( ) Multiple Comparisons on Longitudinal Data Junji Kishimoto SAS Institute Japan / Keio Univ SFC / Univ of Tokyo e-mail address: jpnjak@jpnsascom Dunnett SAS Multiple Comparisons, Repeated 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 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
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) 2 1 2 21 1, 1 data repeated; input id @; do time=0 to 3; input y @; output; end; cards; 301 207 158 184 206 305 186 209 176 243 306 284 213 162 162 307 330 203 288 260 308 173 167 115 121 309 340 216 320 224 310 279 182 238 129 311 273 338 193 261 ; 36
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 0 259000000 1 210750000 02868 2 209500000 02683 3 200750000 01624 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 COL4 1 100000000 2 100000000 3 100000000 4 100000000 13 Dunnett Differences of Least Squares Means Effect TIME _TIME Difference Std Error DF t Pr> t Adjustment Adj P TIME 1 0-048250000 030616143 28-158 01263 Dunnett 02868 TIME 2 0-049500000 030616143 28-162 01171 Dunnett 02683 TIME 3 0-058250000 030616143 28-190 00674 Dunnett 01624 37
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 0 259000000 1 210750000 01189 2 209500000 01073 3 200750000 00502 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 COL4 1 100000000 044604378 049269359 051790125 2 044604378 100000000 034194834 039242273 3 049269359 034194834 100000000 045236522 4 051790125 039242273 045236522 100000000 38
23 Dunnett Differences of Least Squares Means Effect TIME _TIME Difference Std Error DF t Pr> t Adjustment Adj P TIME 1 0-048250000 023037494 21-209 00485 Dunnett-Hsu 01189 TIME 2 0-049500000 023037494 21-215 00435 Dunnett-Hsu 01073 TIME 3 0-058250000 023037494 21-253 00195 Dunnett-Hsu 00502 Dunnett TIME=0 TIME=3 p 01624 00502 ( ) 23 1 2 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 COL4 1 100000000 027679862 007661748 002120761 2 027679862 100000000 027679862 007661748 3 007661748 027679862 100000000 027679862 4 002120761 007661748 027679862 100000000 39
32 AR(1) Hsu Dunnett Differences of Least Squares Means Effect TIME _TIME Difference Std Error DF t Pr> t Adjustment Adj P TIME 1 0-048250000 026019292 21-185 00778 Dunnett-Hsu 01783 TIME 2 0-049500000 029400632 21-168 01071 Dunnett-Hsu 02385 TIME 3 0-058250000 030269911 21-192 00680 Dunnett-Hsu 01575 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 COL4 1 100000000 031304700 084507324 031850774 2 031304700 100000000 009341769 056166162 3 084507324 009341769 100000000 044463342 4 031850774 056166162 044463342 100000000 42 Dunnett Differences of Least Squares Means Effect TIME _TIME Difference Std Error DF t Pr> t Adjustment Adj P TIME 1 0-048250000 024828519 7-194 00931 Simulate 02058 TIME 2 0-049500000 013038405 7-380 00067 Simulate 00162 TIME 3 0-058250000 024890295 7-234 00518 Simulate 01173 TYPE=AR(1) TYPE=UN ( ) 40
3 31 2 2 2 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 0 0 0 ρ 1 ρ 0 0 0 Var( bµ) =σ 2 ρ ρ 1 0 0 0 0 0 0 1 ρ ρ 0 0 0 ρ 1 ρ 0 0 0 ρ ρ 1 1 0 0 1 0 0 L = 0 1 0 0 1 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 4 338 336 331 323 306 278 8 272 270 267 262 251 231 12 254 253 250 246 236 218 16 246 245 242 238 229 212 224 223 221 218 211 196 41
k=3 ν 0 03 05 07 09 1 6 319 316 310 299 280 245 12 275 273 268 261 246 218 18 262 260 256 250 236 210 24 256 254 251 244 231 206 239 238 235 230 219 196 k=4 8 313 309 302 291 269 231 16 278 276 271 262 244 212 24 268 266 261 253 237 206 32 263 261 257 249 234 204 249 247 244 238 224 196 ρ =0 Studentized Maximum Modulus ρ =1 ρ 32 SAS 5 8 4 4 data B; input group $ id $ @; do time=5 to 8; 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 009 201 286 184 250 Vehicle 010 322 336 280 226 Vehicle 011 316 385 342 193 D+P(H) 301 207 158 184 206 D+P(H) 305 186 209 176 243 D+P(H) 306 284 213 162 162 D+P(H) 307 330 203 288 260 D+P(H) 308 173 167 115 121 42
D+P(H) 309 340 216 320 224 D+P(H) 310 279 182 238 129 D+P(H) 311 273 338 193 261 ; 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) 5 259000000 020761292 56 1248 00001 GROUP(TIME) Vehicle 5 285250000 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 GROUP(TIME) Vehicle 7 279500000 020761292 56 1346 00001 GROUP(TIME) D+P(H) 8 200750000 020761292 56 967 00001 GROUP(TIME) Vehicle 8 206250000 020761292 56 993 00001 52 CORR1 CORR2 CORR3 CORR4 CORR5 CORR6 CORR7 CORR8 100-000 030 000 030 000 030 000-000 100-000 030 000 030-000 030 030-000 100 000 030 000 030 000 000 030 000 100 000 030 000 030 030 000 030 000 100 000 030 000 000 030 000 030 000 100 000 030 030-000 030 000 030 000 100 000 000 030 000 030 000 030 000 100 53 Tests of Effect Slices Effect Slice NDF DDF F Pr > F GROUP(TIME) TIME 5 1 56 080 03751 GROUP(TIME) TIME 6 1 56 1201 00010 GROUP(TIME) TIME 7 1 56 568 00205 GROUP(TIME) TIME 8 1 56 004 08521 TIME=6 F=1201, p=00010 43
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 00010 p 00040 030 Bonferroni Frison and Pocock(1992) 10 visit 06 08 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, 1685-1704 [2] Huynh,H and Feldt,LS(1970) Conditions Under Which Mean Square Ratios In Repeated Measures Designs Have Exact F-distributions: JASA 11, 1582-1589 [3] Hsu,JC(1992), The Factor Analytic Approach to Simultaneous Inference in the General Linear Model, Journal of Computatinal Statistics and Graphics, 1, 151-168 [4] (1997) 65 44