Analysis of Variance-ANOVA

ANALYSIS OF VARIANCE
ANOVA
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Presented by: TheCamouflage
Gagan Puri
Rabin BK
Bikram Bhurtel

Two way ANOVA
Design of experiment (DOE)
Principle of DOE
Completely Randomized Design(CRD)
Randomized Block Design
References
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 The systematic process for achieving the variation is called ANOVA
 In a data set if there are more than two treatments we use ANOVA
 ANOVA technique is used to test the significance difference
 Main principle of ANOVA is to analyze the variation
Total variation=variation between sample + variation within the sample(error)
Analysis of variance (ANOVA)
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All the observation should be independent
Additivity
Homogeneity
Linearity
Normality
Assumption of ANOVA
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 To test the homogeneity between three or more groups of
data
 To test the linearity
 To test the significance of observed sample correlation ratio
 To test the significance of equality of two variance
Application of ANOVA
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Sample 1 14 16 18
Sample 2 14 13 15 22
Sample 3 18 16 19 19 20
Ti Ti
2/ni
Sample 1 14 16 18 48 2304/3=768
Sample 2 14 13 15 22 64 1024
Sample 3 18 16 19 19 20 92 1692.8
Total G=204 Ti
2/ni=3484.8
Sample table for one way ANOVA test:
set the hypothesis first:
H0: there is no significance diff. between the three samples
H1: there is significance diff. between the samplesConstructing table:
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TSS= 𝑋𝑖𝑗
2 − Cf = 𝑋𝑖𝑗
2 −
𝐺2
𝑁
═ 84
SSR=
Ti
2
ni
− Cf =
Ti
2
ni
−
𝐺2
𝑁
═ 16.8
SSE=TSS-SSR ═ 67.2
𝑋𝑖𝑗
2=142+162+……+202 = 3552
Source of
variation
Sum of
square
Degree of
freedom
Mean sum of
square
F_ratio
Due to row 16.8 3-1=2 16.8/2=8.4
8.4
7.4667
= 1.125Due to error 67.2 9 67.2/9=7.4667
Total 84 12-1=11
formula and calculations:
ANOVA table:
See tabulated value Ftab(2,9)=4’26
If tabulated value < calculated value we
reject the H0
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We analyze data according to row and column
There will be two null and alternative hypothesis
COMPUTATIONAL FORMULA:
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TWO WAY ANOVA
TSS = 𝑋2
𝑖𝑗 - Cf where, Cf =
G2
𝑁
and G2 = 𝑇𝑖
2
SSR =
𝑇 𝑖
2
𝑁 𝑖
- Cf SSC =
𝑇 𝑗
2
𝑁 𝑗
- Cf
SSE = TSS – SSR - SSC

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Two Way ANOVA table:
Source of
variation
Sum of
square
Degree of freedom Mean sum of square F-ratio
First factor(i) SSR number of -1 = k-1 SSR/(k-1)=a 𝑎
𝑐
Second factor
(j)
SSC Number of column-1= r-1 SSC/ (r-1)=b 𝑏
𝑐
Due to error SSE Total samples –(number of row-1 +
number of column-1)= (k-1)(r-1)
SSE/(k-1)(r-1)=c
Total TSS Total samples-1=N-1

Design:
Planning the experiment after the systematic collection of relevant information for the
problem under study for drawing the inference
Experiment:
Way of getting answer to the question that is in the experimental mind of the problem
under study
Treatment:
It refers to different procedure under comparison in an experiment
Experimental Unit:
Place where different treatments are used
Experimental error:
Error that arise at the time of experiment and cannot be controlled by human hands
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Design Of Experiment (DOE)

Replication:
Repetition of treatments
Used to
 secure more accurate estimate
 Obtain more precise estimate
Randomization:
Allocation of treatments to various plots
Every allotment of treatment have same probability
Local control:
Process of dividing the whole experimental area into homogeneous block row
wise and column wise
Variation between the block is maximum and within the block is minimized.
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Principle of DOE

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Types of DOE
1. Completely Randomized Design (CRD)
2. Randomized Block Design (RBD)

