Overview of this module Course 02429 Analysis of correlated data: - PowerPoint PPT Presentation

Overview of this module Course 02429 Analysis of correlated data: Mixed Linear Models The planks data 1 Module 3: Drying of beech wood - a case study, part I The factor structure 2 Initial explorative analysis 3 Per Bruun Brockhoff Modelling 4 DTU Compute Building 324 - room 220 Post hoc analysis 5 Technical University of Denmark 2800 Lyngby – Denmark The overall approach 6 e-mail: perbb@dtu.dk Testing fixed effects - a recap 7 Per Bruun Brockhoff (perbb@dtu.dk) Mixed Linear Models, Module 3 Fall 2014 1 / 26 Per Bruun Brockhoff (perbb@dtu.dk) Mixed Linear Models, Module 3 Fall 2014 2 / 26 The planks data The planks data The data The data, cont. 20 planks was dryed in a certain period of time. The humidity percentage was measured at: Width 1 Width 2 Width 3 5 depths Depth Depth Depth Planks 1 3 5 7 9 1 3 5 7 9 1 3 5 7 9 1 3.4 4.9 5.0 4.9 4.0 4.1 4.7 5.2 4.6 4.3 4.4 4.8 5.0 4.9 4.2 depth 1: close to the top 2 4.3 5.5 6.2 5.4 4.7 3.9 5.6 5.7 5.5 4.9 4.0 4.7 4.5 3.9 4.0 3 4.2 5.5 5.6 6.3 4.5 5.4 6.2 6.1 6.4 5.2 4.5 4.9 4.9 4.9 4.4 depth 5: in the center 4 4.4 6.0 7.1 6.9 4.6 4.6 6.1 6.6 6.5 4.7 4.9 5.9 5.8 6.4 4.7 5 3.9 4.7 5.2 5.0 3.7 4.2 5.2 5.4 4.8 3.9 4.0 4.4 4.4 4.1 3.5 depth 9: close to the bottom 6 4.6 5.9 6.3 5.8 4.8 5.9 7.3 6.9 6.9 4.4 5.2 5.7 6.6 6.0 4.0 7 3.9 5.6 6.0 5.3 5.0 4.9 6.9 7.1 6.1 4.5 4.3 5.4 5.9 5.5 4.2 depth 3: between 1 and 5 8 3.9 4.5 5.3 5.6 4.7 3.7 4.9 4.8 4.9 4.3 3.8 4.5 5.4 4.8 4.0 9 3.6 4.1 4.0 4.4 3.7 3.8 5.1 5.0 4.6 3.3 3.0 3.9 4.7 4.9 3.8 depth 7: between 5 and 9 10 6.5 8.7 9.5 7.9 6.6 6.9 8.9 7.4 7.0 6.9 5.8 7.5 7.7 7.3 5.9 11 3.7 5.2 5.5 5.9 4.4 4.7 5.8 5.7 4.9 4.2 3.7 5.0 6.3 5.2 4.3 3 widths 12 4.3 5.8 6.2 5.2 4.4 4.8 6.7 7.0 6.1 5.2 5.1 5.7 5.9 6.4 5.1 13 6.5 8.8 9.1 8.9 6.0 5.9 7.5 8.4 7.9 5.7 4.0 4.2 4.9 4.6 3.5 width 1: close to the side 14 4.4 6.2 6.7 6.4 4.3 5.7 7.0 7.4 7.3 5.5 4.6 6.2 6.8 5.8 4.9 15 5.5 7.1 7.5 6.9 5.4 6.4 8.4 8.9 8.1 6.1 6.5 8.4 9.1 9.2 7.5 width 3: in the center 16 5.2 6.0 6.2 6.6 5.3 6.6 7.6 7.8 7.7 5.8 5.9 6.7 6.7 5.0 3.9 17 3.7 4.5 5.0 4.5 3.7 3.7 4.4 4.8 4.4 4.3 3.7 4.5 4.7 5.3 3.9 width 2: between 1 and 3 18 6.0 7.4 7.8 7.5 5.7 6.9 8.6 8.8 7.5 5.4 5.1 6.1 5.2 5.4 4.7 19 3.8 4.6 4.8 4.4 3.8 3.7 4.7 4.7 4.3 3.7 3.3 3.5 3.7 3.4 3.2 20 6.1 7.4 7.7 6.7 4.6 4.7 6.3 7.1 6.5 5.1 4.7 6.0 6.0 6.3 4.2 Aim: To investigate the effect of drying of beech wood on the humidity percentage. Per Bruun Brockhoff (perbb@dtu.dk) Mixed Linear Models, Module 3 Fall 2014 4 / 26 Per Bruun Brockhoff (perbb@dtu.dk) Mixed Linear Models, Module 3 Fall 2014 5 / 26

