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2 I (MCAR, MAR, MNAR) Little and Rubin (2002) MCAR, MAR, MNAR NRC(2010) MCAR MAR MNAR MCAR MAR MNAR MNAR MCAR MAR MNAR MAR

3 Estimand Estimand Estimand Estimand Efficacy effectiveness Mallinckrodt (2013) 6 estimand Estimand Estimand Estimand Estimand 4, Estimand Mallinckrodt et al. (2014) 3 estimand estimand Complete case LOCF BOCF Complete case LOCF BOCF Estimand LOCF BOCF Selection Model SM SM SM estimand MAR SM MAR SM 1 SM MAR SM 2 MMRM MNAR (outcome-dependent dropout) SM SM MNAR SM MNAR SM

4 Pattern-Mixture Model PMM CCMV NCMV ACMV PMM CCMV NCMV ACMV CCMV NCMV ACMV CCMV NCMV ACMV PMM (CCMV, NCMV, ACMV) NFMV PMM (NFMV) NFMV PMM SM PMM ( %PATTERNMIXTURE) Multiple Imuptation MI Multiple Imuptation Multiple Imuptation Rubin Multiple Imuptation Multiple Imputation Marginal treatment effect ( ) PMM pattern imputation Controlled imputation pattern imputation SAS pattern imputation Delta adjustment pattern imputation AE pattern imputation Shared Parameter Model SPM SPM %shared_parameter

5 Inverse Probability Weighted Complete-Case (IPWCC) Estimator µ IPWCC IPWCC estimators for Estimating Equations IPWCC weighted Generalized Estimating Equation (wgee) Doubly Robust (DR) II MMRM 98 9 MMRM MMRM SAS PROC MIXED CLASS MODEL LSMEANS REPEATED RANDOM MMRM MMRM Type (i) Type (ii) SM PMM SPM Estimand

6 11 Estimand Estimand Estimand Estimand Estimand Estimand Estimand Estimand Analytic Road Map

7 Ver Appendix A C Appendix 1, Appendix 2) Appendix A C (Ver1.0 Appendix) Appendix 1 Appendix Appendix 1 Appendix Appendix 1 Appendix 2 Web

8 I

9 1 1.1 III LOCF (Last Observation Carried Forward) LOCF ANCOVA (Tanaka et al., 2014) EMA "Guideline on Missing Data in Confirmatory Clinical Trials" National Research Council "The prevention and treatment of missing data in clinical trials" NRC (2010) EMA FDA JPMA, 2014b JPMA, 2014a; 2014b NRC (2010) estimand estimand estimand NRC (2010) estimand ICH E9 (R1) Final Concept Paper (ICH Steering Committee, 2014) PMDA FDA Lisa LaVange Web, 2015; LaVange, PMDA estimand

10 1. I estimand (1 8 ) 2. II LOCF ANCOVA MMRM (Mixed Model for Repeated Measures) (9 12 ) 3. (Appendix) (LMM) (Appendix A C) LOCF ANCOVA LOCF ANCOVA MMRM Estimand (Linear Model, LM) ANCOVA Linear Mixed Model, LMM LMM Appendix A NRC (2010) JPMA (2014a) Hughes et al. (2012) (2015) Missing At Random "Missing Not At Random (MNAR)" "Not Missing At Random (NMAR)" sensitivity analysis "MNAR" 1 1 MMRM 9 "Mixed" MMRM 1 9

11 1.3.4 (SM) SeM 1.1: ANCOVA ANalysis of COVAriance AR(1) 1 AutoRegressive(1) CS Compound Symmetry iid independent and identically distributed LM Linear Model LMM Linear Mixed Model LOCF MAR Last Observation Carried Forward Missing At Random MI Multiple Imputation MMRM MNAR Mixed Model for Repeated Measures Missing Not At Random PMM Pattern-Mixture Model pmi placebo Multiple Imputation REML REstricted Maximum Likelihood SM Selection Model SPM Shared Paremeter Model i j j j j = 1, 2, j = 1 1 j = 2 2 j = 3 4 i = 1,, N; j = 1,, n 1 n i j M ij 1 0 R ij 10

12 i M i, R i 2 i +1 D i n D i = R ij + 1 j=1 D i n n = SAS SAS Proc MI pmi NRC (2010) NAS NRC MNAR Missingdata.org.uk ( SAS ( ( ) Appendix C [1]. (2015). PMDA. NAS M i1 0 R i1 M i = M i2 M i3 = 0 0, R i = R i2 R i3 = M i4 1 R i

13 EMA estimand ( [2] European Medicines Agency (2010). Guideline on missing data in confirmatory clinical trials. ( [3] Hughes, S., Harris, J., Flack, N., and Cuffe, R. L. (2012). The statistician s role in the prevention of missing data. Pharmaceutical statistics, 11(5), [4] ICH Steering Committee (2014). Final Concept Paper E9(R1): Addendum to Statistical Principles for Clinical Trials dated 22 October [5] JPMA (2014a). - NAS EMA estimand -.. ( [6] JPMA (2014b). - NAS EMA estimand -.. ( [7] LaVange, L. (2015). Missing Data Issues in Regulatory Clinical Trials. NAS EMA estimand ( [8] (2015). 2. NAS EMA estimand ( [9] National Research Council. (2010). The prevention and treatment of missing data in clinical trials. National Academy Press. [10] (2015).. NAS EMA estimand ( [11] Tanaka, S., Fukinbara, S., Tsuchiya, S., Suganami, H., & Ito, Y. M. (2014). Current Practice in Japan for the Prevention and Treatment of Missing Data in Confirmatory Clinical Trials A Survey of Japanese and Foreign Pharmaceutical Manufacturers. Therapeutic Innovation & Regulatory Science, 48(6),

14 NRC (2010) 1 (p8) (missing data) QOL (missing outcome) R ij i j M ij 0 1 R ij

15 i (i = 1,, N) j (j = 1,, n) Y ij i j Y i i Yi o i Yi m i R i i (Y o i, R i) 4 3 i Y i1 1 Y i1 Y i2 ( ) Y i =, Yi o = Y Y i2, 1 Ym i = Y i4, R i = i3 1 Y i3 0 Y i4 (Y o i, R i) (full data 4 ) (Y i, R i ) (observed data) (Y o i, R i) 4 Y i4 R i4 Y i4 0 R i Little and Rubin (2002) R i 0, 1 R i = (1, 1, 1, 0) R i = (1, 0, 0, 1) "complete data" 14

16 (dropout 5 ) 1 (monotone missing) R i = (1, 1, 1, 0), (1, 1, 0, 0), (1, 0, 0, 0) R ij (non-monotone missing) R i = (1, 1, 0, 1), (1, 0, 1, 0) R ij estimand (MCAR, MAR, MNAR) (Y i, R i ) f(y i, R i θ, ψ) = f(y i θ) f(r i Y i, ψ) f(r i Y i, ψ) R i Y i θ ψ Little and Rubin (2002) MCAR, MAR, MNAR MCAR (Missing Completely At Random) MAR (Missing At Random) MNAR (Missing Not At Random) 3 Little and Rubin (2002) 6 (MCAR) : f(r i Y i, ψ) = f(r i ψ) (MAR) : f(r i Y i, ψ) = f(r i Y o i, ψ) (2.1) (MNAR) : f(r i Y i, ψ) f(r i Y o i, ψ) MAR MCAR MNAR MAR 7 6 Pattern-Mixture Model MNAR MAR 2 5 withdrawal attrition 6 Little and Rubin (2002) MNAR NMAR(Not Missing At Random) R i M i 7 (2.1) MNAR MAR 15

17 Little (2008) X i b i (MCAR) : f(r i X i, Y i, b i, ψ) = f(r i X i, ψ) (MAR) : f(r i X i, Y i, b i, ψ) = f(r i X i, Y o i, ψ) (2.2) (MNAR) : f(r i X i, Y i, b i, ψ) f(r i X i, Y o i, ψ) NRC(2010) MCAR MAR MNAR NRC (2010) 2 X 8 V MCAR f(r i X i, V i, Y i ) = f(r i ) (2.3) 9 MAR f(r i X i, V i, Y i ) = f(r i X i, Vi o, Yi o ) Vi o V i MNAR MAR MCAR MAR MNAR (2.1) 10 (MCAR) : f(r i Y i, ψ) = f(r i ψ) (MAR) : f(r i Y i, ψ) = f(r i Yi o, ψ) (MNAR) : f(r i Y i, ψ) f(r i Yi o, ψ) 3 Y i = (Y i1, Y i2, Y i3 ) R i = (R i1, R i2, R i3 ) MCAR MAR MNAR Appendix B 8 9 (2.2) MCAR (2.2) (2.3) 10 Seaman et al. (2013) everywhere MCAR everywhere MAR 16

18 MCAR MCAR 1(j = 1) Y i1 1 Y i 0.1 f(r i1 = 0 Y i ) = 0 (2.4) f(r i2 = 0 Y i, R i1 = 0) = 1 (2.5) f(r i2 = 0 Y i, R i1 = 1) = 0.1 f(r i3 = 0 Y i, R i1 = 0, R i2 = 0) = 1 f(r i3 = 0 Y i, R i1 = 1, R i2 = 0) = 1 (2.6) f(r i3 = 0 Y i, R i1 = 1, R i2 = 1) = 0.1 i j R i1 = = R ij = 0 R i,j+1 = f(r i3 = 0 Y i, R i1 = 0, R i2 = 1) = 1 f(r i3 = 1 Y i, R i1 = 0, R i2 = 1) = MAR MAR j Y ij (j + 1) Y i,j+1 Y ij Y i,j 1 Y i,j+1 f(r i1 = 0 Y i ) = 0 (2.7) f(r i2 = 0 Y i, R i1 = 0) = 1 logitf(r i2 = 0 Y i, R i1 = 1) = Y i1 (2.8) f(r i3 = 0 Y i, R i1 = 0, R i2 = 0) = 1 f(r i3 = 0 Y i, R i1 = 1, R i2 = 0) = 1 (2.9) logitf(r i3 = 0 Y i, R i1 = 1, R i2 = 1) = Y i2 ( ) f(x) logitf(x) = log exp (logitf(x)) = f(x) 1 f(x) 1 f(x) f(x) = exp(logitf(x)) 1 + exp(logitf(x)) (2.8) (2.9) 1 logitf(r i2 = 0 Y i, R i1 = 1) = Y i1 f(r i2 = 0 Y i1, R i1 = 1) = exp( Y i1) 1 + exp ( Y i1 ) 17

19 logitf(r i3 = 0 Y i, R i1 = 1, R i2 = 1) = Y i2 f(r i3 = 0 Y i1, R i1 = 1, R i2 = 1) = exp( Y i2) 1 + exp ( Y i2 ) f(r i3 = 0 Y i, R i1 = 0, R i2 = 1) = 1 f(r i3 = 1 Y i, R i1 = 0, R i2 = 1) = 0 MAR MNAR 1 MNAR 2 1 f(r i1 = 0 Y i ) = 0 (2.10) f(r i2 = 0 Y i, R i1 = 0) = 1 (2.11) logitf(r i2 = 0 Y i, R i1 = 1) = Y i Y i2 f(r i3 = 0 Y i, R i1 = 0, R i2 = 0) = 1 f(r i3 = 0 Y i, R i1 = 1, R i2 = 0) = 1 (2.12) logitf(r i3 = 0 Y i, R i1 = 1, R i2 = 1) = Y i Y i3 f(r i1 = 0 Y i ) = 0 f(r i2 = 0 Y i, R i1 = 0) = 1 f(r i2 = 0 Y i, R i1 = 1) = exp( Y i Y i2 ) 1 + exp( Y i Y i2 ) f(r i3 = 0 Y i, R i1 = 0, R i2 = 0) = 1 f(r i3 = 0 Y i, R i1 = 1, R i2 = 0) = 1 f(r i3 = 0 Y i, R i1 = 1, R i2 = 1) = exp( Y i Y i3 ) 1 + exp( Y i Y i3 ) f(r i3 = 0 Y i, R i1 = 0, R i2 = 1) = 1 f(r i3 = 1 Y i, R i1 = 0, R i2 = 1) = 0 MNAR MNAR 2 MNAR 2 MAR 11 f(r i1 = 0 Y i ) =

20 logitf(r i2 = 0 Y i, R i1 = 0) = Y i1 logitf(r i2 = 0 Y i, R i1 = 1) = Y i1 logitf(r i3 = 0 Y i, R i1 = 0, R i2 = 0) = Y i2 logitf(r i3 = 0 Y i, R i1 = 1, R i2 = 0) logitf(r i3 = 0 Y i, R i1 = 0, R i2 = 1) logitf(r i3 = 0 Y i, R i1 = 1, R i2 = 1) = Y i2 = Y i2 = Y i2 f(r i1 = 0 Y i ) = 0 logitf(r i2 = 0 R i1 Y i ) = Y i1 logitf(r i3 = 0 R i1, R i2, Y i ) = Y i2 f(r i1 = 0 Y i ) = 0 f(r i2 = 0 R i1, Y i ) = f(r i3 = 0 R i1, R i2, Y i ) = exp( Y i1 ) 1 + exp( Y i1 ) exp( Y i2 ) 1 + exp( Y i2 ) MNAR MCAR MAR MNAR MCAR MAR MNAR 1 MCAR MAR 12 MNAR MCAR MAR MNAR 13 MCAR MCAR 1 MAR MCAR MAR 1 MNAR MNAR 2 MAR MNAR Y m (Y o, X, R) MAR MNAR 12 MAR MAR 13 19

21 MAR MNAR MAR MAR MAR (Mallinckrodt et al., 2008) MAR MNAR MNAR Verbeke and Molenberghs(2000) Local Influence MNAR

22 2 7 全データ ( 欠測データ含む ) 応答変数 時点 2.1: MCAR 観測データ ( M C A R ) 応答変数 時点 2.2: MCAR 2.3, 完了例 ( M C A R ) 2 7 中止例 ( M C A R ) 応答変数 応答変数 時点 時点 2.3: MCAR 2.4: MCAR MCAR MCAR

23 MCAR MCAR MCAR 2.1 MAR 観測データ ( M A R ) 応答変数 時点 2.5: MAR 完了例 ( M A R ) 2 7 中止例 ( M A R ) 応答変数 応答変数 時点 時点 2.6: MAR 2.7: MAR 22

24 MNAR 観測データ ( M N A R ) 応答変数 時点 2.8: MNAR 完了例 ( M N A R ) 2 7 中止例 ( M N A R ) 応答変数 応答変数 時点 時点 2.9: MNAR 2.10: MNAR MAR MNAR MAR MAR MAR MAR 16 MAR MNAR 1 MAR MNAR MAR 23

25 2.4.1 Molenberghs and Kenward (2007) Y i = (Yi o, Ym i ) 17 R i R i Y i 18 R i Y i Y i R i (full-data 19 )(Y i, R i ) (Y i, R i ) (full data likelihood) N N L (θ, ψ Y i ) f(y i, R i X i, θ, ψ) = f(yi o, Yi m, R i X i, θ, ψ) i=1 i=1 Yi m Yi m θ, ψ Y i m 20 observed data likelihood) L(θ, ψ Yi o ) L (θ, ψ Y i )dyi m i=1 N i=1 f(y i, R i X i, θ, ψ)dy m i N = f(yi o, Yi m, R i X i, θ, ψ)dyi m 21 (Yi o, R i) θ, ψ Little and Rubin (2002) 2 (missing-data mechanism is ignorable) (i) MAR (ii) θ ψ θ ψ (θ, ψ) Ω θ,ψ θ Ω θ ψ Ω ψ θ ψ f(y i, R i X i, θ, ψ) = f(y i X i, θ) f(r i X i, ψ) 17 Y i Y i = ((Yi o), (Yi m) ) 18 MCAR MCAR 19 "complete-data" 20 (Yi o, R i) 21 Little and Rubin (2002) "full likelihood" 24

26 θ ψ f(y i X i, θ) = ( 1 (2πσ2 ) exp 1 ) 3 2σ 2 (Y i µ) (Y i µ) (µ = (µ 1, µ 2, µ 3 ) ) θ = (µ 1, µ 2, µ 3, σ 2 ) f(r i, Y i, ψ) MAR f(r i1 = 0 Y i, ψ) = 0 f(r i2 = 0 Y i, R i1 = 0, ψ) = 1 f(r i2 = 0 Y i, R i1 = 1, ψ) = exp(ψ 1 + ψ 2 Y i1 ) 1 + exp (ψ 1 + ψ 2 Y i1 ) f(r i3 = 0 Y i, R i1 = 0, R i2 = 0, ψ) = 1 f(r i3 = 0 Y i, R i1 = 1, R i2 = 0, ψ) = 1 f(r i3 = 0 Y i, R i1 = 1, R i2 = 1, ψ) = exp(ψ 1 + ψ 2 Y i2 ) 1 + exp (ψ 1 + ψ 2 Y i2 ) ψ = (ψ 1, ψ 2 ) θ ψ 22 Y ij Y ij 23 θ = (µ 1, µ 2, µ 3, σ 2 ) ψ f(r i Y i, ψ) f(r i1 = 0 Y i, ψ) = 0 f(r i2 = 0 Y i, R i1 = 0, ψ) = 1 ( ) exp ψ 1 + ψ 2 Yi1 µ 1 f(r i2 = 0 Y i, R i1 = 1, ψ) σ = ( ) 1 + exp ψ 1 + ψ 2 Yi1 µ 1 σ 22 µ 1 Ω θ = µ 2 µ 3 R4 µ 1, µ 2, µ 3 R, σ 2 > 0 = R 3 R >0 σ 2 { ( ) } ψ1 Ω ψ = R 2 ψ 1, ψ 2 R = R 2 ψ 2 µ 1 µ 2 µ 3 Ω (θ,ψ) = σ 2 R 4 µ 1, µ 2, µ 3 R, σ 2 > 0, ψ 1, ψ 2 R = (R 3 R >0 ) R 2 ψ 1 ψ 2 Ω (θ,ψ) = Ω θ Ω ψ 23 25

27 f(r i3 = 0 Y i, R i1 = 0, R i2 = 0, ψ) = 1 f(r i3 = 0 Y i, R i1 = 1, R i2 = 0, ψ) = 1 ( ) exp ψ 1 + ψ 2 Yi2 µ 2 f(r i3 = 0 Y i, R i1 = 1, R i2 = 1, ψ) σ = ( ) 1 + exp ψ 1 + ψ 2 Yi2 µ 2 σ ψ = (ψ 1, ψ 2, µ 1, µ 2, σ) ψ θ = (µ 1, µ 2, µ 3, σ 2 ) (µ 1, µ 2 ) θ (σ 2 ) σ θ ψ 24 θ f(r i Y i ) f(r i Y i ) θ f(r i Y i ) Selection Model f(yi o, Ym i θ) Yi m L(θ Yi o ) = f(yi o, Yi m θ)dyi m θ L(θ Yi o) L(θ, ψ Yo i ) R i R i L(θ, ψ Yi o) L(θ Yi o) 5 Little and Rubin (2002) (2002) 2.5 ( PGB) PGB (PLB ) Ω θ = R 3 R >0, Ω ψ = R 4 R >0 ( θ ψ µ 1 µ 2 µ 3 ) σ 2 = ψ 1 ψ 2 µ 1 µ 2 σ Ω (θ, = ψ) x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 R 9 x 1 = x 7, x 2 = x 8, x 4 = x 2 9, x 4, x 9 > 0 Ω (θ, ψ) Ω θ Ω ψ 26

28 2.1: Visit PLB (N = 67) PGB (N = 69) SD SD BL PLB PGB 2.11: 12 PGB 46 (66.7%) PLB 34 (50.7%) 2.2 PGB PLB ( LOCF ) PLB [95%CI] (95%CI: [ 2.150, 0.916]) ( 2.3) 27

29 2.2: PLB PGB (55.2%) 49 (70.0%) (ITT) 67 (100.0%) 69 (98.6%) 30 (44.8%) 21 (30.0%) 9 (13.4%) 15 (21.4%) 20 (29.9%) 5 (7.1%) 1 (1.5%) 1 (1.4%) 2.3: (ITT) [PGB PLB ] a) PLB PGB 95%CI (SE) p N (SD) (0.146) (1.253) N (SE) (0.235) (0.231) [ 2.150, 0.916] < (0.312) a) a) MAR MAR MNAR [1]. (2002)... [2] Little, R. J. (2008). Selection and Pattern-Mixture modes. Cphaiter 18 in Advances in longitudinal data analaysis

30 [3] Little, R. J. A. and Rubin, D. B. (2002). Statistical analysis with missing data. 2nd edition. John Wiley & Sons. New York. [4] Mallinckrodt, C. H., Lane, P. W., Schnell, D., Peng, Y. and Maucuso, J. P. (2008). Recommendations for the primary analysis of continuous endpoints in longitudinal clinical trials. Drug Information Journal, 42, [5] Molenberghs, G., and Kenward, M. (2007). Missing Data in Clinical Studies. Wiley. [6] National Research Council. (2010). The prevention and treatment of missing data in clinical trials. National Academy Press. [7]. ( [8] Seaman, S., Galati, J., Jackson, D., and Carlin, J. (2013). What Is Meant by Missing at Random?. Statistical Science, 28(2), [9] 1991)... [10] Verbeke, G, and Molenberghs, G. (2000). Linear mixed models for longitudinal data. Springer. 29

31 3 Estimand 3.1 NRC (2010) 2 Mallinckrodt (2013) 3 11 Mallinckrodt et al. (2014) ICH Steering Committee (2014) estimand ICH E9 (R1) Estimand Estimand NRC (2010) 2 Recommendation 1 (p26) (a) (b) (c) (d) (the measures of intervention effects) causal estimand Mallicnkrodt (2013) 3 (estimand ) estimand estimand estimand Estimand Mallinckrodt (2013) NRC (2010) estimand NRC (2010) 2 (p22) estimand Mallinckrodt (2013) 3 "What is being estimated" and/or rescue medication ICH E9 (R1) final concept paper (ICH Steering Committee, 2014) "Statement of the Perceived Problem" (p1) (property) rescue medication Estimand Estimand ICH E9 (R1) final concept paper "Issues to be Resolved" (p2) Estimand estimand 30

32 estimand NRC (2010) 2 estimand NRC (2010) 2 Mallinckrodt (2013) 3 estimand estimand 3.3 Efficacy effectiveness estimand 2 Mallinckrodt (2013) 3 efficacy effectiveness 2 Efficacy per-protocol estimand (PPS estimand) "de jure estimand" effectiveness intention-to-treat estimand (ITT estimand) "de facto estimand" rescue medication Mallinckrodt (2013) estimand 1 estimand 6 Mallinckrodt et al. (2013) POC study efficacy effectiveness O Neill and Temple (2012) ITT outcome study 1 studies of symptom Mallinckrodt (2013) 6 estimand Mallinckrodt (2013) (Major Depressive Disorder, MDD) 6 estimand 6 5 NRC (2010) NRC (2010) 2 Mallinckrodt (2013) Estimand 1 Estimand 1 1 OS PFS 2 31

33 estimand effectiveness rescue medication Mallinckrodt et al. (2012) estimand 1 estimand Estimand 2 Estimand 2 estimand run-in phase randomized withdrawal design efficacy NRC (2010) wash-out (2 p24) run-in phase (2 p33) (2 p33) Estimand 3 Estimand 3 estimand efficacy estimand estimand NRC (2010) estimand (available and effective mechanism for ensuring adherence) (2 p25) (rarely used) (2 p24) Mallinckrodt (2013) (14 ) estimand estimand estimand estimand 6 Mallinckrodt et al. (2012) estimand 3 Hypothetical estimand effectiveness Estimand 4, 5 Estimand 4 Area Under the Curve (AUC) Estimand 5 32

34 Estimand 4, 5 effectiveness estimand 4 estimand 5 estimand NRC (2010) 2 responder rate non-responder estimand 5 (p32) Mallinckrodt (2013) 3 estimand (p20) Estimand 6 Estimand 6 estimand effectiveness estimand 1 estimand 6 Estimand 6 rescue medication rescue medication placebo Multiple Imputaion (pmi) 3 BOCF NRC (2010) Reccomendataion 10 (p77) (single imputation) 4 Estimand 3 estimand 3 estimand Mallinckrodt et al. (2014) 3 estimand Mallinckrodt et al. (2014) 3 estimand 6 3 Estimand A Estimand 1 Estimand B Estimand 3 Estimand C Estimand estimand Leuchs et al. (2015) estimand effectiveness efficacy 3 pmi 6 33

35 3.1: Mallinckrodt (2013) MDD 6 estimand Estimand rescue medication 1 effectiveness 2 efficacy 3 efficacy 4 effectiveness 5 effectiveness 6 effectiveness 3.7 estimand Lehmann and Casella (1998) estimand NRC (2010) Rubin (1996) "scientific estimand" NRC (2010) ICH Steering Committee (2014) Mallinckrodt et al. (2012) Mallinckrodt (2013) Mallinckrodt et al. (2014) FDA O Neill and Temple (2012) Soon (2009) PMDA FDA CDER (Center for Drug Evaluation and Research) Lisa LaVange (, 2015; LaVange, 2015) Leuchs et al. (2015) BfArM (Federal Institute for Drugs and Medical Devices) Holzhauer et al. (2015) rescue medication estimand [1]. (2015). PMDA. NAS EMA estimand ( 34

36 [2] Holzhauer, B., Akacha, M., & Bermann, G. (2015). Choice of estimand and analysis methods in diabetes trials with rescue medication. Pharmaceutical statistics. [3] ICH Steering Committee (2014). Final Concept Paper E9(R1): Addendum to Statistical Principles for Clinical Trials dated 22 October [4] LaVange, L. (2015). Missing Data Issues in Regulatory Clinical Trials. NAS EMA estimand ( [5] Lehmann, E. L., & Casella, G. (1998). Theory of point estimation. Springer Science & Business Media. [6] Leuchs, A. K., Zinserling, J., Brandt, A., Wirtz, D., & Benda, N. (2015). Choosing Appropriate Estimands in Clinical Trials. Therapeutic Innovation & Regulatory Science, [7] Mallinckrodt, C. H. (2013). Preventing and treating missing data in longitudinal clinical trials. Cambridge University Press. [8] Mallinckrodt, C. H., Lin, Q., Lipkovich, I., & Molenberghs, G. (2012). A structured approach to choosing estimands and estimators in longitudinal clinical trials. Pharmaceutical statistics, 11(6), [9] Mallinckrodt, C., Roger, J., Chuang-Stein, C., Molenberghs, G., Lane, P. W., O Kelly, M., Ratitch, B., Xu, L., Gilbert, S., Mehrotra, D. V., Wolfinger, R., & Thijs, H. (2013). Missing data: turning guidance into action. Statistics in Biopharmaceutical Research, 5(4), [10] Mallinckrodt, C. H, Roger, J., Chuang-Stein, G., Molenberghs, G., O Kelly, M., Ratitch, B., Janssens, M., and Bunouf, P. (2014). Recent Developments in the Prevention and Treatment of Missing Data. Therapeutic Innovation & Regulatory Science [11] National Research Council. (2010). The prevention and treatment of missing data in clinical trials. National Academy Press. [12] O Neill, R. T., & Temple, R. (2012). The prevention and treatment of missing data in clinical trials: an FDA perspective on the importance of dealing with it. Clinical Pharmacology & Therapeutics, 91(3), [13] Soon, G. G. (2009). Minimizing missing data in clinical trials: Design, operation, and regulatory considerations. Drug Information Journal, 43(4), [14] Rubin, D. B. (1996). Multiple imputation after 18+ years. Journal of the American statistical Association, 91(434),

37 4 Complete case LOCF BOCF 4.1 Complete case LOCF BOCF 3 1. ( ) complete case 2. LOCF (Last Observation Carried Forward) BOCF (Baseline Observation Carried Forward) 3. (GEE ) 9 MMRM (Mixed-effect Models for Repeated Measures) complete case NRC (2010) EMA (2010) Kenward and Molenberghs (2009) LOCF BOCF 4.2 Complete case complete case Complete case MCAR 2 complete case complete case 36

38 4.3 complete case complete case LOCF BOCF NRC (2010) single imputation method ( ) NRC (2010, p77) LOCF BOCF LOCF LOCF NRC (2010, p65-66) LOCF LOCF LOCF complete case ( ) LOCF Molenberghs et al. (2004) LOCF Kenward and Molenberghs (2009) ( ) LOCF LOCF ( ) LOCF LOCF 12 NRC (2010, p65-66) LOCF MCAR MAR LOCF MNAR n n MAR i P (Y n X, Y o, R = 1) = P (Y n X, Y o, R = 0) (4.1) 37

39 R n R = 0 R = 1 (4.1) X Y o = (Y 1,..., Y n 1 ) Y n LOCF Y n 1 Y n 1 P (Y n = Y n 1 X, Y o ) = 1 P (Y n X, Y o, R = 0) P (Y n X, Y o, R = 1) LOCF MCAR MAR NRC (2010) Kenward and Molenberghs (2009) EMA (2010) ( ) LOCF BOCF LOCF BOCF 0 BOCF BOCF EMA (2010) LOCF BOCF BOCF LOCF BOCF LOCF BOCF Estimand LOCF BOCF LOCF BOCF estimand NRC (2010) Ayele et al. (2014) LOCF BOCF efficacy effectiveness estimand LOCF efficacy BOCF effectiveness Ayele et al. (2014) estimand 3 LOCF estimand 6 BOCF estimand LOCF BOCF efficacy effectiveness complete case LOCF BOCF 38

40 Multiple Imputation (MI) MI (M ) LOCF BOCF MAR M M M 1 Pattern Mixture Model (PMM) LOCF BOCF 2 1 MAR MCAR 2 ( ) NRC (2010) EMA (2010) MI MI 6 [1] Ayele, B. T., Lipkovich, I., Molenberghs, G., and Mallinckrodt, C. H. (2014). A Multiple-Imputation-Based Approach to Sensitivity Analyses and Effectiveness Assessments in Longitudinal Clinical Trials. Journal of biopharmaceutical statistics, 24(2), [2] European Medicines Agency. Committee for Medicinal Products for Human Use (CHMP). Guideline on missing data in confirmatory clinical trials. EMA/CPMP/EWP/1776/99 Rev Published July 2, [3] Kenward, M. G., and Molenberghs, G. (2009). Last observation carried forward: a crystal ball?. Journal of biopharmaceutical statistics, 19(5), [4] Molenberghs, G., Thijs, H., Jansen, I., Beunckens, C., Kenward, M. G., Mallinckrodt, C., and Carroll, R. J. (2004). Analyzing incomplete longitudinal clinical trial data. Biostatistics, 5(3), [5] National Research Council. The Prevention and Treatment of Missing Data in Clinical Trials. Panel on Handling Missing Data in Clinical Trials. Committee on National Statistics, Division of Behavioral and Social Sciences and Education. Washington, DC: The National Academies Press;

41 5 Selection Model 5.1 Selection Model (SM) SM 1 MMRM MAR SM 1 MAR SM MMRM MMRM MMRM MAR SM MMRM MNAR SM 5.2 SM i (i = 1,, N) j (j = 1,, n) Y ij i j Y i i Yi o i Yi m i R i i (full data 1 ) (Y i, R i ) (observed data) (Y o i, R i) SM SM (Little and Rubin, 2002) f(y i, R i θ, ψ) = f(y i θ) f(r i Y i, ψ) Y i f(y i θ) R i R i Y i θ ψ θ ψ 2 1 "complete data"

42 X i b i θ ) θ = ( θ1 b i X i Y i f(y i X i, b i, θ 1 ) X i b i f(b i X i, θ 2 ) f(y i X i, θ) = f(y i, X i, b i, θ 1 ) f(b i X i, θ 2 )db i SM f(r i, Y i, b i X i, θ, ψ) = f(y i X i, b i, θ 1 ) f(r i X i, Y i, b i, ψ) f(b i X i, θ 2 ) (5.1) 2 3 θ SM estimand SM estimand efficacy 3 Mallinckrodt 6 estimand 2, MAR SM SM MAR MNAR MAR MAR SM 1 SM MAR θ R i f(r i X i, Y i, b i, ψ) = f(r i X i, Yi o, ψ) 3 ( θ1 θ = θ 2 41 )

43 ( f(y i, R i, b i X i, θ, ψ) = f(y i X i, b i, θ 1 ) f(r i X i, Y o i, ψ) f(b i X i, θ 2 ) (5.2) f(yi o, R i X i, θ, ψ) = f(y i, R i, b i X i, θ, ψ)db i dyi m = f(y i X i, b i, θ 1 ) f(r i X i, Yi o, ψ) f(b i X i, θ 2 )db i dyi m ( (5.2)) = f(r i X i, Yi o, ψ) f(y i X i, b i, θ 1 ) f(b i X i, θ 2 )db i dyi m = f(r i X i, Yi o, ψ) f(yi o, Yi m X i, θ 1, θ 2 )dyi m = f(r i X i, Y o i, ψ) f(y o i X i, θ) log {f(y o i, R i X i, θ, ψ)} = log{f(r i X i, Y o i, ψ) f(y o i X i, θ)} = log{f(r i X i, Y o i, ψ)} + log{f(y o i X i, θ)} θ f(r i X i, Yi o, ψ) θ 0 θ Y i Y i 1 Y i SAS PROC MIXED Y i SM f(y i X i, b i ) = f(yi o, Ym i X i, b i ) 4 IPW GEE wgee Doubly Robust MAR SM 2 MMRM MAR SM MMRM (Mixed Model for Repeated Measures) MMRM 9 "MMRM" Mallinckrodt (2013) 7.2 "MMRM" Direct Likelihood (DL) 4 Type (i) 5 NRC (2010) p73 5 SM 42

