Block Designs RCBD ANOVA
Total Page：16
File Type：pdf, Size：1020Kb
Randomized Complete Block Designs Block Designs Design Control Completely Random Blocked I II III IV I II III IV Trt 2 Trt 1 Trt 4 Trt 1 Trt 2 Trt 1 Trt 4 Trt 1 Trt 3 Trt 3 Trt 1 Trt 4 Trt 1 Trt 3 Trt 3 Trt 4 Trt 3 Trt 2 Trt 1 Trt 4 Trt 3 Trt 4 Trt 1 Trt 2 Trt 4 Trt 3 Trt 2 Trt 2 Trt 4 Trt 2 Trt 2 Trt 3 Yij = + Ti + (i)j Yijk = + Bi + Tj + εij RCBD ANOVA Comparison with CRD CRD RCBD Source df SS Source df SS I I JYY() 2 2 Ti I - 1 i... Bi I - 1 JYY()i... i 1 i 1 IJ J ()YY 2 2 (i)j I(J - 1) ij i. Tj J – 1 IYY()...j ij11 j 1 IJ IJ ()YY 2 ()YYYY 2 Total IJ - 1 ij .. ij (I -1)(J - 1) ij i.. j .. ij11 ij11 IJ 2 Total IJ - 1 ()YYij .. ij11 Agronomy 526 / Spring 2021 1 Randomized Complete Block Designs RCBD Blocking Criteria Spatial • Topography • Edaphic • Windbreak • Flood irrigation (moisture gradient) Temporal • Time Period • Order Other • Greenhouse • Growth chamber • Lab run / Batch RCBD Blocking Patterns Trt 1 Trt 4 Trt 2 Trt 3 Trt 4 Trt 3 Trt 2 Trt 1 Gradient Trt 3 Trt 1 Trt 4 Trt 2 Trt 2 Trt 3 Trt 4 Trt 1 Agronomy 526 / Spring 2021 2 Agronomy 526/ Spring 2021 Randomized CompleteBlock Designs Trt 1 Trt 4 Trt 4 Trt 3 Trt 1 Trt 2 Trt 2 Trt 4 Trt 3 Trt 1 Trt 2 Trt 4 Blocking Blocking Trt 3 Trt 3 Trt 2 RCBD RCBD Gradient Trt 1 Patterns Patterns Trt 3 Trt 1 Trt 3 Trt 4 Trt 1 Trt 2 Trt 2 Trt 4 Trt 3 Trt 2 Trt 2 Trt 4 Trt 3 Trt 1 Trt 4 Trt 1 3 Randomized Complete Block Designs RCBD Linear Additive Model Yij = + Bi + (i) + Tj + BTij Where: th Yij =Observation from the ij plot =Overall mean th Bi =Effect of the i block (i) = Restriction error associated with B th Tj =Effect of the j treatment th BTij =Interaction effect of the i of block with the jth treatment RCBD Expected Mean Squares Blocks Random Source b t EMS R F i j 2 2 2 Bi 1t + t + t B 2 2 (i) 1t + t 2 2 Tj b0 + BT + b(T) 2 2 BTij 10 + BT Agronomy 526 / Spring 2021 4 Randomized Complete Block Designs RCBD Example Kenaf Variety Trial Block Variety1 234Mean Dowling 4.3 3.7 4.0 2.8 3.7 Evergl41 3.4 3.7 3.9 3.3 3.6 Evergl71 3.6 3.0 3.7 3.6 3.5 Gregg 3.7 4.2 4.3 2.8 3.7 PF2 2.6 2.5 3.2 1.7 2.5 SF459 3.2 5.3 2.2 2.2 3.2 T2 5.4 6.1 4.2 4.6 5.1 Whitten 4.5 3.4 3.3 2.6 3.4 Mean 3.8 4.0 3.6 2.9 3.6 RCBD Example Kenaf Variety Trial ANOVA CRD Mean F Source DF SS Square Value Pr > F Variety 7 14.53005 2.075721 3.40 0.0115 Error 24 14.66376 0.61099 RCBD Mean F Source DF SS Square Value Pr > F Block 3 5.161725 1.720575 Var 7 14.53005 2.075721 4.59 0.003 Error 21 9.502032 0.452478 Agronomy 526 / Spring 2021 5 Randomized Complete Block Designs RCBD Example Kenaf Variety Trial CRD 2(.611) LSD 2.063 1.14 .05,24df 4 RCBD 2(.452) LSD 2.079 0.99 .05,21df 4 RCBD Relative Efficiency (1)(3)nnMS RE BC eC (1)(3)nnMSCB eB Where: MSC = Error mean square for CRD MSB = Error mean square for RCBD nC = error df for