The Concept of Analysis of Co-Variance

7/30/2019 The Concept of Analysis of Co-Variance

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The Concept of

Analysis of Co-variance

BY . PRASAD JADHAV


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What exactly a

Analysis of Co-

variance?


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Covariance is a measure of how

much two variables change togetherand how strong the relationship is

between them..


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ANCOVA is really ANOVA withcovariates" or, more simply, a

combination of ANOVA andregression used when you have

some categorical factors and

some quantitative predictors.


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Analysis of covariance may be looked upon as a specialprovision or procedure for exercising necessary statistical

control over the variable or variables that have been left

uncontrolled at the start of the experiment or study on

account of practical limitations and difficulties associated

with the conducting such experiments or studies.


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By using this technique, we try to partial

out the side effects if any on our study dueto lack of exercise proper experimental

control over the intervening variables, after

having conducted the actual study of

covariance.


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Assumptions Underlying of Covariance

The method of analysis of covariance requires some basic

assumptions for its application.

1. The dependent variable which is under measurement

should be normally distributed in the population.

2. The treatment groups should be selected at random

from the same population.

3. Within-groups variances must be approximately equal

4. The regression of the final socres (Y) on initial scores(X) Should be basically the same in all groups

5. There should exist a linear relationship between X & Y.


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We will illustrate ANCOVA with a Example.

Three groups of five students each (Randomlyselected from a Class VIII of a school) were initially

rated for their leadership qualities, and their scores

were recorded. Then they were subjected to different

treatments ( Three different approaches or trainingtechniques for leadership), and after 15 days of such

training, they were again evaluated for their

leadership qualities, and these final scores were also

recorded. The data so collected are given in the next

slide


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Group A Group B Group C

Initial

values

Final

scores

Initial

values

Final

scores

Initial

values

Final

scoresX1 Y1 X2 Y2 X3 Y3

4 6 8 8 7 9

3 5 6 5 9 8

5 6 7 9 10 11

2 4 4 7 12 12

1 4 5 6 12 15

Use this data to compare the relative effectiveness of the treatments given to the groups


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Solution.

There were many observable differences amongst theinitial scores of the three groups, but no attempt was

made to make these groups as equivalent groups at

the start of the study, i.e .before subjecting these to

different treatments.In absence of such an experimental control, the

researcher was forced to exercise statistical control by

applying the technique of analysis of covariance.

The procedure may be understood through

computation . Let us begin with arranging the given

data .


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Group A Group B Group C

X1 Y1 X1Y1 X21 Y2

1 X2 Y2 X2Y2 X2

2 Y2

2 X3 Y3X3Y

3X23 Y

23

4 6 24 16 36 8 8 64 64 64 7 9 63 49 81

3 5 15 9 25 6 5 30 36 25 9 8 72 81 64

5 6 30 25 36 7 9 63 49 81 10 11 110 100 121

2 4 8 4 16 4 7 28 16 49 12 12 144 144 144

1 4 4 1 16 5 6 30 25 36 12 15 180 144 225

SU

M

15 25 81 55 129 30 35 215 190 255 50 55 569 518 635

M

S3 5 6 7 10 11


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For all the three groups,

X = X1 + X2 + X3 = 15 + 30 + 50 = 95

Y = Y1 + Y2 + Y3 = 25 + 35 + 55 = 115

X2 = X21+ X22+ X23= 55 + 190 + 518 = 763

Y2 = Y21+ Y22+ Y23= 129 + 255+ 635 = 1019

XY = X1Y1 + X2Y2 + X3Y3 = 81+ 215+ 569 = 865

After Computing various sums and means thus, the following steps can be adopted


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STEP 1.

Computation of correction terms (Cs) .

Different corrections are applied to different sums of

Sqaures as in the case of analysis of variance. These can

be computed by using the formulae shown in next slide.


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For all the three groups,

i. Cx = ( X)2 95 x 95

N 15

== 601.67

ii. Cy= ( Y)2 115 x 115N 15

== 881.67

iii. Cxy = X Y 95 x 115N 15

== 728.33


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STEP 2.

Computation of total sum of squares (total SS)


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Computation of total sum of squares

(total SS)

SSX = X2 CX = 769 601.67 = 161.33

SSY = Y2 CY = 1019 881.67 = 137.33

SSXY = XY CXY = 865 728.33 = 136.67


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STEP 3.

Computation of Sum of squares (SS) among the means ofthe groups.