44 MMRM SM Appendix A MAR SM MMRM MAR SM MAR SM 6 MMRM 3 σ 2 Y i = X i β + b i ϵ i (b i iid N(0, ν 2 ), ϵ i iid N(0, σ 2 I 3 ), b i ϵ i ) V [Y i ] = V [b i ϵ i ] = 1 3 V [b i ]1 3 + V [ϵ i ] = 1 3 ν σ 2 I = ν σ ν 2 + σ 2 ν 2 ν 2 = ν 2 ν 2 + σ 2 ν 2 ν 2 ν 2 ν 2 + σ 2 (unstructured) MMRM MMRM MMRM (unstructured) 3 unstructured 0 σ1 2 σ 12 σ 13 iid Y i = X i β + b i ϵ i b i N(0, ν 2 iid ), ϵ i N 0, σ 12 σ2 2 σ 23 0 σ 13 σ 23 σ3 2 V [Y i ] = V [b i ϵ i ] (5.3) = 1 3 V [b i ]1 3 + V [ϵ i ] (5.4) σ1 2 σ 12 σ 13 = 1 3 ν σ 12 σ2 2 σ 23 σ 13 σ 23 σ

45 1 1 1 σ1 2 σ 12 σ 13 = ν σ 12 σ 2 2 σ σ 13 σ 23 σ3 2 ν 2 + σ1 2 ν 2 + σ 12 ν 2 + σ 13 = ν 2 + σ 12 ν 2 + σ2 2 ν 2 + σ 23 (5.5) ν 2 + σ 13 ν 2 + σ 23 ν 2 + σ 2 3 ν 2 ν 2, σ 2 1, σ 12, 1 7 unstructured unstructured MMRM Mallinckrodt et al. (2001) MMRM unstructured 1 unstructured ) ν 2 + σ 2 1 ν 2 + σ 12 1 Mallinckrodt (2013) 7.4 unstructured Mallinckrodt et al. (2008) Mallinckrodt et al. (2004) α MMRM Compound Symmetry (parsimonious) unstructured unstructured 1 7 (identifiablity) ν2 + σ 2 1 ν 2 + σ 12 ν 2 + σ 13 ν 2 + σ 12 ν 2 + σ 2 2 ν 2 + σ 23 ν 2 + σ 13 ν 2 + σ 23 ν 2 + σ ν 2 = 0.8 σ 2 1 = σ2 2 = σ2 3 = 0.2, σ 12 = σ 23 = 0.1, σ 13 = 0 2. ν 2 = 0.5 σ 2 1 = σ2 2 = σ2 3 = 0.5, σ 12 = σ 23 = 0.4, σ 13 = 0.3 = 0 < ν 2 <

46 5.3.3 MNAR (outcome-dependent dropout) SM MNAR Little (2008) outcome-dependent dropout 9 f(r i X i, Y i, b i, ψ) = f(r i X i, Y i, ψ) f(y i, R i, b i X i, θ, ψ) = f(y i X i, b i, θ 1 ) f(r i X i, Y i, ψ) f(b i X i, θ 2 ) b i f(y i, R i X i, θ, ψ) = f(y i X i, b i, θ 1 ) f(r i X i, Y i, ψ) f(b i X i, θ 2 )db i = f(y i X i, θ) f(r i X i, Y i, ψ) (5.6) (5.6) b i f(r i X i, Y i, ψ) 10 f(yi o, R i X i, θ, ψ) = f(yi o, Yi m X i, θ) f(r i X i, Yi o, Yi m, ψ)dyi m (5.7) MAR Diggle and Kenward (1994) R i R ij = 1 R ij = 0 logit{p r(r ij = 0 R i1 = 1,, R i,j 1 = 1, Y i, X i, ψ)} = ψ 1 + ψ 2 Y i,j 1 + ψ 3 Y ij (j = 2,, n) (5.8) P r(r ij = 0 R i,j 1 = 0, Y i, X i, ψ) = 1 (j = 2,, n) (5.9) P r(r i1 = 1 Y i, X i, ψ) = 1 (5.10) 11 ψ = (ψ 1, ψ 2, ψ 3 ) (5.8) i j Y ij 1 Y i,j 1 Y ij 2 ψ 3 = 0 Y i,j 1 MAR (5.9) (5.10) (j = 1) 9 NRC (2010) 5 IPW 10 f(y i X i, b i, θ 1 ) f(y i X i, θ) Y i = X i β + Z i b i + ϵ i, b i iid N(0, D), ϵ i iid N(0, Σ) {b i } {ϵ i } f(y i X i, b i, θ 1 ) = Y i b i N(X i β + Z i b i, Σ) f(y i X i, θ) = Y i N(X i β, Z i DZ i + Σ) θ 2 b i D Y i b i Y i 11 X i X i 45

47 1. logistic 2. j j, (j 1) 2 Y 3. j 1 Y 2 1 Y i1 4 3 Y i3 MNAR MNAR NFD (Non Future dependence) NFD 6 Appendix B Diggle and Kenward (1994) Nelder-Mead Nelder and Mead Nelder and Mead (1965a, 1965b) 5.4 SM II MMRM 10 NRC (2010) 5 Type (i) untestable Type (ii) testable 2 Type (ii) Type (i) Type (i) MAR MMRM MAR Type (i), MNAR SM MNAR SM MNAR SM MNAR SM MMRM Diggle and Kenward (1994) logit{p r(r ij = 0 R i1 = 1,, R i,j 1 = 1, Y i, X i, ψ)} = ψ 1 + ψ 2 Y i,j 1 + ψ 3 Y ij (j = 2,, n) (5.11) P r(r ij = 0 R i,j 1 = 0, R i,j 2,, R i1, Y i, X i, ψ) = 1 (j = 2,, n) P r(r i1 = 1 Y i, X i, ψ) = 1 R i,j 2, R i1 ψ 1, ψ 2, ψ 3 ψ 3 = 0 MAR ψ 3 0 MNAR ψ 3 MAR MNAR ψ 3 MAR Verbeke and Molenberghs (2000) H 0 : ψ 3 = 0, H 1 : ψ 3 0 MAR MNAR 12 MAR 12 46

48 MNAR SM 2 2 Type (i) 1 MAR Diggle and Kenward (1994) logit{p r(r ij = 0 R i1 = 1,, R i,j 1 = 1, Y i, X i, ψ)} = ψ 1 + ψ 2 Y i,j 1 + ψ 3 Y ij (j = 2,, n) (5.12) P r(r ij = 0 R i,j 1 = 0, R i,j 2,, R i1, Y i, X i, ψ) = 1 (j = 2,, n) P r(r i1 = 1 Y i, X i, ψ) = 1 R i,j 2, R i1 MAR MNAR ψ 3 ψ 3 MAR ψ 3 MAR ψ 3 MAR NRC (2010) 5 (sensitivity parameter) Delta adjustment 13 (i) ψ (ii) (i) ψ 3 1 MNAR SM (iii) (i) ψ 3 (ii) logistic MNAR MNAR 14 ψ 3 ψ 3 SM MMRM SM MMRM 5.5 Missingdata.org.uk SM SM 1 2 Appendix C 2 1 logit{p r(r ij = 0 R i1 = 1,, R i,j 1 = 1, Y i, X i, ψ)} = ψ 1 + ψ 3 Y i,j 1 + ψ 5 Y ij 2 logit{p r(r ij = 0 R i1 = 1,, R i,j 1 = 1, Y i, X i, ψ)} = ψ 2 + ψ 4 Y i,j 1 + ψ 6 Y ij (5.13) (5.11) (5.12) ψ 1, ψ 2, ψ 3 13 Pattern-Mixture Model SM ψ i 14 47

49 "%Selection_Model2" (5.13) ψ 1 ψ 6 inputds covtype UN TOEP TOEPH ARH AR CSH CS response model clasvar mech MCAR MAR MNAR const 3 8 derivative (0, 1) method NR Newton-Raphoson ridge QN Newton out1 out2 LSMEAN out3 LSMEAN debug "%SM_GridSearch" (5.13) ψ 1, ψ 2, ψ 3, ψ 4 ψ 5, ψ 6 inputds psi5grid 1 ψ 5 (-1 1) psi6grid 2 ψ 6 (-1 1) inputds covtype UN TOEP TOEPH ARH AR CSH CS response model clasvar mech MCAR MAR MNAR const 3 8 derivative (0, 1) method NR Newton-Raphoson ridge QN Newton out1 out2 LSMEAN out3 LSMEAN debug 0 [1] Diggle, P. and Kenward, M. G. (1994). Informative drop-out in longitudinal data analysis. Journal of Applied Statistics, 43,

50 [2] Little, R. J. (2008). Selection and Pattern-Mixture modes. Cphaiter 18 in Advances in longitudinal data analaysis [3] Little, R. J. A. and Rubin, D. B. (2002). Statistical analysis with missing data. 2nd edition. John Wiley & Sons. New York. [4] Mallinckrodt, C. H. (2013). Preventing and treating missing data in longitudinal clinical trials. Cambridge University Press. [5] Mallinckrodt C. H., Clark W. S. and David S. R. (2001). Accounting for dropout bias using mixed-effects models. Journal Biopharmaceutical Statistics, 11, [6] Mallinckrodt, C. H., Kaiser, C. J., Watkin, J. G., Molenberghs, G., & Carroll, R. J. (2004). The effect of correlation structure on treatment contrasts estimated from incomplete clinical trial data with likelihood-based repeated measures compared with last observation carried forward ANOVA. Clinical Trials, 1(6), [7] Mallinckrod, C. H., Lane, P. W., Schnell, D., Peng, Y., & Mancuso, J. P. (2008). Recommendations for the primary analysis of continuous endpoints in longitudinal clinical trials. Drug Information Journal, 42(4), [8] National Research Council. (2010). The prevention and treatment of missing data in clinical trials. National Academy Press. [9] Nelder, J. A., and Mead, R. (1965a). A simplex method for function minimization. The computer journal, 7(4), [10] Nelder, J. A., and Mead, R. (1965b). A simplex method for function minimization - errata. The computer journal, 8(1), 27. [11] Verbeke, G, and Molenberghs, G. (2000). Linear mixed models for longitudinal data. Springer. 49

51 6 Pattern-Mixture Model 6.1 Pattern-Mixture Model PMM PMM PMM Identifying Restrictions PMM Controlled imputation PMM Ratitch et al. (2013) Introduction Mallinckrodt et al. (2013) Jump to reference (J2R) Copy reference (CR), Copy increment reference (CIR) Controlled imputation PMM PMM (Patterm-Mixture Model, PMM) 6.1 PMM PMM estimand PMM efficacy effectiveness Mallinckrodt et al. (2013) 12 Analytic road map PMM estimand efficacy (estimand 3) contorolled imputation placebo multiple imputation effectiveness (estimand 6) Ayele et al. (2014) placebo multiple imputation effectiveness effectiveness 50

52 efficacy efficacy MNAR estimand 6.1: 6.2 PMM 1 R j Y 1,.., Y t 1 Y t f j (Y t Y 1,.., Y t 1 ) := f (Y t Y 1,.., Y t 1, R = j) R 1 R j Y 1,.., Y j Y j+1,.., Y t ( ) NFMV PMM R j (j + 1) j (j + 1) (j + 2) j = 1 Y 3 1 f 1 (Y 3 Y 1, Y 2 ) = f (Y 3 Y 1, Y 2, R = 1) R i i R Molenberghs et al. (1998), Kenward et al. (2003) 51

53 f 1 (3 12) = f (Y 3 Y 1, Y 2, R = 1) 2 1 R 1 Y 1 Y 2 Y 3 Y 3 2 f 2 (Y 3 Y 1, Y 2 ) = f (Y 3 Y 1, Y 2, R = 2) f 2 (3 12) = f (Y 3 Y 1, Y 2, R = 2) R 2 Y 1, Y 2 Y 3 Y 3 3 f 3 (Y 3 Y 1, Y 2 ) = f (Y 3 Y 1, Y 2, R = 3) f 3 (3 12) = f (Y 3 Y 1, Y 2, R = 3) R 3 Y 1, Y 2, Y 3 Y 3 2 R t Y 1,..., Y T f ( t) (Y 1,.., Y T ) = f (Y 1,.., Y T R t) 6.3 PMM PMM Little (1993, 1994, 1995) Selection Model SM PMM ( ) ( f Yi o, Yi m, R i X i, θ, ψ = f R i X i, ψ ) ( f Yi o, Yi m R i, X i, θ ) Y o i : i Y m i : i R i : i X i : i ψ : θ : ψ, θ SM PMM SM Y i R i R i 52

54 Yi o, Ym i X i R i PMM X i Yi m PMM "under-identified" Little (1993, 1994) "Identifying restriction" ( ) 6.4 f Yi o, Ym i, R i X i, θ, ψ ( f Yi o, Yi m, R i X i, ( = f R i X i, ψ ) f ) θ, ψ ( Y o i R i, X i, θ ) f ( Yi m Yi o, R i, X i, θ ) ( f R i X i, ψ ) ( f Yi o R i, X i, θ ) ( f Yi m Yo i, R i, X i, θ ) ( R i 0 1 R i R i R i R i R i PMM (CCMV) (NCMV) (ACMV) : 53

55 NRC , p85 Type (i) Little (1993, 1994) Complete Case Missing Values (CCMV) Neighboring Case Missing Values (NCMV) Available Case Missing Values (ACMV) Thijs et al. (2002) 3 Kenward et al. (2003) Interior family (6.2) PMM (1) (2) (3) (4) (5) PMM (3) PMM 6.5 CCMV NCMV ACMV PMM R = 1,, T R = t t t ( ) f Y 1,..., Y T, R = t X, θ, ψ = f t (Y 1,.., Y t X, θ ) f t (Y t+1, Y t+2,..., Y T Y 1,.., Y t, X, θ ) ( f R = t X, ψ ) X, θ ψ 1 f t (Y 1,.., Y t X, θ ) 2 f t (Y t+1, Y t+2,..., Y T Y 1,.., Y t, X, θ ) X θ ψ (6.1) CCMV, NCMV, ACMV Thijs et al. (2002) (s 1) Y 1,..., Y s 1 54

56 (j = s,..., T ) Y s CCMV NCMV ACMV f t (Y s Y 1,..., Y s 1 ) }{{} = T ω sj f j (Y s Y 1,..., Y s 1 ) j=s, s = t + 1,.., T (6.2) } {{ } = ω ss f s (Y s Y 1,..., Y s 1 ) + + ω st f T (Y s Y 1,..., Y s 1 ) (6.2) s t f s (Y s Y 1,..., Y s 1 ),, f T (Y s Y 1,..., Y s 1 ) R s Y s interior family interior family (6.2) ω sj 6.1: A B C D 6.1 CCMV, NCMV, ACMV ω sj CCMV CCMV (Little, 1993) ω T,T = ω T 1,T = ω T 2,T =... = ω t+1,t = 1 ω sj = 0, j T. Y s f t (Y s Y 1,..., Y s 1 ) = f T (Y s Y 1,..., Y s 1 ), s = t + 1,..., T (6.3) 4 CCMV f 3 (4 123) }{{} = f 4 (4 123) ( ) }{{} f 2 (3 12) = f 4 (3 12), f 2 (4 123) = f 4 (4 123) 55

57 f 1 (2 1) = f 4 (2 1), f 1 (3 12) = f 4 (3 12), f 1 (4 123) = f 4 (4 123) CCMV A CCMV : CCMV ( ) A f 4 (1) f 4 (2 1) f 4 (3 12) f 4 (4 123) B f 3 (1) f 3 (2 1) f 3 (3 12) f 4 (4 123) C f 2 (1) f 2 (2 1) f 4 (3 12) f 4 (4 123) D f 1 (1) f 4 (2 1) f 4 (3 12) f 4 (4 123) NCMV NCMV ω T,T = ω T 1,T 1 = ω T 2,T 2 =... = ω t+1,t+1 = 1 ω sj = 0, j s. Y s f t (Y s Y 1,..., Y s 1 ) = f s (Y s Y 1,..., Y s 1 ), s = t + 1,..., T (6.4) 4 NCMV f 3 (4 123) = f 4 (4 123) f 2 (3 12) = f 3 (3 12), f 2 (4 123) = f 4 (4 123) f 1 (2 1) = f 2 (2 1), f 1 (3 12) = f 3 (3 12), f 1 (4 123) = f 4 (4 123) NCMV 6.1 C D 3 B 3 NCMV : NCMV ( ) A f 4 (1) f 4 (2 1) f 4 (3 12) f 4 (4 123) B f 3 (1) f 3 (2 1) f 3 (3 12) f 4 (4 123) C f 2 (1) f 2 (2 1) f 3 (3 12) f 4 (4 123) D f 1 (1) f 2 (2 1) f 3 (3 12) f 4 (4 123) 56

58 6.5.4 ACMV ACMV(Molenberghs et al., 1998) f t (Y s Y 1,..., Y s 1 ) = f s (Y s Y 1,..., Y s 1 ) = f(y s Y 1,..., Y s 1, R s) = f(y 1,..., Y s 1, Y s, R s) f(y 1,..., Y s 1, R s) = f(y 1,..., Y s 1, Y s R = s)f(r = s) f(y 1,..., Y s 1, Y s R = T )f(r = T ) f(y 1,..., Y s 1 R = s)f(r = s) f(y 1,..., Y s 1 R = T )f(r = T ) = α sf s (Y 1,..., Y s 1, Y s ) α s f T (Y 1,..., Y s 1, Y s ) ( f(r = j) = α j ) α s f s (Y 1,..., Y s 1 ) α T f T (Y 1,..., Y s 1 ) T α j f j (Y 1,..., Y s ) = = = j=s T α j f j (Y 1,..., Y s 1 ) j=s T j=s α j f j (Y 1,..., Y s 1 ) f j (Y s Y 1,.., Y s 1 ) T α l f l (Y 1,..., Y s 1 ) l=s T ω sj f j (Y s Y 1,.., Y s 1 ), s = t + 1,..., T. (6.5) j=s ω sj = α jf j (Y 1,..., Y s 1 ) T α l f l (Y 1,, Y s 1 ) l=s α j j Molenberghs et al., ACMV f 3 (4 123) = f 4 (4 123) f 2 (3 12) = f 3 (3 12), f 2 (4 123) = f 4 (4 123) f 1 (2 1) = f 2 (2 1), f 1 (3 12) = f 3 (3 12), f 1 (4 123) = f 4 (4 123) ACMV 6.1 D 2 2 A B C 2 ACMV : ACMV ( ) A f 4 (1) f 4 (2 1) f 4 (3 12) f 4 (4 123) B f 3 (1) f 3 (2 1) f 3 (3 12) f 4 (4 123) C f 2 (1) f 2 (2 1) f 3 (3 12) f 4 (4 123) D f 1 (1) f 2 (2 1) f 3 (3 12) f 4 (4 123) Molenberghs et al. (1998) 1 Appendix B 1: MAR ACMV 57

59 6.5.5 CCMV NCMV ACMV (6.2) (6.1) t T t 1 f t (Y 1,..., Y T ) = f t (Y 1,.., Y t ) T ω T s,j f j (Y T s Y 1,..., Y T s 1 ) (6.6) s=0 j=t s 4 CCMV NCMV ACMV 4 ω 33 + ω 34 = 1, ω 44 = 1 ω := ω 33 ω 34 = 1 ω f 4 (1234) = f 4 (1234) f 3 (1234) = f 3 (123) f 3 (4 123) = f 3 (123) ω 44 f 4 (4 123) ( CCMV, NCMV, ACMV ) = f 3 (123) f 4 (4 123) ( ω 44 = 1) f 2 (1234) = f 2 (12) f 2 (3 12) f 2 (4 123) = f 2 (12) [ω 33 f 3 (3 12) + ω 34 f 4 (3 12)] ω 44 f 4 (4 123) ( (6.6)) = f 2 (12) [ω f 3 (3 12) + (1 ω) f 4 (3 12)] f 4 (4 123) ( ω 33 = ω, ω 34 = 1 ω, ω 44 = 1) f 1 (1234) = f 1 (1) f 1 (2 1) f 1 (3 12) f 1 (4 123) = f 1 (1) [ω 22 f 2 (2 1) + ω 23 f 3 (2 1) + ω 24 f 4 (2 1)] [ω 33 f 3 (3 12) + ω 34 f 4 (3 12)] ω 44 f 4 (4 123) ( (6.6)) = f 1 (1) [ω 22 f 2 (2 1) + ω 23 f 3 (2 1) + ω 24 f 4 (2 1)] [ω f 3 (3 12) + (1 ω)f 4 (3 12)] f 4 (4 123) ( ω 33 = ω, ω 34 = 1 ω, ω 44 = 1) (6.7) ω Case X Case 1: CCMV ω = ω 33 = ω 22 = ω 23 = 0, ω 44 = ω 34 = ω 24 = 1 (6.7) CCMV 4 f 4 (1234) = f 4 (1234) f 3 (1234) = f 3 (123) f 4 (4 123) f 2 (1234) = f 2 (12) f 4 (3 12) f 4 (4 123) f 1 (1234) = f 1 (1) f 4 (2 1) f 4 (3 12) f 4 (4 123) Case 2: NCMV ω = ω 33 = ω 22 = 1, ω 34 = ω 23 (1) = ω 24 (1) = 0 58

60 CCMV 4 f 4 (1234) = f 4 (1234) f 3 (1234) = f 3 (123) f 4 (4 123) f 2 (1234) = f 2 (12) f 3 (3 12) f 4 (4 123) f 1 (1234) = f 1 (1) f 2 (2 1) f 3 (3 12) f 4 (4 123) Case 3: ACMV α 3 f 3 (12) ω = ω 33 = α 3 f 3 (12) + α 4 f 4 (12), 1 ω = ω α 4 f 4 (12) 34 = α 3 f 3 (12) + α 4 f 4 (12), α 2 f 2 (1) ω 22 = α 2 f 2 (1) + α 3 f 3 (1) + α 4 f 4 (1), ω α 3 f 3 (1) 23 = α 2 f 2 (1) + α 3 f 3 (1) + α 4 f 4 (1), ω 24 = α 4 f 4 (1) α 2 f 2 (1) + α 3 f 3 (1) + α 4 f 4 (1) α j = f(r = j) ACMV 4 Case 3 ω (6.7) MAR Case 3 O Kelly and Ratitch 2014 CCMV NCMV ACMV MAR MAR MAR MMRM O Kelly and Ratitch 2014 MAR MI ACMV PMM, ACMV ACMV NCMV O Kelly and Ratitch CCMV ACMV NCMV NCMV CCMV NCMV ACMV PMM Step1 t (= 1,..., T ) f t (Y 1,.., Y t ) Step2 CCMV NCMV ACMV 59

61 Step3 Step2 f t (Y t+1,..., Y T Y 1,..., Y t ) ω 2 Step Step3-1 (6.2) 1 (6.2) (T s + 1) ω sj (s = t + 1,..., T ; j = s,..., T ) f j (Y s Y 1,..., Y s 1 ) U s1 1 s (T t) (T t) k (= s,..., T ) k 1 k ω sj U ω sj, (k = s,, T, k 2) j=s j=s 0 U ω 11, (k = 1) Step3-2 k Y s Step4 Step3-2 Step5 MMRM Step6 Rubin MAR ACMV 2. MI (CCMV, NCMV, ACMV) 3 CCMV, NCMV, ACMV interior family CCMV NCMV MNAR ACMV MAR CCMV NCMV MNAR MAR O Kelly and Ratitch

62 Interior family Non Future Missing Values (NFMV) Interior family NFMV MNAR 6.7 NFMV PMM CCMV, NCMV MNAR MAR MNAR MNFD (Missing Non Future Dependence) MNFD NFMV PMM MNFD NFD Appendix B NFD MNFD MAR PMM MNAR Kenward et al. (2003) NFMV PMM Interior family CCMV, ACMV, NCMV NFMV MNAR SM R = 1,, T R = t t. SM ( ) ( f Y 1,.., Y T, R = t X, θ, ψ = f Y 1,..., Y T X, θ ) f ( ) R = t Y 1,..., Y T, X. ψ (6.8) PMM ( ) ( f Y 1,..., Y T, R = t X, θ, ψ = f Y 1,..., Y T R = t, X, θ ) ( f R = t X, ψ ) = f t (Y 1,..., Y T X, θ ) ( f R = t X, ψ ) = f t (Y 1,..., Y t X, θ ) ( f t Y t+1 Y 1,..., Y t, X, θ ) f t (Y t+2,..., Y T Y 1,..., Y t+1, X, θ ) ( f R = t X, ψ ) (6.9) (6.9) 2 3 t f t (Y 1,..., Y t X, θ) t + 1 f t (Y t+1 Y 1,..., Y t, X, θ) t + 2 f t (Y t+2,...,( Y T Y 1,..., Y t+1, X, θ) 1 4 (f R = t X, ψ ) ) X θ ψ NFMV SM MNFD Missing Non-Future Dependent MNFD 2 (6.1) (6.1) t (t + 1) (6.9) t (t + 1) (t + 2) (t + 2) NFMV (6.1) 61

63 R = t t Y o = (Y 1,, Y t ) t + 1 Y m = (Y t+1,, Y T ) MNFD (Kenward et al., 2003) f (R = t Y 1,..., Y T ) = f (R = t Y 1,..., Y t+1 ) (6.10) (t + 1) Y t+1 Y t+2,..., Y T Little and Rubin (2002) MNAR MAR Molenberghs et al. (2007) MNAR MAR 6.5 MNAR SM Diggle and Kenward (1994) 3 logit{p r(r i,t+1 = 0 R i1 = 1,, R i,t = 1, Y i, X i, ψ)} = ψ 1 + ψ 2 Y i,t + ψ 3 Y i,t+1 MNFD PMM MNFD Non- Future Missing Value restrictions NFMV s t + 2, f (Y s Y 1,..., Y s 1, R = t) = f (Y s Y 1,..., Y s 1, R s 1) (6.11) Kenward et al. (2003) MNFD NFMV Appendix B t Y t Y t+1, Y t+2,..., Y s 1 Y s (s 1) Y s Y s NFMV s = 5, t = 3, 2, 1 f (Y 5 Y 1, Y 2, Y 3, Y 4, R = 3) = f (Y 5 Y 1, Y 2, Y 3, Y 4, R 4) Y 3 Y 4 f (Y 5 Y 1, Y 2, Y 3, Y 4, R = 2) = f (Y 5 Y 1, Y 2, Y 3, Y 4, R 4) Y 2 Y 3 f (Y 5 Y 1, Y 2, Y 3, Y 4, R = 1) = f (Y 5 Y 1, Y 2, Y 3, Y 4, R 4) Y 1 Y 2 R = 4 Y 5 NFMV D 3 NFMV Missingdata.org.uk 62

64 6.5: NFMV f A f 4 (1) f 4 (2 1) f 4 (3 12) f 4 (4 123) B f 3 (1) f 3 (2 1) f 3 (3 12) C f 2 (1) f 2 (2 1) f 3 (4 123) D f 1 (1) f 2 (3 12) f 3 (4 123) (6.11) f (Y s Y 1,..., Y s 1, R s 1) R = s 1 R s f (Y s Y 1,..., Y s 1, R s 1) = f (Y s Y 1,..., Y s 1, R = s 1) f (R = s 1 R s 1) }{{} + f (Y s Y 1,..., Y s 1, R s) f (R s R s 1) }{{} R = s 1 Y s R s Y s f (Y s Y 1, Y s 1, R = s 1) (6.12) s = 5 f (Y 5 Y 1, Y 2, Y 3, Y 4, R = 4) NFMV NFMV NFMV CCMV NCMV NFMV 1 ACMV ACMV Appendix B ACMV MAR NFMV Kenward et al. (2003) Appendix B 2: MNFD NFMV Molenberghs et al. (1998) MAR ACMV 1 NFMV (6.12) CCMV NCMV MNAR Mallinckrodt et al., 2013 ACMV MAR ACMV PMM NFMV CCMV NFMV NCMV PMM MAR Mallinckrodt et al. (2013) (NFMV) NFMV t f t (Y 1,..., Y T ) = f t (Y 1,..., Y t ) f t (Y t+1 Y 1,..., Y t ) f t (Y t+2,..., Y T Y 1,..., Y t+1 ) T = f t (Y 1,..., Y t ) f t (Y t+1 Y 1,..., Y t ) f t (Y s Y 1,..., Y s 1 ) (6.13) 63 s=t+2

65 (6.11) T = 4 R = 1 f 1 (Y 1, Y 2, Y 3, Y 4 ) = f 1 (Y 1 ) f 1 (Y 2 Y 1 ) 4 f 1 (Y s Y 1, Y 2 ) s=3 = f 1 (Y 1 ) f 1 (Y 2 Y 1 ) f 1 (Y 3 Y 1, Y 2 ) f 1 (Y 4 Y 1, Y 2, Y 3 ) }{{}}{{}}{{}}{{} NFMV 3 NFMV NFMV (6.11) s = t + 2,, T f t (Y s Y 1,..., Y s 1 ) = f (Y s Y 1,..., Y s 1, R s 1) = f s 1 (Y s Y 1,..., Y s 1 ) (6.14) s = t + 2,..., T f s 1 (Y s Y 1,..., Y s 1 ) = = = T j=s 1 T j=s 1 T j=s 1 T j=s 1 α j f j (Y 1,..., Y s ) α j f j (Y 1,..., Y s 1 ) α j f j (Y 1,..., Y s 1 ) f j (Y s Y 1,.., Y s 1 ) T α l f l (Y 1,, Y s 1 ) l=s 1 ω sj f j (Y s Y 1,..., Y s 1 ) (6.15) α j f j (Y 1,..., Y s 1 ) ω sj =, T α l f l (Y 1,, Y s 1 ) l=s 1 α j : j MNAR NFMV (6.12) 4 NFMV 4 (6.15) (6.11) (6.12) g f 2 (1234) ω 21 + ω 22 + ω 23 + ω 23 = 1, ω 32 + ω 33 + ω 34, ω 43 + ω 44 = 1 δ := ω 43 ω 44 = 1 δ f 4 (1234) = f 4 (1234) f 3 (1234) = f 3 (123) f 3 (4 123) = f 3 (123) g 3 (4 123) f 2 (1234) = f 2 (12) f 2 (3 12) f 2 (4 123) 64

66 = f 2 (12) f 2 (3 12) f 3 (4 123) ( NFMV) = f 2 (12) g 2 (3 12)f 3 (4 123) (NFMV g 2 = f 2 (12) g 2 (3 12) [ω 43 f 3 (4 123) + ω 44 f 4 (4 123)] ( (6.15)) [ ] = f 2 (12) g 2 (3 12) δ g 3 (4 123) + (1 δ) f 4 (4 123) ( ω 43 = δ, ω 44 = 1 δ) NFMV g 3 f 1 (1234) = f 1 (1) f 1 (2 1) f 1 (34 12) = f 1 (1) g 1 (2 1) f 1 (3 12) f 1 (4 123) = f 1 (1) g 1 (2 1)f 2 (3 12) f 3 (4 123) ( NFMV) = f 1 (1) g 1 (2 1) [ω 32 f 2 (3 12) + ω 33 f 3 (3 12) + ω 34 f 4 (3 12)] [ ] ω 43 g 3 (4 123) + ω 44 f 4 (4 123) ( (6.15)) [ ] = f 1 (1) g 1 (2 1) ω 32 g 2 (3 12) + ω 33 f 3 (3 12) + ω 34 f 4 (3 12) [ ] δ g 3 (4 123) + (1 δ) f 4 (4 123) ( ω 43 = δ, ω 44 = 1 δ) δ ω 43, ω 44 δ α 3 f 3 (123) δ = ω 43 = α 3 f 3 (123) + α 4 f 4 (123), 1 δ = ω α 4 f 4 (123) 44 = α 3 f 3 (123) + α 4 f 4 (123), α 2 f 2 (12) ω 32 = α 2 f 2 (12) + α 3 f 3 (12) + α 4 f 4 (12), ω α 3 f 3 (12) 33 = α 2 f 2 (12) + α 3 f 3 (12) + α 4 f 4 (12), ω 34 = α 4 f 4 (12) α 2 f 2 (12) + α 3 f 3 (12) + α 4 f 4 (12) NFMV g 1, g 2, g 3. Kenward et al. (2003) NFD1 Case 4 NFD2 Case 5 NFD1 NFD2 interior family CCMV NCMV NFD1 NFD2 6.5 NFMV CCMV NCMV : NFD1 (NFMV + CCMV) A f 4 (1) f 4 (2 1) f 4 (3 12) f 4 (4 123) B f 3 (1) f 3 (2 1) f 3 (3 12) f 4 (4 123) C f 2 (1) f 2 (2 1) f 4 (3 12) f 4 (4 123) D f 1 (1) f 4 (2 1) (1 ω 33 (12)) f 4 (3 12) + ω 33 (12) f 3 (3 12) f 4 (4 123) 6.7: NFD2 (NFMV + NCMV) A f 4 (1) f 4 (2 1) f 4 (3 12) f 4 (4 123) B f 3 (1) f 3 (2 1) f 3 (3 12) f 4 (4 123) C f 2 (1) f 2 (2 1) f 3 (3 12) f 4 (4 123) D f 1 (1) f 2 (2 1) (1 ω 34 (12)) f 3 (3 12) + ω 34 (12) f 4 (3 12) f 4 (4 123) NFMV 4 Case 4: NFMV+CCMV (NFD1) g 1 (2 1) = f 4 (2 1), g 2 (3 12) = f 4 (3 12), g 3 (4 123) = f 4 (4 123) 65

