the concept of analysis of co-variance
TRANSCRIPT
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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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Computation of Sum of squares (SS) among
the means of the groups.
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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Computation of Sum of squares (SS) among
the means of the groups.
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