Agriculture Reference
In-Depth Information
deduce that for the deviance to be equal to
zero all three terms on the right-hand side
must be equal to zero. This occurs when there
is no bias (i.e. the two means μ
1
and μ
2
are
equal), no scale difference (i.e. the standard
deviations σ
1
and σ
2
are equal) and no impre-
cision (i.e. the correlation ρ is equal to 1). The
deviance is an omnibus statistic. The contri-
butions to the sample deviance can be tested
formally in analysis of variance fashion with
partitioning of the sum of squares at the total
level (van Belle, 2002).
size of the average error between
Y
1
and
Y
2
.
Thus, we can conclude that two variables are
concordant, but we can't express the error
of using
Y
2
in place of
Y
1
in an absolute or
even relative scale.
Linear functional relationship
First, we must change our notation so as to
simplify our notation later on. Instead of
using
Y
1
and
Y
2
to represent the two ran-
dom variables, we now annotate them as
X
and
Y
, understanding that we are not mak-
ing any reference to a dependent and an in-
dependent variable. The linear relationship
between the two random variables can be
modelled as follows (Casella and Berger,
1990):
Concordance analysis
We first define a location shift
u
as follows:
(
)
mm
ss
-
(5.13)
1
2
u
=
2
X
Y
ξδ
ηε
ηφβξ
=+
=+
=
…
=+
12
i
i
i
i
1,
,
n
Likewise, a scale shift
v
is defined as:
i
i
i
(5.17)
i
i
(
ss
ss
-
)
(5.14)
v
=
1
2
Where
X
i
and
Y
i
are measurements by the
two methods; ξ
i
and η
i
are the unobserved
true values of
X
i
and
Y
i
; and δ
i
and ε
i
are
measurement errors.
The measurement errors are generally
considered bivariate Gaussian, uncorrelated,
with variance
s
2
12
Lin (1989) used these quantities to identify
a measure of accuracy as:
1
Accuracy =
(
mm
ss
-
)
2
(
ss
ss
-
)
2
1
+
1
2
+
1
2
2 2
and
,
respectively. Thus,
the precision ratio
l
s
d
e
s
2
2
d
e
=
2
/
compares the
relative efficiency in terms of precision. If
the stochastic model has the same precision
as the measurements (i.e. λ = 1), then the
maximum likelihood estimate (MLE) of β in
Eqn 5.17 is:
12
12
(5.15)
Accuracy thus defined is a quantity that var-
ies between 0 and
1,
with a value of
1
when
the location shift and scale shift are 0. The
precision is simply stated as the correlation
coefficient, ρ. Lin (1989) defined the concord-
ance coefficient ρ
c
as:
ρ
c
= accuracy × precision
(
)
+
2
b=
-+ -
SS SS S
S
4
2
yy
xx
yy
xx
xx
(5.18)
(5.16)
2
xy
The concordance coefficient is equal to
1
when there is no location differential, no
scale differential and perfect correlation
between two variables. The concordance co-
efficient ρ
c
is a unit-free, omnibus statistic.
Its unit-free property is both an asset and a
liability. The scale used to express
Y
1
and
Y
2
(e.g. grammes vs kilogrammes) has no effect
on the statistic - this is good. But as with the
correlation coefficient, there is no direct re-
lationship between the value of ρ
c
and the
Where
S
xx
= Σ (
X
i
- X)
2
;
S
yy
= Σ (
Y
i
- Y)
2
;
S
xy
= Σ (
X
i
- X) (Y
i
- Y).
This is also known as orthogonal least-
squares (Tan and Iglewicz, 1999). If the pre-
cision of the two methods is not the same
(i.e. λ ≠ 1), then the MLE becomes:
(
)
+
2
2
SSSS S
S
-
l
+
-
l
4
l
b
yy
xx
yy
xx
xx
=
2
xy
(5.19)
