1. Motivating Example Inferences for Ratios of Normal Means - PowerPoint PPT Presentation

1. Motivating Example Inferences for Ratios of Normal Means Multi-dose experiment including a positive control and placebo (mratios package) (Bauer et al. , 1998) Treatment n mean sd G. Dilba , F. Schaarschmidt , L. A. Hothorn Placebo 62 57.5 75.0 Active C. 59 67.3 90.1 Institute of Biostatistics, University of Hannover Dose 50 60 76.8 75.5 100 60 109.5 87.1 150 62 105.3 85.7 UseR! Conference Difference: H 0 i : µ i − µ 0 ≤ δ i Wien, 15 June 2006 Ratio: H 0 i : µ i /µ 0 ≤ ψ i 1 Merits of the ratio view: Inferences regarding ratios appear in a variety of problems: • Tests for non-inferiority (or superiority) • Easy to specify and interpret thresholds • Calibration • More powerful in some one-sided tests • Relative potency estimation • Comparability across different endpoints 2 3

2. Two-Sample Ratio Problems 3. Simultaneous Inferences: One-way Layout Y ij ∼ N ( µ i , σ 2 ) , Y 1 j ∼ N ( µ 1 , σ 2 Y 2 j ∼ N ( µ 2 , σ 2 i = 1 , . . . , k ; j = 1 , . . . , n i 1 ) , 2 ) , γ = µ 2 /µ 1 γ ℓ = c ′ ℓ µ ℓ µ , ℓ = 1 , . . . , r d ′ • Homogeneous Variances f = � k Distr.: Multivariate t with d 1 n i − k and Corr = R ( γ ) – Test – Confidence Interval for γ (Fieller, 1954) • Multiple Tests: H 0 ℓ : γ ℓ ≤ ψ ℓ versus H 1 ℓ : γ ℓ > ψ ℓ , ℓ = 1 , . . . , r • Heterogeneous Contrasts : User-defined, Dunnett, Tukey, Sequence, etc. – Test (Tamhane and Logan, 2004) • Simultaneous CI: Contrasts : User-defined, Dunnett, Tukey, Sequence, etc. – CI (Plug-in) Methods : Unadjusted, Bonferroni, Sidak, plug-in 4 5 4. General Linear Model Simultaneous CI: � � − B ℓ − � − B ℓ + � B 2 B 2 γ ℓ = c ′ ℓ − 4 A ℓ C ℓ ℓ − 4 A ℓ C ℓ ℓ η ℓ η : , , ℓ = 1 , 2 , . . . , r • Calibration: Y = β 0 + β 1 x + ǫ , γ = y 0 − β 0 d ′ 2 A ℓ 2 A ℓ β 1 where η = Vector of regression coeff . or means η ) 2 − q 2 S 2 d ′ ( d ′ • Multiple Assays (Jensen, 1989) ℓ � A ℓ = ℓ Md ℓ − 2 � ℓ Md ℓ ) � ( c ′ ℓ � η )( d ′ ℓ � η ) − q 2 S 2 c ′ B ℓ = η ) 2 − q 2 S 2 c ′ ( c ′ ℓ � – Parallel line assay C ℓ = ℓ Mc ℓ ( X ′ X ) − 1 M = Y ij = α i + βX ij + ǫ ij , i = 0 , 1 , . . . , m ; j = 1 , . . . , n i and Parameters: γ i = α i /α 0 , i = 1 , . . . , m  t 1 − α 2 ( ν ) , unadjusted   t 1 − α 2 r ( ν ) , Boole’s inequality – Slope ratio assay q = , ˇ c ′  1 − α ( I r ) Sid´ ak inequality  c ′ 1 − α ( R ( � Y ij = α + β i X ij + ǫ ij , i = 0 , 1 , . . . , m ; j = 1 , . . . , n i γ )) , plug-in Parameters: γ i = β i /β 0 , i = 1 , . . . , m (Dilba et al. , 2006) 6 7

5. Sample size calculation: Many-to-One References • Tests for non-inferiority (or superiority) Bauer, P., R¨ ohmel, J., Maurer, W. and Hothorn, L.A. (1998). Testing strategies in multi- dose experiments including active control. Statistics in Medicine 17, 2133-2146. Dilba, G., Bretz, F. and Guiard, V. (2006). Simultaneous confidence sets and confidence intervals for multiple ratios. Journal of Statistical Planning and Inference 136, 2640- • Sample size at LFC 2658. Fieller, E.C. (1954). Some problems in interval estimation. Journal of the Royal Statistical Society. Ser. B 16, 175-185. Jensen, D.R. (1989). Joint confidence sets in multiple dilution assays. Biometrical Journal The mratios package is available at 31 (7), 841-853. Tamhane, A.C. and Logan, B.R. (2004). Finding the maximum safe dose level for het- eroscedastic data. Journal of Biopharmaceutical Statistics 14, 843–856. http://www.bioinf.uni-hannover.de/software/ 8 9