Randomized Complete Block Design (RCBD)RCBD: examples and modelEstimates, ANOVA table and f-testsChecking assumptionsRCBD with subsamplingOutline1Randomized Complete Block Design (RCBD)RCBD: examples and modelEstimates, ANOVA table and f-testsChecking assumptionsRCBD with subsamplingRandomized Complete Block Design (RCBD)Suppose a slope difference in the field is anticipated. We blockthe field by elevation into 4 rows and assign irrigation treatmentrandomly within each block (row). Ex:B A C DD A B CCB D AA C D BAppropriate model:Yi= µ + αj[i]+ βk[i]+ eiwith ei∼ iid N (0, σ2e)µ = population mean across treatments, blocks and plots,αj= deviation of irrigation method j from the mean,constrained toPaj=1αj= 0βk= fixed block effect (elevation), k = 1, . . . , b constrainedtoPbk=1βk= 0, or random effect with βk∼ iid N (0, σ2β)Soil moisture: a = 4, b = 4. Total of kb = 16 observations.Seedling emergence exampleCompare 5 seed disinfectant treatments using RCBD with 4blocks. In each plot, 100 seeds were planted.Response: # plants that emerged in each plot.BlockTreatment 1 2 3 4 Mean (¯yj·)Control 86 90 88 87 87.75Arasan 98 94 93 89 93.50Spergon 96 90 91 92 92.25Semesan 97 95 91 9293.75Fermate 91 93 95 95 93.50Mean (¯y·k) 93.6 92.4 91.6 91.0¯y··= 92.15Model:Yi= µ + αj[i]+ βk[i]+ eiwith ei∼ iid N (0, σ2e)αj: seed treatment effect, βk: block effect.Seedling emergence examplePopulation mean for the trt j and block k: µjk= µ + αj+ βkBlockTrt 1 2 · · · b ¯µj·1 µ + α1+ β1µ + α1+ β2µ + α1+ βbµ + α12 µ + α2+ β1µ + α2+ β2µ + α2+ βbµ + α2· · · · · · · · ·aµ + αa+ β1µ + αa+ β2µ + αa+ βbµ + αa¯µ·kµ + β1µ + β2µ + βbµEstimates:ˆµ =¯y··ˆαj=¯yj·−¯y··ˆβk=¯y·k−¯y··(if fixed block effects)Predicted, or fitted values: ˆµ + ˆαj+ˆβkANOVA table with RCBDSource df SS MS IE(MS)Block b − 1 SSBlk MSBlk σ2e+ aPbk=1β2kb−1(fixed)σ2e+aσ2β(random)Trt a − 1 SSTrt MSTrt σ2e+ bPaj=1α2ja−1Error (b − 1)(a − 1) SSPE MSPE σ2eTotal ab − 1 SSTotWhy not include an interaction Block:Treatment in the model?It would take(b − 1)(a − 1)df and there would remain0df forMSErr.Debate: fixed vs. random block effectsEx: does it make sense to view the 4 specific rows blockedby elevation as randomly selected from a largerpopulation?Ex: 4 dosages of a new drug are randomly assigned to 4mice in each of the 20 litters: RCBD with a = 4 dosagetreatments and b = 20 litters, for a total of ab = 80observations. Here, blocks (litters) can be considered as arandom samples from the population of all litters that couldbe used for the study.In RCBD, the choice fixed vs. random blocks does notaffect the testing of the trt effect. In more complicateddesigns, it could.If we can use the simpler analysis with fixed effects, it isokay to use it!Block variabilityEstimation, if random block effects: ˆσ2β=MSBlk − MSErraTest for the block effects (uncommon):F =MSBlkMSErron df = b − 1, (b − 1)(a − 1)but even if there appears to be non-significant differencesbetween blocks, we would keep blocks into the model, to reflectthe randomization procedure.Other commonly used blocking factors: observers, time, farm,stall arrangement etc. The general guideline to choose blocksis scientific knowledge.F-tests with RCBDTo test H0: αj= 0 for all j (i.e., no treatment effect), use the factthat under H0,F =MSTrtMSErr∼ Fa−1, (b−1)(a−1)Source df SS MS F p-valueTreatments 4 102.30 25.58 3.598 0.038Blocks 3 18.95 6.32 0.889 0.47Error 12 85.30 7.11Total 19 206.55ANOVA 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) # same output either way> anova(fit.aov)Analysis of Variance TableResponse: emergenceDf 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 0.47480Residuals 12 85.300 7.108Model assumptionsThe model assumes:1Additivity: means are µ + αj+ βk, i.e. the trt differencesare the same for every block and the block differences arethe same for every trt. No interaction.2errors eiare independent, have homogeneous variance,and a normal distribution.The ANOVA table and f-test assume completeness: each trtappears at least once in each block. That is n ≥ 1 per trt andblock. Example of an incomplete block design for b = 4, a = 4:B A CD A BC B DA C DModel diagnosticsFor observation yi, define the residual as usual:ri= yi−ˆyiCheck 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 thebalance, all observations have the same leverage. The 4th plotthen plots residuals versus factor levels:plot(fit.lm)Additivity assumptionAdditivity: assumes the blocking factor affects all the trtsuniformly.To assess the absence of interactions visually, use a meanprofile 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 anassumption about the interaction coefficients, if they arenot all 0.If the additivity assumption is violated, how to design anexperiment differently to account for non-additivity of trtand block effects?RCBD with subsamplingYi= µ + αj[i]+ βk[i]+ δj[i]k[i]+ eiwhereµ is a population mean, averaged over all treatments,αjis a fixed trt effect, constrained toPaj=1αj= 0βkis a fixed block effect, k = 1, . . . , b,Pbj=1βj= 0δjk∼ iid N (0, σ2δ) is for variation among samples withinblocks.ei∼ iid N (0, σ2e) is for variation among subsamples.Total of abs observations.ANOVA table and f-test, RCBD with subsamplingSource df SS MS IE(MS)Blocks b − 1 SSBlk MSBlk σ2e+ sσ2δ+ asPbj=1β2kb−1Treatment a − 1 SSTrt MSTrt σ2e+ sσ2δ+ bsPaj=1α2ja−1Plot Error (a − 1)(b − 1) SSPE MSPE σ2e+ sσ2δSubsamp. ab(s − 1) SSSSE MSSSE σ2eTotal abs − 1 SSTotTo test H0: αj= 0 for all j (i.e., no treatment effect), use thefact that under H0,F =MSTrtMSPE∼ Fa−1, (b−1)(a−1).Similarly to CRD with subsampling: we do not use MSSSEat the denominator.We can estimate the overall magnitude of interactioneffects: ˆσ2δ=