67 4 f 4 (1234) = f 4 (1234) f 3 (1234) = f 3 (123) f 3 (4 123) = f 3 (123) f 4 (4 123) f 2 (1234) = f 2 (12) f 2 (3 12) f 2 (4 123) = f 2 (12) f 4 (3 12)f 3 (4 123) ( CCMV, NFMV) = f 2 (12) f 4 (3 12) [ω 43 f 3 (4 123) + ω 44 f 4 (4 123)] ( (6.15)) [ ] = f 2 (12) f 4 (3 12) δ f 4 (4 123) + (1 δ) f 4 (4 123) ( ω 43 = δ, ω 44 = 1 δ, CCMV) = f 2 (12) f 4 (3 12)f 4 (4 123) f 1 (1234) = f 1 (1) f 1 (2 1) f 1 (34 12) = f 1 (1) f 4 (2 1) f 1 (3 12) f 1 (4 123) = f 1 (1) f 4 (2 1)f 2 (3 12) f 3 (4 123) ( NFMV) = f 1 (1) f 1 (2 1) [ω 32 f 2 (3 12) + ω 33 f 3 (3 12) + ω 34 f 4 (3 12)] [ω 43 f 3 (4 123) + ω 44 f 4 (4 123)] [ ( (6.15)) ] = f 1 (1) f 4 (2 1) ω 32 f 4 (3 12) + ω 33 f 3 (3 12) + ω 34 f 4 (3 12) [ ] δ f 4 (4 123) + (1 δ) f 4 (4 123) ( ω 43 = δ, ω 44 = 1 δ, CCMV) [ ] = f 1 (1) f 4 (2 1) (1 ω 33 ) f 4 (3 12) + ω 33 f 3 (3 12) f 4 (4 123) ( ω 32 + ω 33 + ω 34 = 1) 4 Case 1: CCMV R = 4 Case 4: NFD1 f 1 (3 12) R = 3 f 3 (3 12) R = 4 f 4 (3 12) NFD1 CCMV Case 5: NFMV+NCMV (NFD2) g 1 (2 1) = f 2 (2 1), g 2 (3 12) = f 3 (3 12), g 3 (4 123) = f 4 (4 123) 4 f 4 (1234) = f 4 (1234) f 3 (1234) = f 3 (123) f 3 (4 123) = f 3 (123) f 4 (4 123) f 2 (1234) = f 2 (12) f 2 (3 12) f 2 (4 123) = f 2 (12) f 3 (3 12)f 3 (4 123) ( NFMV) = f 2 (12) f 3 (3 12) [ω 43 f 3 (4 123) + ω 44 f 4 (4 123)] ( (6.15)) [ ] = f 2 (12) f 3 (3 12) δ f 4 (4 123) + (1 δ) f 4 (4 123) ( ω 43 = δ, ω 44 = 1 δ, NCMV) = f 2 (12) f 3 (3 12)f 4 (4 123) f 1 (1234) = f 1 (1) f 1 (2 1) f 1 (34 12) = f 1 (1) f 4 (2 1) f 1 (3 12) f 1 (4 123) = f 1 (1) f 2 (2 1)f 2 (3 12) f 3 (4 123) ( NFMV) = f 1 (1) f 2 (2 1) [ω 32 f 2 (3 12) + ω 33 f 3 (3 12) + ω 34 f 4 (3 12)] [ω 43 f 4 (4 123) + ω 44 f 4 (4 123)] ( (6.15)) 66

68 = f 1 (1) f 2 (2 1) [ω 32 f 2 (3 12) + ω 33 f 3 (3 12) + ω 34 f 4 (3 12)] [ ] δ f 4 (4 123) + (1 δ) f 4 (4 123) ( NCMV, ω 43 = δ, ω 44 = 1 δ) [ ] = f 1 (1) f 2 (2 1) ω 32 f 3 (3 12) + ω 33 f 3 (3 12) + ω 34 f 4 (3 12) f 4 (4 123) ( NCMV) [ ] = f 1 (1) f 2 (2 1) (1 ω 34 ) f 3 (3 12) + ω 34 f 4 (3 12) f 4 (4 123) ( ω 32 + ω 33 + ω 34 = 1) 4 Case 2: NCMV Case 5: NFD2 f 1 (3 12) R = 3 f 3 (3 12) R = 4 f 4 (3 12) NFD2 NCMV 4 Case : Case 4 67

69 6.4: Case NFMV PMM Step1 f t (Y 1,..., Y t ) Step2 CCMV NCMV Y t+1 f t (Y t+1 Y 1,..., Y t ) NFMV f (Y s Y 1, Y s 1, R = s 1) g Case 4 (NFMV+CCMV) Case 5 (NFMV+ NCMV) CCMV NCMV Step3 NFMV f t (Y t+2,.., Y T Y 1,.., Y t+1 ) 68

70 Step4 (Multiple Imputaiton) f t (Y t+2,.., Y T Y 1,.., Y t+1 ), f t (Y t+1 Y 1,.., Y t ) 6.8 Step5 MMRM Step6 Rubin (1987). Marginal analysis 6.8 Verbeke and Molenberghs (2009) PMM NFMV 4 g ACMV 10 ( ) PMM Step Step 2 Y t+1 f t (Y t+1 Y 1,..., Y t ) g ACMV Case 6 (NFMV+ACMV) Case 7 (NFMV+ACMV+ ) Case 6: NFMV+ACMV ACMV MAR 2 s T 4 f 4 (1234) = f 4 (1234) g s 1 (Y s Y 1,..., Y s 1 ) = f s (Y s Y 1,..., Y s 1 ) T = ω sj f j (Y s Y 1,.., Y s 1 ) f 3 (1234) = f 3 (123) g 3 (4 123) [ ] f 2 (1234) = f 2 (12) g 2 (3 12) δ g 3 (4 123) + (1 δ) f 4 (4 123) [ ] f 1 (1234) = f 1 (1) g 1 (2 1) ω 32 g 2 (3 12) + ω 33 f 3 (3 12) + ω 34 f 4 (3 12) [ ] δ g 3 (4 123) + (1 δ) f 4 (4 123) g g 1 (2 1) = f 2 (2 1) = ω 22 f 2 (2 1) + ω 23 f 3 (2 1) + ω 24 f 4 (2 1) j=s g 2 (3 12) = f 3 (3 12) = ω 33 f 3 (3 12) + ω 34 f 4 (3 12) g 3 (4 123) = f 4 (4 123) = f 4 (4 123) 69

71 Case 7: NFMV+ACMV+ g s 1 (Y s Y 1,..., Y s 1 ) = f s (Y s Y 1,..., Y s 1 ) T = ω sj f j (Y s Y 1,.., Y s 1 ) Case 6 g g 1 (2 1) = f 2 (2 1) = ω 22 f 2 (2 1) + ω 23 f 3 (2 1) + ω 24 f 4 (2 1) g 2 (3 12) = f 3 (3 12) = ω 33 f 3 (3 12) + ω 34 f 4 (3 12) g 3 (4 123) = f 4 (4 123) = f 4 (4 123) j=s ω sj = α j f j (Y 1,..., Y s 1 ) T α l f l (f 1,..., f s 1 ) l=s ω 22 = ω 24 = ω 34 = α 2 f 2 (1) α 2 f 2 (1) + α 3 f 3 (1) + α 4 f 4 (1), ω α 3 f 3 (1) 23 = α 2 f 2 (1) + α 3 f 3 (1) + α 4 f 4 (1), α 4 f 4 (1) α 2 f 2 (1) + α 3 f 3 (1) + α 4 f 4 (1), ω α 3 f 3 (12) 33 = α 3 f 3 (12) + α 4 f 4 (12), α 4 f 4 (12) α 3 f 3 (12) + α 4 f 4 (12) Case 7 = 0 MAR MAR NRC (2010) SM PMM Molenberghs et al. (1998) ACMV SM Missing At Random (MAR) SM PMM CCMV NCMV Missing Not At Random (MNAR) Kenward et al. (2003) MNAR SM Missing Non-Future Dependent (MNFD NFD) MNAR SM 4 Kenward et al. (2003) MNFD PMM Non-Future Missing Value (NFMV) restriction ACMV Interior family Kenward et al. (2003) MCAR, MAR, MNAR MNAR Non-future dependent Future dependent 6.5 (Molenberghs et al., 2007, p37) 4 Shared parameter models Non-Future Dependent (Wu and Bailey, 1989; Little, 1995) 70

72 6.5: SM PMM 6.5 MAR MNAR PMM ( %PATTERNMIXTURE) %PATTERNMIXTURE 6 %F IRST ANL(constraint =, type =, yvar =, classvars =) %GEN COV (incovmean =, incovvar =, outcovset =, visitval =) MIXED %GENP ARMS(inparm =, incov =, outparm =) %GENV ALS(dataset =, firstvis =, basevis =, yvar =) %MEGACALL(numimp =, Itype =, Iconstraint =, yvar =, firstv =, lastvis =) %GENCOV %GENPARM %GENVALS numimp Itype visit constraint NFMV ALL 2 Iconstraint 2 CCMV NCMV ACMV %ANALY ZE(numimp =, yvar =, model2 =, calssvars2 =, othervars =) MIXED LSMEAN 71

73 6.8 Multiple Imuptation MI Multiple Imuptation MI Rubin 1978, 1987 MAR MAR MI Mallinckrodt MI MI Dmitrienko et al MI PMM Multiple Imuptation Rubin MI 2 θ 1, θ 2 Y f (θ 1, θ 2 Y ) θ 1 θ 2 f (θ 1, θ 2 Y ) = f (θ 1 Y ) f (θ 2 θ 1, Y ) θ 2 f (θ 2 Y ) = E θ1 [f (θ 2 θ 1, Y )] = f (θ 1 Y ) f (θ 2 θ 1, Y ) dθ 1 θ 2 E [θ 2 Y ] = E θ1 [E θ2 [θ 2 θ 1, Y ]] V ar [θ 2 Y ] = E θ1 [V ar θ2 [θ 2 θ 1, Y ]] + V ar θ1 [E θ2 [θ 2 θ 1, Y ]] θ 1 θ (m) 1, m = 1,..., M E [θ 2 Y ] = 1 M V ar [θ 2 Y ] = 1 M M m=1 [ ] E θ2 θ 2 θ (m) 1, Y = θ 2 M m=1 [ ] V ar θ2 θ 2 θ (m) 1, Y + 1 M 1 M m=1 [ ] (E θ2 θ 2 θ (m) 1, Y θ ) 2 2 θ 2 θ 1 θ 2 MI Molenberghs et al, (2007) p

74 6.8.3 Multiple Imuptation PMM ACMV MI MAR MI efficacy estimand 3 ACMV MAR MI Yuan 2011 R = t j = t + 1, t + 2,..., T Y j Y 1,..., Y j 1 Y j f (Y j Y 1,..., Y j 1 ) 1. Y j = β 0 +β 1 Y β j 1 Y j 1 Y 1,..., Y j β = ( β0, β 1,..., β ) j 1 σ 2 j V j = σ j 2 (X X) 1 (X X) Y 1,..., Y j σ 2 j, β σ 2 j = σ 2 j (n j j) /G, G χ 2 n j j β = β + σ j V hjz n j Y j Z j V hj V j = V hj V hj β = (β 0, β 1,..., β j 1 ), σ 2 j Y j Ŷj Ŷ j = β 0 + β 1 Y β j 1 Y j 1 + z i σ j M SAS PROC MI MONOTONE METHOD = REG Multiple Imputation k θ M M M θ (1), θ (2),..., θ (M) θ (m), m = 1, 2,..., M ( ) θ (m) θ N (0, U (m)) θ = 1 M M m=1 Ŵ = θ (m) θ ) ( θ θ N(0, V) ( ) M + 1 V = Ŵ + B M M Û (m) m=1 M 73

75 B = M m=1 ( θ (m) θ ) ( θ(m) θ ) M 1 Ŵ B ( θ θ ) V 1 ( θ θ ) χ 2 k χ 2 k N M Li et al. (1991) F H 0 : θ = θ 0, H 1 : θ θ 0 F p. ( θ θ0 ) V 1 ( θ θ0 ) F = k (R + 1) p = P r ( Fw k > F ) w = 4 + (τ 4) (1 + R = 1 ( k M τ = k (M 1) F k w ) τ (τ > 4 ) R ) tr(bw 1 ) M F F k, = χ 2 /k SAS M PROC MIANALYZE PROC MIANALYZE τ 4 F 2 (Rubin, 1987, p137) w = 1 ( 2 + (k + 1) (M 1) ) R θ θ Lθ L : ( L θ ) Lθ N ( 0, LVL ) ( ) M + 1 LVL = LŴL + L M BL M Ŵ = U (m) m=1 M M ( θ (m) θ ) ( θ(m) ) θ B = m=1 M 1 MI Rubin (1987, p114) Efficiency = ( 1 + γ ) 1 M 74

76 γ γ M γ 6.6 Molenberghs et al, 2007, p : Rubin (1987) 3 5 B k B Graham et al. (2007) 10% 30% 1% 20 Siddiqui (2011) 40% 10 White et al Royston and White ,000 (2015) MI Rubin (1987, p114) % p p 6.9 Marginal treatment effect ( ) Kenward et al PMM t = 1,..., T l = 1,..., g β lt t = 1,..., T π t β l T β l (marginal treatment effect) = β lt π t, l = 1,..., g t=1 V ar ( β1,..., β ) g = AVA V = ) ( βlt Var O O ) (β 1,.., β g ), A = Var ( βt (β 11,.., β T g, π 1,.., π T ) 75

77 H 0 : β 1 =... = β g = 0 Wald β ( β1,..., β ) (AVA ) 1 1 g. χ 2 g β g 6.10 PMM Ratitch et al. (2013) Mallinckrodt et al. (2013) 1. Controlled/Refernece pattern imputation 2. Delta adjustment pattern imputation 3. AE pattern imputation Ratitch et al. (2013), Mallinckrodt et al. (2013) pattern imputation Controlled imputation MI (Controlled imputation) placebo Multiple Imputation (pmi) 5 2 Y 0, Y 1 Y 0 Y 1 Y 1 R = 0( ), 1 T rt = 0( ), 1 θ X MNAR ( f Y 1 T rt = 0, Y 0, R = 0, X, θ ) f or f ( Y 1 T rt = 0, Y 0, R = 1, X, θ ) ( Y 1 T rt = 1, Y 0, R = 0, X, θ ) ( ) f Y 1 T rt = 1, Y 0, R = 1, X. θ MNAR ( 1) Y 1 f ( Y 1 T rt = 1, Y 0, R = 0, X, θ ) ( = f Y 1 T rt = 0, Y 0, R = 1, X, θ ) Mallinkckrodt (2013) 10.5 (p98-99) t = 1,..., T SAS PROC MI Step 1. t = 0 Step 2. t = t + 1 X 1,, t 1 t Step 3. 1 (t 1) t SAS PROC MI t t 5 pmi 76

78 Step 4. t t < T Step 2 t = T Step 5 Step 5. Step 1 Step 4 M M Step 6. M MMRM Step 7. Rubin (1987) p SAS PROC MIANALYIZE Little and Yau (1996) Controlled imputation MNAR MAR controlled imputation 3 estimand 6 "effectiveness" LOCF BOCF reasonable Mallinckrodt (2013) LOCF BOCF 2 / / LOCF BOCF efficacy pattern imputation SAS pattern imputation SAS Ratitch et al. (2011) Yuan (2014) PROC MI (SAS Ver 9.4) PMM MNAR MNAR 2 MODEL ADJUST MODEL controlled imputation SAS code 1 SAS code PROC MI DATA = XXXX; SEED = 14821; NIMPUTE = 100; /* */ OUT = result; CLASS trt; MONOTONE REG(y1); /* */ MNAR MODEL(y1/ MODELOBS = (trt = 0 )); /*trt=0( ) y1 */ VAR x0 y1; /* x0 */ RUN; pattern imputation Mallinckrodt et al. (2013) controlled imputation referenced based imputation Jump to reference (J2R) Copy reference (CR) Copy increment method (CIR) Missingdata.org.uk 77

79 PMM five macro MissingmacroDoc20.docx pattern imputation Kenward estimand MissingmacroDoc20.docx i j (= 1, 2,..., q) k (= 1,..., t) Y ijk r p (= 0,.., t) p MissingmacroDoc20.docx E (Y ijk ) = A jk A jk j k E p (Y ijk ) = B pjk B pjk p j k A jk B pjk J2R CR CIR J2R (Jump to reference) A jk (for k p) B pjk = A rk (for k > p) r MNAR de facto (effectiveness CR (Copy Reference) B pjk = A rk (for all p, j, k) r CIR (Copy Increment Reference) A jk (for k p) B pjk = A jp + A rk A rp (for k > p) r A rk A rp r p k p j A jp r J2R de facto Delta adjustment pattern imputation Delta adjustment pattern imputation (, 10 ) estimand efficacy 78

80 effectiveness MNAR MAR tipping point MAR tipping point analysis Y 0, Y 1 MNAR ( 2) Y 1 Y 1 ( f Y 1 T rt = 1, Y 0, R = 0, X, θ ) ( = f Y 1 + T rt = 1, Y 0, R = 1, X, θ ) MNAR 1 2 SAS ADJUST Delta adjustment pattern imputation SAS 2 SAS code PROC MI DATA = XXXX; SEED = 14821; NIMPUTE = 100; /* */ OUT = result; CLASS trt; MONOTONE REG(y1); /* */ MNAR ADJUST(y1 / SHIFT = ADJUSTOBS = (trt = 1 )); /*trt = 1( ) y1 ( -2-1 ) */ VAR trt x0 y1; /* x0 */ RUN; AE pattern imputation (Ratitch et al., 2013) estimand effectiveness / (CCMV) BOCF BOCF BOCF MAR patter imputaion

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82 [19] Ratitch, B., O Kelly, M., & Tosiello, R. (2013). Missing data in clinical trials: from clinical assumptions to statistical analysis using pattern mixture models.pharmaceutical statistics, 12(6), [20] Royston, P., & White, I. R. (2011). Multiple imputation by chained equations (MICE): implementation in Stata.J Stat Softw, 45(4), [21] Rubin, D.B. (1978), Multiple Imputations in Sample Surveys - A Phenomenological Bayesian Approach to Nonresponse, Proceedings of the Survey Research Methods Section of the American Statistical Association, [22] Rubin, D. B. (1987). Multiple imputation for nonresponse in surveys. John Wiley & Sons. [23] Siddiqui, O. (2011). MMRM versus MI in dealing with missing data a comparison based on 25 NDA data sets. Journal of Biopharmaceutical Statistics. 21, [24] Thijs, H., Molenberghs, G., Michiels, B., Verbeke, G., & Curran, D. (2002). Strategies to fit pattern mixture models. Biostatistics, 3(2), [25] Verbeke, G., & Molenberghs, G. (2009). Linear mixed models for longitudinal data. Springer Science & Business Media. [26] Wu, M. C., & Bailey, K. R. (1989). Estimation and comparison of changes in the presence of informative right censoring: conditional linear model. Biometrics, [27] White, I. R., Royston, P., & Wood, A. M. (2011). Multiple imputation using chained equations: issues and guidance for practice. Statistics in medicine, 30(4), [28] Yuan, Y. (2011). Multiple imputation using SAS software. Journal of Statistical Software, 45(6), [29] Yuan, Y. (2014). Sensitivity Analysis in Multiple Imputation for Missing Data. In Proceedings of the SAS Global Forum 2014 Conference. ( [30]. (2015). (2) (Multiple Imputation MI). SAS. 81

83 7 Shared Parameter Model 7.1 Selection Model (SM) Pattern Mixture Model (PMM) 5 6 MAR MNAR MNAR Wu and Carroll (1988) Wu and Bailey (1989) Follmann and Wu (1995) Shared Parameter Model ( SPM) SPM ( ) SPM (Y ij, R ij, X ij ) i(i = 1,, N) j(j = 1,, n) Y ij i j i Y i o Y i m Y i = (Y o, Y m ) R ij 1 j 0 j f(y i, R i, b i X i, θ, ψ) (7.1) X i Y i R i i θ ψ D i D i = j R ij + 1 D i = n + 1 X i θ ψ i q b i = (b i1, b i2,..., b iq ) 0 G i f(y i, R i, b i ) SPM f(y i, R i, b i ) = f(y i R i, b i )f(r i b i )f(b i ) (7.2) SPM Y ij R ij b i f(y i R i, b i ) = f(y i b i ) (7.3) Y i Y i o Y i m b i (7.2) f(y i b i ) = f(y i o b i )f(y i m b i ) (7.4) f(y i, R i, b i ) = f(y i o b i )f(y i m b i )f(r i b i )f(b i ) (7.5) 82

84 SPM share Fitzmaurice et al. (2008) SPM MNAR Y i = (Yi o,ym i ) R ij f(r ij Y o i, Y m b i ) = i f(r ij b i )f(y o i, Y m i b i )f(b i )db i b i f(y o i, Y m i b i )f(b i )db i = f(r ij b i )f(b i Y o i, Y m i )db i (7.6) b i f(b i Y i o, Y i m ) Y i m Y i m SPM MNAR SPM Selection Model (SM) Pattern Mixture Model (PMM) MNAR 7.2 SPM f(y o i,r i ) f(y o i, R i ) = f(y o i, Y m m i, R i, b i )db i dy i y m b = f(y o i, Y m m i R i, b i )f(r i b i )f(b i )db i dy i y m b = f(y o i, Y m m i b i )f(r i b i )f(b i )db i dy i ( (7.3)) y m b = f(y o i b i )f(y m m i b i )f(r i b i )f(b i )db i dy i ( (7.4)) = = y m b b b ( ) f(y o i b i )f(r i b i )f(b i ) f(y m m i b i )dy i db i y m f(y i o b i )f(r i b i )f(b i )db i (7.7) SPM f(y o i b i ) f(r i b i ) f(y o i b i ) y ij (Group i ) (T ime ij ) (T ime ij Group i ) SPM ( ) 2 (a) (b) Creemers et al. (2010) 83

85 b i0, b i1 (a) f(y o i b i ) (7.8) (b) (7.9) Y ij = β 0 + β 1 T ime ij + β 2 Group i + β 3 T ime ij Group i + b i0 + e ij (7.8) Y ij = β 0 + β 1 T ime ij + β 2 Group i + β 3 T ime ij Group i + b i0 + b i1 T ime ij + e ij (7.9) (b i0, b i1 ) N(0, G) (7.8) b i1 G e ij f(r i b i ) Complementary log-log link (c) (d) Creemers et al. (2010) (c) (7.10) (d) (7.11) α 0j, γ 1, γ 2 P r(d i j) = 1 exp( exp(α 0j + γ 1 b i0 )) (7.10) P r(d i j) = 1 exp( exp(α 0j + γ 1 b i0 + γ 2 b i1 )) (7.11) (7.10) 1 (7.12) P r(d i 1) = 1 exp( exp(α 01 + γ 1 b i0 )) (7.12) 1 2 P r(d i 2) P r(d i 1) = 1 exp( exp(α 02 + γ 1 b i0 )) {1 exp( exp(α 01 + γ 1 b i0 ))} logitp r(d i j) = α 0j + γ 1 b i0 (7.13) logitp r(d i j) = α 0j + γ 1 b i0 + γ 2 b i1 (7.14) 7.3 L = f(y o i, R i ) Monte Carlo EM (McCulloch, 1997) Laplace (Gao, 2004) Gauss Gauss SPM SAS PROC NLMIXED PROC NLMIXED SPM Appendix C PROC NLMIXED 84

86 Gauss Gauss PROC NLMIXED SAS Pinheiro et al. (1995) Unweighted Analysis Approximate Conditional Model Appendix C Fitzmaurice et al. (2008) 7.4 SPM %shared_parameter SPM Little (1995) SPM SPM SM PMM estimand %shared_parameter %shared_parameter %shared_parameter Appendix C MODL, LINK, RANDOM_SLOP E MODL LINK CLOGLOG Complementary log-log link LOGIT RANDOM_SLOP E NONE LINEAR NONE LINEAR %shared_parameter 7.2 SPM %shared_parameter 2 [1] Creemers, A., Hens, N., Aerts, M., Molenberghs, G., Verbeke, G., and Kenward, M. G. (2010). A Sensitivity Analysis for Shared Parameter Models for Incomplete Longitudinal Outcomes. Biometrical Journal, 52(1), [2] Fitzmaurice, G., Davidian, M., Verbeke, G., and Molenberghs, G. (2008). Longitudinal data analysis. CRC Press. [3] Follmann, D., and Wu, M. (1995). An approximate generalized linear model with random effects for informative missing data. Biometrics, 51(1),

87 [4] Gao, S. (2004). A shared random effect parameter approach for longitudinal dementia data with non ignorable missing data. Statistics in Medicine, 23(2), [5] Little, R. J. (1995). Modeling the drop-out mechanism in repeated-measures studies. Journal of the American Statistical Association, 90(431), [6] McCulloch, C. E. (1997). Maximum likelihood algorithms for generalized linear mixed models. Journal of the American statistical Association, 92(437), [7] Pinheiro, J. C., and Bates, D. M. (1995). Approximations to the log-likelihood function in the nonlinear mixedeffects model. Journal of computational and Graphical Statistics, 4(1), [8] Wu, M. C., and Carroll, R. J. (1988). Estimation and comparison of changes in the presence of informative right censoring by modeling the censoring process. Biometrics, 44(1), [9] Wu, M. C., and Bailey, K. R. (1989). Estimation and comparison of changes in the presence of informative right censoring: conditional linear model. Biometrics, 45(3),

88 8 8.1 IPWCC wgee Doubly Robust estimand estimand 3 MAR SM PMM SPM MAR µ µ i Y i R i 1 0 n µ µ = n Y i i=1 n (8.1) µ µ Y i 1 µ µ µ = n R i Y i i=1 n (8.2) R i i=1 8.2 Y i R i = 0 Y i R i Y i = 0 µ MCAR µ µ MNAR 1 Y i R i MCAR MNAR MCAR : P Ri Y i (R i = 1, Y i = y i θ, ψ) = P Ri (R i = 1 ψ) P Yi (Y i = y i θ) 1 MAR Y i 1 87

89 MNAR : P Ri Y i (R i = 1, Y i = y i θ, ψ) = P Ri Y i (R i = 1 Y i = y i, ψ) P Yi (Y i = y i θ) P Ri Y i (R i = 1 Y i = y i, ψ) R i π(y i ) = P Ri Y i (R i = 1 Y i = y i, ψ) (8.3) R i π(y i ) Bernoulli E[R i Y i ] = 1 P (R i = 1 Y i ) + 0 P (R i = 0 Y i ) = P (R i = 1 Y i ) = π(y i ) (8.4) E[R i ] = E[E[R i Y i ]] = E[π(Y i )] (8.5) n 8.2 MCAR : n R i Y i i=1 µ = n = R i i=1 n 1 n i=1 n 1 n R i i=1 R i Y i n E[RY ] E[R] E[Y ] = = E[Y ] = µ (8.6) E[R] E[R] MNAR : (8.4) (8.5) n R i Y i i=1 µ = n = R i i=1 n 1 n i=1 n 1 n R i i=1 R i Y i n E[RY ] E[R] = = E[E[RY Y ]] E[R] E[Y π(y )] E[π(Y )] = E[Y E[R Y ]] E[E[R Y ]] E[Y ] = µ (8.7) µ MCAR µ MNAR µ 8.2 Inverse Probability Weighted Complete-Case (IPWCC) Estimator µ IPWCC (8.2) µ MNAR µ X i W i MAR 88

90 MAR (Tsiatis, 2006) P (R i = 1 Y i = y i, W i = w i, X i = x i ) = P (R i = 1 W i = w i, X i = x i, ψ) (8.8) W i auxiliary variable 2 3 Mallinckrodt, π(w i, X i, ψ) π(w i, X i, ψ) = P (R i = 1 W i = w i, X i = x i, ψ) (8.9) Y i 1/π µ µ IP W = 1 n n R i Y i i=1 π i (W i, X i, ψ) (8.10) ψ µ IP W = 1 n [ ] R i Y i n R i Y i n π i (W i, X i, ψ) E π i (W i, X i, ψ) i=1 [ [ RY ]] = E E Y, W, X π(w, X, ψ) [ ] Y = E E [R Y, W, X] π(w, X, ψ) [ ] Y = E π(w, X, ψ) π(w, X, ψ) = E [Y ] = µ (8.11) π Inverse Probability Weighted Complete-Case (IPWCC) Estimator 8.10 (Robins et al., 1994) IPWCC π MAR IPWCC 0 IPWCC Williamson et al., 2012 P (Y, R) = P (Y )P (R Y ) Selection Model Selection Model IPWCC 2 Collins et al. (2001) a) b) c)

91 1. ψ π(w, X, ψ) = P (R = 1 W, X, ψ) logit {P (R = 1 W, X)} = ψ 0 + Wψ 1 + Xψ 2 2. Y IPWCC µ IP W (Robins et al., 1994) µ IP W = 1 n n R i Y i i=1 π i (W i, X i, ψ) IPWCC estimators for Estimating Equations µ θ n U(Y i, X i, θ) = 0 (8.12) i=1 4 n X i (Y i X iβ) = 0 (8.13) i= Y i U IPW n i=1 n i=1 β IP W R i π i U(Y i, X i, θ) = 0 (8.14) R i π i X i (Y i X iβ) = 0 (8.15) µ IP W i = X i β IP W IPWCC IPWCC 4 Y = X β + ϵ E[ϵ] = 0 ϵ 90

92 ( ) Y1 1. n, Yn n π 1 π n B B 1000 Barker, θ (b) 3. B θ (1), θ (B) θ (2008) Carpenter and Bithell(2000) 8.3 weighted Generalized Estimating Equation (wgee) IPWCC 1 Generalized Estimating Equation GEE GEE IPWCC GEE β Liang and Zeger, 1986; Pan et al., 2000; Pepe and Anderson, 1994 MAR MNAR GEE MAR MNAR "non-ignorable" Hogan et al., 2004; Touloumi et al., 2001 GEE MAR weighted Generalized Estimating Equation (wgee) Robins et al., 1995 observation-specific subject-specific 2 Robins et al., 1995; Preisser et al., 2002; Fitzmaurice et al., 1995; Hogan et al., %WGEE subject-specific Proc GEE observation-specific, 2015) W ij 5 MNAR MAR wgee Mallinckrodt, 2013, 12.4 (2015) wgee wgee wgee observation-specific subject-specific ) Observation-specific (OS) weighted GEE Y i X i β S os (β) = N i=1 µ i β V i(ρ) 1 i (Y i µ i ) = 0 (8.16) i i µ i = E [Y i ] i 5 Inclusive Model 91

93 η i = g(µ i ) = X i V i (ρ) i ρ i Subject-specific (SS) weighted GEE Y o i X i β S ss (β) = N i=1 i (ss) (µo i ) β (Vo i (ρ)) 1 (Y o i µ o i ) = 0 (8.17) i i µ o i = E [Yo i ] i η i = g(µ o i ) = X i V o i (ρ) i (ss) ρ i Observation-specific weighted GEE Observation-specific weighted GEE (8.18) (8.19) i j ϕ ij (W ij, X i, ψ) ψ (j 1) j ϕ ij (W ij, X i, ψ) = P (R ij = 1 R i,j 1 = 1, W ij, X i, ψ) (8.18) logit {P (R ij = 1 R i,j 1 = 1, W ij = w ij, X i = x i, ψ)} = W ij ψ 1 + X i ψ 2 (8.19) i j π ij j = 1 1 j π ij (W i, X i, ψ) = ϕ ik (W ik, X i, ψ) (8.20) k=1 Observation-specific weighted GEE ij = (π ij (W i, X i, ψ)) 1 Y ij i n i n i n i i (j, j) ij 0 MAR (Robins et al., 1995) Subject-specific weighted GEE Subject-specific i observation-specific i subject-specific i (ss) Fitzmaurice et al. (1995) i D i J, i D i = J k=1 R ik + 1 D i k R ik = 1 R ik = 0 D 1 (J + 1) (J + 1) D i d i P (D i = d i W i, X i, ψ) (8.18) (8.19) 92

94 (d i = J + 1) i (ss) = P (D = d i W i, X i, ψ) = P (R ij = 1 W i, X i, ψ) 1 = (π ij (W i, X i, ψ)) 1 (8.21) (d i J) i (ss) = P (D = d i W i, X i, ψ) = P (R idi = 0, R idi 1 = 1 W i, X i, ψ) 1 = ((π idi 1(W i, X i, ψ)) (1 ϕ idi (W idi, X i, ψ))) 1 (8.22) Subject-specific weighted GEE (8.21) (8.22) D i d i P (D = d i W i, X i, ψ) observation-specific weighted GEE MAR (Fitzmaurice et al., 1995) β Subject-specific wgee observation-specific wgee Robin, 1995; Preisser et al., 2002; Fitzmaurice et al., 1995; Hogan et al., 2004; Seaman, 2009 SAS Ver 9.3 PROC GENMOD SS OS Ver 9.4 PROC GEE SS OS (2015) 12 %WGEE subject-specific weighted GEE β 1. (8.18) (8.19) logistic (8.20) (8.21) (8.22) i 2. β β 0 V i (ρ) 4. (8.23) β β r+1 = β r + [ n i=1 µ i β V i(ρ) 1 µ i β ] 1 [ n 5. 3., 4. β i=1 µ i β V i(ρ) 1 i (Y i µ i ) ] (8.23) 6. observation-specific wgee independent PROC GENMOD PROC GEE (Rodriguez, 2014) PROC GENMOD 6 Robin, (1995) Preisser et al., (2002) 8.4 Doubly Robust (DR) IPWCC Y i Y i 6 Proc GENMOD 93

