Geoscience Reference
In-Depth Information
increase was consistent through time as in a linear trend or whether the rate
increased more quickly at the beginning of the study than at the end as in
some curved trends. For this, unit or pooled analyses can be used. The unit
analysis approach is the subject of the next section.
11.3 Unit Analyses of Trends
As mentioned above, in a unit analysis of trends separate trend analyses
are carried out for each of the sampling units. Whenever there is a random
sample of units with repeated measures on each unit, there are generally
two main scopes of inference: inference on the individual units themselves
and inference on the entire study area from which the units were sampled.
In unit analysis, interest is in comparing the results of analyses of each indi-
vidual unit to one another. First, consider the structural form of the trend for
each of the units (i.e., whether the trend is linear or nonlinear). Scatter plots
are a useful graphical summary to begin. In Figure 11.3, a scatter plot graph
with separate panels for each station is provided. Mercury levels appear to
be increasing over time at most of the stations, but the details of the trend
differ among stations. For example, station 89 showed little change in mer-
cury levels over time, whereas station 90 showed a fairly substantial increase
in mercury over the study period. Also, from Figure 11.3 it appears that a
linear trend is an appropriate description of the structural change, so that
the next step is to fit separate linear regressions for each of the stations.
A linear statistical model relating mercury concentration to study year is
given by
merc
_
conc
=β
+β
year
+ε
,
i
= 1, 2, . . . , 10;
j
= 0, 1, . . . , 11,
(11.1)
ij
0,
i
1,
i
j
ij
where
merc_conc
ij
denotes the measured mercury concentration for station
i
in study year
j
; β
0,
i
and β
1,
i
denote the
y
intercept and slope, respectively,
associated with the regression line for station
i
; and ε
ij
denotes the model
errors associated with each station. In a linear regression analysis, the model
errors are assumed to follow a normal distribution with mean 0 and con-
stant variance σ
2
. The use of the subscript
i
indicates that the model allows
for different
y
intercepts and slopes for each of the stations. Practically, the
intercepts for each station represent the mercury concentration in study year
0 or the baseline levels associated with each station. The slopes represent
the expected change in concentration on a per annum basis, with positive
values representing a positive annual change, negative values representing
a negative annual change, and values close to zero representing little to no
annual change.
