Biometry 802 Experimental Design

Biometry 802
Experimental Design
S. D. Kachman
Department of Biometry
University of Nebraska–Lincoln
http://www.ianr.unl.edu/ianr/biometry/faculty/steve/802/2000

Randomized Complete Block Designs –
Introduction
Outline
• Blocking
• Design
• Model
• Estimation
• ANOVA
• Testing
Biometry 802 – Experimental Design – Fall 2000 64

Blocking
• In a CRD assignment takes place completely at random
– Each treatment has an equal chance to be assigned to
a more or less favorable material
– Some treatments will be assigned to more favorable
material
– The role of the analysis is to determine how much of
the observed differences can be accounted for by
chance
• Many times it is possible for the researcher to
systematically group the experimental material into
“blocks” of similar material
– Continuous sources
∗ Light, Temperature, and Moisture
– Discrete Sources
∗ Locations, Days, and Operators
• By assigning each treatment to each block the distribution
of experimental material will be more equitable
– Each treatment will receive an equal allocation of
favorable material.
– The ability of the experimenter to group the material will
determine the effectiveness of blocking
– A poor job of blocking will result in a weaker experiment
Biometry 802 – Experimental Design – Fall 2000 65

Design
Divide Experimental Units into subsets
• Subsets are Different
• Experimental Units within subsets are homogeneous
r Blocks
t Treatments
• Each treatment in every block
• Randomly assign treatments to experimental units within
each block
Biometry 802 – Experimental Design – Fall 2000 66

Block 1 Block 2 Block 3 Block 4
Biometry 802 – Experimental Design – Fall 2000 67

Formula
y ij = µ + τ i + β j + e ij
β j ∼ N(0, σ 2 B )
e ij ∼ N(0, σ 2 )
• β j ’s and e ij ’s are independently distributed
Biometry 802 – Experimental Design – Fall 2000 68

Model
• Expected Value
E(y ij ) = µ + τ i
• Dispersion
var(y ij ) = σ 2 + σ 2 B
cov(y ij , y i ′ j ) = σ 2 B
cov(y ij , y i ′ j ′) = 0
• Distribution – Normal
• Treatment differences
– Estimator ȳ 1· − ȳ 2·
– Standard Error ̂σȳ1·−ȳ 2· =
√
MSE 2 r
Biometry 802 – Experimental Design – Fall 2000 69

ANOVA
Source df SS MS EMS F
Block r − 1 SSBlock MSBlock tσ 2 β + σ2
Treat t − 1 SStrt MStrt rκ 2 τ + σ2 F
Error (r − 1)(t − 1) = rt − r − t + 1 SSerror MSerror σ 2
Of course actual numbers would be used in the df, SS, MS, F, and p-value columns.
SSBlock =
∑ r
j=1 y2·j
t
− y2··
rt
SST rt =
SSError =
∑ t
i=1 y2 i·
r
t∑ r∑
i=1
j=1
− y2··
rt
y 2 ij − ∑ t
i=1 y2 i·
r
−
∑ r
j=1 y2·j
t
+ y2··
rt
Biometry 802 – Experimental Design – Fall 2000 70

Testing
• t-tests for single degree of freedom contrasts
• F-test for multiple degree of freedom contrasts
• Use error degrees of freedom from ANOVA
• Multiple comparison procedures and Confidence Intervals
as before
Biometry 802 – Experimental Design – Fall 2000 71

RCBD–Example Emergence Rate
Outline
• Layout
• Design
– Treatment and Experimental
• Formula
• Analysis
• Conclusions
Biometry 802 – Experimental Design – Fall 2000 72

Layout
Comparing 4 treatments of soybeans and an untreated
control on the emergence rate of soybeans. The following
layout was used:
Rep 1 Rep 2 Rep 3 Rep 4 Rep 5
Biometry 802 – Experimental Design – Fall 2000 73

Design
• Treatment Design
– Treatment factor – Seed treatment
– Treatment levels (5) – Check, Arasan, Spergon,
Semesan Jr., and Fermate
• Experimental Design
– The field is located on a slope
– Form blocks based on elevation
– 5 plots at each elevation
– 5 replications of a RCBD
• Data consists of the number of plants that emerge out of a
total 100 that were planted
– Binomial may be a better choice for the distribution
Biometry 802 – Experimental Design – Fall 2000 74

