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

Applied Statistics and Data Modeling Part 3: Analysis of Variance - Balanced block designs 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 (UH & UG) Applied Statistics and Data Modeling 2020 1 / 56

Balanced block designs Overview Blocking principle Different block designs Anova for RCB and RGCB UGent STATS VM L. Duchateau & P.Janssen (UH & UG) Applied Statistics and Data Modeling 2020 2 / 56

Balanced block designs The blocking principle The blocking principle UGent STATS VM L. Duchateau & P.Janssen (UH & UG) Applied Statistics and Data Modeling 2020 3 / 56

Balanced block designs The blocking principle The blocking principle UGent STATS VM L. Duchateau & P.Janssen (UH & UG) Applied Statistics and Data Modeling 2020 4 / 56

Balanced block designs The blocking principle The blocking principle UGent STATS VM L. Duchateau & P.Janssen (UH & UG) Applied Statistics and Data Modeling 2020 5 / 56

Balanced block designs The blocking principle The blocking principle UGent STATS VM L. Duchateau & P.Janssen (UH & UG) Applied Statistics and Data Modeling 2020 6 / 56

Balanced block designs The blocking principle The blocking principle UGent STATS VM L. Duchateau & P.Janssen (UH & UG) Applied Statistics and Data Modeling 2020 7 / 56

Balanced block designs The blocking principle The blocking principle Group experimental units in homogeneous blocks Reduce background variability Blocking factor is known source of variability that is not of primary interest Attribute treatment at random to experimental units within a block UGent STATS VM L. Duchateau & P.Janssen (UH & UG) Applied Statistics and Data Modeling 2020 8 / 56

Balanced block designs Different block designs Example: Rice plants Effect of insertion of a gene into a rice plant Three different types Wild type (W) Haploid type (H) Diploid type (D) Evaluation of growth for 1000-kernel weight (g) Greenhouse can be split up in different plots and blocks Temperature gradient from the door of the greenhouse at the left towards the right side of the greenhouse UGent STATS VM L. Duchateau & P.Janssen (UH & UG) Applied Statistics and Data Modeling 2020 9 / 56

Balanced block designs Different block designs Randomised design Temperature UGent STATS VM L. Duchateau & P.Janssen (UH & UG) Applied Statistics and Data Modeling 2020 10 / 56

Balanced block designs Different block designs Randomised design Temperature UGent STATS VM L. Duchateau & P.Janssen (UH & UG) Applied Statistics and Data Modeling 2020 11 / 56

Balanced block designs Different block designs Randomised design Temperature UGent STATS VM L. Duchateau & P.Janssen (UH & UG) Applied Statistics and Data Modeling 2020 12 / 56

Balanced block designs Different block designs T emperature UGent STATS VM L. Duchateau & P.Janssen (UH & UG) Applied Statistics and Data Modeling 2020 13 / 56

Balanced block designs Different block designs T emperature UGent STATS VM L. Duchateau & P.Janssen (UH & UG) Applied Statistics and Data Modeling 2020 14 / 56

Balanced block designs Different block designs Randomised complete block design Temperature UGent STATS VM L. Duchateau & P.Janssen (UH & UG) Applied Statistics and Data Modeling 2020 15 / 56

Balanced block designs Different block designs T emperature UGent STATS VM L. Duchateau & P.Janssen (UH & UG) Applied Statistics and Data Modeling 2020 16 / 56

Balanced block designs Different block designs Temperature UGent STATS VM L. Duchateau & P.Janssen (UH & UG) Applied Statistics and Data Modeling 2020 17 / 56

Balanced block designs Different block designs Randomised complete block design with repeated measures Temperature UGent STATS VM L. Duchateau & P.Janssen (UH & UG) Applied Statistics and Data Modeling 2020 18 / 56

Balanced block designs Different block designs Temperature Temperature UGent STATS VM L. Duchateau & P.Janssen (UH & UG) Applied Statistics and Data Modeling 2020 19 / 56

Balanced block designs Different block designs Randomised complete block design with repeated measures Randomised generalised complete block design Temperature Temperature UGent STATS VM L. Duchateau & P.Janssen (UH & UG) Applied Statistics and Data Modeling 2020 20 / 56

Balanced block designs Different block designs Randomised complete block design with repeated measures Randomised generalised complete block design Randomised complete block design Temperature Temperature Temperature UGent STATS VM L. Duchateau & P.Janssen (UH & UG) Applied Statistics and Data Modeling 2020 21 / 56

