Transcript Randomized Block Design

Control of Experimental Error
 Blocking – A block is a group of homogeneous experimental
units
– Maximize the variation among blocks in order to
minimize the variation within blocks
 Reasons for blocking
– To remove block to block variation from the
experimental error (increase precision)
– Treatment comparisons are more uniform
– Increase the information by allowing the researcher
to sample a wider range of conditions
Blocking
 At least one replication is grouped in a
homogeneous area
A B D A
A B D C
C D B C
C D B A
B A D C
B A D C
Just replication
Blocking
Criteria for blocking
 Proximity or known patterns of variation in the field
– gradients due to fertility, soil type
– animals (experimental units) in a pen (block)
 Time
– planting, harvesting
 Management of experimental tasks
– individuals collecting data
– runs in the laboratory
 Physical characteristics
– height, maturity
 Natural groupings
– branches (experimental units) on a tree (block)
Randomized Block Design
 Experimental units are first classified into groups (or
blocks) of plots that are as nearly alike as possible
 Linear Model: Yij =  + i +
– =
mean effect
– βi =
– j =
– ij =
j + ij
ith block effect
jth treatment effect
treatment x block interaction, treated as error
 Each treatment occurs in each block, the same number of
times (usually once)
– Also known as the Randomized Complete Block Design
– RBD = RCB = RCBD
 Minimize the variation within blocks - Maximize the
variation between blocks
Pretty doesn’t count here
Randomized Block Design
Other ways to minimize variation within blocks:
 Field operations should be completed in one
block before moving to another
 If plot management or data collection is handled
by more than one person, assign each to a
different block
Advantages of the RBD
 Can remove site variation from experimental error and
thus increase precision
 When an operation cannot be completed on all plots at
one time, can be used to remove variation between runs
 By placing blocks under different conditions, it can
broaden the scope of the trial
 Can accommodate any number of treatments and any
number of blocks, but each treatment must be replicated
the same number of times in each block
 Statistical analysis is fairly simple
Disadvantages of the RBD
 Missing data can cause some difficulty in the analysis
 Assignment of treatments by mistake to the wrong block
can lead to problems in the analysis
 If there is more than one source of unwanted variation, the
design is less efficient
 If the plots are uniform, then RBD is less efficient than
CRD
 As treatment or entry numbers increase, more
heterogeneous area is introduced and effective blocking
becomes more difficult. Split plot or lattice designs may be
better suited.
Uses of the RBD
 When you have one source of unwanted
variation
 Estimates the amount of variation due to
the blocking factor
Randomization in an RBD
 Each treatment occurs once in each block
 Assign treatments at random to plots within
each block
 Use a different randomization for each block
Analysis of the RBD
 Construct a two-way table of the means and
deviations for each block and each treatment level
 Compute the ANOVA table
 Conduct significance tests
 Calculate means and standard errors
 Compute additional statistics if appropriate:
– Confidence intervals
– Comparisons of means
– CV
The RBD ANOVA
Source
df
SS
MS
Total
rt-1
SSTot =

 i  j Yij  Y
Block
r-1
SSB =

t i Yi  Y
Treatment t-1

SST =

r j Y j  Y
Error
(r-1)(t-1)

2
MSB =
2

F
MSB/MSE
SSB/(r-1)
2
SSE =
SSTot-SSB-SST
MST =
SST/(t-1)
MSE =
SSE/(r-1)(t-1)
MSE is the divisor for all F ratios
MST/MSE
Means and Standard Errors
s Y  MSE r
Standard Error of a treatment mean
Confidence interval estimate
L   i   Y i  t  MSE r
Standard Error of a difference
s Y  Y   2MSE r
1
2
Confidence interval estimate on a difference
L   1   2    Y1  Y 2   t  2MSE r
t to test difference between two means
Y
1  Y2
t
2MSE r
Numerical Example
 Test the effect of different sources of nitrogen on
the yield of barley:
– 5 sources and a control
 Wanted to apply the results over a wide range of
conditions so the trial was conducted on four
types of soil
– Soil type is the blocking factor
 Located six plots at random on each of the four
soil types
ANOVA
Source
df
Total
23
492.36
Soils (Block)
3
192.56
64.19
21.61**
Fertilizer (Trt)
5
255.28
51.06
17.19**
15
44.52
2.97
Error
SS
MS
F
Source (NH4)2SO4 NH4NO3 CO(NH2)2 Ca(NO3)2 NaNO3 Control
Mean
36.25
32.38
29.42
31.02
30.70
25.35
Standard error of a treatment mean = 0.86
CV = 5.6%
Standard error of a difference between two treatment means = 1.22
Confidence Interval Estimates
40
38
36
34
32
30
28
26
24
22
(NH4)2SO4 NH4NO3
Ca(NO3)2
NaNO3
CO(NO2)2
Control
34.41
30.54
29.19
28.86
27.59
23.51
36.25
32.38
31.02
30.70
29.42
25.35
38.09
34.21
32.86
32.54
31.26
27.19
Report of Analysis
 Differences among sources of nitrogen were highly
significant
 Ammonium sulfate (NH4)2SO4 produced the highest mean
yield and CO(NH2)2 produced the lowest
 When no nitrogen was added, the yield was only 25.35
kg/plot
 Blocking on soil type was effective as evidenced by:
– large F for Soils (Blocks)
– small coefficient of variation (5.6%) for the trial
Is This Experiment Valid?
Missing Plots
 If only one plot is missing, you
can use the following formula:
Yij = ( rBi + tTj - G)/[(r-1)(t-1)]



Where:
• Bi = sum of remaining observations in the ith block
• Tj = sum of remaining observations in the jth treatment
• G = grand total of the available observations
• t, r= number of treatments, blocks, respectively
Total and error df must be reduced by 1
Used only to obtain a valid ANOVA
- No change in Error SS
- SS for treatments may be biased upwards
Two or Three Missing Plots
^
Yij = ( rBi + tTj - G)/[(r-1)(t-1)]
 Estimate all but one of the missing values and use the formula
 Use this value and all but one of the remaining guessed values
and calculate again; continue in this manner until you have
resolved all missing plots
 You lose one error degree of freedom for each substituted value
 Better approach: Let SAS account for missing values
– Use a procedure that can accommodate missing values (PROC
GLM, PROC MIXED)
– Use adjusted means (LSMEANS) rather than MEANS
– degrees of freedom are subtracted automatically for each missing
observation
Relative Efficiency
 A way to measure the efficiency of RBD vs CRD
RE = [(r-1)MSB + r(t-1)MSE]/(rt-1)MSE
MSECRD
RE 
MSERBD
Estimated Error for a CRD
Observed Error for RBD

r, t = number of blocks, treatments in the RBD

MSB, MSE = block, error mean squares from the RBD

If RE > 1, RBD was more efficient

(RE - 1)100 = % increase in efficiency

r(RE) = number of replications that would be required in
the CRD to obtain the same level of precision