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

Randomized Block Designs:
RBD and RCBD (§15.2, 15.5)
• Randomized block designs:
– Randomized Complete Block Design
– Randomized Block Design
ExpDes-1
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-2
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-3
Single Replicate RCBD
Design: Complete (every treatment occurs in every block) block
layout with each treatment replicated once in each block
(balanced).
Data:
Treatment
1
2
...
t
1
y11
y21
...
yt1
2
y12
y22
...
yt2
Block
3
y13
y23
...
yt3
...
...
...
...
...
b
y1b
y2b
...
ytb
ExpDes-4
RCBD Soils Example
Design: Complete block layout with each treatment (Solvent)
replicated once in each block (Soil type).
Data:
Treatment
CaCl2
NH4OAc
Ca(H2PO4)2
Water
Block
Troop Lakeland Leon Chipley Norfolk
5.07
3.31
2.54
2.34
4.71
4.43
2.74
2.09
2.07
5.29
7.09
2.32
1.09
4.38
5.70
4.48
2.35
2.70
3.85
4.98
ExpDes-5
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-6
Linear Model: A Two-Factor (Two-Way) AOV
i  1 t
yij    i   j   ij
j  1 b
constraints
treatment i effect
w.r.t. grand mean


block j effect w.r.t.
grand mean
E( yij )    i   j  ij
Treatment
1
2
...
t
mean
1
11
21
...
t1
  1
2
12
22
...
t2
  2
i
0
i
0
i
Block
3
13
23
...
t3
  3
i
...
...
...
...
...
b
1b
2b
...
tb
  b
mean
  1
  2
  t
ExpDes-7
Model Effects
yij    i   j   ij
Linear model
Treatment effects are filtered out from block effects (show on board…)
E( y1  y2 )  (  1 )  (   2 )  1   2
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.
ExpDes-8
RCBD AOV
Source
Treatments
Blocks
Error
Totals
SS
SST
SSB
SSE
TSS
df
t-1
b-1
(b-1)(t-1)
bt-1
MS
F
MST=SST/(t-1)
MST/MSE
MSB=SSB/(b-1) MSB/MSE
MSE=SSE/(b-1)(t-1)
Partitioning of the total sums of squares (TSS)
TSS = SST + SSB + SSE
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!
Regression Sums of Squares
dfTotal = dfTreatment + dfBlock + dfError
ExpDes-9
Sums of Squares - RCBD
t
b
TSS    ( y ij  y  )
i 1 j 1
t
SST  b  ( y i   y  )
i 1
b
SSB  t  ( y  j  y  )
j 1
t b
2
t
b
y
TSS   y ij 
bt
i 1 j 1
t
yi2 y
SST  

bt
i 1 b
b y2
y 
j
SSB  

bt
j 1 t
SSE  TSS  SST  SSB
2
2
2
SSE    ( y ij  y i   y  j  y  )
2
i 1 j 1
Expectation under HaT
Expectation under HaB
Expectation of MST and
MSB under respective
null hypotheses is same
as E(MSE)
E ( MST)   2  bT
E ( MSB)   2  t B
E ( MSE)   2
T 
B 
2

 i
i
t 1

2
j
j
b 1
ExpDes-10
Soils Example in MTB
Stat -> ANOVA
-> Two-Way
Must check “Fit
additive model”
(no interaction).
ExpDes-11
Soils in MTB: Output
Two-way Analysis of Variance
Analysis of Variance for Sulfur
Source
DF
SS
MS
Soil
4
33.965
8.491
Solution
3
1.621
0.540
Error
12
9.642
0.803
Total
19
45.228
Soil
Chipley
Lakeland
Leon
Norfolk
Troop
Mean
3.16
2.68
2.10
5.17
5.27
Solution
Ca(H2PO4
CaCl
NH4OAc
Water
Mean
4.12
3.59
3.32
3.67
F
10.57
0.67
P
0.001
0.585
Individual 95% CI
---+---------+---------+---------+-------(-----*------)
(------*-----)
(-----*------)
(-----*------)
(-----*------)
---+---------+---------+---------+-------1.50
3.00
4.50
6.00
Individual 95% CI
-----+---------+---------+---------+-----(------------*-----------)
(-----------*------------)
(-----------*------------)
(-----------*------------)
-----+---------+---------+---------+-----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
a=0.05.
ExpDes-12
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
Lakeland
CaCl
3.31
Lakeland
NH4OAc
2.74
Lakeland
Ca(H2PO4)2
2.32
Lakeland
Water
2.35
Leon
CaCl
2.54
Leon
NH4OAc
2.09
Leon
Ca(H2PO4)2
1.09
Leon
Water
2.70
Chipley
CaCl
2.34
Chipley
NH4OAc
2.07
Chipley
Ca(H2PO4)2
4.38
Chipley
Water
3.85
Norfolk
CaCl
4.71
Norfolk
NH4OAc
5.29
Norfolk
Ca(H2PO4)2
5.70
Norfolk
Water
4.98
;
proc glm data=soils;
class soil solution;
model sulfur = soil solution ;
title 'RCBD for Sulfur extraction across
different Florida Soils';
ExpDes-13
run;
SAS Output: Soils
RCBD for Sulfur extraction across different Florida Soils
The GLM Procedure
Dependent Variable: Sulfur
Source
Model
Error
Corrected Total
DF
7
12
19
R-Square
0.786822
Sum of
Squares
35.58609500
9.64156000
45.22765500
Coeff Var
24.38083
Mean Square
5.08372786
0.80346333
Root MSE
0.896361
F Value
6.33
Pr > F
0.0028
Sulfur Mean
3.676500
Source
Soil
Solution
DF
4
3
Type I SS
33.96488000
1.62121500
Mean Square
8.49122000
0.54040500
F Value
10.57
0.67
Pr > F
0.0007
0.5851
Source
DF
Type III SS
Mean Square
F Value
Pr > F
4
3
33.96488000
1.62121500
8.49122000
0.54040500
10.57
0.67
0.0007
0.5851
Soil
Solution
ExpDes-14
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-15
SPSS Soils Output
ExpDes-16
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-17
Profile plot: Soils
> interaction.plot(chem,soil,sulf)
ExpDes-18
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:
FR  2,t 1
ExpDes-19
Diagnostics: Soils
> par(mfrow=c(2,2))
> plot(rcbd.fit)
ExpDes-20
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
ExpDes-21