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## Randomized Block Designs: RBD and RCBD ( § 15.2, 15.5)

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**Randomized Block Designs:RBD and RCBD (§15.2, 15.5)**• Randomized block designs: • Randomized Complete Block Design • Randomized Block Design**Randomization in Blocked Designs**• For all one blocking classification designs: • Randomization of treatments to experimental units takes place within each block. • A separate randomization is required for each block. • The design is said to have one restriction on randomization. A completely randomized design requires only one randomization. Note: The randomized block design generalizes the paired t-test to the AOV setting.**Analysis of a RBD**Traditional analysis approach is via the linear (regression on indicator variables) model and AOV. • A RBD can occur in a number of situations: • A randomized block design with each treatment replicated once in each block (balanced and complete). This is a randomized complete block design(RCBD). • A randomized block design with each treatment replicated once in a block but with one block/treatment combination missing. (incomplete). • A randomized block design with each treatment replicated two or more times in each block (balanced and complete, with replication in each block). We will concentrate on 1 and discuss the others.**Single Replicate RCBD**Design: Complete (every treatment occurs in every block) block layout with each treatment replicated once in each block (balanced). Data: Block Treatment 1 2 3 ... b 1 y11 y12 y13 ...y1b 2 y21 y22 y23 ...y2b ... ... ... ... ... ... t yt1 yt2 yt3 ...ytb**RCBD Soils Example**Design: Complete block layout with each treatment (Solvent) replicated once in each block (Soil type). Data: Block Treatment Troop Lakeland Leon Chipley Norfolk CaCl2 5.07 3.312.542.344.71 NH4OAc 4.432.742.092.075.29 Ca(H2PO4)2 7.09 2.32 1.09 4.38 5.70 Water 4.482.352.703.854.98**Minitab**Note: Data must be stacked. From here on out, all statistics packages will require the data to be in a stacked structure. There is no common unstacked format for experimental designs beyond the CRD.**Linear Model: A Two-Factor (Two-Way) AOV**constraints treatment i effect w.r.t. grand mean block j effect w.r.t. grand mean Block Treatment 1 2 3 ... b mean 1 m11 m12 m13 ...m1b m + a1 2 m21 m22 m23 ...m2bm + a2 ... ... ... ... ... ... t mt1mt2mt3...mtb m + at mean m + b1 m + b2 m + b3 m + bb**Model Effects**Linear model Treatment effects are filtered out from block effects (show on board…) H0B: No block effects: b1=b2=b3=...=bb = 0 H0T: No treatment effects: a1=a2=a3=...=at = 0 SAS approach: Test with a multiple regression model with appropriate dummy variables and the F drop tests.**RCBD AOV**Source SS df MS F Treatments SST t-1 MST=SST/(t-1) MST/MSE Blocks SSB b-1 MSB=SSB/(b-1) MSB/MSE Error SSE (b-1)(t-1) MSE=SSE/(b-1)(t-1) Totals TSS bt-1 Usually not of interest! Assessed only to determine if blocking was successful in reducing the variability in the experimental units. This is how/why blocking reduces MSE! Partitioning of the total sums of squares (TSS) TSS = SST + SSB + SSE Regression Sums of Squares dfTotal = dfTreatment + dfBlock + dfError**Sums of Squares - RCBD**Expectation under HaT Expectation underHaB Expectation of MST and MSB under respective null hypotheses is same as E(MSE)**Soils Example in MTB**Stat -> ANOVA -> Two-Way Must check “Fit additive model” (no interaction).