1. Randomized Block Designs: RBD and RCBD (§15.2, 15.5)
Randomized Complete Block Design
Randomized Block Design

2. 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.

3. We will concentrate on 1 and discuss the others.
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.

4. 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 b
1 y11 y12 y y1b
2 y21 y22 y y2b
t yt1 yt2 yt ytb

5. 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
CaCl
NH4OAc
Ca(H2PO4)
Water

6. 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.

7. 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 b mean
1 m11 m12 m m1b m + a1
2 m21 m22 m m2b m + a2
t mt1 mt2 mt mtb m + at
mean m + b1 m + b2 m + b3 m + bb

8. H0B: No block effects: b1=b2=b3=...=bb = 0
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.

9. dfTotal = dfTreatment + dfBlock + dfError
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

10. Sums of Squares - RCBD Expectation under HaT Expectation under HaB
Expectation of MST and MSB under respective null hypotheses is same as E(MSE)

11. Soils Example in MTB Stat -> ANOVA -> Two-Way
Must check “Fit additive model” (no interaction).

12. Soils in MTB: Output Note:
Two-way Analysis of Variance
Analysis of Variance for Sulfur
Source DF SS MS F P
Soil
Solution
Error
Total
Individual 95% CI
Soil Mean
Chipley (-----*------)
Lakeland (------*-----)
Leon (-----*------)
Norfolk (-----*------)
Troop (-----*------)
Solution Mean
Ca(H2PO ( * )
CaCl ( * )
NH4OAc ( * )
Water ( * )
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.

13. Soils in SAS 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 ;
proc glm data=soils;
class soil solution;
model sulfur = soil solution ;
title 'RCBD for Sulfur extraction across
different Florida Soils';
run;
Soils in SAS

14. 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
Error
Corrected Total
R-Square Coeff Var Root MSE Sulfur Mean
Source DF Type I SS Mean Square F Value Pr > F
Soil
Solution
Source DF Type III SS Mean Square F Value Pr > F

15. SPSS Soil Once the data is input use the following commands:
Analyze > General Linear Model > Univariate >
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.

16. SPSS Soils Output

17. > chem <- factor(rep(c("cac","nh4","ca2","h2o"),5))
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 ***
chem
Residuals

18. Profile plot: Soils
> interaction.plot(chem,soil,sulf)

19. 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:

20. > par(mfrow=c(2,2))
> plot(rcbd.fit)
Diagnostics: Soils

21. 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 =
Check group and block means:
> tapply(sulf,chem,mean)
ca2 cac h2o nh4
> tapply(sulf,soil,mean)
Chip Lake Leon Norf Troop