We started by plotting the
measurements over time at several sites.
It was visually apparent that some sites showed dramatic increases and
others, dramatic decreases. To capture
this phenomenon, we used three statistical tests:
1. T-test comparing mean measurement values before and
after a fixed cutoff date;
2. Mann-Kendall nonparametric test for trend;
3. Sen’s nonparametric estimate of slope.
At each site, for each
variable, we computed the mean value before and after a fixed cutoff date and
applied a (two-sided) T test for equality of means from populations of
different variances. The cutoff date
was the same for all sites.
The choice of cutoff date
was, of course, important. We tried to
choose a date that would minimize the effect of known seasonal variations. Soil vapor measurements are affected by both
temperature and soil moisture, and thus fluctuate by seasons. We elected to
compare changes before and after July 1, 1998—one year before the latest data
available for this preliminary report.
The hope is that by including at least one year of data in each
population, the effect of seasonal variations would be minimized as much as
possible. Some trial calculations
convinced us that, in fact, the results described below are relatively
insensitive to the choice of cutoff date.
At each site with at least
two measurements before and two measurements after the cutoff date, we computed
separately the number, the mean, and the variance of measurements before and
after the cutoff date. Considering the
measurements before and after the cutoff date as separate populations with unequal
variance, we computed, using a two-sided T test, the probability of equal
means.
Before applying the T test,
adjustments were made to the data to compensate for a change in measuring
instrument. For reasons explained
below, it also seemed advisable to try applying the T test to the logarithm of
one variable; however, this adjustment did not significantly affect the
results.
At each site, for each
variable, we listed all the pairs of dates for which a measurement was
available. For every pair of dates, we
computed the slope in units per day as
reading2 – reading1 .
date2 – date1
Sen’s estimator is the
median of all such slopes. We computed
approximate two-sided 95% and 99% confidence intervals, using a variant of the
procedure given in [Gilbert, 1987]. The
procedure assumes approximate normality.
We tried to ensure this by only computing the Sen’s estimator for sites
where ten or more observations were available.
Our variant of Gilbert’s procedure was to assume that all tied groups
had size exactly two. The net effect of
this assumption is conservative; some of the confidence intervals may be wider
than necessary. On the other hand, the
assumption is probably reasonable, since the precision of the instrument is high
enough that three or more exactly identical readings at different times are
unlikely.
Sen gives a numerical
estimate of the amount of increase or decrease in units per day at a site. We recorded, but did not use this feature in
the current study. We merely report the
number of sites where the 99% and 95% Sen confidence intervals do not contain
zero (i.e., the sites where there is a high probability of a trend).
The Mann-Kendall statistic
is similar to Sen’s estimator. At each
site, for each pair of dates, compute the slope as above, but only consider the
sign of the slope:
+1 if reading2
> reading1;
Sign of slope = 0 if reading2 = reading1;
-1 if reading2
< reading1.
The Mann-Kendall
statistic is the sum of all signs of slopes.
Using a procedure in [Gilbert, 1987], we computed standardized scores
and two-tailed confidence intervals for the Mann-Kendall number at each
site. These standardized scores can be
used to compute a chi-square test for homogeneity of trend across multiple
sites; the latter test was actually of little use in this case since little
spatial homogeneity was apparent.
Computation of standardized scores assumes approximate normality, which,
as in the Sen estimate, we attempted to ensure by only considering sites with
ten or more measurements.