Formula
y ij = µ + τ i + β j + e ij
• y ij emergence rate of seeds treated with treatment i in
Block j
• µ overall average emergence rate
• τ i increase in emergence rate for seeds treated with
treatment i
• β j increase in emergence rate for seeds planted in block j
• e ij increase in emergence rate for using treatment i in
block j
Biometry 802 – Experimental Design – Fall 2000 75

Analysis
Source df MS EMS F p-value
Rep 4 12.46 σ 2 + 5σ 2 β
2.30 .1032
Treat 4 20.96 σ 2 + 5κ 2 τ
3.87 .0219
Cont vs rest 1 67.41 σ 2 + Q 1 12.46 .0028
Rest 3 5.53 σ 2 + Q 2 1.02 .4087
Error 16 5.41 σ 2
V ar(ȳ Ch· − .25 ∑
i≠Ch
y i·) = V ar(ē Ch· − .25 ∑
= σ2
5 + σ2
20
= σ 2 5
20
= σ 21 + 4 1
16
5
i≠Ch
ē i·)
Biometry 802 – Experimental Design – Fall 2000 76

Check versus rest
̂γ ± t .05,16̂σ̂γ
̂γ = 1(10.8) −
̂σ̂γ =
= −4.1
√
= 1.16
t .05,16 = 2.12
−6.5 < γ < −1.65
6.2 + 8.2 + 6.6 + 5.8
4
5.41 12 + 4 × (1/4) 2
5
We are 95% percent confident that on average the
emergence rate of the treated seeds is between 6.5% and
1.7% more than the emergence rate of the untreated control.
Biometry 802 – Experimental Design – Fall 2000 77

Conclusions
• Treating seeds does have an effect on emergence rate
• There was not a significant difference between the four
treatments
• It might be of interest to check if the four treatment are
“equivalent”
Biometry 802 – Experimental Design – Fall 2000 78

Summary RCBD
• CRD versus RCBD
• Building with blocks
• Analysis
Biometry 802 – Experimental Design – Fall 2000 79

CRD versus RCBD
• The biggest advantage of a CRD is its simplicity
• If we do good job constructing blocks, the advantage of a
RCBD is greater precision
• With a large number of treatments it can be difficult to
form effective blocks
• The CRD and RCBD are both examples of experimental
designs
• At the analysis stage: It is the randomization used that
drives what analysis to use
• There are often times when a CRD is a better design
– Forming blocks for the sake of forming blocks is
generally not a good idea
– Blocking incorrectly will make matters worse
Biometry 802 – Experimental Design – Fall 2000 80

Relative efficiency
• Was blocking effective?
• Based on looking at the number of replications needed to
obtain the same standard for a difference
n CRD = 2σ2 CR
δ 2
n RCBD = 2σ2 RCBD
δ 2
n CRD
n RCBD
= σ2 CRD
σ 2 RCBD
• From the RCBD an ̂σ 2 RCBD = MSE
• After some algebra,
̂σ 2 CRD
=
(r − 1)MSBlock + r(t − 1)MSE
rt − 1
• For the emergence rate experiment
re =
(5 − 1)12.46 + 5(5 − 1)5.41
(5 ∗ 5 − 1)5.41
= 1.22
Biometry 802 – Experimental Design – Fall 2000 81

• Therefore to obtain the same precision we would need
about 6 replications of a CRD versus 5 replications of a
RCBD
Biometry 802 – Experimental Design – Fall 2000 82

Building with blocks
• Subjects/Plots/Animals within a block should be as similar
as possible
• Subjects/Plots/Animals is different blocks should be as
different as possible
• Common blocking factors
– Time
– People/Animals/Plants
– Sources of materials
– Locations
– Machines
Biometry 802 – Experimental Design – Fall 2000 83

Analysis
Biometry 802 – Experimental Design – Fall 2000 84

ANOVA
Source df SS MS EMS F
Block r − 1 SSBlock MSBlock tσ 2 β + σ2
Treat t − 1 SStrt MStrt rκ 2 τ + σ2 F
Error (r − 1)(t − 1) = rt − r − t + 1 SSerror MSerror σ 2
Biometry 802 – Experimental Design – Fall 2000 85

Estimates
• As before
Biometry 802 – Experimental Design – Fall 2000 86