Applied Statistics and Data Modeling Part 3: Analysis of Variance - - PowerPoint PPT Presentation

Applied Statistics and Data Modeling Part 3: Analysis of Variance - Two way ANOVA Luc Duchateau 1 Paul Janssen 2 1 Faculty of Veterinary Medicine Ghent University, Belgium 2 Center for Statistics Hasselt University, Belgium 2020 UGent STATS VM L. Duchateau & P.Janssen (UG & UH) Applied Statistics and Data Modeling 2020 1 / 77

Two-Way Analysis of Variance Overview Overview Introducing two-way data sets Why factorial experiments? Constructing models for two-way data ANOVA for two-way data Testing specific hypothesis for two-way data ANOVA for two-way data with exactly 1 replication UGent STATS VM L. Duchateau & P.Janssen (UG & UH) Applied Statistics and Data Modeling 2020 2 / 77

Two-Way Analysis of Variance Introducing two-way data sets Introducing two-way data sets Example 1: Treatment effect on PCV for Boran and Holstein cows with trypanosomosis Cowid Breed Drug PCV-before PCV-after PCV-difference 1 Boran Berenil 18.4 26.3 7.9 2 Boran Berenil 20.3 28.1 7.8 3 Boran Berenil 22.2 27.8 5.6 4 Boran Samorin 16.3 30.1 13.8 5 Boran Samorin 15.4 27.3 11.9 6 Boran Samorin 19.2 32.7 13.5 7 Holstein Berenil 21.3 28.3 7.0 8 Holstein Berenil 17.4 26.8 9.4 9 Holstein Berenil 18.2 25.8 7.6 10 Holstein Samorin 22.2 38.1 15.9 11 Holstein Samorin 19.8 32.3 12.5 UGent 12 Holstein Samorin 20.4 30.8 10.4 STATS VM L. Duchateau & P.Janssen (UG & UH) Applied Statistics and Data Modeling 2020 3 / 77

Two-Way Analysis of Variance Introducing two-way data sets Example 2: Milk production as a function of parity and inoculation dose Cowid Parity Inoculation dose Milk0 Milk48 Reduction (%) 1 heifer high 32.4 30.2 6.79 2 heifer high 33.6 32.3 3.87 3 heifer medium 29.3 20.5 30.03 4 heifer medium 34.4 21.3 38.08 5 heifer low 31.3 14.5 53.67 6 heifer low 35.3 13.4 62.04 7 multiparous high 42.4 39.5 6.84 8 multiparous high 43.3 39.7 8.31 9 multiparous medium 45.2 23.9 47.12 10 multiparous medium 44.4 24.8 44.14 11 multiparous low 41.5 6.7 83.86 12 multiparous low 45.2 4.1 90.93 UGent STATS VM L. Duchateau & P.Janssen (UG & UH) Applied Statistics and Data Modeling 2020 4 / 77

Two-Way Analysis of Variance Why factorial experiments? Why factorial experiments? Factorial versus ’one at a time’ Investigate two factors separately, e.g. → Compare heifers with multiparous cows at high inoculation dose → Multiparous cows higher reduction, and thus more appropriate as experimental model → Compare inoculation doses for multiparous cows UGent STATS VM L. Duchateau & P.Janssen (UG & UH) Applied Statistics and Data Modeling 2020 5 / 77

Two-Way Analysis of Variance Why factorial experiments? Investigate two factors jointly, e.g. → Take 6 heifers and 6 multiparous cows → Assign inoculation doses at random so that each inoculation dose has 2 heifers and 2 multiparous cows UGent STATS VM L. Duchateau & P.Janssen (UG & UH) Applied Statistics and Data Modeling 2020 6 / 77

Two-Way Analysis of Variance Why factorial experiments? Disadvantages of the ’one factor at a time’ approach We do not evaluate all treatment combinations We cannot evaluate interaction between 2 factors Treatment combinations (of the two factors) of first experiment are not randomly assigned with respect to those of second experiment Logistically more demanding because 2 experiments UGent STATS VM L. Duchateau & P.Janssen (UG & UH) Applied Statistics and Data Modeling 2020 7 / 77

Two-Way Analysis of Variance Why factorial experiments? Advantages of the factorial approach More replications due to ’hidden’ replication In the case of no interaction, all subjects receiving a treatment level of one factor can be considered replication for that factor level Interaction can be evaluated More easily generalizable in the absence of interaction In the case of no interaction, results of one factor are valid at all levels of the other factor UGent STATS VM L. Duchateau & P.Janssen (UG & UH) Applied Statistics and Data Modeling 2020 8 / 77

Two-Way Analysis of Variance Constructing models for two-way data Constructing models for two-way data Assume that all population means are known! Population means of treatment combinations µ ij is population mean for level i of factor A and level j of factor B Factor B: inoculation dose Factor A: Parity j = 1, low j = 2, medium j = 3, high Row mean i = 1, heifer 75 ( µ 11 ) 42 ( µ 12 ) 3 ( µ 13 ) 40 ( µ 1 . ) i = 2, multiparous 75 ( µ 21 ) 42 ( µ 22 ) 3 ( µ 22 ) 40 ( µ 2 . ) Column mean 75 ( µ . 1 ) 42 ( µ . 2 ) 3 ( µ . 3 ) 40 ( µ .. ) UGent STATS VM L. Duchateau & P.Janssen (UG & UH) Applied Statistics and Data Modeling 2020 9 / 77