95 Augmented IPW(AIPW) (Rotnitzky et al., 1998) IPWCC MAR µ 1 µ 0 µ 1, µ 0 AIPW (Tsiatis, 2006) µ 1 = 1 n µ 0 = 1 n n i=1 n i=1 R i1 Y i1 π(w i1, X i1, ψ) 1 n R i0 Y i0 π(w i0, X i0, ψ) 1 n { } n R i1 i=1 π(w i1, X i1, ψ) 1 g(w i1, X i1, β) { } n R i0 π(w i0, X i0, ψ) 1 g(w i0, X i0, β) (8.24) i=1 Y 1 IPWCC g(w ij, X ij, β) β Y g(w ij, X ij, β) = E[Y i W ij, X ij ] 8.24 π(w ij, X ij, ψ) Y g(w ij, X ij, β) β Tsiatis, 2006; Cao et al., 2009; Carpenter et al., 2006 Doubly Robust µ 1 µ 0 µ 1 µ j = 0, 1 µ j = 1 n [ n Y ij + R ij π(w ij, X ij, ψ) i=1 π(w ij, X ij, ψ) { Y ij g(w ij, X ij, β) } ] E[ µ 1 ] E[ µ 0 ] β ψ β ψ 1. π(w ij, X ij, ψ) ψ j = 0, 1 [ Y ij + R ij π(w ij, X ij, ψ) π(w ij, X ij, ψ) ] {Y ij g(w ij, X ij, β )} E[ µ j ] = E [ Rij P (R ij = 1 W ij, X ij, ψ) = E[Y ij ] + E P (R ij = 1 W ij, X ij, ψ) [ E[P (Rij = 1 W ij, X ij, ψ)] P (R ij = 1 W ij, X ij, ψ) = E[Y ij ] + E P (R ij = 1 W ij, X ij, ψ) = E[Y ij ] + E [0 {Y ij g(w ij, X ij, β )}] = E[Y ij ] = µ j ] {Y ij g(w ij, X ij, β )} ( (8.5)) ] {Y ij g(w ij, X ij, β )} 2. Y g(w ij, X ij, β) j = 0, 1 [ Y ij + R ij π(w ij, X ij, ψ ) π(w ij, X ij, ψ ) E[ µ j ] = E = [ Rij π(w ij, X ij, ψ ) E[Y ij ] + E π(w ij, X ij, ψ ) 94 ] {Y ij g(w ij, X ij, β)} ] E [{Y ij g(w ij, X ij, β) W ij, X ij }]

96 [ Rij π(w ij, X ij, ψ ) = E[Y ij ] + E π(w ij, X ij, ψ ) = E[Y ij ] + E = E[Y ij ] = µ j [ Rij π(w ij, X ij, ψ ) π(w ij, X ij, ψ ) ] {E [Y ij W ij, X ij ] g(w ij, X ij, β)} ] 0 µ j IPWCC IPWCC (Seaman and Copas, 2009; Belinda, 2013) Doubly Robust Doubly Robust Tsiatis (2006) (2014) µ = n Y i i=1 n µ = n R i Y i n i=1 R i i=1 MAR MNAR IPWCC µ IP W = 1 n n R i Y i i=1 π(w i, X i, ψ) MAR wgee MAR IPWCC 95

97 Doubly Robust MAR Y Doubly-Robust Double Robust [1] Barker, N. (2005). A practical introduction to the bootstrap using the SAS system. In SAS Conference Proceedings: Phuse 2005: October ; Heidelberg, Germany SAS. [2] Cao, W., Tsiatis, A. A., and Davidian, M. (2009). Improving efficiency and robustness of the doubly robust estimator for a population mean with incomplete data. Biometrika, asp033. [3] Carpenter, J., and Bithell, J. (2000). Bootstrap confidence intervals: when, which, what? A practical guide for medical statisticians. Statistics in medicine, 19(9), [4] Carpenter, J. R., Kenward, M. G., and Vansteelandt, S. (2006). A comparison of multiple imputation and doubly robust estimation for analyses with missing data. Journal of the Royal Statistical Society: Series A (Statistics in Society), 169(3), [5] Collins, L. M., Schafer, J. L., and Kam, C. M. (2001). A comparison of inclusive and restrictive strategies in modern missing data procedures. Psychological methods,6(4), 330. [6] Fitzmaurice, G. M., Molenberghs, G., and Lipsitz, S. R. (1995). Regression models for longitudinal binary responses with informative drop-outs. Journal of the Royal Statistical Society. Series B (Methodological), [7]. (2014).., 62(1), [8] Hogan, J. W., Roy, J., and Korkontzelou, C. (2004). Handling drop out in longitudinal studies. Statistics in medicine, 23(9), [9], (2008), EM MCMC., [10]. (2015). (3)Proc GEE wgee SAS [11] Liang, K. Y., and Zeger, S. L. (1986). Longitudinal data analysis using generalized linear models. Biometrika,73(1), [12] Mallinckrodt, C. H. (2013). Preventing and Treating Missing Data in Longitudinal Clinical Trials. Cambridge Press. [13] Pan, W., Louis, T. A., and Connett, J. E. (2000). A note on marginal linear regression with correlated response data. The American Statistician, 54(3), [14] Pepe, S, M., and Anderson, G. L. (1994). A cautionary note on inference for marginal regression models with longitudinal data and general correlated response data. Communications in Statistics-Simulation and Computation, 23(4),

98 [15] Preisser, J. S., Lohman, K. K., and Rathouz, P. J. (2002). Performance of weighted estimating equations for longitudinal binary data with drop outs missing at random. Statistics in medicine, 21(20), [16] Robins, J. M., Rotnitzky, A., and Zhao, L. P. (1994). Estimation of regression coefficients when some regressors are not always observed. Journal of the American Statistical Association, 89(427), [17] Robins, J. M., Rotnitzky, A., and Zhao, L. P. (1995). Analysis of semiparametric regression models for repeated outcomes in the presence of missing data. Journal of the American Statistical Association, 90(429), [18] Rodriguez, R. N., and Stokes, M. (2014). SAS/STATR 13.1 Round-Up. [19] Rotnitzky, A., Robins, J. M., and Scharfstein, D. O. (1998). Semiparametric regression for repeated outcomes with nonignorable nonresponse. Journal of the American Statistical Association, 93(444), [20] Seaman, S., and Copas, A. (2009). Doubly robust generalized estimating equations for longitudinal data. Statistics in medicine, 28(6), [21] Touloumi, G., Babiker, A. G., Pocock, S. J., and Darbyshire, J. H. (2001). Impact of missing data due to drop outs on estimators for rates of change in longitudinal studies: a simulation study. Statistics in medicine, 20(24), [22] Tsiatis, A. (2006). Semiparametric theory and missing data. Springer. [23] Williamson, E. J., Morley, R., Lucas, A., and Carpenter, J. R. (2012). Variance estimation for stratified propensity score estimators. Statistics in medicine, 31(15),

99 II MMRM

100 9 MMRM 9.1 ( ) LOCF ( ) LOCF complete case (Mallinckrodt et al., 2008) 4 LOCF MMRM(Mixed Model for Repeated Measures) LOCF 2 Rubin(1976) Little & Rubin(2002) 3 MCAR MAR MNAR 2 (Ignorable) (Non-ignorable) Laird(1988) MCAR MAR MNAR Laird & Ware (1982) ( ) ( ) ( Direct Likelifood DL ) MAR( MCAR) MAR Mallinckrodt et al. (2001a; 2001b) "MMRM" MMRM MAR DL MMRM MAR SM 5 99

101 9.2 MMRM MMRM MMRM (Linear Mixed Model LMM) Laird & Ware (1982) LMM 1 Y i = X i β + Z i b i + ϵ i, i = 1,..., N b i N(0, D), ϵ i N(0, Σ i ), (9.1) b 1,..., b N, ϵ 1,..., ϵ N. Y i i n i 2 β p b i q ( ) X i Z i (n i p) (n i q) ϵ i n i D (i, j) d ij d ij = d ji (q q) Σ i (n i n i ) ( 3 ) LMM (General Linear Model LM) ( ) (9.1) Y i N(X i β, V i ), V i = Z i DZ i + Σ i. (9.1) V i Mallinckrodt et al. (2001a; 2001b) MMRM MMRM LMM 4 "MMRM" LMM (Compound Symmetory CS) CS CS CS 1 (1-AutoRegression AR(1) ) Toeplitz ( ) (Unstructured UN ) 2 (Akaike Information Criteria AIC) Bayes (Bayessian Information Criteria BIC) Lu & Mehrotra (2009) Mallinckrodt et al. (2001a;2001b) 1 Laird & Ware (1982) 2 n n i (n i n, i = 1,..., N) 3 4 MMRM 100

102 9.1: MMRM Visit Baseline * * * * * * * Siddiqui (2011) III UN MMRM MMRM (Mallinckrodt et al., 2001b) Laird & Ware (1982) Cnaan et al. (1997) Verbeke & Molenberghs(1997) 9.3 SAS MMRM LMM SAS PROC MIXED MMRM 9.2 MMRM PROC MIXED DATA=datasetName; CLASS TREATMENT TIME_POINT SUBJECT_ID; MODEL RESPONSE = BASELINE TREATMENT TIME_POINT TREATMENT*TIME_POINT / DDFM=KR; LSMEANS TREATMENT*TIME_POINT / ALPHA=0.05 CL DIFF=CONTROL( PLACEBO, tn ); REPEATED TIME_POINT / SUBJECT=SUBJECT_ID TYPE=UN; RUN; 5 UN 101

103 9.2: MMRM SUBJECT_ID TREATMENT BASELINE TIME_POINT RESPONSE A-01 Drug 1234 t A-01 Drug 1234 t A-02 Placebo 1234 t A-03 Placebo 1234 t PROC MIXED SAS PROC MIXED PROC MIXED MMRM EMPIRICAL SCORING EMPRICAL ( ) (Huber, 1967; Liang & Zeger, 1986) UN (CS ) EMPRICAL 6 PROC MIXED (Restricted Maximum Likelihood Method REML) Newton-Raphson UN (Lu & Mehrotra, 2009) PROC MIXED Newton-Raphson Fisher s scoring SCORING SCORING= Fisher s Scoring Newton-Raphson Fisher s Scoring SCORING= CLASS CLASS 6 MCAR (Mallinckrodt, 2013) 102

104 9.3.3 MODEL MODEL ( ) Dinh & Yang (2011) MODEL DDFM PROC MIXED 7 Satterthwaite Kenward-Roger (KR) (Kenward & Roger, 1997) LSMEANS LSMEANS TREATMENT TIME_POINT (ALPHA ) (CL ) (DIFF ) ADJUST REPEATED REPEATED ϵ i Σ i SUBJECT TYPE 8 ( ) REPEATED Σ i R Σ i RCORR RANDOM MMRM RANDOM MMRM MMRM UN UN ( ) 10 REML REPEATED RANDOM 7 8 CS AR(1) Toeplitz UN SAS HELP 9 i Σ i "R=i" "RCORR=i" 10 (Identifiability) 103

105 Hessian Appendix A A MMRM Mallinckrodt et al. (2008) MMRM MAR ( MNAR ) MAR MMRM UN REML MAR (LaVange, 2014) "MMRM" Mallinckrodt et al. (2008) MMRM (REML) ( ) Kenward-Roger α = 0.05 ( ) 2 ( ) MMRM MAR Collins et al. (2001) Mallinckrodt et al. (2013) "Restrictive Model" "Inclusive Model" 10 MAR (Multiple Imputaion MI) (weighted Generalized Estimating Equation wgee) MMRM MMRM Lu et al. (2008) 104

106 J n 1, n 2,..., n J n 1 n 2 n J n J+1 = 0 i Y i = (y i1,..., y ij ) µ = (µ 1,..., µ J ) Σ Σ (j, k) σ jk (j, k = 1,..., J) r j = n j /n 1 (j = 1,..., J) j ( ) r = (r 1,..., r J ) MMRM µ µ µ I 1 ( ) J Σ 1 j 0 j (J j) I = (n j n j+1 ) 0 (J j) j 0 (J j) (J j) j=1 Σ j Σ (j j) 0 kl 0 (k l) (l = 0 ) j j j ( ) J I R 1 j 0 j (J j) = (r j r j+1 ) 0 (J j) j 0 (J j) (J j) j=1 R j R (j j) R (j, k) σ jk /(σ j σ k ) (j, k = 1,..., J) σ j = σ 1/2 jj (9.2) Var( µ J ) = ϕ(σj 2/n 1) ϕ = [I ] 1 JJ µ J ϕ Lu et al. (2008) ϕ "Inflation factor" (9.3) ϕ r R Lu et al. (2008) (9.2) (9.3) 9.6 MAR SM MMRM MNAR MAR Mallinckrodt et al. (2001b) MNAR MMRM (Yuan & Little, 2009) MNAR MMRM MAR MNAR (Mallinckrodt et al., 2004; Shen et al., 2006; Mallinckrodt et al., 2008) [1] Collins, L. M., Schafer, J. L. & Kam, C. M. (2001). A comparison of inclusive and restrictive strategies in modern missing data process. Psychological Methods, 6(4), [2] Cnaan, A., Laird, N. M. & Slasor, P. (1997). Using the general linear mixed model to analysis unbalanced repeated measures and longitudinal data. Statistics in Medicine., 16,

107 [3] Dinh, P. & Yang, P. (2011). Handling baselines in repeated measures analysis with missing data at random. Journal of Biopharmaceutical Statistics, 21, [4] Huber, P. J. (1967). The behavior of maximum likelihood estimation under nonstandard conditions. Proceeding of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, 1, LeCam, L. M. and Neyman, J. editors. University of California Press, [5] Kenward, M. G. & Roger, J. H. (1997). Small sample inference for mixed effects from restricted maximum likelihood. Biometrics, 53(3), [6] Laird N. M. (1988). Missing Data in Longitudinal Studies. Stat. Med., [7] Laird, N. M. & Ware, J. H. (1982). Random-effects models for longitudinal data. Biometrics, 38(4), [8] LaVange, L. M. (2014). The role of statistics in regulatory decision making. Therapeutic Innovation & Regulatory Science, 48(1), [9] Liang, Kung-Yee, & Zeger, S. L. (1986). Longitudinal data analysis using generalized linear models. Biometrika, 73(1), [10] Little, R. J. A. & Rubin, D. B. (2002). Statistical analysis with missing data, 2nded. New York: Wiley. [11] Lu, K., Luo, X. & Chen, P. Y. (2008). Sample size estimation for repeated measures analysis in randomized clinical trials with missing data. The International Journal of Biostatistics, 4(1), [12] Lu, K. & Mehrotra D. V. (2009). Specification of covariance structure in longitudinal data analysis for randomized clinical trials. Statistics in Medicine, 29, [13] Mallinckrodt, C. H. (2013). Preventing and Treating Missing Data in Longitudinal Clinical Trials. Cambridge University press. [14] Mallinckrodt, C. H., Clark, W. S. & David, S. R. (2001a). Accounting for dropout bias using mixed-effects models. Journal Biopharmaceutical Statistics, 11, [15] Mallinckrodt, C. H., Clark, W. S. & David, S. R. (2001b). Type I error rates from mixed effects model repeated measures versus fixed effects anova with missing values imputed via last observation carried forward. Drug Information Journal, 35, [16] Mallinckrodt, C. H., Kaiser, C. J., Watkin, J. G., Molenberghs, G. & Carroll R. J. (2004). The effect of correlation structure on treatment contrasts estimated from incomplete clinical trial data with likelihood-based repeated measures compared with last observation carried forward ANOVA. Clinical trials (London, England), 1, [17] Mallinckrodt, C. H., Lane, P. W., Schnell, D., Peng, Y. & Maucuso, J. P. (2008). Recommendations for the primary analysis of continuous endpoints in longitudinal clinical trials. Drug Information Journal, 42, [18] Mallinckrodt, C. H., Roger, J., Chuang-Stein, C., Molenberghs, G., O Kelly, M., Ratitch, B., Janssens, M. & Bunouf, P. (2013) Recent developments in the prevention and treatment of missing data. Therapeutic Innovation & Regulatory Science, published online. [19] Rubin, D. B. (1976). Inference and missing data. Biometrika, 63, [20] Shen, S., Beunckens, C., Mallinckrodt, C. & Molenberghs, G. (2006). A local influence sensitivity analysis for incomplete longitudinal depression data. J. Biopharm. Stat., 16, [21] Siddiqui, O. (2011). MMRM versus MI in dealing with missing data - A comparison based on 25 NDA data sets. Journal of Biopharmaceutical Statistics, 21,

108 [22] Verbeke, G. & Molenberghs, G. (1997). Linear mixed models in practice: a SAS oriented approach, New York : Springer-Verlag. [23] Yuan, Y. & Little, R. J. A. (2009). Mixed-effect hybrid models for longitudinal data with nonignorable dropout. Biometrics, 65, [24] (2001). ( ).. 107

109 NRC(2010) 3 Mallinckrodt (2013) estimand Analytic Road Map 10.1 Mallinckrodt(2013) Analytic Road Map 2 (1) 9.2 MMRM (2) MAR (3)estimand estimand 3 efficacy (1) (3) (1) (2) MAR (3) estimand effectiveness Analytic Road Map Mallinckrodt (2013) 14 NRC(2010) 5 (1) (2) (3) estimand Mallinckrodt (2013) 10.1: Analytic Road Map 1 ICH E9 (R1) 2 MMRM DL Diagnostics: residuals, influence, correlation, time 108

110 10.2 III HAM-D (12 ) 20% 4 12 estimand estimand : HAM-D SD 10.1: HAM-D SD (3 ) (6 ) (9 ) (12 ) (SD) (SD) (SD) (SD) (SD) (4.1) (4.2) (5.4) (6.3) (6.8) SAS (4.2) (4.2) (4.6) (6.3) (6.1) 10.3: 3 109

111 id : trt : x0 : time : val : 9 MMRM SAS HAM-D Unstructured REML (1) Fisher s scoring (2) Toeplitz, Heterogeneous CS AR(1) CS VC Kenward Roger 4 0 5% ODS OUTPUT DIFFS = diffs; PROC MIXED DATA = inputds; RUN; CLASS trt time id ; * ; MODEL val = x0 trt time trt*time / DDFM = kr; * = ; REPEATED time / TYPE = un SUBJECT = id; LSMEANS trt*time / DIFF; % 10.1 Analytic Road Map : 4 SE p (SE) MMRM (0.69) (0.70)

112 10.3: estimand efficacy 3 (primary) MAR MMRM 1 efficacy MAR MMRM ( ) MNAR SM ( ) efficacy PMM MNAR MMRM 3 pmi (estimand) effectiveness 6 (secondary) MNAR + MMRM Mallinckrodt (2013) 12.3 " " " " " " " " NRC (2010) 4 (p49) Mallinckrodt (2013) (auxiliary variables) MNAR MAR Mallinckrodt (2013) Collins et al. (2001) "Restrictive Model" "Inclusive Model" Restrictive Model Inclusive Model MI wgee MMRM (Unstructured) Unstructured Toeplitz Heterogeneous CS AR(1) CS VC Unstructured AIC 9 111

113 MMRM SAS MIXED PROCEDURE RESIDUAL (1) (2) Mallinckrodt (2013) p134 Student 2 Appendix A Student Student 2 31 MMRM 10.4 SE MMRM Cook D MIXED PROCEDURE INFLUENCE Cook D Cook D 0.03 Cook D 10.4 Cook D MMRM 10.4 local influence Verbeke and Molenberghs (2000) Molenberghs and Kenward (2007) 10.4: Student Cook D 112

114 10.4: SE Student Cook D MCAR MAR MNAR MCAR (Mallinckrodt, 2013) MNAR MAR (Little, 1995) 9 (MMRM) MNAR MAR MAR (Mallinckrodt, 2013) Type (i) Type (ii) Type (i) (ii) (NRC, 2010, p85) Type (ii) (testable) Type (i) (untestable) MAR MNAR Type (i) MAR MNAR NRC(2010) 5 MAR MNAR MAR (Recommendation 15) MNAR PMM SM SPM SAS NRC(2010) Mallinckrodt (2013) SM PMM SPM SM SM (10.1) (10.2) ψ 5 ψ 6 Y ij Y ij ψ 5 ψ 6 0 MAR 0 MNAR ψ 5 ψ 6 Mallinckrodt et al.,

115 SM Type (i) f(y i, R i θ, ψ) = f(y i θ) f(r i Y i, ψ) (10.1) 1 logit{p r(r ij = 0 R i1 = 1,, R i,j 1 = 1, Y i, X i, ψ)} = ψ 1 + ψ 3 Y i,j 1 + ψ 5 Y ij 2 logit{p r(r ij = 0 R i1 = 1,, R i,j 1 = 1, Y i, X i, ψ)} = ψ 2 + ψ 4 Y i,j 1 + ψ 6 Y ij (10.2) SM %SM_GridSearch %SM_GridSearch( psi5grid = , psi6grid = , const = 4, inputds = inputds, * ; covtype response modl clasvar mech = un, = val, derivative = 1, method = x0 trt time trt*time, = trt time, = MNARS, = %str(qn), out1 = &h._sp_estimates, * ; out2 = &h._sp_diffs, * ; out3 = &h._sp_lsmeans, * lsmeans ; debug = 0 ); 10.5: (SM) 4 (SE) SE (0.72) -9.91(0.72) (0.71) -9.84(0.71) (0.70) -9.74(0.71) (0.70) -9.58(0.70) (0.69) -9.31(0.70) SM (0.69) -8.96(0.70) (0.71) -8.60(0.72) (0.73) -8.29(0.74) (0.75) -8.09(0.75) (0.76) -8.00(0.76) (0.76) -7.94(0.76)

116 10.5: SE SM 10.6: SE (SM) PMM PMM Y 1, Y 2, Y 3, Y NRC(2010) 5 PMM (6 ) Case 7 NFMV+ACMV+ g s 1 (Y s Y 1,..., Y s 1 ) = f s (Y s Y 1,..., Y s 1 ) T = ω sj f j (Y s Y 1,.., Y s 1 ) (10.3) j=s +4 NFMV ω 43 + ω 44 = 1, ω 32 + ω 33 + ω 34 = 1 δ = ω 43 ω 44 = 1 δ f 4 (1234) = f 4 (1234) f 3 (1234) = f 3 (123) f 3 (4 123) = f 3 (123) g 3 (4 123) f 2 (1234) = f 2 (12) f 2 (3 12) f 2 (4 123) = f 2 (12) g 2 (3 12)f 3 (4 123) ( NFMV) = f 2 (12) g 2 (3 12) [ω 43 f 3 (4 123) + ω 44 f 4 (4 123)] [ ] = f 2 (12) g 2 (3 12) δ g 3 (4 123) + (1 δ) f 4 (4 123) f 1 (1234) = f 1 (1) f 1 (2 1) f 1 (34 12) = f 1 (1) g 1 (2 1) f 1 (3 12) f 1 (4 123) = f 1 (1) g 1 (2 1)f 2 (3 12) f 3 (4 123) ( NFMV) = f 1 (1) g 1 (2 1) [ω 32 f 2 (3 12) + ω 33 f 3 (3 12) + ω 34 f 4 (3 12)] [ω 43 f 3 (4 123) + ω 44 f 4 (4 123)] = f 1 (1) g 1 (2 1) [ω 32 f 2 (3 12) + ω 33 f 3 (3 12) + ω 34 f 4 (3 12)] 115 ( ω 43 = δ, ω 44 = 1 δ)

117 [ ] δ g 3 (4 123) + (1 δ) f 4 (4 123) ( ω 43 = δ, ω 44 = 1 δ) [ ] = f 1 (1) g 1 (2 1) ω 32 g 2 (3 12) + ω 33 f 3 (3 12) + ω 34 f 4 (3 12) [ ] δ g 3 (4 123) + (1 δ) f 4 (4 123) f 0 (1234) = f 0 (1) f 0 (2 1) f 0 (34 12) = g 0 (1)f 0 (2 1) f 0 (3 12) f 0 (4 123) = g 0 (1)f 1 (2 1)f 2 (3 12)f 3 (4 123) = g 0 (1) [ω 21 f 1 (2 1) + ω 22 f 2 (2 1) + ω 23 f 3 (2 1) + ω 24 f 4 (2 1)] [ ] ω 32 g 2 (3 12) + ω 33 f 3 (3 12) + ω 34 f 4 (3 12) [ ] ω 43 g 3 (4 123) + (1 ω 44 ) f 4 (4 123) ( NFMV) = g 0 (1) [ω 21 f 1 (2 1) + ω 22 f 2 (2 1) + ω 23 f 3 (2 1) + ω 24 f 4 (2 1)] [ ] ω 32 g 2 (3 12) + ω 33 f 3 (3 12) + ω 34 f 4 (3 12) [ ] δ g 3 (4 123) + (1 δ) f 4 (4 123) ( ω 43 = δ, ω 44 = 1 δ) [ ] = g 0 (1) ω 21 g 1 (2 1) + ω 22 f 2 (2 1) + ω 23 f 3 (2 1) + ω 24 f 4 (2 1) [ ] ω 32 g 2 (3 12) + ω 33 f 3 (3 12) + ω 34 f 4 (3 12) [ ] δ g 3 (4 123) + (1 δ) f 4 (4 123) (10.4) α 3 f 3 (123) δ = ω 43 = α 3 f 3 (123) + α 4 f 4 (123), 1 δ = ω α 4 f 4 (123) 44 = α 3 f 3 (123) + α 4 f 4 (123) α 2 f 2 (12) ω 32 = α 2 f 2 (12) + α 3 f 3 (12) + α 4 f 4 (12), ω α 3 f 3 (12) 33 = α 2 f 2 (12) + α 3 f 3 (12) + α 4 f 4 (12), α 4 f 4 (12) ω 34 = α 2 f 2 (12) + α 3 f 3 (12) + α 4 f 4 (12), α 1 f 1 (1) ω 21 = α 1 f 1 (1) + α 2 f 2 (1) + α 3 f 3 (1) + α 4 f 4 (1), ω α 2 f 2 (1) 22 = α 1 f 1 (1) + α 2 f 2 (1) + α 3 f 3 (1) + α 4 f 4 (1), α 3 f 3 (1) ω 23 = α 1 f 1 (1) + α 2 f 2 (1) + α 3 f 3 (1) + α 4 f 4 (1), ω α 4 f 4 (1) 24 = α 1 f 1 (1) + α 2 f 2 (1) + α 3 f 3 (1) + α 4 f 4 (1) g 1 (2 1) g 2 (3 12) g 3 (4 123) ACMV g s 1 (Y s Y 1,..., Y s 1 ) = f s (Y s Y 1,..., Y s 1 ) (10.5) g 0 (1) = f 1 (1 ) g 1 (2 1) = f 2 (2 1) g 2 (3 12) = f 3 (3 12) (10.6) g 3 (4 123) = f 4 (4 123) 116

118 MAR efficacy estimand 3 4 NRC(2010) PMM 5 E[Y 1 R 1] f 1 (1) E[Y 2 Y 1, R 2] f 2 (2 1) R E[Y 1 R 1] = µ 1 E[Y 2 Y 1, R 2] = µ 2 + Y 1 β 2 E[Y 3 Y 2, R 3] = µ 3 + Y 2β 3 (10.7) E[Y 4 Y 3, R = 4] = µ 4 + Y 3β 4 NFMV E[Y 2 Y 1, R = 0] = E[Y 2 Y 1, R 1] E[Y 3 Y 2, R = 0] = E[Y 3 Y 2, R = 1] = E[Y 3 Y 2, R 2] E[Y 4 Y 3, R = 0] = E[Y 4 Y 3, R = 1] = E[Y 4 Y 3, R = 2] = E[Y 4 Y 3, R 3] g (10.6) E[Y 1 R = 0] = µ 1 + E[Y 2 Y 1, R = 1] = µ Y 1 β 2 E[Y 3 Y 2, R = 2] = µ Y 2β 3 (10.8) E[Y 4 Y 3, R = 3] = µ Y 3β 4 E[Y 1 R = 0] g 0 (1) = f 1 (1 ) 6. PMM SM SM PMM SM MAR PMM MAR SM PMM PMM (%delta_pmm) ACMV effectiveness 5 NRC(2010) 6 (10.6) (10.8) ( ) Y2 f 2 Y2 Y 1 dy2 = µ E[Y 2 Y 1, R = 1] g 1 (2 1) E[Y 2 Y 1, R = 1] = = = Y 2 g 1 (2 1) dy ( Y 2 f 2 Y2 ) Y1 dy2 ( (Y 2 )f 2 Y2 ) Y 1 dy2 + f 2 ( Y2 Y 1 ) dy2 = µ

119 %delta_pmm(datain = inputds, trtname = trt, subjname = id, visname = time, basecont = %str(v0), postcont = %str(val), seed = , nimp = 10, deltavis = %str(first), deltacont = %str(&delta), deltacontarm = %str(0,1), deltacontmethod = %str(meanabs), favorcont = %str(&fav), primaryname = val, analcovarcont = %str(v0), analmethod = mmrm, repstr = %str(un), trtref = 0, dataout = data0imputed1, resout = pmmresults1 ); &deltavis 1 ("FIRST") ("ALL") &delta &fav 6 (&delta &fav = (3, high, -3.0) (2, high, -2.0) (1, high, -1.0) (0, high, 0.0) (1, low, 1.0) (2, low, 2.0) (3, low, 3.0) 10.7: SE PMM 10.8: SE (PMM) 118

120 10.6: (PMM) 4 (SE) SE (0.69) -9.18(0.71) (0.69) -9.09(0.71) (0.69) -9.00(0.71) PMM NFV (NCMV) (0.69) -8.91(0.72) (0.69) -8.83(0.72) (0.69) -8.74(0.72) (0.70) -8.65(0.73) SPM 7 SPM (10.9) MNAR SPM MAR f(y i, R i, b i ) = f(y i o b i )f(y i m b i )f(r i b i )f(b i ) (10.9) 7 (10.10) Complementary log-log link Complementary log-log link (10.11 ) Y ij = β 0 + β 1 T ime ij + β 2 Group i + β 3 T ime ij Group i + b i0 + e ij (10.10) P r(d i j) = 1 exp( exp(α 0j + γ 1 b i0 )) (10.11) SPM SE %SHARED_PARAMETER(INPUTDS = inputds, SUBJVAR = id, TRTVAR = trt, TIME = time, MODL = %STR(val = x0 trt time trt*time), LINK = CLOGLOG, RANDOM_SLOPE = %STR(NONE), * ; DEBUG = 0 ); &LINK "CLOGLOG" Complementary log-log link "LOGIT" &RANDOM_SLOPE "NONE" "LINEAR" 119

121 : SPM 2 4 SE (SE) SPM Estimand 3 3 estimand efficacy effectiveness estimand efficacy estimand effectiveness estimand estimand Mallinckrodt (2013) 14 pmi MMRM %cbi_pmm %cbi_pmm(datain = inputds, trtname = trt, subjname = id, visname = time, basecont = %str(v0), baseclass =, postcont = %str(val), postclass =, seed = , nimp = 100, primaryname = val, analcovarcont = %str(v0), trtref = 0, analmethod = mmrm, repstr = un, dataout = data0imputed2, resout = pmmresults2 ); 120

122 10.8: pmi MMRM 4 SE (SE) pmi (0.70) (0.71) MMRM (1) (2) (3)estimand : MMRM estimand MMRM SM PMM+MMRM SPM pmi+mmrm estimand (NRC,2010; Mallinckrodt, 2013) [1] Collins, L. M., Schafer, J. L., and Kam, C. M. (2001). A comparison of inclusive and restrictive strategies in modern missing data procedures. Psychological methods, 6(4), 330. [2] Little, R. J. A. (1995). Modeling the drop-out mechanism in repeated measure studies. Journal of American statistical Association, 90(431): [3] Mallinckrodt, C. H. (2013). Preventing and Treating Missing Data in Longitudinal Clinical Trials. Cambridge Press. [4] Mallinckrodt, C., Roger, J., Chuang-Stein, C., Molenberghs, G., O Kelly, M., Ratitch, B., and Bunouf, P. (2013). Recent developments in the prevention and treatment of missing data. Therapeutic Innovation and Regulatory Science, [5] Molenberghs, G., and Kenward, M. G. (2007). Missing Data in Clinical Studies. Wiley. 121

123 [6] National Research Council. (2010). The Prevention and Treatment of Missing Data in Clinical Trials. The National Academies Press. [7] Verbeke, G., and Molenberghs, G. (2000). Linear Mixed Models for Longitudinal Data. Springer. 122

124 11 Estimand 11.1 Analytic Road Map Estimand 3 Mallinckrodt (2013) 6 estimand 1 estimand estimand estimand Major Depression Disorder, MDD 123

125 11.1: Mallinckrodt (2013) MDD 6 estimand Estimand rescue medication 1 effectiveness MMRM ANCOVA 2 efficacy (MAR) MMRM, PMM (ACMV), MI-ANCOVA, MI-MMRM, wgee (MNAR) SM, PMM (ACMV ), SPM 3 efficacy (MAR) MMRM, PMM (ACMV), MI-ANCOVA, MI-MMRM, wgee (MNAR) SM, PMM (ACMV ), SPM 4 effectiveness t, ANOVA, ANCOVA, LMM 5 effectiveness t, ANOVA, ANCOVA, LMM 6 effectiveness pmi ANCOVA, pmi MMRM BOCF ANCOVA, BOCF MMRM 124

