ExpDes1

1
More Experimental Design ConceptsCRD and
RCBD(Chapter 15)

• Experimental Design Basics.
• Completely randomized design revisited.
• Accounting for more than one factor.
• Blocking
• Randomized Complete Block Design
• Randomized Block Design

2
Experimental Error

• Experimental error is the variation in the
  responses among experimental units (e.u.s) which
  are assigned the same treatment, and are observed
  under the same experimental conditions. It is
  measured by SSE (or MSE).
• Ideally, we would like experimental error to be
  zero! This is impossible because of (at least)
  one or more of the following reasons
• There are inherent differences in the e.u.s
  before they receive treatments.
• There is variation in the devices that record the
  measurements.
• There is variation in applying or setting the
  treatments.
• There are extraneous factors other than the
  treatments which affect the response.
• Control over the magnitude of experimental error
  can be achieved by
• Careful choice of e.u.s
• Taking care with experimental procedures and
  recording of measurements.
• Blocking of e.u.s.
• Choice of experimental design used.
• Using covariates (variables that are related to
  the response).

3
Review Completely Randomized Design (one-factor
design)

• Experimental units are relatively homogeneous.
• Experiment will use very few replicates.
• Treatments are assigned to experimental units at
  random.
• Each treatment is replicated the same number of
  times (balanced design).
• No accommodation made for disturbing variables
  (extraneous sources of variation).
• High probability that a large fraction of the
  experimental units set out at the beginning of
  the study may be lost or unavailable for
  measurement at the appropriate time.

4
Completely Randomized Design

• Experimental Design - Completely randomized
  design (CRD)
• Sampling Design - One-way classification
  design

• Assumptions
• Independent random samples (results of one sample
  do not effect other samples).
• Samples from normal population(s).
• Mean and variance for population i are
  respectively, mi and s2.

Model
AOV model
random error N(0,s2)
Requirement for m to be the overall mean
overall mean
effect due to population i
5
Reference Group Model
Model
This is the model SAS, SPSS and most other
packages use.
random error N(0,s2)
reference group mean
effect due to population i
Mean for the last group (it) is mt. Mean for the
first group (i1) is mt b1 Thus, b1 is the
difference between the mean of the reference
group (cell) and the target group mean. Any
group can be the reference group.
6
Some Practical Problems/Limitations

• In many situations, the researcher
• Does not have sufficient homogeneous experimental
  material in one group (location, batch, etc) to
  effectively use the CRD (i.e. resource
  constraints)
• The study objectives require examining treatments
  over a broad range of experimental units in order
  that results can be extended to more situations
  (i.e. breadth of study objectives).
• The experimental material must be grouped for
  administrative or implementation purposes (i.e.
  implementation constraints).

If the researcher knows something about the
characteristics of the experimental material, it
is often possible to group experimental units
into sets of relatively homogenous material
(blocks), and then compare treatment level means
within these groups.
7
Example

• A scientist was interested in the use of three
  chemicals and water on their effectiveness in
  extracting sulfur from Florida soils. The
  chemicals of interest are
• Calcium Chloride CaCl2
• Ammonium Acetate NH4OAc
• Mono Calcium Phosphate Ca(H2PO4)3
• Water H2O
• Five soils were chosen for this experiment
• Troup Jackson Co. Paleudults soil
• Lakeland Walton Co. Quartzipsamments soil
• Leon Duval Co. Haplaquads soil
• Chipley Jackson Co. Quartzipsamments soil
• Norfolk Alachua Co. Paleudults soil

8
Blocking and Control of Extraneous Variation
The main interest in the experiment is the
comparison of the four extraction methods.

• The variation imposed on the extraction
  procedure by the five different soil types
  represents a source of extraneous variation.
• Unless controlled for in the experiment, this
  variation has the potential to swamp or
  overwhelm the differences among the extraction
  procedure.
• High probability of concluding there are no
  treatment effects when treatment effects are in
  fact present.

Fair comparisons only occur among extractions
within a soil type.
We wish to use the combined experience across
soil types to make a stronger statement about the
extraction procedures.
9
Graphical View
Note The pattern of responses to treatments is
consistent within a given soil type (a block)
but responses vary across soil types.
10
Randomized Block Design (RBD)
Any experimental design in which the
randomization of treatments is restricted to
groups of experimental units within a predefined
block of units assumed to be internally
homogeneous is called a randomized block design.
Blocks of units are created to control known
sources of variation in expected (mean) response
among experimental units. There are two
classifications or factors in an RBD block
effects and treatment effects.

