ST 524 NCSU - Fall 2007 Incomplete block Design Incomplete block designs Example - Balanced Incomplete Block Design y 6 2 3 5 1 7 2 5 4 1 3 6 trt t1 t1 t1 t2 t2 t2 t3 t3 t3 t4 t4 t4 B1 * * * T1 T2 T3 T4 block b1 b2 b3 b1 b2 b4 b1 b3 b4 b2 b3 b4 B2 * * * y 6 2 3 5 1 7 2 5 4 1 3 6 B3 * * * Intercept 1 1 1 1 1 1 1 1 1 1 1 1 T1 1 1 1 0 0 0 0 0 0 0 0 0 B4 * * * T1 T2 T3 T4 T2 0 0 0 1 1 1 0 0 0 0 0 0 B1 6 5 2 13 Source Treatment (Adjusted) Block(Adjusted) Error Total T3 0 0 0 0 0 0 1 1 1 0 0 0 T4 0 0 0 0 0 0 0 0 0 1 1 1 B2 2 1 1 4 B1 1 0 0 1 0 0 1 0 0 0 0 0 B3 3 5 3 11 B2 0 1 0 0 1 0 0 0 0 1 0 0 B3 0 0 1 0 0 0 0 1 0 0 1 0 B4 0 0 0 0 0 1 0 0 1 0 0 1 B4 7 4 6 17 11 13 11 10 45 d.f 4–1 =3 4–1=3 Diff= 11 – 3 -3 = 5 4*3 – 1 = 11 Variance between plots in the same block is smaller than the variance between plots in different blocks. Thursday November 1, 2007 1 t = 4 treatments k = 3 units per block r = 3 replications for each treatment Each treatment occurs together with each other treatment twice. ST 524 NCSU - Fall 2007 Incomplete block Design Linear Model Yijk i j ijk Adjustment factor A Eb Ee k r 1 Eb 10.9722 2.35 0.0982 4 3 1 10.9722 Effective error mean square Ee Ee 1 A =2.35 (1+0.0982)=2.5808 pooled error Erb SSB SSE 32.91666667 11.75000000 44.66667 2.7917 16 k 2 1 r 1 32 1 2 1 % relative precision RP Erb 100 Ee 2.7917 100 108.17% 2.5808 Additive Linear Model Yij j b j ij Treatment means Thursday November 1, 2007 2 ST 524 NCSU - Fall 2007 Incomplete block Design Y11 1 b 1 11 Y 1. 1 Y12 1 b 2 12 3 3 b b b Y 2. 2 1 2 4 21 22 24 3 3 b 3 b 4 11 12 13 21 22 24 Y 1. Y 2. 1 2 3 3 Y13 1 b 3 13 Y1. 3 1 b 1 b 2 b 3 11 12 13 Y 1. 1 b 1 b 2 b 3 11 12 13 3 b 1 b 2 b 3 11 12 13 3 Block Means Y .1 b1 Y11 1 b 1 11 Y21 2 b 1 21 Y .2 b2 Y11 3 b 1 31 Y.1 1 2 3 3b1 11 21 31 Y .1 b1 Y .3 Y .4 b3 b4 b3 b4 3 3 1 2 4 12 22 42 Y .1 Y .2 b1 b2 3 1 2 4 3 3 3 1 3 4 2 3 4 13 33 43 24 34 44 3 1 2 3 3 1 2 4 11 21 31 12 22 42 3 3 b1 b2 3 4 11 21 31 12 22 42 3 3 1 2 3 11 21 31 3 1 2 3 Treatment differences include block effect 31 11 21differences. 3 13 33 43 24 34 44 3 Block differences include treatment effect differences. 3 3 Intra-block analysis: obtains treatment information by comparing block-adjusted least squares estimates Inter-block analysis recovers treatment information contained in the block averages Solution to uncovering the inter-block information: let block effects to be random, with mean zero, model turns to be Yij j b j ij , ij ~ N 0, e2 , b j ~ N 0, b2 , where ij and b j are independent. Unconditional comparison of two mean blocks and treatments: BROAD INFERENCE SPACE E Y 1. Y 2. 1 2 , and E Y .3 Y .4 1 2 Thursday November 1, 2007 3 3 ST 524 NCSU - Fall 2007 Incomplete block Design Conditional comparisons of two block or treatment means : NARROW INFERENCE SPACE b b E Y 1. Y 2. | b 1 2 3 4 , and 3 E Y .3 Y .4 | b b3 b4 1 2 3 Interblock Analysis: In PROC MIXED – RANDOM Block; LSMEANS are estimation of treatment means in the broad inference space Because of non-orthogonality, the estimates of the treatment means as well as their standard errors differ between the fixed effects (PROC GLM) and mixed effects analysis (PROC MIXED). PROC MIXED proc mixed data=b; /* 4 treatments in 4 blocks of size 3 */ class block trt; model y = trt; random block; lsmeans trt / pdiff; estimate 'I trt 1 mean' intercept 1 trt 1 ; estimate 'II trt 1 mean' intercept 3 trt 3 | block 1 1 1 /divisor=3; estimate 'I trt 2 mean' intercept 1 trt 0 1 ; estimate 'II trt 2 mean' intercept 3 trt 0 3 | block 1 1 0 1 estimate 'I trt 3 mean blk1' intercept 1 estimate 'II trt 3 mean' intercept 3 estimate 'I trt 4 mean blk2' intercept 1 estimate 'II trt 4 mean' intercept 3 /* inter -analysis /* intra -analysis - narrow scope */ /* inter -analysis /divisor=3; - broad scope*/ - broad scope*/ /* intra -analysis - narrow scope */ trt 0 0 1 ; trt 0 0 3 | block 1 0 1 1 /divisor=3; /* inter -analysis trt 0 0 