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Analysis Of Variance: ANOVA

Analysis of variance anova

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Page 1: Analysis of variance anova

Analysis Of Variance: ANOVA

Page 2: Analysis of variance anova

contentContent

Introduction of ANOVA Concept and Meaning Definitions of ANOVA Classification Of

ANOVA One-Way ANOVA Two-Way ANOVA Uses of ANOVA Advantages of ANOVA Limitations of ANOVA

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Introduction ANOVA is commonly used to test

hypothesis of equality between two variances and among different means.

It is one of the standard and extremely useful technique for experiments in the field of psychology, Sociology, Education, Commerce and several other disciplines.

It is an advanced technique for the experimental treatment of testing differences among all of the means which is not possible in case of t-test.

This test is known as F test of significance in which we can find out more than two experimental variables and their interaction in the experiment.

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Concept This technique of ANOVA was first developed by Ronald A. Fisher, a

British Scientist in 1923. This method was widely used in the experiments of behavioral and social

sciences to test the significance of differences of means in different group of a varied populations.

For this technique fisher is called the father of Modern Statistics. Through his technique, it is possible to determine the significance of

difference of different means in a single test rather than many. It minimizes the Type 1 error unlike in case of t-test. The value of F ratio is computed through Variance between groups/ Larger estimate of variance Variance within groups/ Smaller estimate of variance The value of F is always more than 1.

F =

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MeaningThe object of F-test is to find out whether the two independent estimates of population variance differ significantly or whether the two sample may be regarded as drawn from the normal population. That’s why also known as Variance Ratio Test.

ANOVA is one of the most powerful techniques available in the field of statistical teaching.

It is a statistical technique in which, significance of more than two mean sample can be analyzed.

This method is mostly used in research process. Because for conclusion or interpretation more than two variables are required.

In earlier times, this method was used in agricultural researches. But in modern times it is also used in natural, social and agricultural researches.

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Definitions of ANOVA “The analysis of variance is

essentially a procedure for testing the difference between different groups of data for homogeneity.”

Yule and Kendal “ Analysis of variance is the

separation of the variance ascribe to one group of cause from the variance ascribe to other group.”

R.A.Fisher

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Some FeaturesA substitute of t-test & z-test.

A Parametric test.

Determining several differences between several means.

Data are treated at once/at the same time.

A general hypothesis among means of different group is tested.

Minimizing type- 1 error.

Breaks the total variance of large sample and makes it two groups.

- Variance within Groups (Average Variance of each & their means)

- Variance Between Groups (Variance of the group means and group mean of all groups)

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Assumptions of Analysis of Variance

Normal distribution

Independent samplesSame population varianceRandom Selection

Additivity of effects

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Techniques of Analysis of Variance

One way classificatio

n

Two way classificatio

n

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ONE-WAY ANOVA

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One Way ANOVAIn one way ANOVA only one factor is computed. It is also called single classification ANOVA. For example, in an experiment, three groups are selected for an experimental treatment on one factor i.e., evaluation of performance of the three groups on the basis of three attitude scales.

In one-way ANOVA, the total variance is equal to the variance within groups and variance between groups.

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Example:-

Set up an analysis of variance table for the following per acre production data for three varieties of wheat, each grown on four plots. Consider variety differences to be significant.

Plot of Land A B C

1 6 5 5

2 7 5 4

3 3 3 3

4 8 7 4

Per Area Production data Variety of Wheat

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AX1 (x1)

BX2 (x2)

CX3 (x3)

6 36 5 25 5 25

7 49 5 25 4 16

3 9 3 9 3 9

8 64 7 49 4 16

∑X =24 ∑x =158 ∑X =20 ∑x =108 ∑X =16 ∑x =661

2

1 2 32

2

2

3

• Solution

2 22

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Step 1:- Null Hypothesis:- Let us take the hypothesis

that there is no significant difference in production of three varieties.

Step 2:- Correlation Factor (c) C.F= (∑X) N C.F= (60) /12=300

2

2

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Step 3:- Sum of Square Total= ∑X – C SSt=(158+108+66) – 300 SSt= 32 Step 4:- Sum of Square Between SSb= (∑x ) (∑x ) (∑x ) n1 n2 n3 SSb= (24)/4 + (20)/4 + (16)/4- 300 SSb= 144+ 100+ 64- 300

2

12

2 3

2 2

+ + - C2 2 2

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SSb=8 Step 5:- Sum of Square With in Groups SSw= SSt- SSb SSw=32-8=24 SSw= 24Step 6:-Degree of Freedom D.O.F for sum of square total= (N-1) = 12-1=11 D.O.F for sum of square between= (k-1) = 3-1=2 D.O.F for sum of square with in groups= (N-k) = 12-3=9

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ANOVA TableSource of variation

Sum of Square

Degree of Freedom

Mean Sum of Square

F-Ratio Table value at 5% level

Between Sample

SSB= 8 K-13-1=2

8/2=4 4/2.67=1.5 F (2,9)=19.45H0= Accepted

Within Sample

SSW= 24 N-K12-3=9

24/9=2.67

Total SST= 32 N-112-1=11

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Step 8:- Interpretation:- The computed value 1.5 is less than table

value 19.45 at 5% level of significance. So the null hypothesis that there is no significant difference in production of three varieties is accepted.