• Rules for blocking
• Carefully examine the situation at hand and
  identify those factors which are known to affect
  the proposed response.
• Choose one or two of these factors as the basis
  for creating blocks.

Blocking factors are sometimes referred to as
disturbing factors.
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Examples of Typical Blocking Factors
Disturbing Variable Experimental Unit Nutrient
gradient Water moisture gradient Field Plot Slope
differences Soil composition Orientation to
sun Flow of air Location in Greenhouse Distributio
n of heat Age Tree Local density Gender Age Person
Socio-demographics
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Blocking Importance

• How blocks are formed is critical to the
  effectiveness of the analysis.
• With field plots, blocks are laid out so that
  they are perpendicular to the maximum direction
  of change in the disturbing factor to be
  controlled.
• Wide border (discard) areas are used to overcome
  interference between neighboring plots (i.e. to
  maintain independence of responses) within blocks
  and between blocks.
• Time blocks may need discard times between
  replications.

This approach maximizes within block homogeneity
while simultaneously maximizing among block
heterogeneity.
13
Blocking Example
moisture gradient
Treatment effects confounded with moisture effect!
14
Blocking Example
moisture gradient
Block effect now removes moisture effect, fair
comparisons among treatments.
15
Advantages and Disadvantages

• Advantages of a Blocked Design
• Controls a single extraneous source of variation
  and removes its effect from the estimate of
  experimental error.
• Allows more flexibility in experimental layout.
• Allows more flexibility in experimental
  implementation and administration.

• Disadvantages of a Blocked Design
• Generally unsuited when there is a large number
  of treatments because of possible loss of within
  block homogeneity.
• Serious problem with the analysis if a block
  factor by treatment interaction effect actually
  exists and no replication within blocks has been
  included. (solution use replication within
  blocks when possible).

16
Complete or Incomplete Designs
Can all treatments be accommodated in each block?
Complete Block Design Every treatment occurs in
each block. Incomplete Block Design Not every
treatment occurs in each block.
Complete
Incomplete
A
B
C
D
A
B
C
B
D
C
A
B
D
A
D
C
A
B
D
C
A
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Balance
Balancing refers to the specific assignment of
treatments to experimental units such that
comparisons of treatment effects are done with
equal precision. This is usually accomplished by
equally replicating each treatment.
Balanced Block Design The variance of the
difference between two treatment means is the
same regardless of which two treatments are
compared. This usually implies that the overall
replication (disregarding which blocks they are
in) for the comparison of two treatments is the
same for all pairs of treatments.
Partially Balanced Design The variance of the
difference between two treatments depends on
which two treatments are being considered. This
usually implies different replication for
different treatments.
Unbalanced Designs Unequal replication in each
block - usually what one ends up with.
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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.
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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 Treatment 1 2 3 ... b 1 y11 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 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
22
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.
23
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
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Model Effects
Linear model
Treatment effects are relative
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.
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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.
Partitioning of the total sums of squares (TSS)
TSS SST SSB SSE
Regression Sums of Squares
dfTotal dfTreatment dfBlock dfError
26
Sums of Squares - RCBD
Expectation under HaT Expectation under HaB
Expectation of MST and MSB under respective null
hypotheses is same as E(MSE)
27
Soils Example in MTB
Stat -gt ANOVA -gt Two-Way
Must check Fit additive model (no interaction).
28
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.
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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
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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 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
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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.
32
SPSS Output
33
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-
rep(c("cac","nh4","ca2","h2o"),5) gt chem 1
"cac" "nh4" "ca2" "h2o" "cac" "nh4" "ca2" "h2o"
"cac" "nh4" "ca2" "h2o" 13 "cac" "nh4" "ca2"
"h2o" "cac" "nh4" "ca2" "h2o" gt soil lt-
c(rep(1,4),rep(2,4),rep(3,4),rep(4,4),rep(5,4)) gt
soil 1 1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 4 5 5 5
5 gt g lt- lm(sulffactor(soil)factor(chem)) gt
anova(g) Analysis of Variance Table Response
sulf Df Sum Sq Mean Sq F value
Pr(gtF) factor(soil) 4 33.965 8.491 10.5683
0.0006629 factor(chem) 3 1.621 0.540
0.6726 0.5851298 Residuals 12 9.642
0.803 --- Signif. codes 0
' 0.001 ' 0.01 ' 0.05 .' 0.1 ' 1