construction of Optimal Designs Results Further Problems The information matrix for ( τ 1 − τ 0 , τ 2 − τ 0 , ..., τ t − τ 0 ) ′ is � 0 1 × t � M d = T ′ C d T T = (7) where I t × t Thus, M d can be simply obtained from C d by deleting the first row and the first column of C d . � t τ i − τ 0 ) (i.e. min d Tr ( M − 1 i =1 Var ( � A-Optimal: min d d )) A.S. Hedayat and Wei Zheng Optimal Crossover Designs

A Taste of Optimal Designs Motivation from Statistics Background Construction of Optimal Designs Results Further Problems The information matrix for ( τ 1 − τ 0 , τ 2 − τ 0 , ..., τ t − τ 0 ) ′ is � 0 1 × t � M d = T ′ C d T T = (7) where I t × t Thus, M d can be simply obtained from C d by deleting the first row and the first column of C d . � t τ i − τ 0 ) (i.e. min d Tr ( M − 1 i =1 Var ( � A-Optimal: min d d )) MV-Optimal: min d max 1 ≤ i ≤ t Var ( � τ i − τ 0 ) Lemma An A-optimal design is also an MV-optimal design if its information matrix, M d , is a completely symmetric matrix. A.S. Hedayat and Wei Zheng Optimal Crossover Designs