u5.2-randomizedblockdesigns.ppt

8/14/2019 U5.2-RandomizedBlockDesigns.ppt

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ExpDes-1

Randomized Block Designs:

RBD and RCBD ( 15.2, 15.5)

Randomized block designs: Randomized Complete Block Design Randomized Block Design


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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 tothe AOV setting.


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ExpDes-3

Analysis of a RBD

Traditional analysis approach is via the linear (regression on indicatorvariables) 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 randomizedcomplete block design (RCBD).

2. A randomized block design with each treatment replicated oncein 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, withreplication in each block).

We will concentrate on 1 and discuss the others.


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ExpDes-4

Single Replicate RCBD

Design: Complete (every treatment occurs in every block) blocklayout with each treatment replicated once in each block(balanced).

Data:

BlockTreatment 1 2 3 ... b1 y 11 y12 y13 ... y1b2 y 21 y22 y23 ... y2b... ... ... ... ... ...t y t1 yt2 yt3 ... ytb


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ExpDes-5

RCBD Soils Example

Design: Complete block layout with each treatment (Solvent)replicated once in each block (Soil type).

Data:

BlockTreatment Troop Lakeland Leon Chipley NorfolkCaCl2 5.07 3.31 2.54 2.34 4.71NH4OAc 4.43 2.74 2.09 2.07 5.29 Ca(H2PO4)2 7.09 2.32 1.09 4.38 5.70Water 4.48 2.35 2.70 3.85 4.98


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ExpDes-6

Minitab

Note: Data must be stacked .From here on out, all statisticspackages will require the data tobe in a stacked structure. Thereis no common unstacked formatfor experimental designs beyondthe CRD.


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ExpDes-7

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

ij jiij y b jt i

1

1

ij jiij y E )( BlockTreatment 1 2 3 ... b mean1 11 12 13 ... 1b 12 21 22 23 ... 2b 2... ... ... ... ... ...t t1 t2 t3 ... tb t mean 1 2 3 b

ii

i

i

0

0

constraintstreatment i effectw.r.t. grand mean

block j effect w.r.t.grand mean


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ExpDes-9

RCBD AOV

Source SS df MS FTreatments SST t-1 MST=SST/(t-1) MST/MSEBlocks SSB b-1 MSB=SSB/(b-1) MSB/MSEError 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

df Total = df Treatment + df Block + df Error

Regression Sums of Squares

Usually not of interest! Assessed only todetermine if blocking wassuccessful in reducingthe variability in the

experimental units. Thisis how/why blockingreduces MSE!


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ExpDes-10

Sums of Squares - RCBD

TSS y y

SST b y y

SSB t y y

SSE y y y y

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SSBSST TSS SSE bt

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MSE E

t MSB E b MST E

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T Expectation under H a T Expectation under Ha B

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t i i

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Expectation of MST andMSB under respective

null hypotheses is sameas E(MSE)


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ExpDes-11

Soils Example in MTB

Must check Fitadditive model(no interaction).

Stat -> ANOVA

-> Two-Way


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ExpDes-12

Soils in MTB: OutputTwo-way Analysis of Variance

Analysis of Variance for SulfurSource 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 whichfactor is the block, thecomputer doesnt knowor care. It simply doessums of squarescomputations.

Conclusion:Block effect is

significant.Treatment effect is

not statisticallysignificant at

=0.05.


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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 2.09

Leon 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 5.29

Norfolk Ca(H2PO4)2 5.70Norfolk Water 4.98;

proc glm data=soils;class soil solution;model sulfur = soil solution ;title 'RCBD for Sulfur extraction acrossdifferent Florida Soils';

run ;


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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 0.80346333

Corrected 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


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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. Solutionwould always be a fixed effect. In somescenarios Soil might be a Random factor(see the Mixed model chapter)

We do a custom model because we only canestimate the main effects of this model andSPSS by default will attempt to estimate theinteraction terms.


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ExpDes-16

SPSS Soils Output


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ExpDes-17

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


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ExpDes-18

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


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ExpDes-19

Nonparametric Analysis of RCBD: Friedmans Test

The RCBD, as in CRD, requires the usual AOV assumptions for theresiduals: Independence; Homoscedasticity; Normality.

When the normality assumption fails, and transformations dont seemto help, Friedmans Test is a nonparametric alternative for the RCBD,

just as Kruskal-Wallis was for the CRD. For example: ratings by apanel of judges (ordinal data).

The procedure is based on ranks (see 15.5 in book), and leads tocalculation of FR statistic.

For large samples, we reject H 0 of equal population medians when:2

1, t FR


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ExpDes-20

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


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ExpDes-21

Friedmans Test: Soils

> friedman.test(sulf, groups=chem, blocks=soil)

Friedman rank sum test

data: sulf, chem and soilFriedman 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

u5.2-randomizedblockdesigns.ppt

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