Introduction and crd

1. !
EXPERIMENTAL DESIGN AND ANALYSIS
OF VARIANCE: BASIC DESIGN
☺
Lecture No: 9-14
Prof. Dr. Md. Ruhul Amin

2. What ? Why ? How?

3. Concept of cause and effect
To
determine/
identify
To
observe/
measure

4. The
preplanne
EXPERIMENTAL DESIGN
d
statistical
procedure
by which
samples
are drawn
EXPERIMENTAL
DESIGN

5. Experimental Design
Experimental design is a set of rules used to choose
samples from populations.
The rules are defined by the researcher himself, and
should be determined in advance.
In controlled experiments, the experimental design
describes how to assign treatments to experimental units,
but within the frame of the design must be an element of
randomness of treatment assignment.

6. Experimental Design..
Treatme
nts
(populat
ion)
Size of
samples
Experime
ntal units
Sample
units
(observati
ons)
Replicatio
n
Experime
ntal error
It is necessary to define

7. Principles of Experimental Design
According to Prof R
A Fisher, the basic
principles of
Experimental
Design are
1.Randomization 2. Replication 3. Error control
Unbiased allocation of
treatments to different
experimental plot
Repetition of the
treatments to more
than one experimental
plot
Measure for
reducing the error
variance

8. The error includes all types of
extraneous variations which are due to
a) Inherent variability in the experimental
material to which the treatments are applied
b) The lack of uniformity in the methodology of
conducting experiment
c) Lack of representativeness of the sample to the
population under study

9. What is Treatment?
Different procedures
under comparison in
an experiment is
called treatment
Example
• Different varieties
of crop
• Different diets
• Different breeds of
animals
• Different dose of
drug/fertilizer
Effec
ts of
treat
ment
s are
comp
ared
in
expt

10. Basic Designs
1. Completely Randomized Design (CRD)
2. Randomized Block design (RBD)
3. Two Factor Factorial Design

11. Types of Analysis of Variance
One way
Data are classified
into groups according
to just one
Lciaftee egoxprieccatla vnacryia ibnl e3
different races in
Malaysia
Here categorical
variable: Races
Level: L1 (Malay), L2
(Chinese), L3 (Indian)
Two way, Three way……..
Data are classified into two or more
categorical variables
CGPA of students of 4 different programmes of
FIAT in different academic years. Two-way..
1. Programmes (4)
2. Academic years (4) ; Design 4x4
Example
Example

12. Designs commonly used in Agricultural /
Biological Science
i) One-way design/single factor
design (no interaction effect)
❑ Fixed effects
❑ Random effects
ii) Factorial design/multifactor
design (interaction effect betn
treatments)
❑ Fixed effects
❑ Interaction effect
❑ Random effects
Both can be fitted into any basic design of experiment ie in CRD, RBD
or LSD

13. Some important definitions
Treatments : Whose effect is to be determined. For
example
i)You are to study difference in lactation milk yield
in different breeds of cows. ….. Treatment is
breed of cows. Breed 1, Breed 2… are levels
(1,2,..)
ii) You intend to see the effect of 3 different diets
on the performance of broilers. ….. Treatment
is diet and diet1, diet2 and diet3 are levels (1,2,3)
iii) You wish to compare the effect of different
seasons on the yield of rubber latex. Season is
treatment and season1, season2 are the levels

14. …..definitions
Experimental units: Experimental material to
which we apply the treatments and on which we
make observations. In the previous two examples
cow and broilers are the experimental materials
and each individual is an experimental unit.
Experimental error: The uncontrolled variations in
the experiment is called experimental error. In
each observation of example(i) there are some
extraneous sources of variation (SV) other than
breed of cow in milk yield. If there is no
uncontrolled SV then all cows in a breed would
give same amount of milk (!!!).

15. …..definitions
Replication (r): Repeated application of
treatment under investigation is known as
replication. In the example (i) no. of cows
under each breed (treatment) constitutes
replication.
Randomization: Independence
(unbiasedness) in drawing sample.
Precision (P): The reciprocal of the variance
of the treatment mean is termed as
P r precision. =
σ 2

16. 1. Completely Randomized Design
(CRD): Fixed Effects One-way
• CRD is the simplest type of experimental
design. Treatments are assigned
completely at random to the
experimental units, with the exception
that the number of experimental units for
each treatment may set by the
researcher.

