ExpDes-1

1
Randomized Block DesignsRBD and RCBD (15.2,
15.5)

• Randomized block designs
• 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
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 1 2 3 ... b 1 y11 y12 y13
... y1b 2 y21 y22 y23 ... y2b ... ... ... ...
... ... t yt1 yt2 yt3 ... 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 Nor
folk 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
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 1 2 3 ... b mean 1 m11 m12
m13 ... m1b m a1 2 m21 m22 m23 ... m2b m
a2 ... ... ... ... ... ... t mt1 mt2 mt3
... mtb m at mean m b1 m b2 m b3 m
bb
8
Model Effects
Linear model
Treatment effects are filtered out from block
effects (show on board)
H0B No block effects b1b2b3...bb 0
H0T No treatment effects a1a2a3...at 0
SAS approach Test with a multiple regression
model with appropriate dummy variables and the F
drop tests.
9
RCBD AOV
Source SS df MS F Treatments SST t-1 MSTSST/(t-1)
MST/MSE Blocks SSB b-1 MSBSSB/(b-1) MSB/MSE Erro
r SSE (b-1)(t-1) MSESSE/(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 -gt ANOVA -gt Two-Way
Must check Fit additive model (no interaction).
12
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
a0.05.
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 L
akeland CaCl 3.31 Lakeland NH4OAc 2.74 Lakelan
d Ca(H2PO4)2 2.32 Lakeland Water 2.35 Leon CaC
l 2.54 Leon NH4OAc 2.09 Leon Ca(H2PO4)2 1.09 L
eon Water 2.70 Chipley CaCl 2.34 Chipley NH4O
Ac 2.07 Chipley Ca(H2PO4)2 4.38 Chipley Water
3.85 Norfolk CaCl 4.71 Norfolk NH4OAc 5.29 Nor
folk Ca(H2PO4)2 5.70 Norfolk Water 4.98 proc
glm datasoils class soil solution model
sulfur soil solution title 'RCBD for Sulfur
extraction across different Florida
Soils' run
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 gt
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 gt 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 gt F Soil
4 33.96488000 8.49122000 10.57
0.0007 Solution 3 1.62121500
0.54040500 0.67 0.5851
15
SPSS Soil
Once the data is input use the following
commands Analyze gt General Linear Model gt
Univariate gt
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
Soils RCBD in R
gt sulf lt-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) gt chem lt-
factor(rep(c("cac","nh4","ca2","h2o"),5)) gt soil
lt- factor(c(rep("Troop",4),rep("Lake",4),rep("Leon
",4),rep("Chip",4),rep("Norf",4))) gt rcbd.fit
aov(sulfsoilchem) gt anova table gt
anova(rcbd.fit) Analysis of Variance
Table Response sulf Df Sum Sq Mean Sq
F value Pr(gtF) 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

18
Profile plot Soils
gt interaction.plot(chem,soil,sulf)
19
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

20
Diagnostics Soils
gt par(mfrowc(2,2)) gt plot(rcbd.fit)

21
Friedmans Test Soils
gt friedman.test(sulf, groupschem, blockssoil)
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 gt tapply(sulf,chem,mean) ca2 cac
h2o nh4 4.116 3.594 3.672 3.324 gt
tapply(sulf,soil,mean) Chip Lake Leon
Norf Troop 3.1600 2.6800 2.1050 5.1700 5.2675