The factor structure The factor structure Identify the factors and their structure Identify the factors and their structure, cont. 5 depth 4 Factor structure: depth × width 8 15 300 3 1 [I] 266 width 2 0 1 Factors: I : 300 experimental units 20 [plank] 19 0 : 1 overall level depth : 5 levels Model: width : 3 levels plank : 20 levels Y i = µ + α ( width i )+ β ( depth i )+ γ ( width i , depth i )+ d ( plank i )+ ǫ i , Crossed "treatment" structure: width × depth : 15 levels plank is a natural block effect. d ( j ) ∼ N (0 , σ 2 Plank ) , ǫ i ∼ N (0 , σ 2 ) . j = 1 , . . . , 20 , i = 1 , . . . , 300 (A larger model is introduced in Module4 (and Module6), but let us do as if we didn’t see that yet) Per Bruun Brockhoff (perbb@dtu.dk) Mixed Linear Models, Module 3 Fall 2014 7 / 26 Per Bruun Brockhoff (perbb@dtu.dk) Mixed Linear Models, Module 3 Fall 2014 8 / 26 Initial explorative analysis Initial explorative analysis Initial explorative analysis - boxplots Initial explorative analysis - boxplots ● 8 ● ● ● 9 9 8 mean of humidity mean of humidity 7 8 8 7 ● 7 7 ● 6 6 6 6 5 5 5 5 4 4 4 4 3 3 1 2 3 1 3 5 7 9 1 2 3 1 3 5 7 9 width depth 6.5 6.5 ● ● ● 9 9 mean of humidity mean of humidity ● 6.0 6.0 8 8 ● 7 7 5.5 5.5 6 6 5.0 5.0 5 5 4 4 4.5 4.5 3 3 1 2 3 1 3 5 7 9 1 4 7 10 14 18 1.1 1.3 1.5 1.7 1.9 Per Bruun Brockhoff (perbb@dtu.dk) Mixed Linear Models, Module 3 Fall 2014 10 / 26 Per Bruun Brockhoff (perbb@dtu.dk) Mixed Linear Models, Module 3 Fall 2014 11 / 26

Modelling Modelling Test of overall effects/model reduction Final model Fixed effects ANOVA tables: 5 depth 4 Source of Numerator Denominator Sums of Mean variation degrees degrees squares squares of freedom of freedom 300 3 1 [I] 274 width 2 0 1 depth 4 266 78.26 < 0.0001 width 2 266 29.65 < 0.0001 20 [plank] 19 depth*width 8 266 1.08 0.3745 Source of Numerator Denominator Sums of Mean Corresponding to; variation degrees degrees squares squares Y i = µ + α ( width i ) + β ( depth i ) + d ( plank i ) + ǫ i of freedom of freedom depth 4 274 78.07 < 0.0001 d ( j ) ∼ N (0 , σ 2 Plank ) , ǫ ijk ∼ N (0 , σ 2 ) . width 2 274 29.57 < 0.0001 Per Bruun Brockhoff (perbb@dtu.dk) Mixed Linear Models, Module 3 Fall 2014 13 / 26 Per Bruun Brockhoff (perbb@dtu.dk) Mixed Linear Models, Module 3 Fall 2014 14 / 26 Post hoc analysis Post hoc analysis Post hoc analysis and summarizing the results Post hoc analysis and summarizing the results Estimates of the fixed parameters: Parameter Estimate SE Lower Upper Estimates of the variance parameters: Depth 1 µ + β (1) 4.7150 0.2361 4.2270 5.2030 σ 2 = 0 . 636 σ 2 ˆ Planks = 0 . 979 , ˆ Depth 3 µ + β (2) 5.9050 0.2361 5.4170 6.3930 Depth 5 µ + β (3) 6.1950 0.2361 5.7070 6.6830 σ Planks = 0 . 990 , ˆ σ = 0 . 4047 ˆ Depth 7 µ + β (4) 5.8633 0.2361 5.3753 6.3514 Depth 9 µ + β (5) 4.6533 0.2361 4.1653 5.1414 2.5 % 97.5 % .sig01 0.72 1.37 Parameter Estimate SE Lower Upper .sigma 0.58 0.69 Width 1 µ + α (1) 5.5140 0.2303 5.0352 5.9928 Width 2 µ + α (2) 5.7860 0.2303 5.3072 6.2648 Width 3 µ + α (3) 5.0990 0.2303 4.6202 5.5778 Per Bruun Brockhoff (perbb@dtu.dk) Mixed Linear Models, Module 3 Fall 2014 16 / 26 Per Bruun Brockhoff (perbb@dtu.dk) Mixed Linear Models, Module 3 Fall 2014 17 / 26