126 Estimand 3 10 primary estimand estimand 3 MAR MMRM 9 PMM (ACMV) 6 MI ANCOVA MI MMRM (6 ) wgee 8 12 MNAR SM 5 PMM ACMV ) 6 SPM 7 LOCF ANCOVA Estimand 2 active run-in estimand 2 estimand 3 estimand 3 LOCF ANCOVA estimand Estimand 1 Estimand 1 MMRM 9 ANCOVA 3 Rescue medication controlled imputation Mallinckrodt (2013) 11 2 LOCF ANCOVA LOCF 3 LOCF LOCF 125

127 Estimand 6 Estimand 6 effectiveness estimand rescue medication rescue medication BOCF (4, 6 ) controlled imputation (pmi) (6 ) Estimand 4 1 AUC 1 2 t ANOVA ANCOVA LMM Appendix A Estimand 5 estimand t ANOVA ANCOVA LMM Appendix A 11.3 Analytic Road Map 10 Analytic Road Map 126

128 11.1: Analytic Road Map estimand efficacy MAR effectiveness estimand MNAR [1] Mallinckrodt, C. H. (2013). Preventing and treating missing data in longitudinal clinical trials. Cambridge University Press. 127

129 III III estimand estimand 3 efficacy MAR 12.1 Analytic Road Map Mallinckrodt, 2013 MMRM MI wgee 3 α 2 LOCF ANCOVA OC ANCOVA 12.1: Analytic Road Map 1 III MAR MNAR MAR MNAR MNAR 2 III MCAR MAR MCAR MNAR 2 128

130 MAR MNAR 10,000 α III HAM-D 2 g g = 1 g = / 4 4 (y 0 y i i i = 1, 2, 3, 4) SD 12.1: SD SD (4.0) 18.0 (5.0) 15.0 (5.0) 12.0 (6.0) 9.0 (6.0) 20.0 (4.0) 18.0 (5.0) 16.0 (5.0) 14.0 (6.0) 12.0 (6.0) 12.2: 129

131 12.2: : % 5% 10% 13% 15% α SD H 0 : µ A4 = µ P 4 µ A4 4 µ P 4 4 α 12.4: α SD : α 130

132 SD : 12.5: SD SD SD SD SD SD (4.1) (4.4) (5.7) (6.6) (6.9) (4.2) (4.5) (4.7) (6.5) (6.2) Bernoulli MAR MNAR y 0 y i i, i = 1, 2, 3, 4 MAR p i = 1 MNAR p i = exp( y i 1 ) exp( y i y i ) 131

133 12.5: MAR MNAR 12.6: MAR MNAR MAR SD SD SD SD SD (4.1) (4.3) (5.5) (6.4) (6.8) (4.2) (4.5) (4.7) (6.7) (6.2) MNAR SD SD SD SD SD (4.1) (4.2) (5.4) (6.3) (6.8) (4.2) (4.2) (4.6) (6.3) (6.1) 10,000 MAR MNAR 10,000 α 12.4 MAR MNAR 10,

134 MAR MNAR 10, α 5% MMRM MI ANCOVA wgee LOCF ANCOVA OC ANCOVA 12.7: Unstructured REML Kenward Roger 5 4 ANCOVA logistic subject-specific EXCH Missingdata.org.uk %WGEE LOCF 4 ANCOVA 4 4 ANCOVA 133

135 4 MAR 12.8: 10,000 MAR % 5.9% 10.5% 13.4% 15.4% 0.0% 6.0% 10.5% 13.8% 16.5% 12.9: MAR α % % 10,000 10,000 MSE SD MMRM MI ANCOVA wgee LOCF ANCOVA OC ANCOVA : MAR 134

136 MNAR 12.10: 10,000 MNAR % 6.0% 10.0% 12.6% 14.3% 0.0% 6.0% 10.4% 13.7% 16.1% 12.11: MNAR α % % 10,000 10,000 MSE SD MMRM MI ANCOVA wgee LOCF ANCOVA OC ANCOVA : MNAR α MAR MNAR wgee 5% wgee MAR MNAR MMRM OC ANCOVA MI ANCOVA LOCF ANCOVA wgee MMRM MI ANCOVA wgee LOCF ANCOVA SD wgee MMRM MI ANCOVA OC ANCOVA MMRM MI ANCOVA MAR MNAR α wgee α 5%

137 wgee (2015) α subject-specific Fitzmaurice et al. (1995) subject-specific (2015) α wgee LOCF ANCOVA α MMRM MI ANCOVA OC ANCOVA MMRM MCAR MAR MCAR MNAR 1 α III g g = 1 g = / 4 4 (y 0 y i i i = 1, 2, 3, 4) SD 12.12: SD SD (1.40) 6.00 (1.60) 5.00 (1.80) 5.00 (2.00) 5.00 (2.00) 7.00 (1.40) 6.75 (1.60) 6.50 (1.80) 6.25 (2.00) 6.00 (2.00) 12.8: 136

138 12.13: : % 20% 26% 30% 34% 0% 18% 30% 40% 48% α SD H 0 : µ A4 = µ P 4 µ A4 4 µ P 4 4 α 12.15: α SD : α 4 estimand α 1 2 estimand α α estimand estimand SD 1 137

139 12.10: 12.16: SD SD SD SD SD SD (1.4) (1.4) (2.0) (2.2) (2.3) (1.5) (1.4) (1.7) (2.2) (2.1) Bernoulli MCAR MAR MCAR+MAR MCAR MNAR MCAR+MNAR MCAR+MAR 100 / 50 MCAR 50 MAR 1 : 1 MCAR+MNAR 1 : 1 MCAR MAR MNAR MCAR g i p gi = exp(γ gi ) γ gi g = 1 g = 2 1 i = i = i = i = MAR i p i = 1 MNAR i p i = exp( y i 1 ) exp( y i y i ) 138

140 12.11: MCAR+MAR MCAR+MNAR 12.17: MCAR+MAR MCAR+MNAR MCAR+MAR SD SD SD SD SD (1.4) (1.4) (2.1) (2.2) (2.2) (1.5) (1.5) (1.7) (2.1) (2.2) MCAR+MNAR SD SD SD SD SD (1.4) (1.4) (2.1) (2.2) (2.1) (1.5) (1.5) (1.7) (2.1) (2.2) 10,000 MCAR+MAR MCAR+MNAR 10,000 α MCAR+MAR MCAR+MNAR 10,

141 MCAR+MAR MCAR+MNAR 10, α 5% MMRM MI ANCOVA wgee LOCF ANCOVA OC ANCOVA 12.18: Unstructured REML Kenward Roger ANCOVA logistic subject-specific EXCH Missingdata.org.uk %WGEE LOCF 4 ANCOVA 4 4 ANCOVA 140

142 4 MCAR+MAR 12.19: 10,000 MCAR+MAR % 20.0% 26.3% 30.3% 34.3% 0.0% 17.6% 30.7% 40.7% 48.9% 12.20: MCAR+MAR α % % 10,000 10,000 MSE SD MMRM MI ANCOVA wgee LOCF ANCOVA OC ANCOVA : MCAR+MAR 141

143 MCAR+MNAR 12.21: 10,000 MCAR+MNAR % 19.1% 24.8% 29.1% 33.2% 0.0% 18.8% 31.9% 41.9% 49.7% 12.22: MCAR+MNAR α % % 10,000 10,000 MSE SD MMRM MI ANCOVA wgee LOCF ANCOVA OC ANCOVA : MCAR+MNAR α MCAR+MAR MCAR+MNAR MMRM MI ANCOVA 5% LOCF ANCOVA MCAR+MAR MCAR+MNAR LOCF ANCOVA MMRM MI ANCOVA OC ANCOVA wgee MMRM MI ANCOVA wgee LOCF ANCOVA SD LOCF ANCOVA wgee MMRM MI ANCOVA MMRM MI ANCOVA MAR MNAR α wgee α 5%

144 LOCF ANCOVA α SD MMRM MI ANCOVA wgee LOCF ANCOVA OC ANCOVA 2 MMRM estimand 3 MAR Mallinckrodt (2013) LOCF ANCOVA α 1 5% 2 α LOCF ANCOVA α 100 / 12.14: 1 α 12.15: 2 α 143

145 [1] Fitzmaurice, G. M., Molenberghs, G., and Lipsitz, S. R. (1995). Regression models for longitudinal binary responses with informative drop-outs. Journal of the Royal Statistical Society. Series B (Methodological), [2],,,,. (2015). (3)Proc GEE wgee. SAS. [3] Mallinckrodt, C. H. (2013). Preventing and treating missing data in longitudinal clinical trials. Cambridge University Press. 144

146 SAS 1, 2 ( ), 3, 5, 11 Appendix A, B (2013 ) 2 (2.5), Appendix C (2014 ) 4, 7 (2014 ) 12 Appendix C Appendix C NRC EMA ( ) RI ( ) FDA, EMA, PMDA ( ) Meiji Seika

147 146

148 Appendix

149 Appendix A 4 A A A A A A A A A AR(1) A A Compound Symmetry A A A A.7 SAS A A A A A A A A A A A A A A A A A A.9.3 (REML) A A A

150 Appendix B 47 B B B B B MCAR B MAR B MNAR B MNAR B.3 5 Selection Model B.3.1 MNAR B i B j (j 3) i B i B.4 6 NFMV NFD ACMV MAR B B B B B B B B.4.2 NFMV ACMV B NFMV B NFMV B NFMV B ACMV B ACMV B ACMV B NFMV ACMV B B B.4.3 NRC (2010) PMM B.4.4 ACMV MAR B B B B B B.4.5 NFMV NFD B B B B B

151 Appendix C SAS 77 C C.2 Selection_Model C C C C C C.3 cbi_pmm C C C C C C.4 delta_pmm C C C C C C.5 Shared_Parameter C C C C

152 Appendix A A SAS MMRM MCAR A.2 i 1 n i 2 Y i = (Y i1,, Y ini ), (i = 1,, N) i (Linear Model, LM) Y i = X i β + ϵ i β X i ϵ i E[ϵ i ] = 0 ϵ i N(0, σ 2 I ni ) (Linear Mixed Model, LMM) b i Y i = X i β + Z i b i + ϵ i (A.1) Z i β b i 1 2 4

153 1 3 i j (j = 1, 2, 3) t j b i Y i1 = µ + t 1 + b i + ϵ i1 Y i2 = µ + t 2 + b i + ϵ i2 (A.2) Y i3 = µ + t 3 + b i + ϵ i3 µ Y i1 Y i = Y i2, β = t 1, ϵ t i = 2 Y i3 t 3 ϵ i1 ϵ i2 ϵ i3 X i = , Z i = (A.2) Y i = X i β + Z i b i + ϵ i Y i, X i, Z i β, b i, ϵ i β (estimation) b i 3 4 (i) (ii) 2 5 (i) 1 6 (ii) Y 1 = µ + b 1 + ϵ 1, Y 2 = µ + b 1 + ϵ 2 b 1, ϵ 1, ϵ 2 V [b 1 ] = σb 2, V [ϵ 1] = V [ϵ 2 ] = σ 2 Y 1, Y 2 Y 1 Y 2 Cov[Y 1, Y 2 ] = Cov[µ + b 1 + ϵ 1, µ + b 1 + ϵ 2 ] = Cov[b 1, b 1 ] = σ 2 b V [Y 1 ] = V [b 1 ] + V [ϵ 1 ] = σ 2 b + σ2, V [Y 2 ] = V [b 1 ] + V [ϵ 2 ] = σ 2 b + σ2 Corr[Y 1, Y 2 ] = Cov[Y 1, Y 2 ] V [Y1 ] V [Y 2 ] = σb 2 σb 2 + σ2 5

154 b i N(0, τ 2 I m ) b i τ 2 b i b i (estimation) (estimator) b i (prediction) (predictor) 7 1 b i = E[b i Y i ] ϵ i b i ϵ j i, j i j b i b j ϵ i ϵ j A (A.1) b i Y i (A.1) Y i = X i β + Z i b i + ϵ i b i N(0, D), ϵ i N(0, Σ i ) b i ϵ i Y i b i N(X i β, Σ i ) Y i N(X i β, Z i DZ i + Σ i) f(y i b i ) f(y i ) g(b i ) f(y i ) = f(y i b i ) g(b i )db i A.4 (i) 7 6

155 (ii) (iii) (i) i t 1, t 2, t 3 3 Y ij t i 1 Y ij = β 0 + β 1 t j + ϵ i 2 Y ij = β 0 + β 1 t j + β 2 t 2 j + ϵ i t 1, t 2, t 3 β 0, β 1, β 2 Y ij = β 0 + t j + ϵ ij t 1, t 2, t 3 t 1, t 2, t 3 8 (ii) 1 MMRM 9 A A.1: 1 15 A.2 A

156 A.2: A.3: A.4: 1 A.5: A.4, A.5 A.1 A.6: A.7:

157 2 3 4 A.6, A.7 A A.8: A.9: 5 6 A.8, A A (iii) LOCF ANCOVA MMRM 0 (Liang and Zeger, 2000) A.5 (A.1) Y i = X i β + Z i b i + ϵ i A β 1, β 2 (t 1, t 2, t 3 ) b i 2 2 i j Y ij i = 1, 2 i = 3, 4 1 Y 11 = µ + β 1 + γt 1 + b 1 + ϵ 11 Y 12 = µ + β 1 + γt 2 + b 1 + ϵ 12 Y 13 = µ + β 1 + γt 3 + b 1 + ϵ 13 Y 21 = µ + β 1 + γt 1 + b 2 + ϵ 21 9

158 Y 22 = µ + β 1 + γt 2 + b 2 + ϵ 22 Y 23 = µ + β 1 + γt 3 + b 2 + ϵ 23 Y 31 = µ + β 2 + γt 1 + b 3 + ϵ 31 Y 32 = µ + β 2 + γt 2 + b 3 + ϵ 32 Y 33 = µ + β 2 + γt 3 + b 3 + ϵ 33 Y 41 = µ + β 2 + γt 1 + b 4 + ϵ 41 Y 42 = µ + β 2 + γt 2 + b 4 + ϵ 42 Y 43 = µ + β 2 + γt 3 + b 4 + ϵ i µ Y i1 Y i = Y i2, β = β 1, ϵ β i = 2 Y i3 γ ϵ i1 ϵ i2 ϵ i3 X 1 = X 2 = t t 2, X 3 = X 4 = t t 2, t 3 1 Z 1 = Z 2 = Z 3 = Z 4 = 1 1 = t 3 11 Y 1 = X 1 β + Z 1 b 1 + ϵ 1 Y 2 = X 2 β + Z 2 b 2 + ϵ 2 Y 3 = X 3 β + Z 3 b 3 + ϵ 3 Y 4 = X 4 β + Z 4 b 4 + ϵ 4 Y i = X i β + Z i b i + ϵ i (i = 1, 2, 3, 4) (A.3) X 1 X 2 X 3 X 4 10

159 Y = Y 1 Y 2 Y 3 Y 4 = Y 11 Y 12 Y 13 Y 21 Y 22 Y 23 Y 31 Y 32 Y 33 Y 41 Y 42 Y 43, b = b 1 b 2 b 3 b 4, ϵ = ϵ 1 ϵ 2 ϵ 3 ϵ 4 = ϵ 11 ϵ 12 ϵ 13 ϵ 21 ϵ 22 ϵ 23 ϵ 31 ϵ 32 ϵ 33 ϵ 41 ϵ 42 ϵ 43 X = X 1 X 2 X 3 X 4 = t t t t t t t t t t t t 3, Z = Z Z Z Z 4 = (A.3) i = 1, 2, 3, 4 Y = Xβ + Zb + ϵ b i ϵ i b i iid N(0, ν 2 ), ϵ i iid N(0, σ 2 I 3 ) b i ϵ i (A.3) Y i V [Y i ] = V [Z i b i + ϵ i ] = Z i V [b i ]Z i + V [ϵ i ] = 1 3 V [b i ]1 3 + σ 2 I 3 11

160 = ν σ ν 2 + σ 2 ν 2 ν 2 = ν 2 ν 2 + σ 2 ν 2 ν 2 ν 2 ν 2 + σ = (A.4) 12 Y i Compound Symmetry ν 2 = 2, σ 2 = V [Y i ] = = j k Y ij Y ik Cov[Y ij, Y ik ] = 2 Cor[Y ij, Y ik ] = Cov[Y ij, Y ik ] V [Yij ] V [Y ij ] = = 0.4 Y = Xβ + Zb + ϵ V [Y] = V [Zb] + V [ϵ] = ZV [b]z + V [ϵ] Z Z σ 2 I Z ( = ν 2 ) 0 Z σ 2 I I 0 0 Z Z σ 2 I Z Z σ 2 I 3 Z 1 Z σ 2 I = ν 2 0 Z 2 Z σ 2 I Z 3 Z σ 2 I Z 4 Z σ 2 I 3 ν 2 Z 1 Z 1 + σ 2 I ν 2 Z 2 Z 2 + σ 2 I = 0 0 ν 2 Z 3 Z 3 + σ 2 I ν 2 Z 4 Z 4 + σ 2 I

161 i = 1, 2, 3, 4 1 ( ) Z i Z i = = = ν 2 Z i Z i + σ 2 I 3 = ν σ = ν 2 + σ 2 ν 2 ν 2 ν 2 ν 2 + σ 2 ν 2 ν 2 ν 2 ν 2 + σ 2 τ 2 = ν 2 + σ 2 τ 2 ν 2 ν ν 2 τ 2 ν ν 2 ν 2 τ τ 2 ν 2 ν ν 2 τ 2 ν ν 2 ν 2 τ V [Y] = τ 2 ν 2 ν ν 2 τ 2 ν ν 2 ν 2 τ τ 2 ν 2 ν ν 2 τ 2 ν ν 2 ν 2 τ 2 A b i ϵ i ϵ i Y i Compound Symmetry (ϵ i ) Compound Symmetry

162 A i = 1, 2 i = 3, 4 k i = 1, 2 k = 1 i = 3, 4 k = 2 14 Y ij = µ + β k + γt j + b i1 + b i2 t j + ϵ ij i b i1, b 2i b 11, b 12, b 21, b 22, b 31, b 32, b 41, b 42 8 i b i = (b i1, b i2 ) b i1 iid N 0, ν2 1 0 b i2 0 0 ν2 2 ϵ iid N(0, σ 2 I 3 ) b 1i b 2i PROC MIXED Variance Components 15 D = ν ν2 2 Y i µ Y i1 Y i = Y i2, β = β 1, b β i = b ϵ i1 i1, ϵ i = ϵ 2 b i2 i2 Y i3 ϵ i3 γ t 1 X 1 = X 2 = t 2, X 3 = X 4 = t 3 1 t 1 Z 1 = Z 2 = Z 3 = Z 4 = 1 t 2 1 t 3 i t t t 3 Y i = X i β + Z i b i + ϵ i (i = 1, 2, 3, 4) E[Y i ] = X i β V [Y i ] = V [Z i b i ] + V [ϵ i ] = Z i V [b i ]Z i + σ 2 I 3 = Z i DZ i + σ 2 I 3 14 β k k = 1, 2 15 ( ) ν 2 ν 12 D = 1 ν 12 ν 12 ν2 2 SAS RANDOM " / TYPE = UN; " 14

163 = 1 t 1 1 t 2 1 t 3 ν ν t 1 t 2 t 3 + σ σ σ 2 = ν t 2 1ν σ 2 ν t 1 t 2 ν 2 2 ν t 1 t 3 ν 2 2 ν t 1 t 2 ν 2 2 ν t 2 2ν σ 2 ν t 2 t 3 ν 2 2 ν t 1 t 3 ν 2 2 ν t 2 t 3 ν 2 2 ν t 2 3ν σ 2 Y = Y 1 Y 2 Y 3 Y 4 = Y 11 Y 12 Y 13 Y 21 Y 22 Y 23 Y 31 Y 32 Y 33 Y 41 Y 42 Y 43, b = b 1 b 2 b 3 b 4 = b 11 b 12 b 21 b 22 b 31 b 32 b 41 b 42, ϵ = ϵ 1 ϵ 2 ϵ 3 ϵ 4 = ϵ 11 ϵ 12 ϵ 13 ϵ 21 ϵ 22 ϵ 23 ϵ 31 ϵ 32 ϵ 33 ϵ 41 ϵ 42 ϵ 43 X = X 1 X 2 X 3 X 4 = t t t t t t t t t t t t 3, Z = Z Z Z Z 4 = 1 t t t t t t t t t t t t 3 Y = Xβ + Zb + ϵ 15

164 E[Y] = Xβ = X 1 X 2 X 3 X 4 β = X 1 β X 2 β X 3 β X 4 β V [Y] = ZV [b]z + V [ϵ] Z D Z Z D Z = 0 0 Z D Z Z D Z 4 σ 2 I σ 2 I σ 2 I σ 2 I 3 Z 1 DZ 1 + σ 2 I Z 2 DZ 2 + σ 2 I = 0 0 Z 3 DZ 3 + σ 2 I Z 4 DZ 4 + σ 2 I 3 A AR(1)

165 AR(1) 1 < ρ < 1 1 ρ ρ 2 iid b i N(0, ν 2 iid ), ϵ ij N 0, σ2 ρ 1 ρ ρ 2 ρ 1 18 Y i Z i = 1 3, b i N(0, σ 2 ) Y i = X i β + Z i b i + ϵ i E[Y i ] = X i β V [Y i ] = 1 3 V [b i ]1 3 + V [ϵ i ] ρ ρ 2 = ν σ2 ρ 1 ρ ρ 2 ρ 1 ν 2 + σ 2 ν 2 + σ 2 ρ ν 2 + σ 2 ρ 2 = ν 2 + σ 2 ρ ν 2 + σ 2 ν 2 + σ 2 ρ ν 2 + σ 2 ρ 2 ν 2 + σ 2 ρ ν 2 + σ 2 (A.5) (A.4) 1 < ρ < 1 AR(1) ν 2 = 1, σ 2 = 2, ρ = V [Y i ] = = j k Cov[Y ij, Y ik ] = j k Corr[Y ij, Y ik ] = Cov[Y ij, Y ik ] V [Yij ] V [Y ik ] = j k = j k Corr[Y i1, Y i2 ] = Corr[Y i2, Y i3 ] = Corr[Y i1, Y i3 ] = = A AR(1) 1 ρ ρ 2 V [ϵ i ] = σ 2 ρ 1 ρ ρ 2 ρ 1 18 ϵ ij (i, j) σ 2 ρ i j 17

166 AR(1) AR(1) k l k, l k l t k t l (t k t l ) ( 1 exp t ) ( 1 t 2 exp t ) 1 t 3 θ θ ( V [ϵ i ] = σ 2 exp t ) ( 1 t 2 1 exp t ) 2 t 3 ( θ θ exp t ) ( 1 t 3 exp t ) 2 t 3 1 θ θ 19 t 1 = 1, t 2 = 2, t 3 = 4, θ = 0.5 ( 1 exp 1 ) ( exp 2 ) ( V [ϵ i ] = σ 2 exp 1 ) ( 1 exp 1 ) ( exp 2 ) ( exp 1 ) exp ( 2) exp ( 4) = σ 2 exp ( 2) 1 exp ( 2) = σ exp ( 4) exp ( 2) SAS PROC MIXED REPEATED Spatial Covariance Structure A Compound Symmetry V [Y i ] 1 Y i = X i β + ϵ i ϵ i iid N 0 0 0, ν 2 + σ 2 ν 2 ν 2 ν 2 ν 2 + σ 2 ν 2 ν 2 ν 2 ν 2 + σ 2 E[Y i ] = X i β (LMM) (LM) MMRM MMRM(Mixed Model for Repeated Measures) 18

167 V [Y i ] = ν 2 + σ 2 ν 2 ν 2 ν 2 ν 2 + σ 2 ν 2 ν 2 ν 2 ν 2 + σ 2 Y i ϵ i 1 Z i b i + ϵ i A.5.7 (A.1) Y i = X i β + Z i b i + ϵ i b i iid N(0, D), ϵ i iid N(0, Σ i ) b i ϵ i E[Y i ] = X i β, V [Y i ] = Z i DZ i + Σ i 22 SAS PROC MIXED b i RANDOM ϵ i REPEATED A i (i) r m,i = Y i X i β (ii) r c,i = Y i X i β Zi bi β b i (i) (ii) X i β Xi β + Zi bi Student (studentized residual) Σ i i D i 23 r m,i, r c,i 24 19

168 r m,i (i) Student V [rm,i] (ii) Student r c,i V [rc,i] V i = Z i DZ i + Σ i, Q i = X i (X i V i X i) X i, K i = I Z i DZ i V 1 i, V [r m,i ] = V i Q i V [r c,i ] = K i ( V i Q i )K i SAS PROC MIXED RANDOM residual ods graphics on html A.7.9 A.10 (i) (ii) (i) (ii) 6 A.10: 2 (a)0 (b)student Y i = X i β + Z i b i + ϵ i E[Z i b i ] = 0, E[ϵ i ] = 0 ϵ i Z i b i + ϵ i 0 Student Student A

169 Cook D (Cook, 1977) Cook D β i β (i) i β 25 i β Cook D D i = 1 rank(x) ( β β (i) ) V [ β] ( β β (i) ) V [ β] β SAS PROC MIXED influence 4 A.7.10 A.11 2 A.11: 3 A.7 SAS 1 5 SAS A.8, A.9 A i j b i N(0, 4 2 ) ϵ ij N(0, 4 2 ) 25 i β β (i) β β (i) β β (i) 21

170 i i = 1 i = 1 j (1, 2, 3) d i 1 2 t i 1, 2, 3 Y i1 = 20 + b i 3t i + ϵ i1 (i = 1,, 10) Y ij = 20 + b i 3d i 3t i + ϵ i1 (i = 1,, 10, j = 2, 3) 3 30% 3 MCAR 3 30% 3 * ; DATA D1; CALL STREAMINIT(16747); DO DRUG = 1 TO 2; DO PATNO = 1 TO 10; PAT = RAND( NORM, 0, 4); * ; TIME = 1; * 1; TIMECLASS = TIME; E = RAND( NORM, 0, 4); * ; Y = 20 + PAT - 3*TIME + E ; * ; OUTPUT; DO TIME = 2, 3; * 2, 3; TIMECLASS = TIME; E = RAND( NORM, 0, 4); * ; Y = 20 + PAT - 3* DRUG - 3*TIME + E ; * 2,3 ; OUTPUT; END; END; END; RUN; 22

171 * ; DATA D2; SET D1; CALL STREAMINIT(51885); IF TIME = 3 THEN DO; IF RAND( UNIF ) < 0.3 THEN DELETE; * 3 30% ; END; RUN; D2 S A S システム O B S D R U G P A T N O T I M E T I M E C L A S S Y A.12: D2 3 DRUG PATNO TIME TIMECLASS Y 23

172 A.13: A.14: 1 5 Y ij = µ + b i + β di + γt i + ϵ ij β di d i = 1 β di = β 1 d i = 2 β di = β 2 A PROC MIXED DATA = D2; CLASS DRUG PATNO; MODEL Y = DRUG TIME / S ; RANDOM INT / SUB = PATNO(DRUG) ; RUN; SE WORK.D2 Y Variance Components PATNO(DRUG) REML Profile Model Based Containment 24

173 DRUG 2 PATNO PATNO 10 PATNO(DRUG) 10 2 X 4 Z 1 20 Obs MMRM 25

174 Intercept PATNO(DRUG) Residual b i N(0, ν 2 ), ϵ i N(0, σ 2 I 3 ) ν 2 = , σ 2 = Y i V [Y i ] = ν 2 + σ 2 ν 2 ν 2 ν 2 ν 2 + σ 2 ν 2 ν 2 ν 2 ν 2 + σ 2 = Ĉorr[Y ij, Y ik ] = ν 2 ν2 + σ 2 ν 2 + σ 2 = (j k) AIC ( ) AICC ( ) BIC ( ) t Pr > t Intercept <.0001 DRUG DRUG TIME < V [Y i ] V [Y i ] Ĉorr ŜE 26

175 β = µ β 1 β 2 γ = Containment µ : 18, β 1 : 18, γ : 32 ŜE[ µ] = , ŜE[ β 1 ] = , ŜE[ γ] = Type 3 Type 3 F Pr > F DRUG TIME <.0001 F = ( β1 ŜE[ β 1 ] ) 2 = ( ) p % A Kenward-Roger PROC MIXED DATA = D2 ; CLASS DRUG PATNO; MODEL Y = DRUG TIME / S DDFM = KR; RANDOM INT / SUB = PATNO(DRUG) ; RUN; 27

176 Intercept PATNO(DRUG) Residual β 27 t Pr > t Intercept <.0001 DRUG DRUG TIME <.0001 µ : 44.5, β 1 : 16.9, γ : ŜE[ µ] = , ŜE[ β 1 ] = , ŜE[ γ] = Containment 1-1 A A

177 PROC MIXED DATA = D2 EMPIRICAL; CLASS DRUG PATNO; MODEL Y = DRUG TIME / S; RANDOM INT / SUB = PATNO(DRUG) ; RUN; t Pr > t Intercept <.0001 DRUG DRUG TIME <.0001 Containment 1-1 µ : 18, β 1 : 18, γ : 32 ŜE[ µ] = , ŜE[ β 1 ] = , ŜE[ γ] = A Containment 29

178 PROC MIXED DATA = D2 EMPIRICAL; CLASS DRUG PATNO; MODEL Y = DRUG TIME / S ; RANDOM INT TIME / SUB = PATNO(DRUG) ; RUN; Intercept PATNO(DRUG) TIME PATNO(DRUG) Residual D = ν ν2 2 b i = b i1 b i2 iid N(0, D), ϵ i N(0, σ 2 I 3 ) ν 2 1 = , ν 2 2 = , σ 2 = V [Y i ] = Z i DZ i + σ 2 I 3 ν t 2 1 ν 2 2 ν t 1 t 2 ν 2 2 ν t 1 t 3 ν = ν t 1 t 2 ν 2 2 ν t 2 2 ν 2 2 ν t 2 t 3 ν σ ν t 1 t 3 ν 2 2 ν t 2 t 3 ν 2 2 ν t 2 3 ν =

179 = t Pr > t Intercept <.0001 DRUG DRUG TIME <.0001 β = µ β 1 β 2 γ = Containment 1-1 ŜE[ µ] = , ŜE[ β 1 ] = , ŜE[ γ] = A AR(1) REPEATED TIME TIME TIMECLASS TIMECLASS PROC MIXED DATA = D2 EMPIRICAL; CLASS DRUG PATNO TIMECLASS; MODEL Y = DRUG TIME / S; RANDOM INT / SUB = PATNO(DRUG); REPEATED TIMECLASS / SUB = PATNO(DRUG) TYPE = AR(1) ; RUN; 31

180 Intercept PATNO(DRUG) AR(1) PATNO(DRUG) Residual b i N(0, ν 2 ), ϵ i N 0, σ2 1 ρ ρ 2 ρ 1 ρ ρ 2 ρ 1 ν 2 = , ρ = , σ 2 = ρ ρ 2 V [Y i ] = ν σ2 ρ 1 ρ ρ 2 ρ 1 ν 2 + σ 2 ν 2 + σ 2 ρ ν 2 + σ 2 ρ 2 = ν + σ 2 ρ ν 2 + σ 2 ν 2 + σ 2 ρ ν 2 + σ 2 ρ 2 ν 2 + σ 2 ρ ν 2 + σ = =

181 t Pr > t Intercept <.0001 DRUG DRUG TIME <.0001 β = µ β 1 β 2 γ = ŜE[ µ] = , ŜE[ β 1 ] = , ŜE[ γ] = A PROC MIXED DATA = D2 EMPIRICAL; CLASS DRUG PATNO TIMECLASS; MODEL Y = DRUG TIME / S ; RANDOM INT / SUB = PATNO(DRUG); REPEATED TIMECLASS / SUB = PATNO(DRUG) TYPE = SP(EXP)(TIME); RUN;

182 Intercept PATNO(DRUG) SP(EXP) PATNO(DRUG) Residual b i N(0, ν 2 ), ϵ i N 0, σ 2 ( exp exp ( 1 exp t ) 1 t 2 θ t ) 1 t 2 1 exp ( θ t ) ( 1 t 3 exp t ) 2 t 3 θ θ ( exp t ) 1 t 3 ( θ t ) 2 t 3 θ 1 ν 2 = , θ = , σ 2 = V [Y i ] = ν σ exp = ( exp ν 2 + σ 2 ( ν 2 + σ 2 exp t ) 1 t 2 ( θ ν 2 + σ 2 exp t ) 1 t 3 θ ( 1 exp t ) 1 t 2 θ t ) 1 t 2 1 exp ( θ t ) ( 1 t 3 exp t ) 2 t 3 θ θ ( ν 2 + σ 2 exp t ) 1 t 2 θ ν 2 + σ 2 ( ν 2 + σ 2 exp t ) 2 t 3 θ ( exp t ) 1 t 3 ( θ t ) 2 t 3 θ 1 ( ν 2 + σ 2 exp ν 2 + σ 2 exp ( exp ( ) = exp exp ( ) = t ) 1 t 3 ( θ t ) 2 t 3 θ ν 2 + σ 2 ) ( ) exp ( ) exp ( ) exp

183 t Pr > t Intercept <.0001 DRUG DRUG time <.0001 β = µ β 1 β 2 γ = ŜE[ µ] = , ŜE[ β 1 ] = , ŜE[ γ] = A CS PROC MIXED DATA = D2 EMPIRICAL; CLASS DRUG PATNO TIMECLASS; MODEL Y = DRUG TIME / S; REPEATED TIMECLASS / SUB = PATNO(DRUG) TYPE = CS; RUN;