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TWO-WAYANOVA

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Two-Way ANOVAIn two-way ANOVA there is more than one experimental factor and one or more control factors. In case of the above one way ANOVA the three attitude scales for the test of performance are experimental factor where as the examiner is the control factor.

If the tests are conducted by two examiners so the over all factors will be 3×2=6.

It is also called double classification ANOVA. This procedure can be used for the application of three-way or larger class of ANOVA.

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Illustration:-A farmer applied three types of fertilizers on 4 separate plots. The figure on yield per acre are tabulated below:

Fertilizers Plots

A B C D Total

Nitrogen 6 4 8 6 24

Potash 7 6 6 9 28

phosphates 8 5 10 9 32

Total 21 15 24 24 84

Yield

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Plots

Fertilizer

A

X1

B

X2

C

X3

D

X4

RowTotal

(x1) (x2) (x3) (x4)

Nitrogen 6 4 8 6 24 36 15 64 36

Potash 7 6 6 9 28 49 36 36 81

Phosphates

8 5 10 9 32 64 25 100 81

Column Total

21 15 24 24 T=84 149 77 200 198

2 2 2 2

Yield Square of Data

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Find out if the plots are materially different in fertility, as also, if the three fertilizers make any material difference in yields.

Solution:- Step 1:- Null Hypothesis:- Let us take the hypothesis that: 1. All plots are not significantly differ in fertility (Column

wise analysis) 2. All the fertilizers are not significantly differ in yields.

(Row wise analysis) Step 2:- Correlation Factor (c) C= (∑X) N C= (84) /12= 588

2

2

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Step 3:- Sum of Square Total= ∑X – C ∑X=149+77+200+198=624 SSt= 624- 588 SSt= 36 Step 4:- Sum of Square Between Columns: SSc= (∑x ) (∑x ) (∑x ) (∑x ) n1 n2 n3 n4

2

2

1 2 3 4

2 2 2 2

+ + + - C

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SSc= (21)/3+ (15)/3 + (24)/3 + (24)/3-588 SSc= 147+75+192+192-588=18 SSc=18 Step 5:- Sum of Square Between Rows: SSr= (∑x ) (∑x ) (∑x ) n1 n2 n3 SSr=(24)/4+ (28)/4+ (32)/4- 588 SSr=144+196+256-588=8 Step 6:- Sum of Square With in Groups:

2 2 2 2

1 2 32 2 2

+ + - c

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SSw= SSt- (SSc+SSr) SSw= 36- (18+8)=10 SSw= 10 Step 7:-Degree of Freedom:D.O.F for total sum of square= (N-1) = 12-1=11D.O.F for sum of square between columns= c-1 = 4-1=3D.O.F for sum of square between rows= r-1 = 3-1=2D.O.F for sum of square with in groups= (r-1) (c-1) = 2×3=6Here: r= no. of rows c= no. of columns

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Source of Variance

Sum of Square

Degree of Freedom

Mean Sum of Square

F-Ratio Table value

Between Columns

SSc=18 c-14-1=3

18/3=6 SSc/SSw6/1.667=3.6

F (3,6)=8.94H0= Accepted

Between Rows

SSr= 8 r-13-1=2

8/2=4 SSr/SSw4/1.667=2.4

F (2,6)=19.33H0= Accepted

With in Groups

SSw= 10 (r-1) (c-1) 3×2=6

10/6=1.667

Total SSt= 36 N-1= 11

ANOVA Table

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Step 9:-Interpretation1. Columns wise analysis:- The computed value

of F=3.6 is less than table value 8.94, hence the null hypothesis is accepted, it means the plots are not significantly differ in fertility.

2. Row wise analysis:- the calculated value of F=2.4 is less than table value 19.33, hence the null hypothesis is accepted. It means the fertilizers are alike so far as productivity concern.

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Uses of ANOVATo test the significance between variance of two samples.

Used to study the homogeneity in case of Two-way classification.

It is used in testing of correlation & regression.

ANOVA is used to test the significance of multiple correlation coefficient.

The linearity of regression is also tested with the help of Analysis of Variance.

Interpretation of significance of means & their interactions.

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Advantages of ANOVAIt is improved technique over t-test & z-test.

Suitable for multi-dimensional variables.

Analysis various factors at a time.

Can be used in three and more than three groups.

Economical and good method of Parametric testing.

It involve more than one independent variables in studying the main impact & interaction effect.

The experimental design ( simple random design & level treatment design) are based on one way ANOVA technique.

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Limitations of ANOVAIt is difficult to analyze ANOVA under strict assumptions regarding the nature of data.

It is not so helpful in comparison with t-test that there is no special interpretation of the significance of two means.

It is not always easy to interpret the cases of multiple interactions and their significance level.

It has a fixed and difficult set for designing experiments for the researcher.

Requirement of post- ANOVA t-test for further testing.

Sometimes, time consuming & also time requires knowledge & skills for solving numerical problems.

It provides no additional information as compared to t-test.

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