Balanced block designs Different block designs Randomised complete block design with repeated measures Randomised generalised complete block design Randomised complete block design Temperature Temperature Temperature UGent STATS VM L. Duchateau & P.Janssen (UH & UG) Applied Statistics and Data Modeling 2020 22 / 56

Balanced block designs Different block designs Temperature UGent STATS VM L. Duchateau & P.Janssen (UH & UG) Applied Statistics and Data Modeling 2020 23 / 56

Balanced block designs Different block designs Balanced incomplete block design Temperature UGent STATS VM L. Duchateau & P.Janssen (UH & UG) Applied Statistics and Data Modeling 2020 24 / 56

Balanced block designs Different block designs T emperature UGent STATS VM L. Duchateau & P.Janssen (UH & UG) Applied Statistics and Data Modeling 2020 25 / 56

Balanced block designs Different block designs Balanced block design with repetition of W T emperature UGent STATS VM L. Duchateau & P.Janssen (UH & UG) Applied Statistics and Data Modeling 2020 26 / 56

Balanced block designs Different block designs UGent STATS VM L. Duchateau & P.Janssen (UH & UG) Applied Statistics and Data Modeling 2020 27 / 56

Balanced block designs Different block designs Balanced block design with different block sizes UGent STATS VM L. Duchateau & P.Janssen (UH & UG) Applied Statistics and Data Modeling 2020 28 / 56

Balanced block designs Different block designs Summary RCB and GCB restrictive designs Each block needs to contain exactly the same number of experimental units Number of experimental units needs to be a multiple of number of treatments Balanced designs Whenever we can not deduce information for the treatment effect by comparing blocks This is ok when the proportion of occurence of each treatment is the same in each block UGent STATS VM L. Duchateau & P.Janssen (UH & UG) Applied Statistics and Data Modeling 2020 29 / 56

Balanced block designs ANOVA for balanced block designs ANOVA for balanced block designs UGent STATS VM L. Duchateau & P.Janssen (UH & UG) Applied Statistics and Data Modeling 2020 30 / 56

Balanced block designs ANOVA for balanced block designs Factor effects model for RCB Assume no interaction between treatment and block Model Y ij = µ .. + α i + β j + e ij where the overall population mean (constant) µ .. main effect of level i of factor A (treatment) , i = 1 , . . . , a α i a constants with restriction � α i = 0 i =1 β j effect of block j , j = 1 , . . . , b b � constants with restriction β j = 0 j =1 UGent STATS VM independent random error term ∼ N (0 , σ 2 ) e ij L. Duchateau & P.Janssen (UH & UG) Applied Statistics and Data Modeling 2020 31 / 56

Balanced block designs ANOVA for balanced block designs Factor effects model for RCB Decomposition of sum of squares SS tot = SS trt + SS block + SS err with a b � 2 � � Y ij − ¯ � SS tot = Y .. i =1 j =1 a � ¯ � 2 � Y i . − ¯ SS trt = b Y .. i =1 b � ¯ � 2 � Y . j − ¯ SS block = a Y .. j =1 a b � 2 Y ij − ¯ Y i . − ¯ Y . j + ¯ UGent � � � SS err = Y .. STATS VM i =1 j =1 L. Duchateau & P.Janssen (UH & UG) Applied Statistics and Data Modeling 2020 32 / 56

Balanced block designs ANOVA for balanced block designs Factor effects model for RCB Mean sum of squares SS tot MS tot = ab − 1 SS trt MS trt = a − 1 SS block MS block = b − 1 SS err MS err = ( a − 1)( b − 1) UGent STATS VM L. Duchateau & P.Janssen (UH & UG) Applied Statistics and Data Modeling 2020 33 / 56

Balanced block designs ANOVA for balanced block designs Factor effects model for RCB Expected values a 1 σ 2 + � b α 2 E(MS trt ) = i a − 1 i =1 b 1 σ 2 + � a β 2 E(MS block ) = j b − 1 j =1 σ 2 E(MS err ) = test statistic to test H 0 : α 1 = α 2 = . . . = α a = 0 F trt* = MS trt ∼ F a − 1 , ( a − 1)( b − 1) MS err UGent P-value STATS VM P( F trt* ≥ f trt* ) L. Duchateau & P.Janssen (UH & UG) Applied Statistics and Data Modeling 2020 34 / 56