184 CS PATNO(DRUG) Residual ϵ i N 0, ν 2 + σ 2 ν 2 ν 2 ν 2 ν 2 + σ 2 ν 2 ν 2 ν 2 ν 2 + σ 2 ν 2 = , σ 2 = t Pr > t Intercept <.0001 DRUG DRUG TIME <.0001 β = µ β 1 β 2 γ = ŜE[ µ] = , ŜE[ β 1 ] = , ŜE[ γ] = A Model Residual 36

185 PROC MIXED DATA = D2 ; CLASS DRUG PATNO; MODEL Y = DRUG TIME / S RESIDUAL; RANDOM INT / SUB = PATNO(DRUG) ; RUN; A.10 A.7.10 PROC MIXED DATA = D2 ; CLASS DRUG PATNO; MODEL Y = DRUG TIME / S INFLUENCE (EFFECT = PATNO(DRUG) ITER = 5); RANDOM INT / SUB = PATNO(DRUG) ; RUN; β 1 β (i) iter i β (i) iter = 5 5 β A.11 A CS 5 Y i 0 σ 2 + η 2 η 2 η 2 iid b i N(0, ν 2 iid ), ϵ i N 0, η 2 σ 2 + η 2 η 2 0 η 2 η 2 σ 2 + η 2 V [Y i ] = 1 3 V [b i ]1 3 + V [ϵ i ] σ 2 + η 2 η 2 η 2 = ν η 2 σ 2 + η 2 η η 2 η 2 σ 2 + η 2 σ 2 + (ν 2 + η 2 ) (ν 2 + η 2 ) (ν 2 + η 2 ) = (ν 2 + η 2 ) σ 2 + (ν 2 + η 2 ) (ν 2 + η 2 ) (ν 2 + η 2 ) (ν 2 + η 2 ) σ 2 + (ν 2 + η 2 ) ν 2 η 2 (ν 2 + η 2 ) ν 2 + η 2 = 3 ν 2 η 2 ν 2 = 0 η 2 37

186 SAS PROC MIXED PROC MIXED DATA = D2 EMPIRICAL; CLASS TIMECLASS DRUG PATNO; MODEL Y = DRUG TIME / S; RANDOM INT /SUB = PATNO(DRUG) ; REPEATED TIMECLASS / SUB = PATNO(DRUG) TYPE = CS; RUN; Hessian 2 y ij = µ + α i + ϵ ij (i = 1, 2, j = 1,, n i ) SAS 0 28 β 1, β 2 µ γt i A β 2 =

187 Box and Draper (1976) "Esentially, all models are wrong, but some are useful" (sensitivity analysis) 30 MMRM MMRM LOCF ANCOVA unstructured 31 III 32 A.8.1 SAS PROC MIXED RANDOM Containment REPEATED RANDOM Between-within Satterthwaite Kenward-Roger RANDOM REPEATED Type = UN SAS Kenward-Roger χ 2 Kenward-Roger SAS/STAT 14.1 KenwardRoger II 39

188 A β β V m [ β] = (X V 1 X) 1 V r [ β] = (X V ( α) 1 X) 1 ( n i=1 ) 1 1 X i V i (Y i X i β)(yi X i β) V i X i (X V ( α) 1 X) 1 SAS PROC MIXED PROC MIXED EMPIRICAL SAS Ver 9.3 Satterthwaite Kenward-Roger A.9 Y i = X i β + Z i b i + ϵ i (b i N(0, D), ϵ i N(0, Σ)) β D Σ D = ν2 1 0, Σ = σ ν (ν1, 2 ν2, 2 σ 2 ) ρ ρ 2 ( ) D = ν 2, Σ = σ 2 ρ 1 ρ ρ 2 ρ 1 (ν 2, σ 2, ρ) 3 α β D, Σ D, Σ α α β D, Σ 40

189 α Σ, D 33 (ML) (REML) 2 2 A.9.1 (ML) (REML) i (i = 1,, N) n i 34 Y i = X i β + Z i b i + ϵ i b i iid N(0, D), ϵ i iid N (0, Σ i ) b i ϵ i β Y i iid N(X i β, Z i DZ i + Σ i ) i 1 f(y i β, α) = ( (2π) n i Zi DZ i + Σ i exp 1 ) 2 (Y i X i β) (Z i DZ i + Σ i ) 1 (Y i X i β) (A.6) N L(β, α) = f(y i β, α) i=1 N l(β, α) = log f(y i β, α) i=1 β, α Newton-Raphson Fisher s Scoring SAS PROC MIXED ridge-stabilized Newton-Raphson Fisher s scoring PROC MIXED scoring 35 A.9.2 i b i Y i f(y i β, b i, Σ i ) = 1 ( (2π) n i Σi exp 1 ) 2 (Y i X i β Z i b i ) Σ 1 i (Y i X i β Z i b i ) 33 Σ, D α = (α 1, α 2 ) 34 n i (A.7)

190 Y i, X i, Z i β, α b i V i, X i V ix i V i 36 b i b i b i N(0, D) b i β 37 b i D b i D b i f(y i β, b i, Σ i ) : Y i f(b i D) : b i b i (integrate out) f(y i β, α) = f(y i β, b i, Σ i ) f(b i D)db i (A.8) b i D f(y i β, α) (A.8) 1 f(y i β, α) = ( (2π) n i Zi DZ i + Σ i exp 1 ) 2 (Y i X i β) (Z i DZ i + Σ i ) 1 (Y i X i β) 38 V i = Z i DZ i + Σ i V i α V i (α) l(α, β) = N i=1 { 1 ( ) 2 log (2π) n i V i + 1 } 2 (Y i X i β) V 1 i (Y i X i β) (A.9) β β = N i=1 ( X i V 1 i ) 1 X i X V 1 i i Y i β V i β α α β β (A.9) l ML (α) = N i=1 α { 1 2 log ( (2π) 3 V i ) (Y i X i β) V 1 i (Y i X i β) } Lee and Nelder (1996) 38 42

191 A.9.3 (REML) (ML) (REML) α β α ML REML β N(µ, σ 2 ) x 1,, x N µ, σ 2 µ = 1 N N x i, σ 2 = 1 N i=1 N (x i µ) 2 i=1 E[ σ 2 ] = N 1 N σ2 σ 2 = 1 N 1 N (x i µ) 2 σ 2 (REML) (N 1) (N 1) E[ σ 2 ] (N 1) i=1 1 Ñ rank(x) = p X Ñ X Ñ (Ñ p) a 1,, añ p Ñ (Ñ p) A 39 ) A = (a 1,, añ p Y A W = A Y Ñ = N n i W (Ñ p) i=1 E[W] = E[A Y] = A E[Y] = A Xβ = 0 ( A X ) V [W] = V [A Y] = A V [Y]A = A VA W 0 α W α REML REML rank(x) = p, dim(y i ) = n i, Ñ = N l REML (α) = Ñ p N i=1 log(2π) + 1 N 2 log X i X i 1 2 log N ( Y i X i β ) V 1 i i=1 ( ) Y i X i β 39 A X 43 i=1 X i V 1 i X i 1 2 N log V i i=1 i=1 n i

192 ML l REML (α) = 1 N 2 log X i V 1 i X i + l ML(α) + C i=1 (C α, β ) A.9.4 α β α β SAS PROC MIXED REML l REML (α) = 1 2 log N i=1 X i V 1 i X i + l ML(α) + C (C α, β ) A.9.5 b i b i b i D b i β, α b i prediction (estimatior) (estimated value) (predictor) (predicted value) EBLUP(Empirical Best Linear Unbiased Predictor) b = E[b Y] b Mixed Model Equation X Σ 1 X Z Σ 1 X b X Σ 1 Z Z Σ 1 Z + D β b b = DZ V 1 (Y X β) = X V 1 Y Z V 1 Y PROC MIXED DATA = D2 ; CLASS DRUG PATNO; MODEL Y = DRUG TIME / S ; RANDOM INT / SUB = PATNO(DRUG) S; RUN; 44

193 PATNO t Pr > t Intercept Intercept Intercept Intercept Intercept Intercept Intercept Intercept Intercept Intercept Intercept Intercept Intercept Intercept Intercept Intercept Intercept Intercept Intercept Intercept A.10 SAS (2001) (2009) (2011) Verbeke and Molenberghs (2000) Molenberghs Web SAS SAS Little et al. (2006) [1] Box,G.E.P. and Draper,N.R. (1987). Empirical Model-Building and Response Surfaces. Wiley. [2] Cook, R. D. (1977), Detection of Influential Observations in Linear Regression, Technometrics, 19,

194 [3]. (2009)... [4] Liang, K. Y., & Zeger, S. L. (2000). Longitudinal data analysis of continuous and discrete responses for pre-post designs. Sankhyā: The Indian Journal of Statistics, Series B, [5] McCulloch, C. E., Searle, S. R. and Neuhaus, J. M. (2008). Generalized, Linear, and Mixed Models (2nd Ed.), Wiley ( (2011). CAC). [6] Little, R, C, Milliken, G, A, Stroup, Wolfinger, R, D, and Schabenberber, O. (2006). SAS for Mixed Models. SAS Institute. [7] Verbeke, G, and Molenberghs, G. (1997). Linear Mixed Models in Practice: A SAS-oriented approach. Springer., (2001) -SAS -.. [8] Verbeke, G, and Molenberghs, G. (2000). Linear Mixed models for longitudinal data. Springer. 46

195 Appendix B B.1 2 MCAR MAR MNAR 5 MNAR SM 6 (i) NFMV ACMV ACMV (ii) ACMV MAR (iii) NFMV NFD B.2 2 B.2.1 Little and Rubin (2002) MCAR MAR MNAR (MCAR) : f(r i Y i, ψ) = f(r i ψ) (MAR) : f(r i Y i, ψ) = f(r i Y o i, ψ) (B.1) (MNAR) : f(r i Y i, ψ) f(r i Y o i, ψ) 2 MCAR MAR MNAR MCAR MAR MNAR B i j Y ij Y i = (Y i1, Y i2, Y i3 ) R i = (R i1, R i2, R i3 ) R ij = 1 R ij = 0 B MCAR 2 MCAR f(r i1 = 0 Y i ) = 0 (B.2) 1 Seaman et al. (2013) everywhere MCAR MAR 47

196 f(r i2 = 0 Y i, R i1 = 0) = 1 f(r i2 = 0 Y i, R i1 = 1) = 0.1 f(r i3 = 0 Y i, R i1 = 0, R i2 = 0) = 1 f(r i3 = 0 Y i, R i1 = 1, R i2 = 0) = 1 f(r i3 = 0 Y i, R i1 = 1, R i2 = 1) = 0.1 (B.3) (B.4) i j R ij = 0 R i,j+1 = 0 1 f(r i3 = 0 Y i, R i1 = 0, R i2 = 1) = 1 f(r i3 = 1 Y i, R i1 = 0, R i2 = 1) = 0 1 R ij (B.1) R i = (R i1, R i2, R i3 ) (B.2) (B.5) (B.1) 2 R i MCAR (B.2) (B.5) f(r i1 = 1 Y i ) = 1 f(r i2 = 1 Y i, R i1 = 0) = 0 f(r i2 = 1 Y i, R i1 = 1) = 0.9 f(r i3 = 1 Y i, R i1 = 0, R i2 = 0) = 0 f(r i3 = 1 Y i, R i1 = 1, R i2 = 0) = 0 f(r i3 = 1 Y i, R i1 = 1, R i2 = 1) = 0.9 R i = (R i1, R i2, R i3 ) 3 f(r i1 = 1, R i2 = 1, R i3 = 1 Y i ) = f(r i3 = 1 Y i, R i1 = 1, R i2 = 1) f(r i2 = 1, R i1 = 1 Y i ) (B.5) = f(r i3 = 1 Y i, R i1 = 1, R i2 = 1) f(r i2 = 1 Y i, R i1 = 1) f(r i1 = 1 Y i ) = = 0.81 f(r i1 = 0, R i2 = 1, R i3 = 1 Y i ) = f(r i3 = 1 Y i, R i1 = 0, R i2 = 1) f(r i2 = 1 Y i, R i1 = 0) f(r i1 = 0 Y i ) = = 0 f(r i1 = 1, R i2 = 0, R i3 = 1 Y i ) = f(r i3 = 1 Y i, R i1 = 1, R i2 = 0) f(r i2 = 0 Y i, R i1 = 1) f(r i1 = 1 Y i ) = = 0 f(r i1 = 0, R i2 = 0, R i3 = 1 Y i ) = f(r i3 = 1 Y i, R i1 = 0, R i2 = 0) f(r i2 = 0 Y i, R i1 = 0) f(r i1 = 0 Y i ) = = 0 f(r i1 = 1, R i2 = 1, R i3 = 0 Y i ) = f(r i3 = 0 Y i, R i1 = 1, R i2 = 1) f(r i2 = 1 Y i, R i1 = 1) f(r i1 = 1 Y i ) 2 Y i R i 3 R i1 = 0, 1, R i2 = 0, 1, R i3 = 0, = 8 48

197 = = 0.09 f(r i1 = 0, R i2 = 1, R i3 = 0 Y i ) = f(r i3 = 0 Y i, R i1 = 0, R i2 = 1) f(r i2 = 1 Y i, R i1 = 0) f(r i1 = 0 Y i ) = = 0 f(r i1 = 1, R i2 = 0, R i3 = 0 Y i ) = f(r i3 = 0 Y i, R i1 = 1, R i2 = 0) f(r i2 = 0 Y i, R i1 = 1) f(r i1 = 1 Y i ) = = 0.1 f(r i1 = 0, R i2 = 0, R i3 = 0 Y i ) = f(r i3 = 0 Y i, R i1 = 0, R i2 = 0) f(r i2 = 0 Y i, R i1 = 0) f(r i1 = 0 Y i ) = = 0 Y i f(r i Y i ) = f(r i ) (B.1) MCAR B MAR MAR f(r i1 = 0 Y i ) = 0 f(r i2 = 0 Y i, R i1 = 0) = 1 (B.6) logitf(r i2 = 0 Y i, R i1 = 1) = Y i1 (B.7) f(r i3 = 0 Y i, R i1 = 0, R i2 = 0) = 1 f(r i3 = 0 Y i, R i1 = 1, R i2 = 0) = 1 logitf(r i3 = 0 Y i, R i1 = 1, R i2 = 1) = Y i2 (B.8) ( ) f(x) logitf(x) = log exp (logitf(x)) = f(x) 1 f(x) 1 f(x) f(x) = exp(logitf(x)) 1 + exp(logitf(x)) (B.7) (B.8) 1 logitf(r i2 = 0 Y i, R i1 = 1) = Y i1 f(r i2 = 0 Y i1, R i1 = 1) = exp( Y i1) 1 + exp ( Y i1 ) logitf(r i3 = 0 Y i, R i1 = 1, R i2 = 1) = Y i2 f(r i3 = 0 Y i1, R i1 = 1, R i2 = 1) = exp( Y i2) 1 + exp ( Y i2 ) (B.6) (B.8) f(r i1 = 0 Y i ) = 0 49

198 f(r i2 = 0 Y i, R i1 = 0) = 1 f(r i2 = 0 Y i, R i1 = 1) = exp( Y i1) 1 + exp ( Y i1 ) f(r i3 = 0 Y i, R i1 = 0, R i2 = 0) = 1 f(r i3 = 0 Y i, R i1 = 1, R i2 = 0) = 1 f(r i3 = 0 Y i, R i1 = 1, R i2 = 1) = exp( Y i2) 1 + exp ( Y i2 ) f(r i3 = 0 Y i, R i1 = 0, R i2 = 1) = 1 f(r i3 = 1 Y i, R i1 = 0, R i2 = 1) = 0 f(r i1 = 1 Y i ) = 1 f(r i2 = 1 Y i, R i1 = 0) = 0 1 f(r i2 = 1 Y i, R i1 = 1) = 1 + exp ( Y i1 ) f(r i3 = 1 Y i, R i1 = 0, R i2 = 0) = 0 f(r i3 = 1 Y i, R i1 = 1, R i2 = 0) = 0 1 f(r i3 = 1 Y i, R i1 = 1, R i2 = 1) = 1 + exp ( Y i2 ) MAR Y o i ϕ f(r i1 = 1, R i2 = 1, R i3 = 1 Y i ) = f(r i3 = 1 Y i, R i1 = 1, R i2 = 1) f(r i2 = 1 Y i, R i1 = 1) f(r i1 = 1 Y i ) 1 = 1 + exp ( Y i2 ) exp ( Y i1 ) 1 1 = {1 + exp ( Y i2 )}{1 + exp ( Y i1 )} (Y o i = (Y i1, Y i2, Y i3 ) ) f(r i1 = 0, R i2 = 1, R i3 = 1 Y i ) = f(r i3 = 1 Y i, R i1 = 0, R i2 = 1) f(r i2 = 1 Y i, R i1 = 0) f(r i1 = 0 Y i ) = = 0 (Y o i = (Y i2, Y i3 ) ) f(r i1 = 1, R i2 = 0, R i3 = 1 Y i ) = f(r i3 = 1 Y i, R i1 = 1, R i2 = 0) f(r i2 = 0 Y i, R i1 = 1) f(r i1 = 1 Y i ) = 0 exp( Y i1 ) 1 + exp ( Y i1 ) 1 = 0 (Yo i = (Y i1, Y i3 )) f(r i1 = 0, R i2 = 0, R i3 = 1 Y i ) = f(r i3 = 1 Y i, R i1 = 0, R i2 = 0) f(r i2 = 0 Y i, R i1 = 0) f(r i1 = 0 Y i ) = = 0 (Y o i = (Y i3 )) f(r i1 = 1, R i2 = 1, R i3 = 0 Y i ) = f(r i3 = 0 Y i, R i1 = 1, R i2 = 1) f(r i2 = 1 Y i, R i1 = 1) f(r i1 = 1 Y i ) = exp( Y i2) 1 + exp ( Y i2 ) exp ( Y i1 ) 1 exp( Y i2 ) = {1 + exp ( Y i2 )}{1 + exp ( Y i1 )} (Y o i = (Y i1, Y i2 ) ) f(r i1 = 0, R i2 = 1, R i3 = 0 Y i ) = f(r i3 = 0 Y i, R i1 = 0, R i2 = 1) f(r i2 = 1 Y i, R i1 = 0) f(r i1 = 0 Y i ) = = 0 (Y o i = (Y i2 )) 50

199 f(r i1 = 1, R i2 = 0, R i3 = 0 Y i ) = f(r i3 = 0 Y i, R i1 = 1, R i2 = 0) f(r i2 = 0 Y i, R i1 = 1) f(r i1 = 1 Y i ) = 1 exp( Y i1 ) 1 + exp ( Y i1 ) 1 = exp( Y i1) 1 + exp ( Y i1 ) (Y o i = (Y i1 )) f(r i1 = 0, R i2 = 0, R i3 = 0 Y i ) = f(r i3 = 0 Y i, R i1 = 0, R i2 = 0) f(r i2 = 0 Y i, R i1 = 0) f(r i1 = 0 Y i ) = = 0 (Y o i = ϕ) Y ij Y o i f(r i Y i ) = f(r i Y o ) (B.1) MAR B MNAR 1 2 MNAR 1 f(r i1 = 0 Y i ) = 0 f(r i2 = 0 Y i, R i1 = 0) = 1 (B.9) logitf(r i2 = 0 Y i, R i1 = 1) = Y i Y i2 (B.10) f(r i3 = 0 Y i, R i1 = 0, R i2 = 0) = 1 f(r i3 = 0 Y i, R i1 = 1, R i2 = 0) = 1 logitf(r i3 = 0 Y i, R i1 = 1, R i2 = 1) f(r i1 = 0 Y i ) = 0 f(r i2 = 0 Y i, R i1 = 0) = 1 = Y i Y i3 f(r i2 = 0 Y i, R i1 = 1) = exp( Y i Y i2 ) 1 + exp( Y i Y i2 ) f(r i3 = 0 Y i, R i1 = 0, R i2 = 0) = 1 f(r i3 = 0 Y i, R i1 = 1, R i2 = 0) = 1 f(r i3 = 0 Y i, R i1 = 1, R i2 = 1) = exp( Y i Y i3 ) 1 + exp( Y i Y i3 ) f(r i3 = 0 Y i, R i1 = 0, R i2 = 1) = 1 f(r i3 = 1 Y i, R i1 = 0, R i2 = 1) = 0 f(r i1 = 1 Y i ) = 1 f(r i2 = 1 Y i, R i1 = 0) = 0 1 f(r i2 = 1 Y i, R i1 = 1) = 1 + exp ( Y i Y i2 ) f(r i3 = 1 Y i, R i1 = 0, R i2 = 0) = 0 f(r i3 = 1 Y i, R i1 = 1, R i2 = 0) = 0 1 f(r i3 = 1 Y i, R i1 = 1, R i2 = 1) = 1 + exp ( Y i Y i3 ) 51 (B.11)

200 MNAR Y o i ϕ f(r i1 = 1, R i2 = 1, R i3 = 1 Y i ) = f(r i3 = 1 Y i, R i1 = 1, R i2 = 1) f(r i2 = 1 Y i, R i1 = 1) f(r i1 = 1 Y i ) 1 = 1 + exp ( Y i Y i3 ) exp ( Y i Y i2 ) 1 1 = {1 + exp ( Y i Y i3 )}{1 + exp ( Y i Y i2 )} (Y o i = (Y i1, Y i2, Y i3 ) ) f(r i1 = 0, R i2 = 1, R i3 = 1 Y i ) = f(r i3 = 1 Y i, R i1 = 0, R i2 = 1) f(r i2 = 1 Y i, R i1 = 0) f(r i1 = 0 Y i ) = = 0 (Y o i = (Y i2, Y i3 ) ) f(r i1 = 1, R i2 = 0, R i3 = 1 Y i ) = f(r i3 = 1 Y i, R i1 = 1, R i2 = 0) f(r i2 = 0 Y i, R i1 = 1) f(r i1 = 1 Y i ) = 0 exp( Y i Y i2 ) 1 + exp ( Y i Y i2 ) 1 = 0 (Yo i = (Y i1, Y i3 )) f(r i1 = 0, R i2 = 0, R i3 = 1 Y i ) = f(r i3 = 1 Y i, R i1 = 0, R i2 = 0) f(r i2 = 0 Y i, R i1 = 0) f(r i1 = 0 Y i ) = = 0 (Y o i = (Y i3 )) f(r i1 = 1, R i2 = 1, R i3 = 0 Y i ) = f(r i3 = 0 Y i, R i1 = 1, R i2 = 1) f(r i2 = 1 Y i, R i1 = 1) f(r i1 = 1 Y i ) = exp( Y i Y i3 ) 1 + exp ( Y i Y i3 ) exp ( Y i Y i2 ) 1 exp( Y i Y i3 ) = {1 + exp ( Y i Y i3 )}{1 + exp ( Y i Y i2 )} (Y o i = (Y i1, Y i2 ) ) f(r i1 = 0, R i2 = 1, R i3 = 0 Y i ) = f(r i3 = 0 Y i, R i1 = 0, R i2 = 1) f(r i2 = 1 Y i, R i1 = 0) f(r i1 = 0 Y i ) = = 0 (Y o i = (Y i2 )) f(r i1 = 1, R i2 = 0, R i3 = 0 Y i ) = f(r i3 = 0 Y i, R i1 = 1, R i2 = 0) f(r i2 = 0 Y i, R i1 = 1) f(r i1 = 1 Y i ) = 1 exp( Y i Y i2 ) 1 + exp ( Y i Y i2 ) 1 = exp( Y i Y i2 ) 1 + exp ( Y i Y i2 ) (Y o i = (Y i1 )) f(r i1 = 0, R i2 = 0, R i3 = 0 Y i ) = f(r i3 = 0 Y i, R i1 = 0, R i2 = 0) f(r i2 = 0 Y i, R i1 = 0) f(r i1 = 0 Y i ) = = 0 (Y o i = ϕ) f(r i1 = 1, R i2 = 1, R i3 = 0) Y o i Y ij MNAR B MNAR 2 MNAR 2 MAR R ij f(r i1 = 0 Y i ) = 0 logitf(r i2 = 0 Y i, R i1 = 0) logitf(r i2 = 0 Y i, R i1 = 1) = Y i1 = Y i1 52

201 logitf(r i3 = 0 Y i, R i1 = 0, R i2 = 0) logitf(r i3 = 0 Y i, R i1 = 1, R i2 = 0) = Y i2 = Y i2 logitf(r i3 = 0 Y i, R i1 = 0, R i2 = 1) logitf(r i3 = 0 Y i, R i1 = 1, R i2 = 1) = Y i2 = Y i2 f(r i1 = 0 Y i ) = 0 logitf(r i2 = 0 Y i, R i1 ) = Y i1 logitf(r i3 = 0 Y i, R i1, R i2 ) = Y i2 f(r i1 = 0 Y i ) = 0 f(r i2 = 0 Y i, R i1 ) = f(r i3 = 0 Y i, R i1, R i2 ) = exp( Y i1 ) 1 + exp( Y i1 ) exp( Y i2 ) 1 + exp( Y i2 ) f(r i1 = 1 Y i ) = 1 f(r i2 = 1 Y i, R i1 ) = f(r i3 = 1 Y i, R i1, R i2 ) = exp( Y i1 ) exp( Y i2 ) f(r i1 = 1, R i2 = 1, R i3 = 1 Y i ) = f(r i3 = 1 Y i, R i1 = 1, R i2 = 1) f(r i2 = 1 Y i, R i1 = 1) f(r i1 = 1 Y i ) 1 = 1 + exp( Y i2 ) exp( Y i1 ) 1 1 = (Yi o = (Y i1, Y i2, Y i3 ) ) {1 + exp( Y i2 }{1 + exp( Y i1 )} f(r i1 = 0, R i2 = 1, R i3 = 1 Y i ) = f(r i3 = 1 Y i, R i1 = 0, R i2 = 1) f(r i2 = 1 Y i, R i1 = 0) f(r i1 = 0 Y i ) = exp( Y i2 ) exp( Y i1 ) 0 = 0 (Y o i = (Y i2, Y i3 ) ) f(r i1 = 1, R i2 = 0, R i3 = 1 Y i ) = f(r i3 = 1 Y i, R i1 = 1, R i2 = 0) f(r i2 = 0 Y i, R i1 = 1) f(r i1 = 1 Y i ) 1 = 1 + exp( Y i2 ) exp( Y i1 ) 1 + exp( Y i1 ) 1 exp( Y i1 ) = {1 + exp( Y i2 )}{1 + exp( Y i1 )} (Y o i = (Y i1, Y i3 ) ) f(r i1 = 0, R i2 = 0, R i3 = 1 Y i ) = f(r i3 = 1 Y i, R i1 = 0, R i2 = 0) f(r i2 = 0 Y i, R i1 = 0) f(r i1 = 0 Y i ) = exp( Y i2 ) exp( Y i1 ) 1 + exp( Y i1 ) 0 = 0 (Y o i = (Y i3 )) f(r i1 = 1, R i2 = 1, R i3 = 0 Y i ) = f(r i3 = 0 Y i, R i1 = 1, R i2 = 1) f(r i2 = 1 Y i, R i1 = 1) f(r i1 = 1 Y i ) = exp( Y i2) 1 + exp( Y i2 ) exp( Y i1 ) 1 exp( Y i2 ) = {1 + exp( Y i2 )}{1 + exp( Y i1 )} 53 (Y o i = (Y i1, Y i2 ) )

202 f(r i1 = 0, R i2 = 1, R i3 = 0 Y i ) = f(r i3 = 0 Y i, R i1 = 0, R i2 = 1) f(r i2 = 1 Y i, R i1 = 0) f(r i1 = 0 Y i ) = exp( Y i2) 1 + exp( Y i2 ) exp( Y i1 ) 0 = 0 (Y o i = (Y i2 )) f(r i1 = 1, R i2 = 0, R i3 = 0 Y i ) = f(r i3 = 0 Y i, R i1 = 1, R i2 = 0) f(r i2 = 0 Y i, R i1 = 1) f(r i1 = 1 Y i ) = exp( Y i2) 1 + exp( Y i2 ) exp( Y i1 ) 1 + exp( Y i1 ) 1 exp( Y i1 ) exp( Y i2 ) = {1 + exp( Y i2 )}{1 + exp( Y i1 )} (Y o i = (Y i1 )) f(r i1 = 0, R i2 = 0, R i3 = 0 Y i ) = f(r i3 = 0 Y i, R i1 = 0, R i2 = 0) f(r i2 = 0 Y i, R i1 = 0) f(r i1 = 0 Y i ) = exp( Y i2) 1 + exp( Y i2 ) exp( Y i1 ) 1 + exp( Y i1 ) 0 = 0 (Y o i = ϕ) f(r i1 = 1, R i2 = 0, R i3 = 1, Y i ) Y o i Y ij MNAR B.3 5 Selection Model 5 MNAR B.3.1 MNAR MNAR Selection Model logit{p r(r ij = 0 R i1 = 1,, R i,j 1 = 1, Y i, X i, ψ)} = ψ 1 + ψ 2 Y i,j 1 + ψ 3 Y ij (j = 2,, n) P r(r ij = 0 R i,j 1 = 0, R i,j 2, R i1, Y i, X i, ψ) = 1 (j = 2,, n) P r(r i1 = 1 Y i, X i, ψ) = 1 (B.12) R i,j 2, R i1 MNAR 1 6 NFD (Non-Future Dependence) 3 B i Yi o = Y i, R i1 = R i2 = R in = 1 f(yi o, R i X i, θ, ψ) = f(y, R i X i, θ, ψ) = f(y i X i, θ) f(r i Y i, X i, ψ) = f(y i X i, θ) P r(r i1 = 1,, R in = 1 Y i, ψ) = f(y i X i, θ) P r(r i1 = 1 X i, Y i, ψ) 54

203 P r(r i2 = 1 R i1 = 1, X i, Y i, ψ) P r(r in = 1 R i1 = = R i,n 1 = 1, Y i, X i, ψ) (B.13) f(y i X i, θ, ψ) (B.12) θ, ψ 4 (B.12) P r(r ij = 0 R i1 = = R i,j 1 = 1, Y i, X i, ψ) = exp(ψ 1 + ψ 2 Y i,j 1 + ψ 3 Y ij ) 1 + exp(ψ 1 + ψ 2 Y i,j 1 + ψ 3 Y ij ) P r(r ij = 1 R i1 = = R i,j 1 = 1, Y i, X i, ψ) =1 P r(r ij = 0 R i1 = = R i,j 1 = 1, Y i, X i, ψ) 1 = 1 + exp(ψ 1 + ψ 2 Y i,j 1 + ψ 3 Y ij ) 1 f(y i X i, θ) = ( (2π)n Σ exp 1 ) 2 (Y i µ i ) Σ 1 (Y i µ i ) 5 Y i = Yi o (B.13) f(yi o 1, R i X i, θ, ψ) = ( (2π)n Σ exp 1 ) 2 (Y i µ i ) Σ 1 (Y i µ i ) 1 i exp(ψ 1 + ψ 2 Y i1 + ψ 3 Y i2 ) exp(ψ 1 + ψ 2 Y i,n 1 + ψ 3 Y in ) B j (j 3) i R i1 = = R i,j 1 = 1, R ij = = R in = 0 Yi o = (Y i1,, Y i,j 1 ), Yi m = (Y ij,, Y in ) f(yi o, R i X i, θ, ψ) = f(yi o, Yi m, R i X i, θ, ψ)dyi m = f(yi o, Yi m X i, θ) f(r i X i, Yi o, Yi m, ψ)dyi m = f(y m i Y o i, X i, θ) f(y o i X i, θ) P r(r ij = = R in = 0 R i1 = = R i,j 1 = 1, X i, Y i, ψ) P r(r i1 = = R i,j 1 = 1 X i, Y o i, ψ)dy m i = f(yi o X i, θ) P r(r i1 = = R i,j 1 = 1 X i, Yi o, ψ) f(yi m Yi o, X i, θ) P r(r ij = = R in = 0 R i1 = = R i,j 1 = 1, X i, Y i,j 1,, Y in, ψ)dy ij dy in ( Y m i = (Y ij,, Y in ) ) µ i i µ i = X i β X i 55