Two-Way Analysis of Variance Constructing models for two-way data Population means of factor levels µ i . is population mean for observations of level i of factor A b � µ ij j =1 µ i . = b µ . j is population mean for observations of level j of factor B a � µ ij i =1 µ . j = a µ .. is the overall population mean a b b a � � � µ ij µ . j � µ i . i =1 j =1 j =1 i =1 µ .. = = = UGent ab a b STATS VM L. Duchateau & P.Janssen (UG & UH) Applied Statistics and Data Modeling 2020 10 / 77

Two-Way Analysis of Variance Constructing models for two-way data Translation to factor effects Main effect of level i of factor A: α i = µ i . − µ .. Main effect of level j of factor B: β j = µ . j − µ .. a � a µ i . i =1 � As µ .. = ⇒ α i = 0 a i =1 b � µ . j b j =1 As µ .. = ⇒ � β j = 0 b j =1 ⇒ Sum Restrictions UGent STATS VM L. Duchateau & P.Janssen (UG & UH) Applied Statistics and Data Modeling 2020 11 / 77

Two-Way Analysis of Variance Constructing models for two-way data What is α 1 , α 2 , β 1 , β 2 and β 3 in example below? Factor B: inoculation dose Factor A: Parity j = 1, low j = 2, medium j = 3, high Row mean i = 1, heifer 75 ( µ 11 ) 42 ( µ 12 ) 3 ( µ 13 ) 40 ( µ 1 . ) i = 2, multiparous 75 ( µ 21 ) 42 ( µ 22 ) 3 ( µ 22 ) 40 ( µ 2 . ) Column mean 75 ( µ . 1 ) 42 ( µ . 2 ) 3 ( µ . 3 ) 40 ( µ .. ) UGent STATS VM L. Duchateau & P.Janssen (UG & UH) Applied Statistics and Data Modeling 2020 12 / 77

Two-Way Analysis of Variance Constructing models for two-way data What is α 1 , α 2 , β 1 , β 2 and β 3 in example below? Factor B: inoculation dose Factor A: Parity j = 1, low j = 2, medium j = 3, high Row mean i = 1, heifer 75 ( µ 11 ) 42 ( µ 12 ) 3 ( µ 13 ) 40 ( µ 1 . ) i = 2, multiparous 75 ( µ 21 ) 42 ( µ 22 ) 3 ( µ 22 ) 40 ( µ 2 . ) Column mean 75 ( µ . 1 ) 42 ( µ . 2 ) 3 ( µ . 3 ) 40 ( µ .. ) α 1 = 0, α 2 = 0 UGent STATS VM L. Duchateau & P.Janssen (UG & UH) Applied Statistics and Data Modeling 2020 12 / 77

Two-Way Analysis of Variance Constructing models for two-way data What is α 1 , α 2 , β 1 , β 2 and β 3 in example below? Factor B: inoculation dose Factor A: Parity j = 1, low j = 2, medium j = 3, high Row mean i = 1, heifer 75 ( µ 11 ) 42 ( µ 12 ) 3 ( µ 13 ) 40 ( µ 1 . ) i = 2, multiparous 75 ( µ 21 ) 42 ( µ 22 ) 3 ( µ 22 ) 40 ( µ 2 . ) Column mean 75 ( µ . 1 ) 42 ( µ . 2 ) 3 ( µ . 3 ) 40 ( µ .. ) α 1 = 0, α 2 = 0 β 1 = µ . 1 − µ .. = 75 − 40 = 35 UGent STATS VM L. Duchateau & P.Janssen (UG & UH) Applied Statistics and Data Modeling 2020 12 / 77

Two-Way Analysis of Variance Constructing models for two-way data What is α 1 , α 2 , β 1 , β 2 and β 3 in example below? Factor B: inoculation dose Factor A: Parity j = 1, low j = 2, medium j = 3, high Row mean i = 1, heifer 75 ( µ 11 ) 42 ( µ 12 ) 3 ( µ 13 ) 40 ( µ 1 . ) i = 2, multiparous 75 ( µ 21 ) 42 ( µ 22 ) 3 ( µ 22 ) 40 ( µ 2 . ) Column mean 75 ( µ . 1 ) 42 ( µ . 2 ) 3 ( µ . 3 ) 40 ( µ .. ) α 1 = 0, α 2 = 0 β 1 = µ . 1 − µ .. = 75 − 40 = 35 β 2 = 2, β 3 = − 37 UGent STATS VM L. Duchateau & P.Janssen (UG & UH) Applied Statistics and Data Modeling 2020 12 / 77

Two-Way Analysis of Variance Constructing models for two-way data Additive factor effects We say that factor effects are additive if we only need the factor effects to obtain the population means, i.e., µ ij = µ .. + α i + β j This corresponds to → Absence of interaction → Effect of one factor does not depend on the level of the other factor UGent STATS VM L. Duchateau & P.Janssen (UG & UH) Applied Statistics and Data Modeling 2020 13 / 77

Two-Way Analysis of Variance Constructing models for two-way data Are the two factors additive in the example below? Factor B: inoculation dose Factor A: Parity j = 1, low j = 2, medium j = 3, high Row mean i = 1, heifer 75 ( µ 11 ) 42 ( µ 12 ) 3 ( µ 13 ) 40 ( µ 1 . ) i = 2, multiparous 75 ( µ 21 ) 42 ( µ 22 ) 3 ( µ 22 ) 40 ( µ 2 . ) Column mean 75 ( µ . 1 ) 42 ( µ . 2 ) 3 ( µ . 3 ) 40 ( µ .. ) UGent STATS VM L. Duchateau & P.Janssen (UG & UH) Applied Statistics and Data Modeling 2020 14 / 77