**Soils in MTB: Output**Two-way Analysis of Variance Analysis of Variance for Sulfur Source DF SS MS F P Soil 4 33.965 8.491 10.57 0.001 Solution 3 1.621 0.540 0.67 0.585 Error 12 9.642 0.803 Total 19 45.228 Individual 95% CI Soil Mean ---+---------+---------+---------+-------- Chipley 3.16 (-----*------) Lakeland 2.68 (------*-----) Leon 2.10 (-----*------) Norfolk 5.17 (-----*------) Troop 5.27 (-----*------) ---+---------+---------+---------+-------- 1.50 3.00 4.50 6.00 Individual 95% CI Solution Mean -----+---------+---------+---------+------ Ca(H2PO4 4.12 (------------*-----------) CaCl 3.59 (-----------*------------) NH4OAc 3.32 (-----------*------------) Water 3.67 (-----------*------------) -----+---------+---------+---------+------ 2.80 3.50 4.20 4.90 Note: You must know which factor is the block, the computer doesn’t know or care. It simply does sums of squares computations. Conclusion: Block effect is significant. Treatment effect is not statistically significant at a=0.05.**data soils;**input Soil $ Solution $ Sulfur; datalines; Troop CaCl 5.07 Troop NH4OAc 4.43 Troop Ca(H2PO4)2 7.09 Troop Water 4.48 Lakeland CaCl 3.31 Lakeland NH4OAc 2.74 Lakeland Ca(H2PO4)2 2.32 Lakeland Water 2.35 Leon CaCl 2.54 Leon NH4OAc 2.09 Leon Ca(H2PO4)2 1.09 Leon Water 2.70 Chipley CaCl 2.34 Chipley NH4OAc 2.07 Chipley Ca(H2PO4)2 4.38 Chipley Water 3.85 Norfolk CaCl 4.71 Norfolk NH4OAc 5.29 Norfolk Ca(H2PO4)2 5.70 Norfolk Water 4.98 ; procglm data=soils; class soil solution; model sulfur = soil solution ; title 'RCBD for Sulfur extraction across different Florida Soils'; run; Soils in SAS**SAS Output: Soils**RCBD for Sulfur extraction across different Florida Soils The GLM Procedure Dependent Variable: Sulfur Sum of Source DF Squares Mean Square F Value Pr > F Model 7 35.58609500 5.08372786 6.33 0.0028 Error 12 9.64156000 0.80346333 Corrected Total 19 45.22765500 R-Square Coeff Var Root MSE Sulfur Mean 0.786822 24.38083 0.896361 3.676500 Source DF Type I SS Mean Square F Value Pr > F Soil 4 33.96488000 8.49122000 10.57 0.0007 Solution 3 1.62121500 0.54040500 0.67 0.5851 Source DF Type III SS Mean Square F Value Pr > F Soil 4 33.96488000 8.49122000 10.57 0.0007 Solution 3 1.62121500 0.54040500 0.67 0.5851**Once the data is input use the following commands:**Analyze > General Linear Model > Univariate > SPSS Soil Sulfur is the response (dependent variable) Both Solution and Soil are factors. Solution would always be a fixed effect. In some scenarios Soil might be a Random factor (see the Mixed model chapter) We do a custom model because we only can estimate the main effects of this model and SPSS by default will attempt to estimate the interaction terms.**Soils RCBD in R**> sulf <-c(5.07,4.43,7.09,4.48,3.31,2.74,2.32,2.35,2.54,2.09,1.09,2.70,2.34, 2.07,4.38,3.85,4.71,5.29,5.70,4.98) > chem <- factor(rep(c("cac","nh4","ca2","h2o"),5)) > soil <- factor(c(rep("Troop",4),rep("Lake",4),rep("Leon",4),rep("Chip",4),rep("Norf",4))) > rcbd.fit = aov(sulf~soil+chem) > # anova table > anova(rcbd.fit) Analysis of Variance Table Response: sulf Df Sum Sq Mean Sq F value Pr(>F) soil 4 33.965 8.491 10.5683 0.0006629 *** chem 3 1.621 0.540 0.6726 0.5851298 Residuals 12 9.642 0.803**> interaction.plot(chem,soil,sulf)**Profile plot: Soils**Nonparametric Analysis of RCBD: Friedman’s Test**• The RCBD, as in CRD, requires the usual AOV assumptions for the residuals: • Independence; • Homoscedasticity; • Normality. • When the normality assumption fails, and transformations don’t seem to help, Friedman’s Test is a nonparametric alternative for the RCBD, just as Kruskal-Wallis was for the CRD. For example: ratings by a panel of judges (ordinal data). • The procedure is based on ranks (see §15.5 in book), and leads to calculation of FR statistic. • For large samples, we reject H0 of equal population medians when:**> par(mfrow=c(2,2))**> plot(rcbd.fit) Diagnostics: Soils**Friedman’s Test: Soils**> friedman.test(sulf, groups=chem, blocks=soil) Friedman rank sum test data: sulf, chem and soil Friedman chi-squared = 1.08, df = 3, p-value = 0.7819 Check group and block means: > tapply(sulf,chem,mean) ca2 cac h2o nh4 4.116 3.594 3.672 3.324 > tapply(sulf,soil,mean) Chip Lake Leon Norf Troop 3.1600 2.6800 2.1050 5.1700 5.2675