Agriculture Reference
In-Depth Information
11.1.1 Analysis of Covariance for
One-Way Classified Data with
One Covariate
2. The residuals are independently and normally
distributed with mean 0 and the common
variance.
3. The covariate
x
is fixed and is measured
Let the criterion variate be denoted by
y
and the
covariate by
without error.
The model is often written in the form
. Thus, for each experimental unit,
there is a pair of observations (
x
x
,
y
).
þ e
ij
;
y
ij
¼ μ
þ α
i
þ β x
ij
x
th observation of the
variate and covariate, respectively, for the
Let
y
ij
and
x
ij
denote the
j
i ¼
1
;
2
;
3
:::: t
j ¼
1
;
2
;
3
; :::: n
i
th
treatment in a one-way classified data. Then the
linear model is given by
i
where
μ
¼ μ þ βx
y
ij
¼ μ þ α
i
þ βx
ij
þ e
ij
;
i ¼
1
;
2
;
3
:::: t
The least square estimates of
μ; α
i
;
and
β
are
j ¼
1
;
2
;
3
; :::: n
i
y
ij
μ
L ¼
P
i
P
obtained by minimizing
j
α
i
βx
ij
2
. The normal equations are
XX
y
ij
¼ nμ þ
X
n
i
α
i
þ β
XX
x
ij
;
X
where
μ ¼
general effect,
α
i
¼
effect due to
i
th treatment,
X
β ¼
regression coefficient of
y
on
x
,
y
ij
¼ n
i
μ þ n
i
α
i
þ β
x
ij
;
e
ij
¼
independent normal variate with mean
zero
ð
0
Þ
and variance
; σ
j
j
XX
XX
X
X
X
2
:
2
ij
:
x
ij
y
ij
¼ μ
x
ij
þ
α
i
x
ij
þβ
x
i
j
j
Since
X
i
μ ¼ y βx;
n
i
α
i
¼
0
;
The above model is based on the following
assumptions:
1. The regression of
α
i
¼ y
i
y βðx
i
xÞ
is linear and indepen-
dent of the treatment so that the treatment and
regression effects are additive.
y
on
x
XX
X
XX
nx þ
n
i
x
i
þ β
x
ij
y
ij
¼ y βx
y
i
y β x
i
x
2
ij
ð
Þ
x
i
X
X
XX
¼ ny x nβx
n
i
x
i
y
i
ny x β
Þn
i
x
i
þ β
2
2
ij
þ
ð
x
i
x
x
h
i
XX
X
XX
X
β
2
ij
2
i
n
i
x
i
y
i
or
x
n
i
x
¼
x
ij
y
ij
PP
x
ij
x
i
y
ij
y
i
β ¼
or
:
2
PP
x
ij
x
i
From the analysis of variance, we know
that
We want to test
H
01
: β ¼
α
i
¼ y
i
y
, whereas from the analysis of
;
H
02
: α
1
¼ α
2
¼ α
3
¼ ......¼ α
t
¼
0
α
i
¼ y
i
y β x
i
x
:
To test the above, we are to find out the resid-
ual sum of squares for the full model. The residual
or the minimum error sum of squares under the
full model
0
covariance,
ð
Þ
. The factor
β x
i
x
is called the adjustment factor which
is the amount of the linear effect of
ð
Þ
th
treatment effect. The adjustment is zero for the
treatment for which
x
on the
i
y
ij
¼ μ þ α
i
þ βx
ij
þ e
ij
is given by
