Geoscience Reference
In-Depth Information
is estimated to be increasing at a rate of 0.0131 ppm per year, so that for a
10-year period, the expected increase in concentration would be 0.131 ppm.
Certainly, there is sampling variability associated with the estimate pro-
vided in Equation (11.2), and a 95% confidence interval on this trend could
be computed using
(
ˆ
sd
β
s
)
ˆ
..(
ˆ
)
ˆ
i
β±
t
s e
β=β±
2.306
8,0.025
10
0.0131 2.306
0.0094
10
(11.3)
=
±
0.01311 0.0069
=
±
(
ˆ
i
denotes the standard deviation of the sample of observed slope
coefficients associated with the 10 stations. In summary, it is estimated that
the expected per annum increase in mercury concentration across the entire
study area is between 0.0062 and 0.0200 ppm. Note from Equation (11.3) that
inference about the entire study area involves a
t
statistic with 8 degrees of
freedom. The 8 degrees of freedom are calculated as 10 − 2 because there are
10 sampling units and 2 regression parameters.
It should be noted that the data from each of the stations are not indepen-
dent in this study because each station was repeatedly surveyed, although
there is independence among the sampling units because the units were ran-
domly selected. If the data were analyzed as if there are
n
− 2 degrees of
freedom and the repeated measures structure of the data were ignored, the
120 observations would be pooled and one regression model fitted as
where
sd
β
s
merc
_
conc
=β +β
year
+ε
.
(11.4)
ij
j
ij
0
1
In Equation (11.4), there is a common population intercept β
0
and slope β
1
; the
resulting analysis is given in Table 11.1. The overall estimates of the popula-
tion intercept and slope are identical to the sample means of the estimated
intercepts and slopes, respectively, from the unit analysis, but the reported
standard errors,
t
values, and
p
values are all based on an assumed sample
size of 120 instead of the true sample size of 10. When this sort of mistake is
made, confidence intervals will be much narrower than they should be, and
there will be a high type I error rate when reporting results (i.e., your chance
of concluding effects to be statistically significant when they are not truly
significant will be much higher than your stated value of α).
