Outline 1 Randomized Complete Block Design RCBD RCBD examples and model Estimates ANOVA table and f tests Checking assumptions RCBD with subsampling 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 B A C D D A B C C B D A Appropriate model A C D B Yi j i k i ei with ei iid N 0 e2 population mean across treatments blocks and plots j deviation of Pirrigation method j from the mean constrained to aj 1 j 0 k P fixed block effect elevation 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 kb 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 the trt j and block k jk j k Trt 1 2 a k 1 1 1 2 1 a 1 1 Block 2 1 2 2 2 a 2 2 Estimates y j y j y k y k y if fixed block effects Predicted or fitted values j k b 1 b 2 b a b b j 1 2 a ANOVA table with RCBD Source df SS MS Block b 1 SSBlk MSBlk e2 a e2 a 2 Trt Error Total a 1 b 1 a 1 ab 1 SSTrt SSPE SSTot MSTrt MSPE e2 b e2 IE MS Pb 2 k 1 k b 1 fixed random Pa 2 j 1 j a 1 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 a 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 Block variability Estimation if random block effects 2 MSBlk MSErr a 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 with RCBD 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 F 3 598 0 889 p value 0 038 0 47 ANOVA in R with RCBD emerge read table seedEmergence txt header T emerge block factor emerge block fit lm lm emergence treatment block data emerge fit aov aov emergence treatment block data emerge anova fit lm anova fit aov same output either way Analysis of Variance Table Response emergence 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 Model assumptions The model assumes 1 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 2 errors ei are independent have homogeneous variance and a normal distribution The ANOVA table and f test 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 For observation yi define the residual as usual ri yi y i Check that They approximately have a normal distribution no pattern trend unequal variance across blocks no pattern trend unequal variance across treatments Because both predictors are factors and because of the balance all observations have the same leverage The 4th plot then plots residuals versus factor levels plot fit lm Additivity assumption Additivity assumes the blocking factor affects all the trts uniformly To assess the absence of interactions visually use a mean profile plot Additivity should show up as parallelism with emerge interaction plot treatment block emergence col 1 4 Tukey s additivity test can be used but it still makes an assumption about the interaction coefficients if they are not all 0 If the additivity assumption is violated how to design an experiment differently to account for non additivity of trt and block effects RCBD with subsampling Yi j i k i j i k i ei where is a population mean averaged over all treatments P j is a fixed trt effect constrained to aj 1 j 0 P k is a fixed block effect k 1 b bj 1 j 0 jk iid N 0 2 is for variation among samples within blocks ei iid N 0 e2 is for variation among subsamples Total of abs observations ANOVA table and f test RCBD with subsampling Source df Blocks Treatment Plot Error Subsamp Total SS MS b 1 SSBlk MSBlk a 1 a 1 b 1 ab s 1 abs 1 SSTrt SSPE SSSSE SSTot MSTrt MSPE MSSSE IE MS e2 e2 e2 e2 s 2 s 2 s 2 Pb as bs 2 j 1 k b 1 Pa 2 j 1 j a 1 To test H0 j 0 for all j i e no treatment effect use the fact that under H0 F MSTrt Fa 1 b 1 a 1 MSPE Similarly to CRD with subsampling we do not use MSSSE at the denominator We can estimate the overall magnitude of interaction effects 2 MSSSE