This document outlines key concepts in standard experimental design. It defines experimental design as assigning experimental units to treatment conditions to measure and compare treatment effects. Sample design selects units for measurement from a population. The document discusses necessary steps like replication and randomization. It presents linear statistical models including fixed, random, and mixed effects models. It also explains analysis of variance and standard designs like completely randomized design, randomized block design, and Latin square design, including their analysis of variance tables. The conclusion compares the efficiency of these standard designs.

1. Basic Concepts of Standard
Experimental Designs
Presented by:
Hasnat Israq
Islamic University , Bangladesh
On behalf of group E
1

2. OUTLINE
1. Concept of experimental design
2. Experimental design vs sample design
3. Necessary steps of an experiment
4. Basic principles of experimental design
5. Linear statistical model
6. Analysis of Variance (ANOVA)
7. Standard designs
8. Conclusion
2

3. EXPERIMENTAL DESIGN
Experimental design refers to a powerful statistical
plan for assigning experimental units to treatments
condition where treatment is the procedure whose
effects is to be measured and compare with similar
others in an experiment.
A releated point,
Sample design methods are the technique used to
select sample units for measurement.
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4. Experimental Design VS
Sample Design:
 Sample design concerns with collection of
sample from finite and existent population
with absolute experiment, where
experiment design is concerned with data
from nonexistent and infinite population
with comparative experiment.

5. NECESSARY STEPS OF AN
EXPERIMENT

6. BASIC OF PRINCIPLES
EXPERIMENTAL DESIGN
Three basic principles:
(i) Replication:
Replication means the repeatition of basic
treatments on several experimental units
under investigation.
(ii) Randomisation:
Randomisation is the process of distributing the
treatments to experimental units by chance
mechanism.
(iii) Local control:
Local control is the procedure of reducing and
controlling error variation by blocking the subjects.
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7. LINEAR STATISTICAL
MODEL
An equation displaying the relation of the
effects with response variable along with error
term in any experiment
7
Classifying this we get:
 Fixed effects model
 Random effects model
 Mixed effects model

8. FIXED EFFECTS MODEL
Fixed effect model is model in which all the
assignable factors have fixed effects and
only the error effect is random.
The model is,
𝑦𝑖=𝛼 𝑖
+ 𝛽𝑗 𝑥𝑖+ 𝜀𝑖
Here,
𝑦𝑖= dependent variable
𝛼𝑖= intercept (fixed)
𝛽𝑗 = slope coefficient (fixed)
𝑥𝑖= independent variable
𝜀𝑖= random error
𝜀𝑖~ iid N(0, 𝜎2)
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9. RANDOM EFFECTS MODEL
In statistics, a random effects model is a
statistical model where the model parameters
are random variables.
The model is,
𝑦𝑖= 𝛽0
+ 𝛽𝑗 𝑥𝑖+ 𝜀𝑖
Here,
𝑦𝑖= dependent variable
𝛼𝑖= intercept (random)
𝛽𝑗=slope coefficient (random)
𝑥𝑖= independent variable
𝜀𝑖= random error
𝛼𝑖~
iid N(0, 𝜎 𝛼
2)
𝛽𝑗 ~ iid N(0, 𝜎𝛽
2)

10. MIXED EFFECTS MODEL
A model in which some factors fixed effects while
others have random effects is mixed effect model.
The model is,
𝑦𝑖=𝛼 𝑖
+ 𝛽𝑗 𝑥𝑖+ 𝜀𝑖
Here, 𝑦𝑖= dependent variable
𝛼𝑖= intercept
𝛽𝑗=slope coefficient
𝑥𝑖= independent variable
𝜀𝑖= random error
Where 𝛽𝑗 are random variable while at least one 𝛽𝑗 is a
constant.
10

11. ANALYSIS OF
VARIANCE(ANOVA)
According to R.A Fisher “ANOVA is the
separation of variance ascribable to one group
of causes from the variance ascribable to other
group.
Analysis of variance is classified by two types
,they are
1.One-way classification and
2.Two-way classification

12. CONTINUE…
One-way classification
Arrangement the observation in various classes
on the basis of a single factor.
Two-way classification
Arrangement of the observation according to
two factors.

13. LAYOUT OF ONE-WAY
CLASSIFICATION
1 2 … … … … … …
k
𝑦11 𝑦21 … … … … … … … … 𝑦 𝑘1
𝑦12 𝑦22 … … … … … … … … 𝑦 𝑘2
.
.
.
𝑦1𝑟1 𝑦2𝑟2 … … … … … … … … 𝑦 𝑘𝑟𝑘
Total: 𝑇1 𝑇2 … … … … … … … …
𝑇𝑘
Mean:𝑦1 𝑦2 … …. …. … … … … …
𝑦 𝑘

14. LAYOUT OF TWO WAY
CLASSIFICATION
𝑩 𝟏 𝑩 𝟐 … … … 𝑩 𝒌 Total
mean
𝐴1 𝑦11 𝑦12 … … … 𝑦1𝑘 𝑅1
𝐴2 𝑦21 𝑦22 … … … 𝑦2𝑘 𝑅2
𝑦2
.
.
.
𝐴 𝑟 𝑦 𝑟1 𝑦 𝑟2 … … … 𝑦 𝑟𝑘 𝑅 𝑟
𝑦 𝑟
Total : 𝐶1 𝐶2 … … … 𝐶 𝑘
Mean: 𝑦.1
𝑦.2
… … … 𝑦.𝑘

