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

Randomized block designs:Randomized Complete Block DesignRandomized 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

Treatment123...b

1y11 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

TreatmentTroopLakelandLeonChipleyNorfolk

CaCl25.07 3.312.54 2.344.71

NH4OAc4.43 2.742.09 2.075.29

Ca(H2PO4)27.092.321.094.385.70

Water4.482.35 2.703.854.98

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

Block

Treatment123...bmean

1m11 m12 m13 ... m1bm + a1

2 m21 m22 m23 ...m2b m + a2

..................

t mt1 mt2 mt3 ...mtb m + at

mean m + b1 m + b2 m + b3 m + bb

constraints

treatment i effect w.r.t. grand mean

block j effect w.r.t. grand mean

Model Effects

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.

Linear model

Treatment effects are filtered out from block effects (show on board)

RCBD AOV

SourceSSdfMSF

TreatmentsSSTt-1MST=SST/(t-1)MST/MSE

BlocksSSBb-1MSB=SSB/(b-1)MSB/MSE

ErrorSSE(b-1)(t-1)MSE=SSE/(b-1)(t-1)

TotalsTSSbt-1

Partitioning of the total sums of squares (TSS)

TSS = SST + SSB + SSE

dfTotal = dfTreatment + dfBlock + dfError

Regression Sums of Squares

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!

Sums of Squares - RCBD

Expectation under HaT

Expectation under HaB

Expectation of MST and MSB under respective null hypotheses is same as E(MSE)

Soils Example in MTB

Must check Fit additive model (no interaction).

Stat -> ANOVA

-> Two-Way

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 doesnt 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.

Soils in SAS

data soils;

input Soil $ Solution $ Sulfur;

datalines;

TroopCaCl5.07

TroopNH4OAc4.43

TroopCa(H2PO4)27.09

TroopWater4.48

LakelandCaCl3.31

LakelandNH4OAc2.74

LakelandCa(H2PO4)22.32

LakelandWater2.35

LeonCaCl2.54

LeonNH4OAc2.09

LeonCa(H2PO4)21.09

LeonWater2.70

ChipleyCaCl2.34

ChipleyNH4OAc2.07

ChipleyCa(H2PO4)24.38

ChipleyWater3.85

NorfolkCaCl4.71

NorfolkNH4OAc5.29

NorfolkCa(H2PO4)25.70

NorfolkWater4.98

;

proc glm data=soils;

class soil solution;

model sulfur = soil solution ;

title 'RCBD for Sulfur extraction across

different Florida Soils';

run;

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

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.

SPSS Soils Output

Soils RCBD in R

> sulf chem soil 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

Profile plot: Soils

> interaction.plot(chem,soil,sulf)

Nonparametric Analysis of RCBD: Friedmans Test

The RCBD, as in CRD, requires the usual AOV assumptions for the residuals:

Independence; Homoscedasticity; Normality.

When the normality assumption fails, and transformations dont seem to help, Friedmans 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:

Diagnostics: Soils

> par(mfrow=c(2,2))

> plot(rcbd.fit)

Friedmans 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

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