ExpDes-1

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

• Randomized block designs:– Randomized Complete Block Design– Randomized Block Design

ExpDes-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.

ExpDes-3

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:

1. A randomized block design with each treatment replicated once

in each block (balanced and complete). This is a randomized

complete block design (RCBD).

2. A randomized block design with each treatment replicated once

in a block but with one block/treatment combination missing.

(incomplete).

3. 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.

ExpDes-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 1 2 3 ... b

1 y11 y12 y13 ... y1b

2 y21 y22 y23 ... y2b

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

t yt1 yt2 yt3 ... ytb

ExpDes-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

CaCl2 5.07 3.31 2.54 2.34 4.71

NH4OAc 4.43 2.74 2.09 2.07 5.29

Ca(H2PO4)2 7.09 2.32 1.09 4.38 5.70

Water 4.48 2.35 2.70 3.85 4.98

ExpDes-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.

ExpDes-7

Linear Model: A Two-Factor (Two-Way) AOV

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treatment i effect w.r.t. grand mean

block j effect w.r.t. grand mean

ExpDes-8

Model Effects

ijjiijy

H0B: No block effects: 1=2=3=...=b = 0

H0T: No treatment effects: 1=2=3=...=t = 0

SAS approach: Test with a multiple regression model with appropriate dummy variables and the F drop tests.

212121 )()()( yyE

Linear model

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

ExpDes-9

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

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!

ExpDes-10

Sums of Squares - RCBD

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

ExpDes-11

Soils Example in MTB

Must check “Fit additive model” (no interaction).

Stat -> ANOVA

-> Two-Way

ExpDes-12

Soils in MTB: Output

Two-way Analysis of Variance

Analysis of Variance for Sulfur Source DF SS MS F PSoil 4 33.965 8.491 10.57 0.001Solution 3 1.621 0.540 0.67 0.585Error 12 9.642 0.803Total 19 45.228

Individual 95% CISoil 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% CISolution 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 =0.05.

ExpDes-13

Soils in SAS

data soils;input Soil $ Solution $ Sulfur;datalines;Troop CaCl 5.07Troop NH4OAc 4.43Troop Ca(H2PO4)2 7.09Troop Water 4.48Lakeland CaCl 3.31Lakeland NH4OAc 2.74Lakeland Ca(H2PO4)2 2.32Lakeland Water 2.35Leon CaCl 2.54Leon NH4OAc 2.09Leon Ca(H2PO4)2 1.09Leon Water 2.70Chipley CaCl 2.34Chipley NH4OAc 2.07Chipley Ca(H2PO4)2 4.38Chipley Water 3.85Norfolk CaCl 4.71Norfolk NH4OAc 5.29Norfolk Ca(H2PO4)2 5.70Norfolk Water 4.98;proc glm data=soils; class soil solution; model sulfur = soil solution ; title 'RCBD for Sulfur extraction across different Florida Soils';run;

ExpDes-14

RCBD for Sulfur extraction across different Florida Soils

The GLM ProcedureDependent Variable: Sulfur

Sum ofSource DF Squares Mean Square F Value Pr > FModel 7 35.58609500 5.08372786 6.33 0.0028Error 12 9.64156000 0.80346333Corrected Total 19 45.22765500

R-Square Coeff Var Root MSE Sulfur Mean0.786822 24.38083 0.896361 3.676500

Source DF Type I SS Mean Square F Value Pr > FSoil 4 33.96488000 8.49122000 10.57 0.0007Solution 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 0.0007Solution 3 1.62121500 0.54040500 0.67 0.5851

SAS Output: Soils

ExpDes-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.

ExpDes-16

SPSS Soils Output

ExpDes-17

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

ExpDes-18

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

ExpDes-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:2

1, tFR

ExpDes-20

Diagnostics: Soils

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

> plot(rcbd.fit)

ExpDes-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 = 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