0 1 ; trt 0 0 0 3 | block 0 1 1 1 /divisor=3; /* inter -analysis - broad scope*/ /* intra -analysis - narrow scope */ - broad scope*/ /* intra -analysis - narrow scope */ estimate 'trt1 vs trt3' trt 1 0 -1; estimate 'trt1 vs trt3 blk1' trt 1 0 -1 | block 1; run; The Mixed Procedure Model Information Data Set Dependent Variable Covariance Structure Estimation Method Residual Variance Method Fixed Effects SE Method Degrees of Freedom Method Thursday November 1, 2007 WORK.B y Variance Components REML Profile Model-Based Containment 4 ST 524 NCSU - Fall 2007 Incomplete block Design Class Level Information Class Levels block trt 4 4 Values b1 b2 b3 b4 t1 t2 t3 t4 Dimensions Covariance Parameters Columns in X Columns in Z Subjects Max Obs Per Subject 2 5 4 1 12 Number of Observations Number of Observations Read Number of Observations Used Number of Observations Not Used 12 12 0 Iteration History Iteration Evaluations -2 Res Log Like Criterion 0 1 1 1 40.85575344 38.55564281 0.00000000 Convergence criteria met. Covariance Parameter Estimates Cov Parm block Residual Estimate 3.2333 2.3500 Fit Statistics -2 Res Log Likelihood AIC (smaller is better) AICC (smaller is better) BIC (smaller is better) Thursday November 1, 2007 38.6 42.6 45.0 41.3 5 ST 524 NCSU - Fall 2007 Incomplete block Design Solution for Fixed Effects Effect Intercept trt trt trt trt trt Estimate 3.4643 0.7590 0.9018 -0.5180 0 t1 t2 t3 t4 Standard Error 1.2843 1.3117 1.3117 1.3117 . DF 3 5 5 5 . t Value 2.70 0.58 0.69 -0.39 . Pr > |t| 0.0739 0.5879 0.5224 0.7092 . Solution for Random Effects Effect block block block block block b1 b2 b3 b4 Estimate Std Err Pred DF t Value Pr > |t| 0.3929 -2.1610 0.09823 1.6699 1.1523 1.1523 1.1523 1.1523 5 5 5 5 0.34 -1.88 0.09 1.45 0.7470 0.1196 0.9354 0.2070 Type 3 Tests of Fixed Effects Effect trt Num DF Den DF F Value Pr > F 3 5 0.52 0.6893 Estimates Label I trt 1 mean II trt 1 & Block mean I trt 2 mean II trt 2 & Block mean I trt 3 mean blk1 II trt 3 & Block mean I trt 4 mean blk2 II trt 4 & Block mean trt1 vs trt3 trt1 vs trt3 in blk1 II trt 1 in Block 1 pred II trt 3 in Block 1 pred Thursday November 1, 2007 Estimate Standard Error DF t Value Pr > |t| 4.2233 3.6667 4.3661 4.3333 2.9463 3.6667 3.4643 3.3333 1.2770 1.6699 4.6162 3.3392 1.2843 0.8851 1.2843 0.8851 1.2843 0.8851 1.2843 0.8851 1.3117 1.7459 1.1158 1.1158 5 5 5 5 5 5 5 5 5 5 5 5 3.29 4.14 3.40 4.90 2.29 4.14 2.70 3.77 0.97 0.96 4.14 2.99 0.0218 0.0090 0.0193 0.0045 0.0703 0.0090 0.0429 0.0131 0.3750 0.3828 0.0090 0.0304 6 ST 524 NCSU - Fall 2007 Incomplete block Design Least Squares Means Effect trt trt trt trt trt t1 t2 t3 t4 Estimate Standard Error DF t Value Pr > |t| 4.2233 4.3661 2.9463 3.4643 1.2843 1.2843 1.2843 1.2843 5 5 5 5 3.29 3.40 2.29 2.70 0.0218 0.0193 0.0703 0.0429 Differences of Least Squares Means Effect trt _trt trt trt trt trt trt trt t1 t1 t1 t2 t2 t3 t2 t3 t4 t3 t4 t4 Estimate Standard Error DF t Value Pr > |t| -0.1428 1.2770 0.7590 1.4197 0.9018 -0.5180 1.3117 1.3117 1.3117 1.3117 1.3117 1.3117 5 5 5 5 5 5 -0.11 0.97 0.58 1.08 0.69 -0.39 0.9176 0.3750 0.5879 0.3285 0.5224 0.7092 Predicted values Obs y trt block Pred 1 2 3 4 5 6 7 8 9 10 11 12 6 2 3 5 1 7 2 5 4 1 3 6 t1 t1 t1 t2 t2 t2 t3 t3 t3 t4 t4 t4 b1 b2 b3 b1 b2 b4 b1 b3 b4 b2 b3 b4 4.61620 2.06228 4.32152 4.75899 2.20506 6.03595 3.33924 3.04456 4.61620 1.30329 3.56253 5.13418 Thursday November 1, 2007 StdErr Pred DF Alpha 1.11580 1.11580 1.11580 1.11580 1.11580 1.11580 1.11580 1.11580 1.11580 1.11580 1.11580 1.11580 5 5 5 5 5 5 5 5 5 5 5 5 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 Lower Upper Resid 1.74795 -0.80597 1.45327 1.89074 -0.66319 3.16770 0.47099 0.17631 1.74795 -1.56496 0.69428 2.26593 7.48445 4.93053 7.18977 7.62724 5.07331 8.90420 6.20749 5.91281 7.48445 4.17154 6.43078 8.00243 1.38380 -0.06228 -1.32152 0.24101 -1.20506 0.96405 -1.33924 1.95544 -0.61620 -0.30329 -0.56253 0.86582 7