204 = f(yi o X i, θ) P r(r i1 = = R i,j 1 = 1 X i, Yi o, ψ) f(y ij,, Y in Y i1,, Y i,j 1, X i, θ) P r(r ij = 0 R i1 = = R i,j 1 = 1, X i, Y i,j 1, Y ij, ψ) P r(r i,j+1 = 0 R i1 = = R i,j 1 = 1, R ij = 0, X i, Y ij, Y i,j+1, ψ) P r(r in = 0 R i1 = = R i,j 1 = 1, R ij = = R i,n 1 = 0, X i, Y i,n 1, Y in, ψ)dy ij dy in ( NFD) = f(yi o X i, θ) P r(r i1 = 1 X i, Y 1, ψ) P r(r i2 = 1 R i1 = 1, X i, Y 1, Y 2, ψ) P r(r i,j 1 = 1 R i1 = = R i,j 2 = 1, Y i,j 2, Y i,j 1, X i, ψ) f(y ij,, Y in Y i1,, Y i,j 1, X i, θ) P r(r ij = 0 R i1 = = R i,j 1 = 1, X i, Y i,j 1, Y ij, ψ) 1 1 dy ij dy i,j+1 dy in = f(yi o X i, θ) P r(r i2 = 1 R i1 = 1, X i, Y i1, Y i2, ψ) P r(r i,j 1 = 1 R i1 = = R i,j 2 = 1, X i, Y i,j 2, Y i,j 1, ψ) f(y ij Yi o, X i, θ) P r(r ij = 0 R i1 = = R i,j 1 = 1, X i, Y i,j 1, Y ij, ψ)dy ij (B.14) NFD 1 Diggle and Kenward (1994) Nelder-Mead 6 %Selection_Model2, %SM_GridSearch, Newton-Raphson ridge Newton ( ) f(y ij Yi o 1, θ) = exp (Y ij µ ij ) 2 2πσij 2 2σij 2 µ ij, σij 2 Yo i = (Y i1,, Y i,j 1 ) Y ij Y ij (B.12) P r(r ij = 0 R i1 = = R i,j 1 = 1, X i, Y i, ψ) = exp(ψ 1 + ψ 2 Y i,j 1 + ψ 3 Y ij ) 1 + exp(ψ 1 + ψ 2 Y i,j 1 + ψ 3 Y ij ) (B.14) ( ) 1 exp (Y ij µ ij ) 2 exp(ψ 1 + ψ 2 Y i,j 1 + ψ 3 Y ij ) 2πσij 2 2σij exp(ψ 1 + ψ 2 Y i,j 1 + ψ 3 Y ij ) dy ij Y i,j 1 i B i 3 Yi o f(yo i X i, θ) P r(r i2 = 1 R i1 = 1, X i, Y i1, Y i2, ψ) Nelder-Mead SAS Proc IML NLPNRR Nelder-Mead (1965a, 1965b) 56

205 3 R i1 = 1, R i2 = R i3 = = R in = 0, Yi o = (Y i1), Yi m = (Y i2,, Y in ) f(yi o, R i X i, θ, ψ) = f(y i1, R i X i, θ, ψ) = f(y i1, Yi m, R i X i, θ, ψ)dyi m = f(y i1, Yi m X i, θ) P r(r i1 = 1, R i2 = = R in = 0 X i, Y i1, Yi m, ψ)dyi m = f(y i2,, Y in Y i1, X i, θ) f(y i1 X i, θ) P r(r i2 = = R in = 0 R i1 = 1, X i, Y i, ψ) P r(r i1 = 1 X i, Y i1, ψ)dy i2 dy in = f(y i1 X i, θ) P r(r i1 = 1 X i, Y i1, ψ) f(y i2,, Y in Y i1, X i, θ) P r(r i2 = = R in = 0 R i1 = 1, X i, Y i1,, Y in, ψ)dy i2 dy in = f(yi o X i, θ) 1 f(y i2,, Y in Y i1, X i, θ) P r(r i2 = 0 R i1 = 1, X i, Y i1, Y i2, ψ) P r(r i3 = 0 R i1 = 1, R i2 = 0, X i, Y i2, Y i3, ψ) P r(r in = 0 R i1 = 1, R i,2 = = R i,n 1 = 0, X i, Y i,n 1, Y in, ψ)dy i2 dy in = f(yi o X i, θ) f(y i2, Y i3,, Y in Y i1, X i, θ) P r(r i2 = 0 R i1 = 1, X i, Y i1, Y i2, ψ) 1 1 dy i2 dy i3 dy in = f(yi o X i, θ) f(y i2 Y i1, X i, θ) P r(r i2 = 0 R i1 = 1, X i, Y i1, Y i2, ψ)dy i2 j (j 3) B.4 6 NFMV NFD ACMV MAR 6 1. NFMV ACMV ACMV 2. ACMV MAR 3. NFMV NFD 2 3 SM PMM 1. NRC (2010) 5 PMM 3 NRC (2010) 5 PMM MAR B

206 B "ACMV MAR" "NFMV NFD" 6 i, j B i 1 i T i = 1 R Y i i 1 Y i i i > 1 f(y i Y i 1, R = j) j Y i Y j 1 f(y 3 Y 2, R = 1) 1 3 f(y 3 Y 2, R 3) f j (Y i Y i 1 ) f(y i Y i 1, R = j) B P r(r = t 1 R t 1) 0 ( t 2) i 1 Y i f(y i ) > 0 B f(y i Y i 1, R j) =f(y i Y i 1, R = j) P (R = j Y i 1, R j) + f(y i Y i 1, R j + 1) P (R j + 1 Y i 1, R j) (B.15). f(y i Y i 1, R j) = f(y i, Y i 1, R j) f(y i 1, R j) = f(y i, Y i 1, R = j) + f(y i, Y i 1, R j + 1) f(y i 1, R j) (T 1) 58

207 = f(y i, Y i 1, R = j) f(y i 1, R = j) f(y i 1, R = j) f(y i 1, R j) + f(y i, Y i 1, R j + 1) f(y i 1, R j + 1) f(y i 1, R j + 1) f(y i 1, R j) =f(y i Y i 1, R = j) P (R = j Y i 1, R j) + f(y i Y i 1, R j + 1) P (R j + 1 Y i 1, R j) B t 2, j t 1 f(y t Y t 1, R t) f(y t Y t 1, R = j) f(y t Y t 1, R t) R t t t j t 1 R = j < t f(y t Y t 1, R = j) j (< t) t B f(y 4 Y 3, R = 4), f(y 3 Y 2, R = 4), f(y 3 Y 2, R = 3), f(y 2 Y 1, R = 4), f(y 2 Y 1, R = 3), f(y 2 Y 1, R 2) 8 f(y 4 Y 3, R = 3), f(y 4 Y 3, R = 2), f(y 4 Y 3, R = 1), f(y 3 Y 2, R = 2), f(y 3 Y 2, R = 1), f(y 2 Y 1, R = 1) 1 4 B.1: f(y 3 Y 2, R 3) R f(y 3 Y 2, R = 3), f(y 3 Y 2, R = 4) 6 interior family 59

208 B.2: f(y 1 R = 4) f(y 2 Y 1, R = 4) f(y 3 Y 2, R = 4) f(y 4 Y 3, R = 4) 2 f(y 1 R = 3) f(y 2 Y 1, R = 3) f(y 3 Y 2, R = 3) f(y 4 Y 3, R = 3) 4 3 f(y 1 R = 2) f(y 2 Y 1, R = 2) f(y 3 Y 2, R = 2) f(y 4 Y 3, R = 2) 3 4 f(y 1 R = 1) f(y 2 Y 1, R = 1) f(y 3 Y 2, R = 1) f(y 4 Y 3, R = 1) f(y 4 Y 3, R = 4) complete case 2 4 f(y 4 Y 3, R = 3), f(y 4 Y 3, R = 2), f(y 4 Y 3, R = 1) B.4.2 NFMV ACMV PMM 1 CCMV NCMV ACMV NFMV 6 NFMV ACMV 2 B NFMV B NFMV t 3, j t 2 f(y t Y t 1, R = j) = f(y t Y t 1, R t 1) (B.16) NFMV 9 f(y t Y t 1, R = 1) = f(y t Y t 1, R = 2) = = f(y t Y t 1, R = t 2) B NFMV 4 NFMV t = 4, j = 1 ( 2 = t 2) (B.16) f(y 4 Y 3, R = 1) = f(y 4 Y 3, R 3) (B.17) 9 Kenward et al. (2003) t = 2 t = 2 f(y 2 Y 1, R = 0) 0 1 t 3 60

209 10 t = 4, j = 2 t = 3, j = 1 f(y 4 Y 3, R = 2) = f(y 4 Y 3, R 3) f(y 3 Y 2, R = 1) = f(y 3 Y 2, R 2) (B.18) (B.19) t 3, j t 2 NFMV (B.16) NFMV (B.17) (B.19) f(y 4 Y 3, R = 1) = f(y 4 Y 3, R = 2) = f(y 4 Y 3, R 3) f(y 3 Y 2, R = 1) = f(y 3 Y 2, R 2) 2 (i) f(y 4 Y 3, R = 1), f(y 4 Y 3, R = 2), f(y 3 Y 2, R = 1) 11 (ii) f(y 4 Y 3, R = 3), f(y 3 Y 2, R = 2), f(y 2 Y 1, R = 1) 12 NFMV 4 B.3: B.4: NFMV f(y 1 R = 4) f(y 2 Y 1, R = 4) f(y 3 Y 2, R = 4) f(y 4 Y 3, R = 4) 2 f(y 1 R = 3) f(y 2 Y 1, R = 3) f(y 3 Y 2, R = 3) 3 f(y 1 R = 2) f(y 2 Y 1, R = 2) f(y 4 Y 3, R 3) 4 f(y 1 R = 1) f(y 3 Y 2, R 2) f(y 4 Y 3, R 3) B ACMV B ACMV t 2, j t 1 f(y t Y t 1, R = j) = f(y t Y t 1, R t) (B.20) ACMV 10 4 R = 4 R 4 11 f(y 4 Y 3, R 3) f(y 4 Y 3, R 3) = f(y 4 Y 3, R = 3) P (R = 3 Y 3, R 3) + f(y 4 Y 3, R = 4) P (R = 4 Y 3, R 3) }{{}}{{} 12 61

210 B ACMV NFMV 4 f(y 4 Y 3, R = 1) = f(y 4 Y 3, R = 4) f(y 4 Y 3, R = 2) = f(y 4 Y 3, R = 4) f(y 4 Y 3, R = 3) = f(y 4 Y 3, R = 4) f(y 3 Y 2, R = 1) = f(y 3 Y 2, R 3) f(y 3 Y 2, R = 2) = f(y 3 Y 2, R 3) f(y 2 Y 1, R = 1) = f(y 2 Y 1, R 2) (B.21) (B.22) (B.23) (B.24) (B.25) (B.26) f(y 4 Y 3, R = 1) = f(y 4 Y 3, R = 2) = f(y 4 Y 3, R = 3) =f(y 4 Y 3, R = 4) f(y 3 Y 2, R = 1) = f(y 3 Y 2, R = 2) =f(y 3 Y 2, R 3) f(y 2 Y 1, R = 1) =f(y 2 Y 1, R 2) NFMV NFMV B.5: B.6: ACMV f(y 1 R = 4) f(y 2 Y 1, R = 4) f(y 3 Y 2, R = 4) f(y 4 Y 3, R = 4) 2 f(y 1 R = 3) f(y 2 Y 1, R = 3) f(y 3 Y 2, R = 3) f(y 4 Y 3, R = 4) 3 f(y 1 R = 2) f(y 2 Y 1, R = 2) f(y 3 Y 2, R 3) f(y 4 Y 3, R = 4) 4 f(y 1 R = 1) f(y 2 Y 1, R 2) f(y 3 Y 2, R 3) f(y 4 Y 3, R = 4) 13 B NFMV ACMV NFMV ACMV ACMV NFMV ACMV f(y 3 Y 2, R 3) f(y 3 Y 2, R = 3), f(y 3 Y 2, R = 4) 62

211 NFMV ACMV NFMV ACMV NFMV 4 B.4 14 B ACMV NFMV ACMV NFMV (B.21) (B.26) (B.17) (B.19) (B.21) (B.23) (B.24) (B.25) (B.17) (B.18) (B.19) (B.21) (B.23) (B.17), (B.18) (B.23) f(y 4 Y 3, R = 3) = f(y 4 Y 3, R = 4) 2 f(y 4 Y 3, R 3) 15 f(y 4 Y 3, R = 4) = f(y 4 Y 3, R = 3) = f(y 4 Y 3, R 3) (B.27) (B.21) f(y 4 Y 3, R = 1) = f(y 4 Y 3, R = 4) = f(y 4 Y 3, R 3) (B.17) (B.22) (B.27) f(y 4 Y 3, R = 2) = f(y 4 Y 3, R = 4) = f(y 4 Y 3, R 3) (B.18) (B.21) (B.23) (B.17) (B.18) (B.24), (B.25) (B.19) (B.25) f(y 3 Y 2, R = 2) = f(y 3 Y 2, R 3) f(y 3 Y 2, R 2) 16 (B.24) f(y 3 Y 2, R = 1) = f(y 3 Y 2, R 3) = f(y 3 Y 2, R 2) (B.19) (B.24) (B.25) (B.19) (B.21) (B.26) (B.17) (B.19) (B.26) ACMV NFMV f(y 4 Y 3, R 3) = f(y 4 Y 3, R = 3) P (R = 3 Y 3, R 3) + f(y 4 Y 3, R = 4) P (R = 4 Y 3, R 3) = f(y 4 Y 3, R = 4) P (R = 3 Y 3, R 3) + f(y 4 Y 3, R = 4) P (R = 4 Y 3, R 3) ( (B.23)) { } = f(y 4 Y 3, R = 4) P (R = 3 Y 3, R 3) + P (R = 4 Y 3, R 3) = f(y 4 Y 3, R = 4) P (R 3 Y 3, R 3) = f(y 4 Y 3, R = 4) (B.23) f(y 4 Y 3, R = 3) 16 f(y 3 Y 2, R 2) = f(y 3 Y 2, R = 2) P (R = 2 Y 2, R 2) + f(y 3 Y 2, R 3) P (R 3 Y 2, R 2) = f(y 3 Y 2, R 3) P (R = 2 Y 2, R 2) + f(y 3 Y 2, R 3) P (R 3 Y 2, R 2) ( (B.25)) { } = f(y 3 Y 2, R 3) P (R = 2 Y 2, R 2) + P (R 3 Y 2, R 2) = f(y 3 Y 2, R 3) 63

212 NFMV ACMV 17 (B.21) (B.23) (B.17) (B.22) (B.23) (B.18) (B.24) (B.25) (B.19) (B.21) (B.17) (B.22) (B.18) (B.24) (B.19) (B.23) (B.25) (B.26) NFMV f(y t Y t 1, R = t 1) 18 NFMV (B.23), (B.25), (B.26) ACMV (B.17) (B.19) (B.23) (B.25) (B.26) (B.21) (B.26) (1) (B.17) (B.23) (B.21) (2) (B.18) (B.23) (B.22) (3) (B.23) (B.23) (4) (B.19) (B.25) (B.24) (5) (B.25) (B.25) (6) (B.26) (B.26) (1) (B.17) (B.23) (B.21) (B.17) (B.15) f(y 4 Y 3, R = 1) = f(y 4 Y 3, R 3) = f(y 4 Y 3, R = 3) P (R = 3 Y 3, R 3) + f(y 4 Y 3, R = 4) P (R = 4 Y 3, R 3) = f(y 4 Y 3, R = 4) P (R = 3 Y 3, R 3) + f(y 4 Y 3, R = 4) P (R = 4 Y 3, R 3) ( (B.23)) { } = f(y 4 Y 3, R = 4) P (R = 3 Y 3, R 3) + P (R = 4 Y 3, R 3) = f(y 4 Y 3, R = 4) P (R 3 Y 3, R 3) = f(y 4 Y 3, R = 4) f(y 4 Y 3, R = 1) = f(y 4 Y 3, R = 4) (B.21) (B.17) (B.23) (B.21) (2) (B.18) (B.23) (B.22) (B.18) (B.23) f(y 4 Y 3, R = 2) = f(y 4 Y 3, R 3) = f(y 4 Y 3, R = 3) P (R = 3 Y 3, R 3) + f(y 4 Y 3, R = 4) P (R = 4 Y 3, R 3) = f(y 4 Y 3, R = 4) P (R = 3 Y 3, R 3) + f(y 4 Y 3, R = 4) P (R = 4 Y 3, R 3) ( (B.23)) { } = f(y 4 Y 3, R = 4) P (R = 3 Y 3, R 3) + P (R = 4 Y 3, R 3) = f(y 4 Y 3, R = 4) P (R 3 Y 3, R 3) = f(y 4 Y 3, R = 4) (B.22) (B.18) (B.23) (B.22) (3) (B.23) (B.23) (4) (B.19) (B.25) (B.24) f(y 3 Y 2, R = 1) = f(y 3 Y 2, R 2) B.4 64

213 = f(y 3 Y 2, R = 2) P (R = 2 Y 2, R 2) + f(y 3 Y 2, R 3) P (R 3 Y 2, R 2) = f(y 3 Y 2, R 3) P (R = 2 Y 2, R 2) + f(y 3 Y 2, R 3) P (R 3 Y 2, R 2) ( (B.25)) { } = f(y 3 Y 2, R 3) P (R = 2 Y 2, R 2) + P (R 3 Y 2, R 2) = f(y 3 Y 2, R 3) (B.24) (B.19) (B.25) (B.24) (5) (B.25) (B.25) (6) (B.26) (B.26) NFMV (B.23), (B.25), (B.26) ACMV (1) (6) NFMV (B.23), (B.25), (B.26) ACMV (B.23), (B.25), (B.26) NFMV B.4 B.7 (B.23), (B.25), (B.26) f(y 4 Y 3, R = 3) = f(y 4 Y 3, R = 4) f(y 3 Y 2, R = 2) = f(y 3 Y 2, R 3) f(y 2 Y 1, R = 1) = f(y 2 Y 1, R 2) B.7 B.7: NFMV B f(y 1 R = 4) f(y 2 Y 1, R = 4) f(y 3 Y 2, R = 4) f(y 4 Y 3, R = 4) 2 f(y 1 R = 3) f(y 2 Y 1, R = 3) f(y 3 Y 2, R = 3) 3 f(y 1 R = 2) f(y 2 Y 1, R = 2) f(y 4 Y 3, R 3) 4 f(y 1 R = 1) f(y 3 Y 2, R 2) f(y 4 Y 3, R 3) NFMV t 3, j t 1 f(y t Y t 1, R = j) = f(y t Y t 1, R t 1) 3 f(y t Y t 1, R = t 1) = f(y t Y t 1, R t) ( t 2) NFMV ACMV ACMV ACMV t 3, j t 2 f(y t Y t 1, R = j) = f(y t Y t 1, R t 1) t 2 f(y t Y t 1, R = t 1) = f(y t Y t 1, R t) ACMV 19 B.8 19 NFMV t, j (NFMV t 3, j t 2) 65

214 B.8: ACMV f(y 1 R = 4) f(y 2 Y 1, R = 4) f(y 3 Y 2, R = 4) f(y 4 Y 3, R = 4) 2 f(y 1 R = 3) f(y 2 Y 1, R = 3) f(y 3 Y 2, R = 3) f(y 4 Y 3, R = 4) 3 f(y 1 R = 2) f(y 2 Y 1, R = 2) f(y 3 Y 2, R 3) f(y 4 Y 3, R 3) 4 f(y 1 R = 1) f(y 2 Y 1, R 2) f(y 3 Y 2, R 2) f(y 4 Y 3, R 3) B (B.23) f(y 4 Y 3, R = 3) = f(y 4 Y 3, R = 4) (B.25) f(y 3 Y 2, R = 2) = f(y 3 Y 2, R 3) (B.26)f(Y 2 Y 1, R = 1) = f(y 2 Y 1, R 2) f(y t Y t 1, R t) = f(y t Y t 1, R = t 1) ( t 2) (B.28) NFMV (B.28) ACMV 4 ACMV NFMV (B.28) (B.28) ACMV ACMV NFMV ACMV t 3 f(y t Y t 1, R t) = f(y t Y t 1, R = t 1) = f(y t Y t 1, R t 1) (B.20) t 3, j t 2 f(y t Y t 1, R = j) = f(y t Y t 1, R t) = f(y t Y t 1, R t 1) (B.16) NFMV ACMV NFMV ACMV NFMV (B.28) NFMV (B.28) ACMV t 3, j t 2 NFMV (B.16) f(y t Y t 1, R = j) = f(y t Y t 1, R t 1) = f(y t Y t 1, R = t 1) P (R = t 1 Y t 1, R t 1) + f(y t Y t 1, R t) P (R t Y t 1, R t 1) = f(y t Y t 1, R t) P (R = t 1 Y t 1, R t 1) + f(y t Y t 1, R t) P (R t Y t 1, R t 1) ( (B.28)) { } = f(y t Y t 1, R t) P (R = t 1 Y t 1, R t 1) + P (R t Y t 1, R t 1) = f(y t Y t 1, R t) t 3, j t 2 (B.20) ACMV t 2 j = t 1 (B.20) (B.28) NFMV (B.28) ACMV NFMV (B.28) ACMV NFMV f(y t Y t 1, R = t 1) = f(y t Y t 1, R t) ( t 2) 66 ACMV

215 NFMV ACMV (B.28) f(y t Y t 1, R = t 1) = f(y t Y t 1, R t) ( t 2) NFMV ACMV (B.28) ACMV MAR 20 (B.28) = MAR = MAR NFMV f(y t Y t 1, R = t 1) B.7 (B.28) MAR NRC (2010) 5 PMM B.4.3 NRC (2010) PMM NRC (2010) 5 PMM NRC (2010) p98 NFMV 21 NRC (2010) p99 f(y t Y t 1, R = t 1) f(y t Y t 1, R t) (NRC 22) NRC (2010) (22) (NRC 22) (NRC 32) 1 4 E[Y 1 ] = µ 1 E[Y 2 Y 1, R 2] = µ 2 + Y 1 β 2 E[Y 3 Y 2, R 3] = µ 3 + Y 2β 3 (B.29) E[Y 4 Y 3, R = 4] = µ 4 + Y 3β 4 (NRC 33) E[Y 2 Y 1, R = 1] = µ 2 + Y 1 β 2 E[Y 3 Y 2, R = 2] = µ 3 + Y 2β 3 (B.30) E[Y 4 Y 3, R = 3] = µ 4 + Y 3β 4 NFMV (B.28) ACMV (B.29) (B.30) ACMV ( MAR) 22 NRC (2010) 5 β 2 = β2, β 3 = β 3, β 4 = β 4, NFD NFD NFMV 22 (B.28) E[Y 2 Y 1, R = 1] = E[Y 2 Y 1, R 2] E[Y 3 Y 2, R = 2] = E[Y 3 Y 2, R 3] E[Y 4 Y 3, R = 3] = E[Y 4 Y 3, R = 4] (B.28) NRC (2010) 67

216 µ 2 = µ 2 + µ2 µ 3 = µ 3 + µ3 µ 4 = µ 4 + µ4 (B.30) E[Y 2 Y 1, R = 1] = µ 2 + µ2 + Y 1 β 2 (= E[Y 2 Y 1, R 2] + µ2 ) E[Y 3 Y 2, R = 2] = µ 3 + µ3 + Y 2β 3 (= E[Y 3 Y 2, R 3] + µ3 ) E[Y 4 Y 3, R = 3] = µ 4 + µ4 + Y 3β 4 (= E[Y 4 Y 3, R = 4] + µ4 ) µ2 = µ3 = µ4 = 0 ACMV ( MAR) 23 µ2, µ3, µ4 0 MAR µ2, µ3, µ4 MAR NRC (2010) 5 PMM MI missingdata.org.uk %delta_pmm %cbi_pmm Appendix C B.4.4 ACMV MAR ACMV MAR NFMV NFD ACMV MAR B.4.1. ACMV MAR Molenberghs et al. (1998) Verbeke and Molenberghs (2000) B MAR t 1 P (R = t Y T ) = P (R = t Y t ) ACMV t 2, j t 1 f(y t Y t 1, R = j) = f(y t Y t 1, R t) f(y t Y t 1, R = 1) = f(y t Y t 1, R = 2) = = f(y t Y t 1, R = t 1) 1 i j T i, j Y j i = (Y i,, Y j ) Y i T T i = j Y j i = Y i (= Y j ) B j T, 1 t T, P (R = j Y t ) > 0 Y j f(y j ) > 0 23 NFMV ACMV MAR 68

217 B B.4.1. AMCV t 2, j t 1 f(y t Y t 1, R = j) = f(y t Y t 1 ) (B.31). ACMV (B.31) t j t, j 1 t 1 f(y t Y t 1 ) = f(y t Y t 1, R = i) P (R = i Y t 1 ) + f(y t Y t 1, R t) P (R t Y t 1 ) i=1 (B.32) ACMV (B.31) (B.32) ACMV t 1 f(y t Y t 1 ) = f(y t Y t 1, R = i) P (R = i Y t 1 ) + f(y t Y t 1, R t) P (R t Y t 1 ) i=1 t 1 = f(y t Y t 1, R = j) P (R = i Y t 1 ) i=1 + f(y t Y t 1, R = j) P (R t Y t 1 ) ( ACMV ) t 1 = f(y t Y t 1, R = j) P (R = i Y t 1 ) + f(y t Y t 1, R = j) P (R t Y t 1 ) = f(y t Y t 1, R = j) = f(y t Y t 1, R = j) i=1 { t 1 } P (R = i Y t 1 ) + P (R t Y t 1 ) i=1 (B.31) (B.31) ACMV (B.32) (B.31) f(y t Y t 1, R = 1) = f(y t Y t 1, R = 2) = = f(y t Y t 1, R = t 1) = f(y t Y t 1 ) j t 1 t 1 f(y t Y t 1 ) = f(y t Y t 1, R = i) P (R = i Y t 1 ) + f(y t Y t 1, R t) P (R t Y t 1 ) i=1 t 1 f(y t Y t 1, R = j) = f(y t Y t 1, R = j) P (R = i Y t 1 ) + f(y t Y t 1, R t) P (R t Y t 1 ) i=1 i=1 ( j t 1, (B.31)) t 1 = f(y t Y t 1, R = j) P (R = i Y t 1 ) + f(y t Y t 1, R t) P (R t Y t 1 ) t 1 f(y t Y t 1, R = j) f(y t Y t 1, R = j) P (R = i Y t 1 ) = f(y t Y t 1, R t) P (R t Y t 1 ) i=1 { } t 1 f(y t Y t 1, R = j) 1 P (R = i Y t 1 ) = f(y t Y t 1, R t) P (R t Y t 1 ) i=1 f(y t Y t 1, R = j) P (R t Y t 1 ) = f(y t Y t 1, R t) P (R t Y t 1 ) 69

218 t 2 P (R t Y t 1 ) > 0 f(y t Y t 1, R = j) = f(y t Y t 1, R t) t 2, j t 1 ACMV (B.31) ACMV B.4.2. MAR 1 t u T t, u P (R = t Y u ) = P (R = t Y t ). u = t u = T MAR 1 t < u < T P (R = t Y u ) = f(r = t, Y u) f(y u ) f(r = t, Yu, Y T = u+1)dyu+1 T f(y u ) f(r = t, YT )dyu+1 T = f(y u ) P (R = t YT )f(y T )dyu+1 T = f(y u ) = P (R = t Yt )f(y T )dy T u+1 f(y u ) = P (R = t Y t) f(y T )dy T u+1 f(y u ) ( MAR) ( u > t) = P (R = t Y t) f(y u ) f(y u ) = P (R = t Y t ) 1 t u T P (R = t Y u ) = P (R = t Y t ) B < i < j T i, j j 1 f(y j i+1 Y i) = f(y t+1 Y t ). 3 X, Y, Z t=i f(x, Y Z) = f(x Y, Z) f(y Z) f(x, Y Z) = = f(y j i+1 Y i) = f(y j 1 i+1, Y j Y i ) f(x, Y, Z) f(z) f(x, Y, Z) f(y, Z) f(y, Z) f(z) = f(x Y, Z) f(y Z) 70

219 = f(y j Y j 1 i+1, Y i) f(y j 1 i+1 Y i) = f(y j Y j 1 ) f(y j 2 i+1, Y j 1 Y i ) = f(y j Y j 1 ) f(y j 1 Y j 2 i+1, Y i) f(y j 2 i+1 Y i) = f(y j Y j 1 ) f(y j 1 Y j 2 ) f(y j 2 i+1 Y i). = f(y j Y j 1 ) f(y j 1 Y j 2 ) f(y i+1 Y i ) j 1 = f(y t+1 Y t ) t=i B B.4.1 B.4.1. B.4.1 ACMV (B.31) MAR (B.31) MAR (B.31) MAR j 1 f(y T, R = j) = P (R = j Y T ) f(y T ) = P (R = j Y j ) f(y T ) ( MAR) (B.33) MAR 1 j < t T B.4.2 f(y T, R = j) = f(y t, Yt+1, T R = j) = f(yt+1 T Y t, R = j) P (R = j Y t ) f(y t ) = f(yt+1 T Y t, R = j) P (R = j Y j ) f(y t ) ( t > j B.4.2 ) (B.34) (B.33) = (B.34) f(yt+1 T Y t, R = j) P (R = j Y j ) f(y t ) = P (R = j Y j ) f(y T ) f(yt+1 T Y t, R = j) = f(y T ) f(y t ) = f(y t, Yt+1) T = f(yt+1 T Y t ) ( P (R = j Y j ) > 0) f(y t ) Y T t+2 f(y t+1 Y t, R = j) = f(y t+1 Y t ) t = T 1 t = T (B.34) (B.33) f(y T, R = j) = f(y T 1, Y T, R = j) = f(y T Y T 1, R = j) P (R = j Y T 1 ) f(y T 1 ) = f(y T Y T 1, R = j) P (R = j Y j ) f(y T 1 ) f(y T Y T 1, R = j) P (R = j Y j ) f(y T 1 ) = P (R = j Y j ) f(y T ) 71

220 f(y T Y T 1, R = j) = f(y T ) f(y T 1 ) = f(y T Y T 1 ) ( P (R = j Y j ) > 0) MAR (B.31) (B.31) MAR t 2, j t 1 f(y T, R = j) = f(y j, R = j) f(y T j+1 Y j, R = j) T = f(y j, R = j) f(y t Y t 1, R = j) t=j+1 ( B.4.3) T = f(y j, R = j) f(y t Y t 1 ) t=j+1 ( (B.31)) = f(y j, R = j) f(y j+1 Y j ) f(y j+2 Y j+1 ) f(y T Y T 1 ) ( B.4.3) = f(y j, R = j) f(yj+1 T Y j ) = P (R = j Y j ) f(y j ) f(yj+1 T Y j ) = P (R = j Y j ) f(y T ) (B.35) f(y T, R = j) = P (R = j Y T ) f(y T ) (B.36) (B.35) (B.36) P (R = j Y j ) f(y T ) = P (R = j Y T ) f(y T ) f(y T ) > 0 P (R = j Y j ) = P (R = j Y T ) t 2, j t 1 1 j T 1 j = T (B.31) MAR MAR (B.31) B.4.1 MAR ACMV B.4.5 NFMV NFD ACMV MAR NFMV NFD B.4.2. NFMV NFD Kenward (2003) B NFD (Non Future Dependence) t 1 P (R = t Y T ) = P (R = t Y t+1 ) NFMV t 3, j t 2 f(y t Y t 1, R = j) = f(y t Y t 1, R t 1) f(y t Y t 1, R = 1) = f(y t Y t 1, R = 2) = = f(y t Y t 1, R = t 2) 72

221 B B j T, 1 t T, P (R = j Y t ) > 0 Y j f(y j ) > 0 B ACMV MAR 2 B.4.4. NFMV t 3, j t 2 f(y t Y t 1, R = j) = f(y t Y t 1 ) (B.37). NFMV (B.37) t j t, j 1 t 2 f(y t Y t 1 ) = f(y t Y t 1, R = i) P (R = i Y t 1 ) + f(y t Y t 1, R t 1) P (R t 1 Y t 1 ) i=1 NFMV (B.37) (B.38) NFMV t 2 f(y t Y t 1 ) = f(y t Y t 1, R = i) P (R = i Y t 1 ) + f(y t Y t 1, R t 1) P (R t 1 Y t 1 ) i=1 t 2 = f(y t Y t 1, R = j) P (R = i Y t 1 ) i=1 + f(y t Y t 1, R = j) P (R t 1 Y t 1 ) ( (NFMV )) t 2 = f(y t Y t 1, R = j) P (R = i Y t 1 ) + f(y t Y t 1, R = j) P (R t 1 Y t 1 ) = f(y t Y t 1, R = j) = f(y t Y t 1, R = j) NFMV (B.37) i=1 { t 2 } P (R = i Y t 1 ) + P (R t 1 Y t 1 ) i=1 (B.38) (B.37) NFMV (B.38) (B.37) f(y t Y t 1, R = 1) = f(y t Y t 1, R = 2) = = f(y t Y t 1, R = t 2) = f(y t Y t 1 ) t 2 f(y t Y t 1 ) = f(y t Y t 1, R = i) P (R = i Y t 1 ) + f(y t Y t 1, R t 1) P (R t 1 Y t 1 ) i=1 t 2 f(y t Y t 1, R = j) = f(y t Y t 1, R = j) P (R = i Y t 1 ) i=1 + f(y t Y t 1, R t 1) P (R t 1 Y t 1 ) ( (B.37)) t 2 = f(y t Y t 1, R = j) P (R = i Y t 1 ) + f(y t Y t 1, R t 1) P (R t 1 Y t 1 ) i=1 t 2 f(y t Y t 1, R = j) f(y t Y t 1, R = j) P (R = i Y t 1 ) = f(y t Y t 1, R t 1) P (R t 1 Y t 1 ) i=1 73

222 { } t 2 f(y t Y t 1, R = j) 1 P (R = i Y t 1 ) = f(y t Y t 1, R t 1) P (R t 1 Y t 1 ) i=1 f(y t Y t 1, R = j) P (R t 1 Y t 1 ) = f(y t Y t 1, R t 1) P (R t 1 Y t 1 ) t 2 P (R t 1 Y t 1 ) > 0 f(y t Y t 1, R = j) = f(y t Y t 1, R t 1) (B.37) NFMV (B.37) NFMV B.4.5. NFD 1 t < u T t, u P (R = t Y u ) = P (R = t Y t+1 ). u = T NFD 1 t < u < T P (R = t Y u ) = f(r = t, Y u) f(y u ) f(r = t, Yu, Y T = u+1)dyu+1 T f(y u ) f(r = t, YT )dyu+1 T = f(y u ) P (R = t YT )f(y T )dyu+1 T = f(y u ) = P (R = t Yt+1 )f(y T )dy T u+1 f(y u ) = P (R = t Y t+1) f(y T )dy T u+1 f(y u ) ( NFD) ( u > t) = P (R = t Y t+1) f(y u ) f(y u ) = P (R = t Y t+1 ) 1 t < u T P (R = t Y u ) = P (R = t Y t+1 ) B B.4.2 B.4.2. B.4.4 NFMV (B.37) NFD (B.37) NFD (B.37) NFD j 1 f(y T, R = j) = P (R = j Y T ) f(y T ) = P (R = j Y j+1 ) f(y T ) ( NFD) (B.39) 74

