( ) Multiple Comparisons on Longitudinal Data Junji Kishimoto SAS Institute Japan / Keio Univ. SFC / Univ. of Tokyo address: jpnjak@jpn.sas.

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1 ( ) Multiple Comparisons on Longitudinal Data Junji Kishimoto SAS Institute Japan / Keio Univ SFC / Univ of Tokyo address: jpnjak@jpnsascom Dunnett SAS Multiple Comparisons, Repeated Measures, Quadrature t 0 t 1 t 2 t k 1 y 10 y 11 y 12 y 1k 2 y 20 y 21 y 22 y 2k n y n0 y n1 y n2 y nk µ 0 µ 1 µ 2 µ k 0 H 01 : µ 0 = µ 1, H 02 : µ 0 = µ 2,,H 0p : µ 0 = µ k (1) t Dunnett 35

Measures, Quadrature 1 1 11 t 0 t 1 t 2 t k 1 y 10 y 11 y 12 y 1k 2 y 20 y 21 y 22 y 2k n y n0 y

2 2 22 t 1 t 2 t k 1 y 11 y 12 y 1k 1 m y m1 y m2 y mk µ 11 µ 12 µ 1k m+1 y m+11 y m+12 y m+1k 2 n y n1 y n2 y nk µ 21 µ 22 µ 2k t H 01 : µ 11 = µ 21, H 02 : µ 12 = µ 22,,H 0k : µ 1k = µ 2k (2) , 1 data repeated; input do time=0 to 3; input output; end; cards; ; 36

repeated; input id @; do time=0 to 3; input y @; output; end; cards; 301 207 158 184 206 305 186 209 176 243

3 Dunnett PROC GLM proc glm data=repeated; class time; model y = time; lsmeans time / adjust=dunnett pdiff=control( 0 ); 11 Dunnett General Linear Models Procedure Least Squares Means Adjustment for multiple comparisons: Dunnett TIME Y Pr > T H0: LSMEAN LSMEAN=CONTROL PROC MIXED TYPE=VC proc mixed data=repeated; class time; model y = time / ddfm=residual; repeated time / type=vc subject=id rcorr; lsmeans time / adjust=dunnett pdiff=control( 0 ); 12 R Correlation Matrix for Subject 1 Row COL1 COL2 COL3 COL Dunnett Differences of Least Squares Means Effect TIME _TIME Difference Std Error DF t Pr> t Adjustment Adj P TIME Dunnett TIME Dunnett TIME Dunnett

y = time / ddfm=residual; repeated time / type=vc subject=id rcorr; lsmeans time / adjust=dunnett pdiff=control( 0 ); 12 R Correlation Matrix for Subject 1 Row COL1 COL2 COL3 COL4 1 100000000 2

4 12 22 F Huyhn and Feldt(1970) F Tukey Dunnett PROC GLM proc glm data=repeated; class id time; model y = id time; lsmeans time / adjust=dunnett pdiff=control( 0 ); 21 Dunnett General Linear Models Procedure Least Squares Means Adjustment for multiple comparisons: Dunnett TIME Y Pr > T H0: LSMEAN LSMEAN=CONTROL PROC MIXED proc mixed data=repeated; class time; model y = time; repeated time / type=hf subject=id rcorr; lsmeans time / adjust=dunnett pdiff=control( 0 ); 22 R Correlation Matrix for Subject 1 Row COL1 COL2 COL3 COL

PROC MIXED proc mixed data=repeated; class time; model y = time; repeated time / type=hf subject=id rcorr; lsmeans time / adjust=dunnett pdiff=control( 0 ); 22 R Correlation Matrix for

5 23 Dunnett Differences of Least Squares Means Effect TIME _TIME Difference Std Error DF t Pr> t Adjustment Adj P TIME Dunnett-Hsu TIME Dunnett-Hsu TIME Dunnett-Hsu Dunnett TIME=0 TIME=3 p ( ) Hsu(1992) 3 t p Release 610 PROC GLM LSMEANS ADJUST=DUNNETT Hsu AR(1) Hsu proc mixed data=repeated; class time; model y = time; repeated time / type=ar(1) subject=id rcorr; lsmeans time / adjust=dunnett pdiff=control( 0 ); 31 AR(1) R Correlation Matrix for Subject 1 Row COL1 COL2 COL3 COL

ADJUST=DUNNETT Hsu AR(1) Hsu proc mixed data=repeated; class time; model y = time; repeated time / type=ar(1) subject=id rcorr; lsmeans time / adjust=dunnett pdiff=control( 0 ); 31 AR(1) R

