ExpDes1

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Experimental Design

• Dr. Imad Khamis
• Southeast Missouri State University

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Expanding on Experimental Design

• Accounting for more than one factor.
• Blocking
• Randomized Complete Block Design
• Randomized Block Design
• Latin Square Design

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Completely Randomized Design
Different Names for the Same Design

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

Experimental Units
Sampling Units

• Assumptions
• Independent random samples (response/observation
  of one sample unit/individual do not effect other
  sample units).
• Responses follow a normal distribution.
• Common true variance, s2, across all
  groups/treatments.
• True mean for population i is mi.
• Interest in is comparing means.

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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 replicated the same number of
  times (balance).
• No accommodation made for disturbing variables
  (other 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.

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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
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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.
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Randomized Block Designs

• Two classifications in the experimental design.
• Concept of control in experimentation
• Block effects and treatment effects.

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Practical Situations

• In many situations, the researcher
• Does not have sufficient homogeneous experimental
  material in one group (location, batch, etc) to
  effectively use the completely randomized design
  (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 homogeneous material and
then compare treatment level means within these
groups.
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Example

• A scientist was interested in the use of three
  chemicals and water on their effectiveness in
  extracting sulfur from Florida soils. The
  chemical 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

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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,
resulting in the high probability of concluding
there are no treatment effects when in fact there
actually are treatment effects 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.
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Graphical View
Note The pattern of responses to treatments is
consistent within soil groups (Blocks), but soil
groups give overall higher or lower responses.
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Randomized Block Design
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.

• Rules for blocking
• Carefully examine the situation at hand and
  identify those factors which are know 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.
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Blocking Example
moisture gradient
Treatment effects confounded with moisture effect!
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Blocking Example
moisture gradient
Block effect now removes moisture effect, fair
comparisons among treatments.
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Advantages and Disadvantages

• Advantage of a Blocked Design
• To control a single extraneous source of
  variation and remove its effect from the estimate
  of experimental error.
• Allows more flexibility in experimental layout.
• Allows more flexibility in experimental
  implementation and administration.

• Disadvantage 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).

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Complete or Incomplete Designs
Can all treatments be accommodated in each block?
Complete Block Design Every treatment occurs in
each block. Incomplete Block Design Every
treatment does not occur 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 the 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 Usually what you end up with
- not a design.
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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 complete block design with each treatment
  replicated once in each block (balanced).
• A complete block design with each treatment
  replicated once in a block but with one
  block/treatment combination missing.
• A complete block design with each treatment
  replicated two or more times in each block
  (balanced with rep in each block).
• A complete block design with subsamples within
  each replicate of the treatment (balanced)

We will concentrate on number 1 and discuss the
others.
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Single Replicate RCBD
Design Complete block layout with each treatment
replicated once in each block.
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
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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.
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Linear Analysis Model
constraints
Block Treatment 1 2 3 ... b sum 1 m11 m12
m13 ... m1b m a1 2 m21 m22 m23 ... m2b m
a2 ... ... ... ... ... ... t mt1 mt2 mt3
... mtb m at sum 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
H0A 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 SSQ df MS F Treatments SST t-1 MSTSST/(t-1
) MST/MSE Blocks SSB b-1 MSBSSB/(b-1) MSB/MSE Err
or SSE (b-1)(t-1) MSESSE/(b-1)(t-1) Totals TSS bt
-1
Partitioning of the total sums of squares (TSS)
TSS SST SSB SSE
Regression Sums of Squares
dfTotal dfTreatment dfBlock dfError
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Sums of Squares - RCBD
Expectation under the alternative hypothesis.
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Soils Example
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 Lakeland
CaCl 3.31 Lakeland NH4OAc 2.74 Lakeland Ca(H2PO4)
2 2.32 Lakeland Water 2.35 Leon CaCl 2.54 Leon NH4
OAc 2.09 Leon Ca(H2PO4)2 1.09 Leon Water 2.70 Chip
ley CaCl 2.34 Chipley NH4OAc 2.07 Chipley Ca(H2PO4
)2 4.38 Chipley Water 3.85 Norfolk CaCl 4.71 Norfo
lk NH4OAc 5.29 Norfolk Ca(H2PO4)2 5.70 Norfolk Wat
er 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 9 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.
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SPSS Output