Overview of this module Course 02429 Analysis of correlated data: Mixed Linear Models Module 7: The analysis of split-plot design data Simple Split-plot designs 1 Example: Tenderness of pork. Per Bruun Brockhoff Split-plot with blocks 2 Example: Split-plot with blocks, Yield of oats DTU Compute Building 324 - room 220 Technical University of Denmark 2800 Lyngby – Denmark Split-plot in perspective 3 e-mail: perbb@dtu.dk Per Bruun Brockhoff (perbb@dtu.dk) Mixed Linear Models, Module 7 Fall 2014 1 / 16 Per Bruun Brockhoff (perbb@dtu.dk) Mixed Linear Models, Module 7 Fall 2014 2 / 16 Simple Split-plot designs Simple Split-plot designs Example: Tenderness of pork. Basic structure Example: Tenderness of pork. 24 porks ( pork ) in 2 treatment groups (low pH/high pH).( pH ) Each pork cut in half (right/left side). a 3 b 1 a 1 b 4 a 1 b 1 a 2 b 3 a 3 b 3 a 2 b 2 2 cooling methods: One for each side. ( C ) a 3 b 3 a 1 b 3 a 1 b 3 a 2 b 2 a 3 b 1 a 2 b 1 48 observations of tenderness. a 3 b 2 a 1 b 1 a 1 b 4 a 2 b 1 a 3 b 2 a 2 b 4 Factor structure: a 3 b 4 a 1 b 2 a 1 b 2 a 2 b 4 a 3 b 4 a 2 b 3 24 2 [P] 22 Ph 1 Treatments are given on different levels 48 1 [I] 22 0 1 Treatment effects are more easily found on finer levels For B: Randomized block setting with whole-plots as blocks. Ph × C 1 4 2 C 1 For A: Completely randomized design with whole-plots as observational unit. Per Bruun Brockhoff (perbb@dtu.dk) Mixed Linear Models, Module 7 Fall 2014 4 / 16 Per Bruun Brockhoff (perbb@dtu.dk) Mixed Linear Models, Module 7 Fall 2014 5 / 16
Simple Split-plot designs Example: Tenderness of pork. Simple Split-plot designs Example: Tenderness of pork. The split-plot mixed model for the example Example, test results Source of Numerator Denominator F P-value The model: variation degrees degrees Y i = α ( pH i ) + β ( C i ) + γ ( pH × C i ) + d ( P i ) + ε i of freedom of freedom phgroup 1 22 8.67 0.0075 pH tested versus the pork (whole plot) variation: cooling 1 22 2.25 0.1479 phgroup*cooling 1 22 0.18 0.6790 F = MS pH MS P Source of Numerator Denominator F P-value variation degrees degrees C and pH × C tested versus the residual error: of freedom of freedom , F = MS C × pH MS C phgroup 1 22 8.67 0.0075 F = MS Error MS Error cooling 1 22 2.33 0.1403 Per Bruun Brockhoff (perbb@dtu.dk) Mixed Linear Models, Module 7 Fall 2014 6 / 16 Per Bruun Brockhoff (perbb@dtu.dk) Mixed Linear Models, Module 7 Fall 2014 7 / 16 Simple Split-plot designs Example: Tenderness of pork. Split-plot with blocks Example: Split-plot with blocks, Yield of oats Example, results summary Example: Split-plot with blocks, Yield of oats 156 118 109 99 n 3 n 2 n 2 n 3 v 3 v 3 n 1 140 n 0 105 n 0 63 n 1 70 n 0 111 n 1 130 n 0 80 n 2 94 v 1 v 2 174 157 126 82 Variances: n 3 n 2 n 3 n 1 117 114 90 100 n 0 n 1 n 1 n 2 v 2 v 1 n 2 161 n 3 141 n 3 116 n 0 62 σ 2 ˆ = 1 . 2463 , W σ 2 ˆ = 0 . 4725 . n 2 104 n 0 70 n 3 96 n 0 60 v 3 v 2 89 117 89 102 n 1 n 3 n 2 n 1 Fixed effects: 122 74 112 86 n 3 n 0 n 2 n 3 v 1 v 1 n 1 89 n 2 81 n 0 68 n 1 64 n 1 103 n 0 64 n 2 132 n 3 124 v 2 v 3 α ( low ) ˆ = 5 . 6529 , ([4 . 9240 , 6 . 3819]) 132 133 129 89 n 2 n 3 n 1 n 0 α ( high ) ˆ = 7 . 1163 , ([6 . 3873 , 7 . 8452]) n 1 108 n 2 126 n 2 118 n 0 53 v 2 v 1 149 70 113 74 n 3 n 0 n 3 n 1 144 124 104 86 n 3 n 1 n 3 n 2 v 3 v 2 n 2 121 n 0 96 n 0 89 n 1 82 n 0 61 n 3 100 n 0 97 n 1 99 v 1 v 3 91 97 119 121 n 1 n 2 n 2 n 3 Per Bruun Brockhoff (perbb@dtu.dk) Mixed Linear Models, Module 7 Fall 2014 8 / 16 Per Bruun Brockhoff (perbb@dtu.dk) Mixed Linear Models, Module 7 Fall 2014 10 / 16
Split-plot with blocks Example: Split-plot with blocks, Yield of oats Split-plot with blocks Example: Split-plot with blocks, Yield of oats Example: Yield of oats Yield of oats, exploration Average yield as a function of nitrogen level for each variety. Factors and their levels: v 1 , v 2 , v 3 V n 0 , n 1 , n 2 , n 3 N 1 , 2 , . . . , 18 P 1 , 2 , . . . , 6 B Factor structure: 18 6 [P] 10 [B] 5 72 3 1 [I] 45 V 2 0 1 V × N 6 12 4 N 3 Per Bruun Brockhoff (perbb@dtu.dk) Mixed Linear Models, Module 7 Fall 2014 11 / 16 Per Bruun Brockhoff (perbb@dtu.dk) Mixed Linear Models, Module 7 Fall 2014 12 / 16 Split-plot with blocks Example: Split-plot with blocks, Yield of oats Split-plot in perspective Yield of oats, testing Split-plot in perspective Source of Numerator Denominator F P-value variation degrees degrees Treatments on different levels occur very often in practice of freedom of freedom More complicated and more than two levels may occur: fertil 3 45 37.69 < .0001 Split-plot with correlated whole plots variety 2 10 1.49 0.2724 Whole plot conducted as an incomplete latin square fertil*variety 6 45 0.30 0.9322 A strip-split-split-plot design The above are case studies in Littel et al. (1999). Source of Numerator degrees Denominator degrees F P-value Serves as a kind of basis for repeated measures analysis. variation of freedom of freedom fertil 3 51 41.05 < .0001 variety 2 10 1.49 0.2724 Per Bruun Brockhoff (perbb@dtu.dk) Mixed Linear Models, Module 7 Fall 2014 13 / 16 Per Bruun Brockhoff (perbb@dtu.dk) Mixed Linear Models, Module 7 Fall 2014 15 / 16
Split-plot in perspective Overview of this module Simple Split-plot designs 1 Example: Tenderness of pork. Split-plot with blocks 2 Example: Split-plot with blocks, Yield of oats Split-plot in perspective 3 Per Bruun Brockhoff (perbb@dtu.dk) Mixed Linear Models, Module 7 Fall 2014 16 / 16
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