The simplest design based upon the principles randomization and
replication
Treatment are allocated in a block in completely random way
Study of the effect of treatment only, so this is the case of one way
ANOVA
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Let there are four treatments A,B,C,D each repeated 3 times.
The layout of CRD may be,
A B C D
C D B A
C D A B
C D B A
B A C D
A B C D
Or,
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Let us assume that there are altogether ‘t’ treatments and each treatments occur the same number, say ‘r’.
Therefore, the total no. of observation is, N=tr.
Let the linear model be , xij = 𝜇ij + eij [ i=1,2,…..,t j=1,2,…..,r ]
Where uij is called Fixed effect and eij is called Even effect
xij = 𝜇 + (𝜇 i – 𝜇) +eij
= 𝜇 + 𝛼 + eij…..............(1)
Where, 𝛼i = 𝜇i – 𝜇 [ effect due to it’s treatment ]
The value of general mean (𝜇) and a can be estimated by method of least square as,
𝜇 = x’.. and 𝛼i = xi – x’.. Where , x’.. = combined mean
x’x’i. = mean of it’s treatment
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 TSS = 𝑋2
G2
𝑁
and G2 = 𝑇𝑖
2
 SST =
𝑇 𝑖
2
𝑁 𝑖
- Cf SSE = TSS – SSR - SSC
Decision: if cal. F-ratio < tab. F-ratio, we accept Ho.
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Source of variation d.f. SS MSS F-ratio
Due to treatment t-1 SST (SST/t-1) = s
Due to error t(r-1) SSF {SSE/(t(r –
1))}
= e
s/e
Total N – 1
ANOVA table
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1. The design is very simple and is easily laid out
2. It has the simplest statistical analysis
3. It provides the maximum number of degree of freedom for the error sum of
squares
4. The design is flexible i.e. any number of treatments and of replications may be
used
5. The design is applicable only to a small number of treatment
6. The main demerit lies in the assumption of homogeneity. If the whole
experiment material is not homogeneous, there may be more error.
Advantages and Disadvantages of CRD
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Design which uses all the principles of design of experiment.
Treatments are allocated at random within the block
Each treatment must once occur either in each row or in each
column
This is the case of two way ANOVA
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Let there be four treatments A,B,C,D, then layout RBD is,
A B C D
D C B A
C A B D
A D B
B C C
C B D
D A A
Or,
Layout
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Let us assume t treatment allocated in the ‘r’th block. So the number of experimental unit, N =
rt. The mathematical model will be
Xij = 𝜇 = 𝛼i + 𝛽j + eij………………….(1)
Where,
𝜇 = general term
𝛼 i = effect due to it’s treatment
𝛽 j = effect due to jth block
The value of u, B, a, are estimated by the method of least square as
𝜇 = x’..
𝛼i = x’i. – x’..
𝛽j = x’.j – x’..
Statistical Analysis
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 TSS = 𝑋2
G2
𝑁
and G2 = 𝑇𝑖
2
 SST =
𝑇 𝑖
2
𝑁 𝑖
- Cf SSB =
𝑇 𝑗
2
𝑁 𝑗
- Cf
 SSE = TSS – SSR - SSC
Decision: If cal. F-ratio < tab. F-ratio, we accept Ho
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Source of variation d.f. SS MSS F-ratio
Due to treatment t-1 SST (SST/t-1) = s1
Due to block r – 1 SSB (SSB/r-1) = s2
s1/e
Due to error t(r-1) SSF {SSE/((t-1)(r – 1))}
= e
s2/e
Total N - 1 TSS
ANOVA table
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Advantages
• This is a simple design with one local control for more efficient utilization of
the experimental units
• Reduces the experimental error and the test of significance becomes more
efficient
• Any number of treatments and any number of repliations may be included but
each treatment should have same number of replications
Disadvantages
• When the data from some experimental units are missing the “Missing plot
technique” has to be used
• If the missing observations are more, this design is less convenient than CRD25

References
1. Statistics II, Vikash Raj Satyal, Bijaya Lal Pradhan
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Analysis of Variance-ANOVA

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