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Computation of Sum of squares (SS) among

the means of the groups.

X21 X22 X23

N1 N2 N3

++ -Cxi) SS AmongstMeans for X =

15

2

+ 30

2

+ 50

2

5

-= 601.67

123.33=


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y21 y22 y23

N1 N2 N3

++ -Cyi) SS AmongstMeans for y =

25

2

+ 35

2

+ 55

2

5

-= 881.67

93.33=


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x1y1 x2y2 x3y3

N1 N2 N3

++ - Cxyi) SS AmongstMeans for xy =

15x25 +30x35

+ 50 x 55

5-

=

728.33

106.67=


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STEP 4

Computation of Sum of squares (SS) with in group


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STEP 5

Computation of the number of degree of freedom


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Computation of the number of degree of

freedom

i. Amongstmeans (df) = K-1 = no of groups = 3 1 = 2

ii. Within groups (df) = N-K = 15 3 = 12


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STEP 6

Analysis of variance of X & Y scores taken saperately


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Source

of

variation

Df SSx (

Sum of

squares

for X)

Ssy (

Sum of

squares

for Y)

MSx (

Mean

square

variance

for X orX

MSy (

Mean

square

variance

for y ory

Amongst

-means

2 123.33 93.33 123.33/

2 =

61.66

93.33/2

= 46.66

Within-

groups

12 38 44 38/12 =

3.17

44/12 =

3.67

total 14 161.33 137.33


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Fy =

Mean square variance of among-groups (for y)

Mean square variance of within-groups

61.66

3.17= = 19.45

Fx =

Mean square variance of among-groups (for x)

Mean square variance of within-groups

46.66

3.67

= = 12.71

Where Fx = F ratio for X Scores

Fy= F Ratio for Y Scores

From Table R of the Appendix for df (2, 12), We can have the critical value of F

At 0.05 level = 3.88 and at 0.01 level = 6.93


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Computation of adjusted sum of squares (SS for y i.e.

SS for x)

The initial differences in the groups X scores may

cause variablility in their final scores measured after

giving treatment. It needs to be checked and

controlled. For this purpose, necessary adjustments

are made in various sum of squares (SS) for Y by using

the following general formula.


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STEP 7

Computation of adjusted sum of squares (SS for y i.e. SSfor x)


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Rule : If Fx is significant the H0 is to be rejected

showing that initially groups were different. Hence

covariance is needed. If not significant we can have

only analysis of variance.

The computed value of of F for X scores is significant

at both the levels, and similar is the case with the

computed F for Y Scores. Hence H0 for X Scores as wellas Y scores are rejected, leading to the conclusion that

(i) There is significant difference in intial (X) Scores and

(ii) there are significant difference in final (Y) Scores

(SS xy)2

SSx= SS YX= Ssy -

(Here SSyx stands for the sum of squares of Y adjusted for X difference.)


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The Specific adjusted SS for Y May be computed as

follows :(a)Adjusted Sum of squares for total , i.e.

,

Total (SS xy)2

total SSxSSYx (Total) = SSy -

(136.67)2

161.33

= 137.33-

= 137.33 115.77 = 21.56

(b)Adjusted Sum of squares for within-group means, i.e.,

within (SS xy)2

within SSxSSYx (within-means) = SSy -

(30)2

38= 44 -

= 44 23.68= 20.32


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(c)Adjusted Sum of squares for among-group means, i.e.,

Ssyx (within)SSYx (among-means) = Ssyx (Total ) -

= 21.56 20.32 = 1.24


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STEP 8

Computation of Analysis of Covariance :


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Computation of Analysis of Covariance :

It is carried out as shown in the table

Source of

variationdf SSx SSy SSxy SSyx

Msyx or

VyxSdyx

Among-

Group

means

2 123.33 93.33 106.67 0.44 0.22

Within

group

means

11* 38 44 30 20.32 1.761.76

=1.32

Total 13 161.33 137.33 136.67 20.76

(*1 df is lost because of regression of Y on X)


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(

Fyx = Vyx (among)

Vyx (within)

= 0. 221.76

=0.125

From Table R of the appendix for df (2,11)

Critical F at 0.05 level = 3.98

AndCritical F at 0.01 level = 7.20

The computed value of F(Fyx) is not significant at both the levels. Hence H0 is

to be accepted with the conclusion that groups do not different significantly

after giving treatments . Hence out of the three treatments cannot be termed

better than another

The Concept of Analysis of Co-Variance

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