Geoscience Reference
In-Depth Information
For example, in the initial year of the study (i.e., year j = 0), the estimated pro-
portion of response variation caused by site-to-site differences is
2
0.0294
(0.0294
=
0.1487
.
2
2
+
0.0703 )
Knowing the proportions of variation that are attributable to site-to-site dif-
ferences versus within-site variation is helpful in planning future allocation
of sampling effort.
Recall that one of the major goals in trend analysis is to describe the overall
trend for a population as well as the individual observed trends associated
with each sampling unit. From the model given in Equation (11.5), the pre-
dicted mercury concentration for the i th station is given by
( ˆ
ˆ )( ˆ
ˆ
merc
_
conc
=β+δ
+β+δ
)
year
.
(11.10)
ij
0
0,
i
1
1,
i
j
The ^ notation in Equation (11.10) suggests that replacing the unknown
parameters from Equation (11.5) with their estimated values from the anal-
ysis. Recall that the values of ˆ 0 and ˆ 1 represent the estimated mercury
concentration in the initial study year and the yearly rates of change, respec-
tively, associated with the entire study area (i.e., the population from which
the sampling units were drawn). The values of ˆ 0 and ˆ 1 for the mercury
concentration data are found in Table 11.3 in the section “Fixed Effects” with
β=
ˆ 0.013 1 . Note that these estimates are precisely the same
estimates observed when taking the sample means of the unit intercepts and
slopes in Equation (11.2). Also, note that the estimated standard error of ˆ 1
is reported to be 0.0030, and that this value is the same standard error com-
puted for ˆ 1 in Equation (11.3) when conducting the unit analysis.
To obtain the individual prediction equations for each sampling unit
requires these predicted values of ˆ 0 and ˆ 1 and the predicted values of
δ
ˆ
0.1864
and β=
0
ˆ
and ˆ
δ
,1 . The predicted trend equation for station 90 is given by
i
,0
i
( ˆ
ˆ
)( ˆ
ˆ
merc
_
conc
=β+δ
+β+δ
)
year
,
90
0
0,1 90
=
1
1,90
i
=
and the predicted trend equation for the station with the least trend (station
81) is given by
( ˆ
ˆ
)( ˆ
ˆ
merc
_
conc
=β+δ
+β+δ
)
year
.
81
0
0,81
i
=
1
1, 81
i
=
With the information from Table 11.4,
merc
_
conc
=
0.1488 0.0244
+
year
,
90
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