17. are
describ
ed
with
depend
ent
variabl
e, and
the
way of
groupin
1. Completely Randomized Design
(CRD): Fixed Effects One-way ANOVA
1. Testing
hypothesis to
examine
differences
between two or
2. Each
treatm
ent
group
repres
ents a
popula
tion.
more
categorical
treatment groups.
Milk
yield
Feed

18. Designing a simple CRD
experiment
For example, an agricultural scientists wants to
study the effect of 4 different fertilizers
(A,B,C,D) on corn productivity. 4 replicates of
the 4 treatments are assigned at random to the
16 experimental units
!
➢Treatment : Types of fertilizer (A,B,C,D)
➢Experimental unit : Corn tree
➢Dependent variable : Production of corn

19. Steps
1
• Label the experimental units with number 1 to 16
2
• Find 16 three digit random number from random number table
3
• Rank the random number from smallest to largest
4
• Allocate Treatment A to the first 4 experimental units, treatment B to the next 4
experimental units and so on.

20. Random
Number
Ranking
(experimental unit)
Treatment
104 4 A
223 5 A
241 6 A
421 9 A
375 8 B
779 12 B
995 16 B
963 15 B
895 14 C
854 13 C
289 7 C
635 11 C
094 2 D
103 3 D
071 1 D
510 10 D

21. • The following table shows the plan of
experiment with the treatments have
been allocated to experimental units
according to CRD
!
!
experimental
unit number
!
!!
Treatment
A 4 5 6 9
B 8 12 16 15
C 14 13 7 11
D 2 3 1 10

22. Fixed effects one-way ANOVA..
In applying a CRD or when groups indicate a
natural way of classification, the objectives
can be
1. Estimating the mean
2. Testing the difference between
groups

23. Fixed effects one-way ANOVA..
Model
!
ij i ij Y = μ +T + e
Where
Yij = Observation of ith treatment in jth replication
= Overall mean
μ
Ti = the fixed effect of treatment i (denotes an unknown
parameter)
eij = random error with mean ‘0’ and variance σ 2
‘ ‘
!
The factor or treatment influences the value of observation

24. Designing ANOVA Table
• Suppose we have a treatment or different level of a single factor.
The observed response from each of the “a” treatments is a
random variable, as shown in the table:
Treatment
(level)
Observations Totals Mean
1 y y … y y
2 y y … y y
.
.
.
.
a y y … y y
1. y
2. y
a. y
y.. y..

25. Cont..
Source SS df MS F
Between
SSTrt a-1
treatment
Error
(within trt)
SSE N-a
Total SST N-1
❖ a= level of treatment
❖ N= number of population
❖ SS = Sum of Squares
❖ SST = Sum of Square Total
= the sample variance of the y’s
❖ SSE = Sum of Square Error
❖ SST = SSTrt + SSE
= (total variability between treatment)
+ total variability within treatment)
MS SSA
−1
MSE SSE
F MSTRT =
If the calculated value of
F with (a-1) and (N-a) df
is greater than the
tabulated value of F with
same df at 100α % level
of significance, then the
hypothesis may be
rejected
=
a
TRT
−1
=
a
MSE

26. Cont..
y y
SSTrt i
SST y y ij
2 ..
= ΣΣ −
2
N
= Σ −
N
n
2
.. 2.
1
SSE = SST – SSTrt

27. Fixed effects one-way ANOVA..
Problem 1:
An expt. was conducted to investigate the
effects of 3 different rations on post
weaning daily gains (g) on beef calf. The
diets are denoted with T1, T2, and T3. Data,
sums and means are presented in the
following table.

28. Fixed effect one-way ANOVA..: Post
weaning daily gains (g)
T T T
270 290 290
300 250 340
280 280 330
280 290 300
270 280 300
Total
1400 1390 1560 4350
n 5 5 5 15
280 278 312 290
Yi
y

29. One-way ANOVA: Hypothesis
Null hypothesis
!
Ho: There is no significant
difference between the
effect of different rations
on the daily gains in beef
calves ie Effects of all
treatments are same.
Alternative hypothesis
!
H1: There is significant
difference between the
effect of different rations
on the daily gains in beef
calves ie Effect of all
treatments are not same.
Ho: μ = μ = μHa : μ ≠ μ ≠
μ1 2 3 1 2 3

30. Level of significance or confidence interval
Commonly used level of significances (in biology/
agric)
α=0.05
• True in 95% cases
• p<0.05
α=0.01
• True in 99% cases
• p<0.01
p< 0.05, conf. interval = 95% ; p< 0.01, conf. interval =
99%

