The Big PictureRandomized Block DesignsExampleSeed Germination ExperimentRandomized Complete Block DesignBret LargetDepartments of Botany and of StatisticsUniversity of Wisconsin—MadisonMarch 22, 2007Statistics 572 (Spring 2007) March 22, 2007 1 / 13The Big Picture Randomized Block DesignsThe Big PictureA blocking variable is a categorical variable that is not the primaryvariable of interest where observations within each level ought to behomogeneous except for treatment.In a randomized complete block design, each treatment is applied toindividuals selected at random within each block.In a randomized incomplete block design, treatments are assigned atrandom within blocks, but every treatment may not be represented inevery block.Statistics 572 (Spring 2007) March 22, 2007 2 / 13Example Seed Germination ExperimentSeed Germination ExperimentIn an experiment, four sites were selected where the soil and climateconditions were expected to be very similar within the site.Here we will treat each site as a block.Within each block, five plots were identified.The treatment was applying a seed disinfectant to seeds. There werefour different treatments (brands) plus a control.The researchers planted 100 seeds from a single treatment in eachplot.The response is the number of seeds that germinated.We will also analyze this data using a generalized linear model later inthe semester. The response variable is better modeled as binomialthan as normal.Statistics 572 (Spring 2007) March 22, 2007 3 / 13Example Seed Germination ExperimentDataBlockTreatment 1 2 3 4Control 86 90 88 87Arasan 98 94 93 89Spergon 96 90 91 92Semesan 97 95 91 92Fermate 91 93 95 95Lines show treatmentBlockTreatment 1 2 3 4Control 86 90 88 87Arasan 98 94 93 89Spergon 96 90 91 92Semesan 97 95 91 92Fermate 91 93 95 95Lines show blockingStatistics 572 (Spring 2007) March 22, 2007 4 / 13Example Seed Germination ExperimentCrossed VariablesA categorical variable A iscrossed with a categoricalvariable B if there is one ormore observations for eachcombination of levels of A andB.Here, treatment and blockingare crossed.Both treatment and blocking arenested within the grand mean.Individual error is nested withineach other variable.Block could be random or fixed.Grand MeanTreatment BlockErrorStatistics 572 (Spring 2007) March 22, 2007 5 / 13Example Seed Germination ExperimentModelA model isyij= µ + αi+ βj+ eijwhere:µ is an overall mean,αiis the effect of treatment i wherePiαi= 0, for i = 1, . . . , 5,βjis the effect in block j wherePjβi= 0, for j = 1, . . . , 4,and eij∼ iid N(0, σ2e) is individual random error.Note that µij= µ + αi+ βj.Instead of the sum constraints, we could instead assume α1= β1= 0(the default in R).Statistics 572 (Spring 2007) March 22, 2007 6 / 13Example Seed Germination ExperimentData> seed = read.table("seed.txt", header = T)> attach(seed)> str(seed)'data.frame': 20 obs. of 3 variables:$ count : int 86 90 88 87 98 94 93 89 96 90 ...$ treatment: Factor w/ 5 levels "AControl","Arasan",..: 1 1 1 1 2 2 2 2 5 5 ...$ block : Factor w/ 4 levels "b1","b2","b3",..: 1 2 3 4 1 2 3 4 1 2 ...> seedcount treatment block1 86 AControl b12 90 AControl b23 88 AControl b34 87 AControl b45 98 Arasan b16 94 Arasan b27 93 Arasan b38 89 Arasan b49 96 Spergon b110 90 Spergon b211 91 Spergon b312 92 Spergon b413 97 Semesan b114 95 Semesan b215 91 Semesan b316 92 Semesan b417 91 Fermate b118 93 Fermate b219 95 Fermate b320 95 Fermate b4Statistics 572 (Spring 2007) March 22, 2007 7 / 13Example Seed Germination ExperimentPlot of Data> library(lattice)> fig1 = xyplot(count ~ treatment | block, pch = 16)> print(fig1)The block effect seems prettyweak.treatmentcount86889092949698AControlArasanFermateSemesanSpergon●●●●●b1●●●●●b2●●●●●b3AControlArasanFermateSemesanSpergon86889092949698●●●●●b4Statistics 572 (Spring 2007) March 22, 2007 8 / 13Example Seed Germination ExperimentAlternative Plot of Data> library(lattice)> fig1 = xyplot(count ~ treatment, pch = 16)> print(fig1)It looks like each treatmentimproves germination rate overthe control.The spread is similar withintreatments.treatmentcount86889092949698AControl Arasan Fermate Semesan Spergon●●●●●●●●●●●●●●●●●●●●Statistics 572 (Spring 2007) March 22, 2007 9 / 13Example Seed Germination ExperimentLinear Model Analysis> seed.lm = lm(count ~ treatment + block)> anova(seed.lm)Analysis of Variance TableResponse: countDf 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 7.108---Signif. codes: 0'***'0.001'**'0.01'*'0.05'.'0.1' '1> summary(seed.lm)Call:lm(formula = count ~ treatment + block)Residuals:Min 1Q Median 3Q Max-3.950 -1.113 0.325 1.850 3.050Coefficients:Estimate Std. Error t value Pr(>|t|)(Intercept) 89.200 1.686 52.899 1.37e-15 ***treatmentArasan 5.750 1.885 3.050 0.01008 *treatmentFermate 5.750 1.885 3.050 0.01008 *treatmentSemesan 6.000 1.885 3.183 0.00788 **treatmentSpergon 4.500 1.885 2.387 0.03433 *blockb2 -1.200 1.686 -0.712 0.49028blockb3 -2.000 1.686 -1.186 0.25854blockb4 -2.600 1.686 -1.542 0.14904---Signif. codes: 0'***'0.001'**'0.01'*'0.05'.'0.1' '1Residual standard error: 2.666 on 12 degrees of freedomMultiple R-Squared: 0.587, Adjusted R-squared: 0.3461F-statistic: 2.437 on 7 and 12 DF, p-value: 0.08395Statistics 572 (Spring 2007) March 22, 2007 10 / 13Example Seed Germination ExperimentResidual Plot> plot(fitted(seed.lm), residuals(seed.lm))> abline(h = 0)Residual plot looks okay.Normal approximation tobinomial is still not too bad asthere are sufficient numbers ofsuccesses and failures in eachtreatment group.●●●●●●●●●●●●●●●●●●●●88 90 92 94−4 −3 −2 −1 0 1 2 3fitted(seed.lm)residuals(seed.lm)Statistics 572 (Spring 2007) March 22, 2007 11 / 13Example Seed Germination ExperimentRandom Effects ModelThere are very few differences in a random effects model (treatingblock as random).The variance of the block variable is essentially 0.It is more difficult to carry out hypothesis tests.Statistics 572 (Spring 2007) March 22, 2007 12 / 13Example Seed Germination ExperimentComparison> library(lme4)> seed.lmer = lmer(count ~ treatment + (1 | block))> summary(seed.lmer)Linear mixed-effects model fit by REMLFormula: count ~ treatment + (1 | block)AIC BIC logLik MLdeviance REMLdeviance90.58 96.56 -39.29 89.78 78.58Random effects:Groups Name Variance Std.Dev.block (Intercept)