6 32 AR(1) Hsu Dunnett Differences of Least Squares Means Effect TIME _TIME Difference Std Error DF t Pr> t Adjustment Adj P TIME Dunnett-Hsu TIME Dunnett-Hsu TIME Dunnett-Hsu p proc mixed data=repeated; class time; model y = time; repeated time / type=un subject=id rcorr; lsmeans time / adjust=simulate pdiff=control( 0 ); 41 R Correlation Matrix for Subject 1 Row COL1 COL2 COL3 COL Dunnett Differences of Least Squares Means Effect TIME _TIME Difference Std Error DF t Pr> t Adjustment Adj P TIME Simulate TIME Simulate TIME Simulate TYPE=AR(1) TYPE=UN ( ) 40

subject=id rcorr; lsmeans time / adjust=simulate pdiff=control( 0 ); 41 R Correlation Matrix for Subject 1 Row COL1 COL2 COL3 COL4 1 100000000 031304700 084507324 031850774 2 031304700 100000000

7 k (1997) t p Y ihr Y ihr = µ ih + ε ihr, i =1, 2, h =1,,k, r =1,,n ih, (3) Y ihr bµ =(bµ 11,,bµ 1k, bµ 21,,bµ 2k ) k=3 1 ρ ρ ρ 1 ρ Var( bµ) =σ 2 ρ ρ ρ ρ ρ 1 ρ ρ ρ L = (5) (4) Var(Lbµ) =2σ 2 1 ρ ρ ρ 1 ρ ρ ρ 1 (6) t SAS PROBMC Dunnett 35% k=2 ρ ν

0 (5) 0 0 1 0 0 1 (4) Var(Lbµ) =2σ 2 1 ρ ρ ρ 1 ρ ρ ρ 1 (6) t SAS PROBMC Dunnett 35% k=2 ρ ν 0 03 05 07 09 1 4 338 336

8 k=3 ν k= ρ =0 Studentized Maximum Modulus ρ =1 ρ 32 SAS data B; input group $ id do time=5 to 8; input output; end; cards; Vehicle Vehicle Vehicle Vehicle Vehicle Vehicle Vehicle Vehicle D+P(H) D+P(H) D+P(H) D+P(H) D+P(H)

input y @; output; end; cards; Vehicle 001 325 355 344 178 Vehicle 003 259 280 418 263 Vehicle 004 309 329 205 128 Vehicle 007 266 286 227 182 Vehicle 008 284 243 236 230 Vehicle 009 201 286 184

9 D+P(H) D+P(H) D+P(H) ; proc mixed data=b; class group time id; model y = group(time) / ddfm=residual; repeated time / type=cs subject=id RCORR; lsmeans group(time) / corr slice=time; 51 Least Squares Means Level LSMEAN Std Error DDF T Pr > T GROUP(TIME) D+P(H) GROUP(TIME) Vehicle GROUP(TIME) D+P(H) GROUP(TIME) Vehicle GROUP(TIME) D+P(H) GROUP(TIME) Vehicle GROUP(TIME) D+P(H) GROUP(TIME) Vehicle CORR1 CORR2 CORR3 CORR4 CORR5 CORR6 CORR7 CORR Tests of Effect Slices Effect Slice NDF DDF F Pr > F GROUP(TIME) TIME GROUP(TIME) TIME GROUP(TIME) TIME GROUP(TIME) TIME TIME=6 F=1201, p=

020761292 56 1374 00001 GROUP(TIME) D+P(H) 6 210750000 020761292 56 1015 00001 GROUP(TIME) Vehicle 6 312500000 020761292 56 1505 00001 GROUP(TIME) D+P(H) 7 209500000 020761292 56 1009 00001

10 F (t ) p data mult_t; k = 4; t = sqrt(1201); rho = 030; nu = 56; array lambda{4}; do i=1 to k; lambda{i} = sqrt(rho); end; p = 1 - probmc("dunnett2", t,, nu, k, of lambda1-lambda4); put p= 64; P=00040 TIME=6 p p Bonferroni Frison and Pocock(1992) 10 visit Bonferroni 4 SAS [1] Frison,L, and Pocock,SJ(1992) Repeated Measures In Clinical Trials: Analysis Using Mean Summary Statistics And Its Implications For Design, Statistics In Medicine, 11, [2] Huynh,H and Feldt,LS(1970) Conditions Under Which Mean Square Ratios In Repeated Measures Designs Have Exact F-distributions: JASA 11, [3] Hsu,JC(1992), The Factor Analytic Approach to Simultaneous Inference in the General Linear Model, Journal of Computatinal Statistics and Graphics, 1, [4] (1997) 65 44

Summary Statistics And Its Implications For Design, Statistics In Medicine, 11, 1685-1704 [2] Huynh,H and Feldt,LS(1970) Conditions Under Which Mean Square Ratios In Repeated Measures Designs Have

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