Download - U5.2-RandomizedBlockDesigns
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Randomized Block Designs:
Randomized block designs:Randomized Complete Block DesignRandomized Block Design
RBD and RCBD (15.2, 15.5) -
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.
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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.
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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
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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
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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.
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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
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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)
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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!
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Sums of Squares - RCBD
Expectation under HaT
Expectation under HaB
Expectation of MST and MSB under respective null hypotheses is same as E(MSE)
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Soils Example in MTB
Must check Fit additive model (no interaction).
Stat -> ANOVA
-> Two-Way
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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.
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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;
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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
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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.
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SPSS Soils Output
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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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Profile plot: Soils
> interaction.plot(chem,soil,sulf)
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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:
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Diagnostics: Soils
> par(mfrow=c(2,2))
> plot(rcbd.fit)
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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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