31. One-way ANOVA…
!
1. SST =
( ... ... ) (4350)
270 300 300
y
= 1268700 – 1261500 = 7200
!
2. SSTr=
( ) 2 2 2 2 2
1400 1390 1560 (4350)
y
Σ
i Σ y
N
− = + + −
5 5 5 15
..
2
i i n
!
3. SSE = SST – SSTr
15
= 7200-3640 = 3560
2
2 2 2
2
2 ..
ΣΣ − = + + − N
i j
ij y
1265140 1261500 3640
= − =

32. ANOVA for Problem 1.
Source SS df MS F
Treatment 3640 3-1=2 1820 6.13
Error
3560 15-3=12 296.67
(residual)
Total 7200 15-1=14
The critical value of F for 2 and 12 df at α = 0.05 level of
significance is
F0.05 (2,12 )= 3.89. Since the calculated F (6.13) > tabulated F or
critical value of F(3.89), Ho is rejected. It means the experiments
concludes that there is significant difference ANOVA (p<is 0.05) significant
between
the effect of different rations (at least in two) on calves’ daily
gain.
!
Now the question of difference between Difference any two betn means any two will means be solved ?????
by
MULTIPLE COMPARISON TEST(S).

33. Multiple Comparison among Group Means
(Mean separation) or Post hoc tests
There are many
post hoc tests
such as
• Least significant
difference (LSD)

34. Multiple comparison: Least Significant
Difference(LSD) test
LSD compares treatment means to see whether
the difference of the observed means of
treatment pairs exceeds the LSD numerically.
LSD is calculated by
!
t 2
MSE !
α / 2, N −
a
n
!
where is the value of Student’s t (2-tail)with
error df t α / at 2 100 α
% level of significance, n is the
no. of replication of the treatment. For unequal
replications, n1 and n2 LSD=
( 1 1 )
/ 2, t MSE r r N a × + α −
1 2

36. Multiple comparison: Tukey’s test
Compares treatment means to see whether
the difference of the observed means of
treatment pairs exceeds the Tukey’s
numerically. Tukey’s is calculated by
T α =
q (a, f ) MSE
!
α
n
Where f is df error .

38. Multiple comparison: Duncan’s multiple
range test

39. Based on problem 1
Using Tukey’s test, the mean comparison as
follows (which treatment means are differ).

41. Random Effects One-way ANOVA: Difference between fixed and random effect
Fixed effect Random effect
Small number (finite)of groups or
treatment
Large number (even infinite) of
groups or treatments
Group represent distinct
populations each with its own mean
The groups investigated are a
random sample drawn from a single
population of groups
Variability between groups is not
explained by some distribution
Effect of a particular group is a
random variable with some
probability or density distribution.
Example: Records of milk production
in cows from 5 lactation order viz.
Lac 1, Lac 2, Lac 3, Lac 4, Lac 5.
Example: Records of first lactation
milk production of cows constituting
a very large population.

42. Advantages of One-Popular way analysis(CRD)
design for
its
simplicity
,
flexibility
and
validity
Can be
applied
with
moderate
Any
number
number
of
of
treatmen
ts and
treatmen
ts (<10)
any
number
of
replicatio
ns can be
Analysis
is straight
forward
even one
or more
observati
ons are
missing

43. A practical example of one-way ANOVA
Problem: Adjusted weaning weight (kg) of
lambs from 3 different breeds of sheep are
furnished below. Carry out analysis for i)
descriptive Statistics ii) breed difference.
Suffolk: 12.10, 10.50, 11.20, 12.00, 13.20,
10.90,10.00
Dorset: 11.50, 12.80, 13.00, 11.20, 12.70
Rambuillet: 14.20, 13.90, 12.60, 13.60,
15.10, 14.70, 13.90, 14.50

44. Analysis by using SPSS 14
Descriptive Statistics
N minimum maximum mean Std. dev
Suff 7 10.00 13.20 11.4143 1.09153
Dors 5 11.20 13.00 12.2400 .82644
Ramb 8 12.60 15.10 14.0625 .76520
Valid N
5
(list wise)
Mean is expressed as : X ± SD

45. ANOVA (F test)
a) One-Way ANOVA
Sum of
squares
df Means
Squares
F Sig.
Between
groups
27.473 2 13.736 16.705 .000
Within groups 13.979 17 .822
Total 41.452 19
Since the significance level of F is far below than 0.01 so breed
effect is highly significant (p<0.01)

46. Mean Separation
Post hoc tests
Homogenous subsets
Wean
Duncan
3 N Subset for alpha
=0.05
Suff 7 11.414
Dors 5 12.240
Ramb 8
14.063
Sig. .121
1.000