223 B.4.5 t 3, j t 2 t f(y T, R = j) = f(y t, Y T t+1, R = j) = f(y T t+1 Y t, R = j) P (R = j Y t ) f(y t ) = f(y T t+1 Y t, R = j) P (R = j Y j+1 ) f(y t ) ( t > j B.4.5 ) (B.40) (B.39) = (B.40) f(y T t+1 Y t, R = j) P (R = j Y j+1 ) f(y t ) = P (R = j Y j+1 ) f(y T ) f(yt+1 T Y t, R = j) = f(y T ) f(y t ) = f(y t, Yt+1) T = f(yt+1 T Y t ) ( P (R = j Y j+1 ) > 0) f(y t ) Y T t+2 f(y t+1 Y t, R = j) = f(y t+1 Y t ) t = T 1 t = T f(y T, R = j) = f(y t, Y T t+1, R = j) NFD (B.37) = f(y T Y T 1, R = j) P (R = j Y T 1 ) f(y T 1 ) = f(y T Y T 1, R = j) P (R = j Y j+1 ) f(y T 1 ) (B.37) NFD t 3, j t 2 f(y T, R = j) = f(y j+1, R = j) f(y T j+2 Y j+1, R = j) T = f(y j+1, R = j) f(y t Y t 1, R = j) t=j+2 ( B.4.3) T = f(y j+1, R = j) f(y t Y t 1 ) t=j+2 ( (B.37)) = f(y j+1, R = j) f(y j+2 Y j+1 ) f(y j+3 Y j+2 ) f(y T Y T 1 ) = f(y j+1, R = j) f(yj+2 T Y j+1 ) ( B.4.3) = P (R = j Y j+1 ) f(y j+1 ) f(yj+2 T Y j+1 ) = P (R = j Y j+1 ) f(y T ) (B.41) f(y T, R = j) = P (R = j Y T ) f(y T ) (B.42) (B.41) (B.42) P (R = j Y j+1 ) f(y T ) = P (R = j Y T ) f(y T ) f(y T ) > 0 P (R = j Y j+1 ) = P (R = j Y T ) t 3, j t 2 1 j T 2 j = T 1 (B.37) NFD NFD (B.37) B.4.4 NFD NFMV 75

224 [1] Diggle, P. and Kenward, M. G. (1994). Informative drop-out in longitudinal data analysis. Journal of Applied Statistics, 43, [2] Kenward, M. G., Molenberghs, G., and Thijs, H. (2003). Pattern-mixture models with proper time dependence. Biometrika, 90(1), [3] Mallinckrodt, C. H. (2013). Preventing and treating missing data in longitudinal clinical trials. Cambridge University Press. [4] Molenberghs, G., Michiels, B., Kenward, M. G., and Diggle, P. J. (1998). Monotone missing data and patternmixture models. Statistica Neerlandica, 52(2), [5] Nelder, J. A., and Mead, R. (1965a). A simplex method for function minimization. The computer journal, 7(4), [6] Nelder, J. A., and Mead, R. (1965b). A simplex method for function minimization - errata. The computer journal, 8(1), 27. [7] National Research Council. (2010). The prevention and treatment of missing data in clinical trials. National Academy Press. [8] Seaman, S., Galati, J., Jackson, D., and Carlin, J. (2013). What Is Meant by Missing at Random?. Statistical Science, 28(2), [9] Verbeke, G., and Molenberghs, G. (2000). Linear mixed models for longitudinal data. Springer. 76

225 Appendix C SAS C.1 DIA (Drug Information Association) Scientific Working Group on Missing Data DIA working group Mallinckrodt et al. (2013) DIA working group SAS DIA working group C.1 DIA working group SAS C.1 SAS C.1: DIA working group SAS 77

226 C.1: DIA working group Descriptive Summaries Inclusive Modeling Approaches SAS Code for Descriptive Summaries and Plot Vansteelandt s Doubly Robust Estimation Method Direct Likelihood Approach Including Influence and Residual Diagnosis Inclusive Multiple Imputation (MI) and Weighted Generalized Estimation Equation (wgee) Descriptive Statistics Dbrobust Vansteelandt Doubly Robust Direct Likelihood MMRM MI Inclusive and WGEE MI, WGEE Missing Not at Random (MNAR) Methods Pattern Mixture Model PMM Identifiability Constraints CCMV NCMV Shared Parameter Model Selection Model SPM Shared Parameter Model Selection Model Control- Based Multiple Imputation Reference-based MI via Multivariate Normal Repeated measures (MNRM) Five Macros 5 CIR, J2R, CR, ALMCF OLMCF Placebo MI including Tipping Point Analysis with Delta-Adjusting Imputation PMM Delta Tipping Point and CBI Placebo multiple imputation Example Datasets Mallinckrodt, C. H., Roger, J., Chuang-Stein, C., Molengerghs, G., Lane, P. W., O Kelly, M., Ratitch, B., Xu, Lei., Gilbert, S., Mehrotra, D. V., Wolfinger, R., and Thijs, H. (2013), Missing Data: Turning Guidance into Action. Statistics in Biopharmaceutial Research. 5,

227 C.2 Selection_Model2 C.2.1 SAS Selection_Model2 Diggle and Kenward (1994) MAR, MNAR 5 Selection Model DIA working group Missing Not at Random (MNAR) Methods Selection Model Base SAS, SAS/Stat, SAS/IML %Selection_Model2( INPUTDS = HMGR, COVTYPE =UN, response = change, MODL = seq therapy seq*therapy basval basval*seq, CLASVAR = seq therapy, mech= MNAR, psi5=-.02, psi6=-.02, const= 8, derivative= 1, out1 = HMGR_estimates, out2 = HMGR_lsmeandifferences, out3 = HMGR_lsmeans ); Out1: Out2: Out3: C.2.2 INPUTDS INPUTDS=HMGR COVTYPE COVTYPE=UN UN, TOEP, TOEPH, ARH, AR, CSH, CS Proc mixed response Response= change 79

228 MODL model = seq therapy seq*therapy basval basval*seq Proc mixed CLASVAR CLASVAR = seq therapy mech mech= MNAR MCAR MAR MNAR: ψ 5, ψ 6 MNARS: MNAR ψ 5, ψ 6 psi5 psi5 = ψ 5 mech=mnars psi6 psi6 = ψ 6 mech=mnars const const= (µ-8σ, µ+8σ) 3 8 µ σ derivative derivative = 1 1: Loglikderivative 0: out1 out1 = HMGR_estimates out2 out2 = HMGR_lsmeandifferences out3 out3 = HMGR_lsmeans C Selection_Model2 SM_GridSearch ψ5 ψ6 80

229 Selection_Model2:. Selection_Model "initial" "SeM" "initial" "SeM" Initial: (1)proc logistic (2) proc mixed MMRM Selection Model (3) visit SeM2: proc iml loglik loglikderivative Loglik Loglikderivative SAS nlpnrr derivative nlpnrr loglikderivate SAS nlpfdd proc print SM_GridSearch psi5grid psi6grid psi5 psi6 Selection_Model2 C Visit "seq" 3. "therapy" 4. ID "patient" SAS "derivative = 1" "derivative = 0" (10 30 ), "derivative=0" C Diggle, P. & Kenward, M. G. (1994). Informative drop-out in longitudinal data analysis. Applied Statistics. 43(1),

230 C.3 cbi_pmm C.3.1 cbi_pmm Pattern mixture model (Control based imputation) 6 Pattern Mixture Model 10 SAS DIA working group Control-Based Multiple Imputation Placebo MI including Tipping Point Analysis with Delta-Adjusting Imputation Base SAS and SAS/STAT %cbi_pmm( datain=data0s, trtname=trt, subjname=id, visname=visit, basecont=%str(v0), baseclass=, postcont=%str(val), postclass=, seed= , nimp=100, primaryname=val, analcovarcont=%str(v0), trtref=0, analmethod=mmrm, repstr=un, dataout=data0imputed, resout=pmmresults ); Resout SAS Output C.3.2 datain 82

231 trtname trtname=trt subjname subjname=subjid ID visname visname=timeptn Visit ID Visit 0 (0,1,2, ) poolsite poolsite=invid ID basecont baseclass postcont postcont=%str(scorepr SCORESC) postclass postclass=%str(scoretr SCOREOT) seed seed= nimp nimp= primaryname analcovarcont analcovarclass trtref 0 analmethod analmethod=%str(mmrm) ancova Impute ancova mmrm ancova repstr repstr=un analmethod=mmrm type=un DDFM=KR dataout pmmimputations resout pmmresults C.3.3 cbi_pmm 83

232 C.3.4 proc MI MAR proc MI MCMC C Ratitch B, O Kelly M. Implementation of Pattern-Mixture Models Using Standard SAS/STAT Procedures. PharmaSUG Available at 84

233 C.4 delta_pmm C.4.1 delta_pmm Pattern mixture model 6 Pattern Mixture Model 10 SAS DIA working group Missing Not at Random (MNAR) Methods Placebo MI including Tipping Point Analysis with Delta-Adjusting Imputation Base SAS, SAS/Stat %delta_pmm( datain=data0, trtname=trt, subjname=patient, visname=visit, poolsite=poolinv, basecont=%str(basval), baseclass=%str(gendern), postcont=%str(hamdtl17 hamatotl), postclass=%str(pgiimp), seed= , nimp=100, deltavis=all, deltacont=%str(1 0.2), deltacontarm=%str(1 1), deltacontmethod=%str(meanabs meanper), favorcont=%str(low low), deltacontlimit=%str(), deltaclass=%str(0.5), deltaclassarm=%str(1), favorclass=%str(low), deltaclasslimit=%str(7), primaryname=hamdtl17, analcovarcont=%str(basval basval*visit), analcovarclass=%str(poolinv), trtref=0, analmethod=mmrm, repstr=un, dataout=data0imputed2, resout=pmmresults2 ); 85

234 Resout SAS Output C.4.2 datain trtname trtname=trt subjname subjname=subjid ID visname visname=timeptn Visit ID Visit 0 (0,1,2, ) poolsite poolsite=invid ID basecont baseclass postcont postcont=%str(scorepr SCORESC) postclass postclass=%str(scoretr SCOREOT) seed seed= nimp nimp= deltavis deltavis=%str(all) first Delta adjustment visit visit visit all first NRC(2010) first 86

235 deltacont deltacont=%str(1 0.1) postcont delta MAR favorcont Mean delta adjustment (deltacontmethod=meanabs) delta Mean delta adjustment deltacontmethod=meanper 0 1 MAR deltacontmethod=slopeper 0 1 visit deltacontarm deltacontarm=%str(1 0,1) delta adjustment postcont postcont 1 1 delta adjustment 2 (0,1) delta adjustment deltacontmethod deltacontmethod=%str(meanabs slopeper) Delta adjustment Postcont meanabs, meanper, slopeper 3 favorcont favorcont=%str(low high) Postcont Low high 87

236 deltacontlimit deltacontlimit=%str(0 10) Postcont deltaclass deltaclass=%str( ) postclass delta MAR favorclass deltaclassarm deltaclassarm=%str(1 0,1) delta adjustment postclass postclass 1 1 delta adjustment 2 (0,1) delta adjustment favorclass favorclass=%str(low high) Postclass Low high deltaclasslimit deltaclasslimit=%str(0 5) Postclass primaryname analcovarcont analcovarclass trtref analmethod analmethod=%str(mmrm) Impute ancova mmrm ancova 88

237 repstr repstr=unr analmethod=mmrm dataout myimputations resout pmmresults C.4.3 delta_pmm C.4.4 (1) MAR (2) visit C National Research Council. Panel on Handling Missing Data in Clinical Trials. Committee on National Statistics, Division of Behavioral and Social Sciences and Education. The Prevention and Treatment of Missing Data in Clinical Trials. The National Academies Press: Washington, DC; Van Buuren S., Boshuizen, H.C., and Knook, D.L. Miltiple Imputation of Missing Blood Pressure Covariates in Survival Analysis. Statistics in Medicine, 1999 (18), Bohdana Ratitch, Michael O Kelly and Robert Tosiello. Missing Data in Clinical Trials: from Clinical Assumptions to Statistical Analysis Using Pattern Mixture Models. Pharmaceutical Statistics, 2013, (12)

238 C.5 Shared_Parameter1 C.5.1 SAS Shared_Parameter1 Shared Parameter Model 7 Shared Parameter Model DIA working group Missing Not at Random (MNAR) Methods Shared Parameter Model Base SAS SAS/Stat SAS/IML %shared_parameter( INPUTDS=TEMP, SUBJVAR=PATIENT, TRTVAR=DRUG, TIME=SWK, Additional_Class=%STR(POOLINV GENDER), MODL=%STR(CHANGE=POOLINV BASE DRUG SWK SWK*DRUG), LINK=%STR(CLOGLOG), RANDOM_SLOPE=%STR(LINEAR), DEBUG=0 ); _SP_OUT_FINAL C.5.2 INPUTDS INPUTDS=TEMP SUBJVAR SUBJVAR=PATIENT ID TRTVAR TRTVAR=DRUG 0 1 TIME TIME=SWK 90

239 QUADTIME QUADTIME=QUADWEEK Additional_Class Additional_Class=%STR(POOLINV GENDER) MODL PROC MIXED CLASS SUBJVAR TRTVAR LINK LINK=%STR(CLOGLOG) CLOGLOG 2 CLOGLOG LOGIT MODL MODL=%STR(CHANGE=POOLINV BASE DRUG SWK SWK*DRUG) PROC MIXED MODEL RANDOM_SLOPE RANDOM_SLOPE=%STR(LINEAR) 3 NONE LINEAR QUAD CRIT CRIT= DEBUG DEBUG=0 0 DEBUG=1 91

240 C.5.3 shaped parameter model Mixed Y i = X i β + Z i U i + ϵ i (C.1) Complementary log-log link (C.2) C.3) log( log(1 P rob(d i = j D i j))) = W i α + Γ i U i log( log(1 P rob(d i j)) = W i α + Γ i U i (C.2) (C.3) (C.4) LINK=%STR(CLOGLOG) logit(p rob(d i j)) = W i α + Γ U i (C.5) U i RANDOM_SLOPE=STR(XXXX) 3 NONE LINEAR 1 2 QUAD endpoint endpoint endpoint PROC NLMIXED 3 MIXED 1 MAR 2 3 C RANDOM_SLOPE=STR(QUAD) 2. PROC NLMIXED 3. 92

241 SAS 1, 2 ( ), 3, 5, 11 Appendix A, B (2013 ) 2 (2.5), Appendix C (2014 ) 4, 7 (2014 ) 12 Appendix C Appendix C NRC EMA ( ) RI ( ) FDA, EMA, PMDA ( ) Meiji Seika

242 94

243 Appendix 2 欠測のあるデータに関する感度分析の事例 日本製薬工業協会医薬品評価委員会データサイエンス部会 2015 年度タスクフォース 4 欠測のあるデータの解析チーム 1

244 目次 1. はじめに 事例調査の方法 各事例 Brexpiprazole(REXULTI ) 背景 統合失調症の治療 大うつ病の補助療法 参考文献 Nintedanib(OFEV ) 背景 INPULSIS 試験 参考文献 Riociguat(ADEMPAS ) 背景 試験 (PATENT-1) 感度分析 Reviewer の見解 参考文献 Vedolizumab(Entyvio ) 背景 試験デザイン 試験結果 参考文献

245 1. はじめに 医薬品評価委員会データサイエンス部会 2013 年度タスクフォース 2 では, 医薬品開発において欠測データによって生じる問題を把握するために, 承認審査において欠測データの影響について議論された事例を調査した. 当時の調査対象は,2010 年 1 月から 2013 年 6 月の期間に承認された品目の FDA,EMA および PMDA の審査報告書であった ( 詳細な調査範囲については 2014 年 7 月に公開された 審査資料で欠測データが議論された事例 を参照 ).NRC レポートは 2010 年に公開されたことから, 対象とした試験のほとんどは NRC レポートが公開される前, または公開されて間もない段階で第 3 相試験が計画, 実施されていたと考えられ, 欠測メカニズムに MNAR を仮定した pattern-mixture model に基づく感度分析を実施した事例は見られなかった. そのため今回の調査は前調査のフォローアップとして,FDA の審査報告書 ( 特に statistical review) において欠測データの取り扱いが議論になっており, さらに欠測メカニズムに MNAR を仮定した下で pattern-mixture model などを利用した感度分析を実施している試験に注目した. 今回注目した品目の事例については,2015 年 12 月 22 日に実施された 2015 年計量生物セミナーにて口頭発表またはスライド * の公開により紹介を行った. 本 appendix 2 は, その内容を報告書として再構成したものである. * 本調査の方法の詳細を 2 章に示した. 取り上げた事例は,FDA の審査報告書および関連する論文での記述を基に 3 章に記載した. なお,3 章での記載は FDA の審査報告書や論文の内容を日本語化し, タスクフォースメンバーにより再構成したものであるため, 記載した文章および用語等が元の文献の逐語的な翻訳には必ずしもなっていないことに留意していただきたい. 3

246 2. 事例調査の方法 FDA の web* 上に公開されている承認品目一覧 (New Molecular Entity and New Therapeutic Biological Product Approvals) から, 承認日が 2013 年 7 月 1 日から 2015 年 7 月 31 日までの品目を調査対象として特定した ( 診断薬及び抗悪性腫瘍薬を除いた 57 件 ). 次に, 該当品目の審査報告書, 特に statistical review を調査し, 欠測メカニズムに MNAR を仮定した下で pattern-mixture model などを利用した感度分析を実施しているものとして,Brexpiprazole(REXULTI ),Nintedanib (OFEV ),Riociguat(ADEMPAS ), および Vedolizumab(ENTYVIO ) の 4 品目を注目する事例として取り上げることとした. なお,ANORO TM ELLIPTA は該当品目の一つであり,jump to reference による解析を感度分析として実施しているが,2015 年 2 月 13 日に開催された, 日本製薬工業協会データサイエンス部会シンポジウム 臨床試験の欠測データの取り扱いに関する最近の展開と今後の課題について NAS レポート,EMA ガイドライン,estimand と解析方法の概説 で紹介された事例であるため本 Appendix 2 では取り上げない. * 4

247 3. 各事例 3.1 Brexpiprazole(REXULTI ) 背景 Brexpiprazole( 商品名 REXULTI ) は, 統合失調症及び大うつ病 ( 補助療法 ) の治療薬として 2015 年 7 月に米国で承認された. 申請データパッケージにおいては, 各適応症に対して, それぞれ 2 本ずつの第 3 相試験の結果が提出された. 統合失調症の治療 : Study 230,231 大うつ病の補助療法 : Trial 227,228 以下では, それぞれの適応症ごとに,statistical review に記載のあった欠測データの取扱いに関する議論について要約する 統合失調症の治療 試験デザイン 統合失調症の治療 に対しては,2 本の第 3 相試験が実施された (Study 230,231). いずれの試験もランダム化二重盲検プラセボ対照並行群間比較試験であり, 試験デザインについては, 群の構成及びその割付比を除いて, ほとんどの部分で共通していた ( 表 3.1.1). 表 Study 230 と 231 の群構成及びその割付比 Study 230 Study 231 群 プラセボ,1 mg,2 mg,4 mg プラセボ,0.25 mg,2 mg,4 mg 割付比 3:2:3:3 2:1:2:2 観察期間は 6 週間で, 主要評価項目は 6 週時点の PANSS total score の変化量, 主解析には MMRM, その感度分析として multiple imputation を利用した pattern-mixture approach が設定された. 次項で感度分析の詳細について説明する 感度分析の詳細 主解析の MMRM に対する感度分析として,multiple imputation を利用した pattern-mixture approach が実施された. 当該感度分析では, 以下に示した 2 つのシナリオが申請者により検討されている. シナリオ 1 : Brexpiprazole 投与群において lack of efficacy で中止した被験者に MNAR を仮定し,Brexpiprazole 投与群におけるその他の理由による中止, 及びプラセボ投与群における中止に対しては MAR を仮定したもの. シナリオ 2 : Brexpiprazole 投与群において lack of efficacy または adverse event で中止した被験者に MNAR を仮定し,Brexpiprazole 投与群におけるその他の理由による中止, 及びプラセボ投与群における中止に対しては MAR を仮定したもの. 5

248 MNAR を仮定した被験者に対しては以下の手順で補完を行う. まず,MAR を仮定した multiple imputation によって欠測データを補完する. 次に, その補完した値から, 主解析の MMRM で推定された群間差の k% だけマイナスする.k を 0,10,20,,100,, と動かしていき, 主解析の結論が逆転する k, すなわち,tipping point を探索する (tipping point analysis) 試験結果 中止割合被験者の内訳を表 3.1.2, 表 に示す. いずれの試験も中止割合は概ね 30~40% 程度であった. 表 Study 230 の被験者の内訳 (Brexpiprazole FDA statistical review. (2015). p. 23. Table 11 より改変 ) Brexpiprazole プラセボ 1 mg 2 mg 4 mg ランダム化例数 120 (100%) 186 (100%) 184 (100%) 184 (100%) 有効性解析対象例数 117 (97.5%) 179 (96.2%) 181 (98.4%) 180 (97.8%) 試験完了例数 81 (67.5%) 129 (69.4%) 130 (70.7%) 118 (64.1%) 試験中止例数 39 (32.5%) 57 (30.6%) 54 (29.3%) 66 (35.9%) 追跡不能 0 (0%) 0 (0%) 0 (0%) 0 (0%) 有害事象 11 (9.2%) 11 (5.9%) 13 (7.1%) 22 (12.0%) 中止基準に該当 0 (0%) 0 (0%) 2 (1.1%) 1 (0.5%) 治験責任医師による 2 (1.7%) 0 (0%) 0 (0%) 1 (0.5%) 同意撤回 被験者による 15 (12.5%) 25 (13.4%) 23 (12.5%) 21 (11.4%) 同意撤回 プロトコールからの 2 (1.7%) 1 (0.5%) 0 (0%) 0 (0%) 逸脱 有効性の欠如 9 (7.5%) 20 (10.8%) 16 (8.7%) 21 (11.4%) 6

249 表 Study 231 の患者の内訳 (Brexpiprazole FDA statistical review. (2015). p. 23. Table 12 より改変 ) Brexpiprazole プラセボ 1 mg 2 mg 4 mg ランダム化例数 90 (100%) 182 (100%) 180 (100%) 184 (100%) 有効性解析対象例数 87 (96.7%) 180 (98.9%) 178 (98.9%) 178 (96.7%) 試験完了例数 56 (62.2%) 124 (68.1%) 121 (67.2%) 109 (59.2%) 試験中止例数 34 (37.8%) 58 (31.9%) 59 (32.8%) 75 (40.8%) 追跡不能 0 (0%) 0 (0%) 0 (0%) 1 (0.5%) 有害事象 12 (13.3%) 15 (8.2%) 17 (9.4%) 32 (17.4%) 中止基準に該当 1 (1.1%) 0 (0%) 1 (0.6%) 0 (0%) 治験責任医師による 0 (0%) 1 (0.5%) 1 (0.6%) 3 (1.6%) 同意撤回 被験者による 13 (14.4%) 24 (13.2%) 31 (17.2%) 21 (11.4%) 同意撤回 プロトコールからの 1 (1.1%) 1 (0.5%) 2 (1.1%) 0 (0%) 逸脱 有効性の欠如 7 (7.8%) 17 (9.3%) 7 (3.9%) 18 (9.8%) 主解析の結果 Study 230 では,4 mg で有意差が認められ,Study 231 では,4 mg に加えて 2 mg においても有意差が認められた ( 表 3.1.4). 7

250 表 Study 230 と 231 の主解析の結果 (Brexpiprazole FDA statistical review. (2015). p. 13. Table 3 より改変 ) Study 230 Brex. 1 mg Brex. 2 mg Brex. 4 mg Placebo Number of patients N=117 N=179 N=181 N=180 Baseline Mean (SD) 93.2 (12.7) 96.3 (12.9) 95.0 (12.4) 94.6 (12.8) Mean Change at Week 6 (SE) (1.9) (1.5) (1.5) (1.5) Treatment Difference % Confidence Interval (-8.1, 1.3) (-7.2, 1.1) (-10.6, -2.3) - p-value Average Effect (2mg & 4mg) LS Mean Difference=-4.78, p-value= versus Placebo Study 231 Brex mg Brex. 2 mg Brex. 4 mg Placebo Number of patients N=87 N=180 N=178 N=178 Baseline Mean (SD) 93.6 (11.5) 95.9 (13.7) 94.7 (12.1) 95.7 (11.5) Mean Change at Week 6 (SE) (2.2) (1.5) (1.5) (1.6) Treatment Difference % Confidence Interval (-8.3, 2.5) (-13.1, -4.4) (-12.0, -3.3) - p-value 0.29 < Average Effect (2mg & 4mg) LS Mean Difference=-8.18, p-value< versus Placebo N=number of patients, SD=Standard Deviation, SE=Standard Error 感度分析の結果感度分析の結果を表 3.1.5(Study 230) と表 3.1.6(Study 231) に示す. 先に挙げた 2 つのシナリオに加えて, Brexpiprazole 投与群において中止したすべての被験者に MNAR を仮定したもの であるシナリオ 3 が reviewer によって実施された. Study 230 では, シナリオ 1,2,3 の tipping point はそれぞれ,260(%),130(%),80(%) であった.Study 231 では, シナリオ 2 と 3 の tipping point はそれぞれ,300(%),160(%) であったが, シナリオ 1 においては,k を 500(%) まで動かしても主解析の結論が逆転することはなかった. 8

251 表 Study 230 の感度分析の結果 (Brexpiprazole FDA statistical review. (2015). p. 15. Table 4 より改変 ) Percentage of Treatment Effect Subtracted from MAR imputed missing data Treatment Comparison Treatment Difference 95% Confidence Interval p-value Lack of Efficacy in Brex Group as MNAR( シナリオ 1) k=100(%) Brex.2mg vs Placebo -2.8 (-7.3, 1.7) 0.22 Brex.4mg vs Placebo -5.2 (-9.7, -0.7) k=260(%) Brex.2mg vs Placebo -2.3 (-6.7, 2.1) 0.31 (tipping point) Brex.4mg vs Placebo -4.3 (-8.8, 0.1) Lack of Efficacy and AE in Brex Group as MNAR( シナリオ 2) k=100(%) Brex.2mg vs Placebo -2.7 (-7.2, 1.9) 0.25 Brex.4mg vs Placebo -4.8 (-9.3, -0.2) k=130(%) Brex.2mg vs Placebo -2.7 (-7.2, 1.8) 0.25 (tipping point) Brex.4mg vs Placebo -4.4 (-8.9, 0.1) Dropout due to Any Reason in Brex Group as MNAR( シナリオ 3) k=80(%) Brex.2mg vs Placebo -2.8 (-7.6, 2.0) 0.25 (tipping point) Brex.4mg vs Placebo -4.4 (-8.9, 0.1)

252 表 Study 231 の感度分析の結果 (Brexpiprazole FDA statistical review. (2015). p. 16. Table 5 より改変 ) Percentage of Treatment Effect Subtracted from MAR imputed missing data Treatment Comparison Treatment Difference Lack of Efficacy in Brex Group as MNAR( シナリオ 1) 95% Confidence Interval p-value k=100(%) Brex.2mg vs Placebo -9.1 (-14.0, -4.3) Brex.4mg vs Placebo -8.4 (-13.0, -3.9) k=500(%) (tipping Brex.2mg vs Placebo -5.9 (-11.1, -0.8) point not reached) Brex.4mg vs Placebo -7.2 (-12.4, -2.0) Lack of Efficacy and AE in Brex Group as MNAR( シナリオ 2) k=100(%) Brex.2mg vs Placebo -8.6 (-13.4, -3.9) Brex.4mg vs Placebo -7.9 (-12.5, -3.2) k=300(%) Brex.2mg vs Placebo -5.0 (-10.1, 0.2) (tipping point) Brex.4mg vs Placebo -5.3 (-10.7, 0.2) Dropout due to Any Reason in Brex Group as MNAR( シナリオ 3) k=100(%) Brex.2mg vs Placebo -7.2 (-12.3, -2.1) Brex.4mg vs Placebo -6.2 (-11.0, -1.4) k=160(%) (tipping point) Brex.2mg vs Placebo -5.8 (-10.8, -0.9) Brex.4mg vs Placebo -5.1 (-10.4, 0.2) Statistical review では, 申請者が実施したシナリオ 1 と 2 では tipping point が 100(%) を超 えていたこと,reviewer が実施したシナリオ 3 においても tipping point が 80(%) に届いていた ことを受けて, 主解析の結果と感度分析の結果は一貫していた と結論付けている 大うつ病の補助療法 試験デザイン 大うつ病の補助療法 に対しても,2 本の第 3 相試験が実施された (Trial 227,228). いずれの試験もランダム化二重盲検プラセボ対照並行群間比較試験であり, 試験デザインについては, 群の構成を除いて, ほとんどの部分で共通していた. 群の構成は,Trial 227 がプラセボ, Brexpiprazole 1 mg,3 mg の 3 群,Trial 228 ではプラセボと Brexpiprazole 2 mg の 2 群であった. 観察期間は, 既存の抗うつ治療で効果が不十分な患者 を選択するための 8 週間 ( プラセボ単盲検,Phase A) と, 有効性および安全性評価のための 6 週間 ( 二重盲検,Phase B) から成 10

253 っており, 主要評価項目には Phase A 終了時から Phase B 終了時まで (6 週間 ) の MADRS total score の変化量が設定された. 主解析には MMRM, その感度分析には multiple imputation を利用した pattern-mixture approach の他に,shared parameter model,random coefficient pattern-mixture model,observed case における ANCOVA や,LOCF で欠測データを補完した ANCOVA が設定された 試験結果 中止割合 いずれの試験も 10% に満たない中止割合であった. 主解析の結果 Trial 227 及び 228 の主解析の結果を表 3.1.7, 表 に示す. 表 Trial 227 の主解析の結果 (Brexpiprazole FDA statistical review. (2015). p. 41. Table 5 より改変 ) Variable 1mg Brex+ADT* 3mg Brex+ADT* Placebo+ADT* MADRS Total Score, MMRM N=225 N=226 N=218 Mean (SD) End of Phase A (5.61) (5.24) (5.27) LS Mean (SE) Change At Week (0.50) (0.51) (0.51) LS Mean Difference (95%CI) (-2.58, 0.20) (-2.92, -0.13) - P-value * ADT: Antidepressant treatment 表 Trial 228 の主解析の結果 (Brexpiprazole FDA statistical review. (2015). p. 48. Table 12 より改変 ) Endpoint Parameter Primary Efficacy Endpoint 2 mg/day Brex+ADT* Placebo+ADT* MADRS Total Score, MMRM by Randomized Treatment N=187 N=191 Group (Efficacy Sample - Primary Analysis) Mean (SD) end of Phase A (Week 8) (5.79) (5.60) LS mean change (SE) at end of Phase B (Week 14) (0.61) (0.60) LS mean difference (95%CI) (-4.70, -1.54) - P-value * ADT: Antidepressant treatment 11

254 Trial 227 では, どちらの用量もプラセボに対する優越性を示すことはできなかった. (Brexpiprazole 3 mg では多重性を調整しない場合 p= であったが, 事前に規定された Hochberg の方法によって多重性を調整すると有意ではなかった.) 一方,Trial 228 では Brexpiprazole 2 mg のプラセボに対する優越性が示された. 感度分析の結果 感度分析の結果は主解析と同様であった と記載があったのみで,statistical review には, 感度分析の方法も含めて, 詳細な記述がなかった. 上記のように結論付けるにあたって, 中止割合が小さく, 欠測値による問題はわずかなものであった との記述があった 参考文献 [1] Brexpiprazole FDA statistical review. (2015). 12

255 3.2 Nintedanib(OFEV ) 背景 Nintedanib( 商品名 OFEV ) は, 特発性肺線維症 (idiopathic pulmonary fibrosis, 以下 IPF) の治療薬として 2014 年 10 月に米国で承認され,2015 年 7 月に日本においても承認された. その申請データパッケージの中で 2 つの第 3 相試験の結果が提出された 試験 (INPULSIS 1):IPF 患者を対象に, 有効性及び安全性の検討 試験 (INPULSIS 2):IPF 患者を対象に, 有効性及び安全性の検討 INPULSIS 1,INPULSIS 2 試験は国際共同第 3 相試験として実施され, 試験デザインが類似している sister trial であった. 試験デザインが類似していることより, 本項では INPULSIS 試験として記載を統一し,FDA の statistical review を基に内容を記載する INPULSIS 試験 試験デザイン INPULSIS 試験は二重盲検ランダム化並行群間プラセボ対照比較試験であった. 主要評価項目は FVC( 努力肺活量 ) の年間減少率 (ml/year) が設定されており, 図 において丸で囲まれた時点が測定時点であった.INPULSIS 試験は投与中止した場合でも, 来院を依頼し, データを収集するプロトコールであった. 図 INPULSIS 試験デザイン (Nintedanib statistical review (2014) p7. Figure 1 を改変 ) 13