Post hoc analysis Post hoc analysis Comparisons of the fixed parameters Comparisons of the fixed parameters, results t-test: Depth Parameter Estimate SE Lower Upper P-value β (1) − ˆ ˆ difference β (2) t = 1-3 β (1) − β (2) -1.1900 0.1162 -1.5090 -0.8710 < 0.0001 � � β (1) − ˆ ˆ β (2) SE 1-5 β (1) − β (3) -1.4800 0.1162 -1.7990 -1.1610 < 0.0001 1-7 β (1) − β (4) -1.1483 0.1162 -1.4673 -0.8294 < 0.0001 1-9 β (1) − β (5) 0.06167 0.1162 -0.2573 0.3806 0.9841 95% confidence interval: 3-5 β (2) − β (3) -0.2900 0.1162 -0.6090 0.02896 0.0943 3-7 β (2) − β (4) 0.04167 0.1162 -0.2773 0.3606 0.9964 � � β (1) − ˆ ˆ β (1) − ˆ ˆ β (2) ± t . 975 , 274 SE β (2) 3-9 β (2) − β (5) 1.2517 0.1162 0.9327 1.5706 < 0.0001 5-7 β (3) − β (4) 0.3317 0.1162 0.01271 0.6506 0.0370 5-9 β (3) − β (5) 1.5417 0.1162 1.2227 1.8606 < 0.0001 Warning: ONLY make comparisons decided for in advance (before 7-9 β (4) − β (5) 1.2100 0.1162 0.8910 1.5290 < 0.0001 seeing the data) this way. For multiple comparisons between all levels of a factor: Use an Width Parameter Estimate SE Lower Upper P-value "adjustment" method: difference 1-2 α (1) − α (2) -0.2720 0.08997 -0.4840 -0.05998 0.0077 Bonferroni 1-3 α (1) − α (3) 0.4150 0.08997 0.2030 0.6270 < 0.0001 Tukey-Kramer 2-3 α (2) − α (3) 0.6870 0.08997 0.4750 0.8990 < 0.0001 Or other methods Per Bruun Brockhoff (perbb@dtu.dk) Mixed Linear Models, Module 3 Fall 2014 18 / 26 Per Bruun Brockhoff (perbb@dtu.dk) Mixed Linear Models, Module 3 Fall 2014 19 / 26 Post hoc analysis The overall approach Comparisons of the fixed parameters, in summary The overall approach - REALLY: Identify factors and their structure leading to starting model. Estimate Explorative analysis Depth 9 4 . 6533 a Decide on which effects to be random (consider if residual error Depth 1 4 . 7150 a structure is needed) Depth 7 5 . 8633 b FIRST: Test of RANDOM effects/model reduction leading to final RANDOM model part Depth 3 5 . 9050 bc REML based test (always OK) or: Depth 5 6 . 1950 c ML based test (always OK) or: F-tests coming from a model where a random effect is considered fixed (sometimes OK) Estimate NOTE: Sometimes this part is skipped because the structure is simple. E.g. for the planks data! Width 3 5 . 0990 a Width 1 5 . 5140 b THEN: Test of FIXED effects/model reduction of leading to final model Width 2 5 . 7860 c Post hoc analysis of fixed effects (as before) Model Diagnostics is still missing: (Module 6) Per Bruun Brockhoff (perbb@dtu.dk) Mixed Linear Models, Module 3 Fall 2014 20 / 26 Per Bruun Brockhoff (perbb@dtu.dk) Mixed Linear Models, Module 3 Fall 2014 22 / 26