15. STANDARD DESIGNS
15
 Commonly used experimental designs are :
I. Completely Randomized Design (CRD)
II. Randomized (Complete) Block Design (RBD)
III. Latin Square Design (LSD)

16. COMPLETELY RANDOMIZED
DESIGN(CRD)
A CRD is a design in which the selected treatments are
allocated or distributed to the experimental units
completely at random.
Fixed effect model for CRD:
𝑦𝑖𝑗= 𝜇 + 𝜏𝑖+ 𝜀𝑖𝑗 𝑖 = 1,2,……,k
𝑗 =1,2,…,𝑟𝑖
Where, 𝑦𝑖𝑗= observation in the 𝑗 𝑡ℎ
unit under 𝑖 𝑡ℎ
treatment
𝜇 = general mean
𝜏𝑖= fixed effect of 𝑖 𝑡ℎ
𝜀𝑖𝑗= random error component in the (i.𝑗)th unit
Total SS = Treatment SS+ Error SS16

17. ANOVA TABLE FOR CRD
Source of
Variation
DF SS MS 𝒇 𝒄
Treatment
s
Error
K-1
n-k
𝑖=1
𝑘
𝑟𝑖(𝑦𝑖-y̅)²
𝑖=1
𝑘
𝑗=1
𝑟 𝑖
(𝑦𝑖𝑗-𝑦𝑖
)²
𝑖=1
𝑘 𝑟 𝑖( 𝑦̅ 𝑖−y̅) ²
𝑘−1
= 𝑠𝑡
2
𝑖=1
𝑘
𝑗=1
𝑟 𝑖 (𝑦𝑖𝑗−𝑦𝑖
)²
𝑛 − 𝑘
= 𝑠 𝑒
2
𝑠 𝑡
2
𝑠 𝑒
2
Total n-1

18. RANDOMISED BLOCK DESIGN (RBD)
A RBD is a design in which the subjects are
arranged in several blocks which are internally
homogeneous and externally heterogeneous.
Fixed effect model for RBD:
𝑦𝑖𝑗= 𝜇 + 𝛽𝑖 +𝜏𝑖+ 𝜀𝑖𝑗 𝑖 = 1,2,……,r
𝑗 =1,2,…,𝑘
Here, 𝑦𝑖𝑗= observation in the 𝑗 𝑡ℎ treatment under
𝑖 𝑡ℎ
block
𝜇 = general mean
𝛽𝑗= fixed effect of the 𝑖 𝑡ℎ block
𝜏𝑖= fixed effect of the 𝑗 𝑡ℎ
treatment
𝜀𝑖𝑗= random error component in the (i.𝑗)th
unit

19. ANOVA TABLE FOR RBD
Source
of
variatio
n
DF SS MS
F
Blocks
Treatme
nt
Error
(r-1)
(k-1)
(r-1)(k-
1)
k 𝑖=1
𝑟
(𝑦𝑖.−y̅)² =𝑠 𝑏
2
r 𝑗=1
𝑘
(𝑦.𝑗
−y̅)² =𝑠𝑡
2
𝑖=1
𝑟
𝑗=1
𝑘
(𝑦𝑖𝑗 − 𝑦𝑖.
-𝑦.𝑗
+ y)²=𝑠 𝑒
2
𝑆 𝑏
2
𝑟−1
= 𝑠 𝑏
2
𝑠 𝑡
2
𝑘−1
= 𝑠𝑡
2
𝑠 𝑒
2
(𝑟−1)(𝑘−1)
=𝑠𝑡
2
𝑠𝑡
2
𝑠 𝑒
2
Total rk-1

20. LATIN SQUARE DESIGN (LSD)
LSD IS A DESIGN IN WHICH EXPERIMENTAL UNITS ARE ARRANGED
IN COMPLETE BLOCKS IN ROWS AND COLUMNS.
Latin square design is a design in which experimental
units are arranged in complete blocks in rows and
columns.
Linear model for LSD:
𝑦𝑖𝑗𝑘= 𝜇 + 𝛼𝑖 + 𝛽𝑗 + 𝜏 𝑘+ 𝜀𝑖𝑗𝑘 𝑖, 𝑗, 𝑘 =
1,2,……,r
Where, 𝑦𝑖𝑗= observation in the 𝑖 𝑡ℎ
row, 𝑗 𝑡ℎ
column,
𝑘 𝑡ℎ
treatment
𝜇 = general mean
𝛼𝑖= fixed effect of 𝑖 𝑡ℎ
row
𝛽𝑗= fixed effect of the 𝑗 𝑡ℎ column
𝜏 𝑘= fixed effect of the 𝑘 𝑡ℎ
treatment
𝜀𝑖𝑗= random error term

21. ANOVA TABLE FOR LSD
Source of
variation
DF SS MS F
Rows
Columns
Treatments
Error
(r-1)
(r-1)
(r-1)
(r-1)(r-
2)
R
C
T
E
𝑅
𝑟−1
= 𝑠𝑟
2
𝑅
𝑟−1
=𝑠𝑐
2
𝑇
𝑟−1
= 𝑠𝑡
2
𝐸
(𝑟−1)(𝑟−2)
=𝑠𝑒
2
𝑠𝑡
2
𝑠 𝑒
2
Total 𝑟2
− 1

22. Conclusion:
CRD is the basic and simplest design. RBD is more
efficient than CRD and thus provides more accurate and
precise results than CRD.
LSD is more efficient than RBD and CRD.

23. THANK YOU TO
ALL
For More
Email : www.hasnat.rbm@gmail.com
LinkedIn : hasnatrbm