Outline 1 Randomized Complete Block Design RCBD RCBD examples and model Estimates ANOVA table and f tests Checking assumptions RCBD with subsampling Model 2 Latin square design Design and model ANOVA table Multiple Latin squares Randomized Complete Block Design RCBD Suppose a slope difference in the field is anticipated We block the field by elevation into 4 rows and assign irrigation treatment randomly within each block row Ex sample c A B C D 1 D A B C B D C A A A B C C B D D D C A B RCBD model response treatment block error Here block and error variation at the no treatment block interaction Treatments and blocks are crossed factors level RCBD model Model response treatment block error Yi j i k i ei with ei iid N 0 e2 population mean across treatments j deviation of Pairrigation method j from the mean constrained to j 1 j 0 Fixed treatment effects k fixed blockPeffect categorical k 1 b constrained to bk 1 k 0 or random effect with k iid N 0 2 Soil moisture a 4 b 4 Total of ab 16 observations Seedling emergence example Compare 5 seed disinfectant treatments using RCBD with 4 blocks In each plot 100 seeds were planted Response plants that emerged in each plot Treatment Control Arasan Spergon Semesan Fermate Mean y k 1 86 98 96 97 91 93 6 Block 2 3 90 88 94 93 90 91 95 91 93 95 92 4 91 6 4 87 89 92 92 95 91 0 Mean y j 87 75 93 50 92 25 93 75 93 50 y 92 15 Model Yi j i k i ei with ei iid N 0 e2 j seed treatment effect k block effect Seedling emergence example Population mean for trt j and block k jk j k Predicted means or fitted values jk j k How Trt 1 2 a k 1 1 1 2 1 a 1 1 Block 2 1 2 2 2 a 2 2 b 1 b 2 b a b b Estimated coefficients balance 1 obs trt block y j y j y k y k y if fixed block effects j 1 2 a ANOVA table with RCBD Source df SS MS Block b 1 SSBlk MSBlk k 1 k e2 a b 1 fixed 2 2 e a random f test Trt Error Total a 1 b 1 a 1 ab 1 SSTrt SSErr SSTot MSTrt MSErr e2 b e2 IE MS Pb 2 Pa 2 j 1 j a 1 f test SSBlk involves y k y 2 over all blocks k SSTrt involves y j y 2 over all treatments j SSErr involves yij ij 2 from all residuals SSTot involves yij y 2 Why not include an interaction Block Treatment in the model It would take MSErr df and there would remain df for Debate fixed vs random block effects Ex does it make sense to view the 4 specific rows blocked by elevation as randomly selected from a larger population Ex 4 dosages of a new drug are randomly assigned to 4 mice in each of the 20 litters RCBD with a 4 dosage treatments and b 20 litters for a total of ab 80 observations Here blocks litters can be considered as random samples from the population of all litters that could be used for the study In RCBD the choice fixed vs random blocks does not affect the testing of the trt effect In more complicated designs it could If we can use the simpler analysis with fixed effects it is okay to use it F test for block variability Estimation if random block effects 2 MSBlk MSErr a ANOVA table Test for the block effects uncommon F MSBlk on df b 1 b 1 a 1 MSErr but even if there appears to be non significant differences between blocks we would keep blocks into the model to reflect the randomization procedure Other commonly used blocking factors observers time farm stall arrangement etc The general guideline to choose blocks is scientific knowledge F tests for treatment effects To test H0 j 0 for all j i e no treatment effect use the fact that under H0 F Source Treatments Blocks Error Total MSTrt Fa 1 b 1 a 1 MSErr df 4 3 12 19 SS 102 30 18 95 85 30 206 55 MS 25 58 6 32 7 11 ANOVA table F 3 598 0 889 p value 0 038 0 47 ANOVA in R with RCBD emerge read table seedEmergence txt header T str emerge data frame 20 obs of 3 variables treatment Factor w 5 levels Arasan Control 2 1 5 4 block int 1 1 1 1 1 2 2 2 2 2 emergence int 86 98 96 97 91 90 94 90 95 93 emerge block factor emerge block Make sure blocks are treated as categorical They should be associated with b 1 3 df in the ANOVA table or LRT ANOVA in R with RCBD fit lm lm emergence treatment block data emerge anova fit lm Df Sum Sq Mean Sq F value Pr F treatment 4 102 300 25 575 3 5979 0 03775 block 3 18 950 6 317 0 8886 0 47480 Residuals 12 85 300 7 108 fit lm lm emergence block treatment data emerge anova fit lm Df Sum Sq Mean Sq F value Pr F block 3 18 95 6 3167 0 8886 0 47480 treatment 4 102 30 25 5750 3 5979 0 03775 Residuals 12 85 30 7 1083 drop1 fit lm Single term deletions Df Sum of Sq RSS AIC F value Pr F none 85 30 45 009 block 3 18 95 104 25 43 021 0 8886 0 47480 treatment 4 102 30 187 60 52 772 3 5979 0 03775 ANOVA in R with RCBD Here the output of anova does not depend on the order in which treatment and block are given Here type I sums of squares sequential anova and type III sums of squares drop1 are equal Because the design is balanced Significant effect of treatments Non significant differences between blocks but still keep blocks in the model Note aov could have been used in place of lm Model assumptions The model assumes 1 Errors ei are independent have homogeneous variance and a normal distribution 2 Additivity means are j k i e the trt differences are the same for every block and the block differences are the same for every trt No interaction Extra assumption for the ANOVA table and f test balance In particular they assume completeness each trt appears at least once in each block That is n 1 per trt and block Example of an incomplete block design for b 4 a 4 B D C A A A B C C B D D Model diagnostics Check that residuals ri yi y i approximately have a normal distribution no pattern trend unequal variance across blocks no pattern trend unequal variance across treatments plot fit lm Residuals vs Fitted Constant Leverage Residuals vs Factor Levels Normal …