Environmental Engineering Reference
In-Depth Information
Solution
10.10.1 Correlation
The correlation coefficient,
ρ
xy
, between two variables,
x
and
y
, is defined as
From the given data,
N
= 10, and the following slope
estimates are determined by differencing the data in
accordance with Equation (10.118):
σ
σσ
xy
ρ
=
(10.120)
xy
x
y
j
where
σ
xy
is the covariance between
x
and
y
, and
σ
x
and
σ
y
are the standard deviations of
x
and
y
, respectively.
The sample estimate of
ρ
xy
is commonly denoted by
r
xy
,
which is calculated from the sample data using the
relation
2
3
4
5
6
7
8
9
10
k
1 −3.29 −1.10 −0.42 0.26 −0.39 0.33 0.31 0.28 0.38
2
1.10
1.01 1.44
0.33 1.05 0.91 0.79 0.84
3
0.92 1.61
0.07 1.04 0.88 0.74 0.80
4
2.30 −0.35 1.08 0.87 0.70 0.78
5
−3.00 0.48 0.39 0.31 0.48
N
∑
6
3.95 2.08 1.41 1.35
(
)
(
)
x
−
x y
−
y
i
i
7
0.22 0.14 0.49
i
=
1
r
=
(10.121)
8
0.05 0.62
xy
1
2
1
2
N
N
9
1.19
∑
∑
(
)
2
(
)
2
x
−
x
y
−
y
i
i
i
=
1
i
=
1
These results for
Q
i
yield
M
= 45 estimates of the the
slope with a median value of
Q
med
= 0.70 (mg/l)/yr. To
determine the 95% confidence interval, take
α
= 0.05,
which corresponds to a standard normal deviate
z
α
/2
= 1.960. The following parameters can be calculated
in accordance with the Sen method:
The correlation coefficient,
r
xy
, is usually denoted
simply by
r
, and is sometimes referred to as the
Pearson
product moment correlation coefficient.
. Values of
r
xy
can
be anywhere in the range of [−1, 1]. When the popula-
tion correlation coefficient,
ρ
xy
, is zero, it can be shown
that the statistic
t
* defined as
1
18
1
18
σ
S
=
N N
(
−
1 2
)(
N
+
5
)
=
(
10 10 1 2 10
)[
−
][ (
)
+
5
]
N
−
1
t
*
=
r
(10.122)
xy
2
=
11 18
.
1
−
r
xy
C z
=
α
σ
=
( .
1 960 11 18
)(
.
)
=
21 91
.
/
2
S
has a
t
distribution with
N
− 2 degrees of freedom, pro-
vided that both
x
and
y
are normally distributed. limit-
ing values of
r
xy
corresponding to various
N
values for
α
= 0.05 are given in Table 10.9, where it is apparent that
values of
r
xy
much higher than zero are usually neces-
sary to show significant correlation. For large values of
N
, the
t
distribution closely approximates the normal
distribution and limiting values of
r
xy
at
α
= 0.05 can be
approximated by
1
2
1
2
(
)
=
(
)
=
M
=
M C
−
45 21 91
−
.
11 54
.
1
α
1
2
1
2
(
)
=
(
)
=
M
=
M C
+
45 21 91
+
.
33 46
.
2
α
The 11th and 12th ranked values of
Q
i
are 0.26 and 0.28,
respectively, so the value of
Q
i
with a rank of 11.54 is
interpolated as 0.27. Similarly, the 33rd and 34th ranked
values of
Q
i
are 1.01 and 1.04, respectively, so the value
of
Q
i
with a rank of 33.46 is interpolated as 1.03. There-
fore, the 95% confidence interval of the estimated slope
(= 0.70) is [0.27,1.03]. This confidence interval further
supports the assertion that the slope is significantly
nonzero.
xy
= ±
1 96
.
r
(10.123)
N
TABLE 10.9. Limiting
Values of
r
xy
for Zero
Correlation at α = 0.05
N
r
xy
5
±0.75
10.10 RELATIONSHIPS BETWEEN
VARIABLES
10
±0.58
20
±0.42
30
±0.35
It is sometimes necessary to assess and evaluate the
relationships between variables. This is commonly done
using correlation and regression analyses.
50
±0.27
100
±
0.20
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