CRD nB = error df for RCBD Agronomy 526 / Spring 2021 6 Randomized Complete Block Designs RCBD Example Kenaf Variety Trial (21 1)(24 3)0.611 RE 1.34 (24 1)(21 3)0.453 Where: MSC = 0.611 MSB = 0.453 nC = 24 nB = 21 RCBD ANOVA Analysis –Barley Example Petersen, 1994 Blocks (Soil Types) Treatment Mean 1234 1(NH4)2SO4 32.1 35.6 41.9 35.4 36.25 2NH4NO3 30.1 31.5 37.1 30.8 32.38 3CO(NH2)2 25.4 27.1 33.8 31.1 29.35 4Ca(NO3)2 24.1 33.0 35.6 31.4 31.03 5NaNO3 26.1 31.0 33.8 31.9 30.70 6 Control 23.2 24.8 26.7 26.7 25.35 Agronomy 526 / Spring 2021 7 Randomized Complete Block Designs RCBD ANOVA Analysis –Barley Example RCBD Source DF SS MS FP>F Block 3 192.748 64.249 21.1 <.0001 Treatment 5 256.153 51.231 16.9 <.0001 Error 15 45.617 3.041 Total 23 494.518 CRD Source DF SS MS FP>F Treatment 5 256.153 51.231 3.9 0.0148 Error 18 238.365 13.243 Total 23 494.518 RCBD ANOVA Analysis –Barley Example RCBD 2(3.041) LSD 2.132 2.63 .05,15 df 4 CRD 2(13.243) LSD 2.101 5.41 .05,18 df 4 Agronomy 526 / Spring 2021 8 Randomized Complete Block Designs RCBD ANOVA Analysis –Barley Example (15 1)(18 3)13.243 Efficiency 4.28 (18 1)(15 3)3.041 Where: MSC = 13.243 MSB = 3.041 nC = 18 nB = 15 RCBD Expected Mean Squares Blocks Fixed Source b t EMS F F i j 2 2 Bi 0t + t + t(B) 2 2 (i) 1t + t 2 Tj b0 + b(T) 2 BTij 00 + BT) Agronomy 526 / Spring 2021 9 Randomized Complete Block Designs Generalized RCBD Field Layout Trt 1 Trt 1 Trt 2 Trt 3 Trt 2 Trt 3 Trt 4 Trt 4 Trt 4 Trt 3 Trt 2 Trt 1 Trt 1 Trt 4 Trt 2 Trt 3 Gradient Trt 2 Trt 3 Trt 4 Trt 1 Trt 3 Trt 1 Trt 4 Trt 2 Trt 2 Trt 1 Trt 4 Trt 1 Trt 4 Trt 3 Trt 2 Trt 3 Generalized RCBD Model Yijk = + Bi + (i) + Tj + BTij + (ij)k Where: th Yijk =Observation from the ijk plot =Overall mean th Bi =Effect of the i block. (i) = Restriction error associated with B th Tj =Effect of the j treatment th BTij =Interaction effect of the i of block with the jth treatment. th th (ij)k =Effect of the k eu within i block and jth treatment. Agronomy 526 / Spring 2021 10 Randomized Complete Block Designs Generalized RCBD Expected Mean Squares Blocks Random Source b t r EMS R F R i j k 2 2 2 Bi 1tr + tr + tr B 2 2 (i) 1tr + tr 2 2 Tj b0r + r BT + br(T) 2 2 BTij 10r + r BT 2 (ij)k 111 Generalized RCBD Expected Mean Squares Blocks Fixed Source b t r EMS F F R i j k 2 2 Bi 0tr + tr + tr(B) 2 2 (i) 1tr + tr 2 Tj b0r + br(T) 2 BTij 00r + rBT) 2 (ij)k 111 Agronomy 526 / Spring 2021 11 Randomized Complete Block Designs Fields as Blocks pp. 95‐103, Lorenzen and Anderson Layout: Field 12 Fertilizer 12345 12345 16... 2631... Plot 27 . 27. 38... ..... 49..24....49 510. 2025 . 4550 Fields as Blocks pp. 95‐103, Lorenzen and Anderson Yijk = + Fi + (i) + Nj + FNij + (ij)k 2 5 5 R F R Source i j kEMS 2 2 2 Fi 155 + 25 + 25 F 2 2 (i) 155 + 25 2 2 Nj 205 + 5 FN + 10(F) 2 2 FNij 105 + 5 FN 2 (ij)k 111 Agronomy 526 / Spring 2021 12.