Analysis of Trends in Low-Flow Time Series of Canadian Rivers - Hydrologic Time Series Analysis: Theory and Practice - page 206

trends, while it is true (Douglas et al., 2000). The term 'field significance', in
contrast to 'point or local significance', has been brought into hydrology from
climatology/meteorology. This approach was introduced in the work of Livezey
and Chen (1983). When the cross correlation in a gauging network is negligible,
the theory of binomial distribution can be used to identify field significance.
However, when the cross correlation cannot be ignored (e.g. observed and
climate model simulated rainfall fields), the methods based on the Monte
Carlo simulations and vector block bootstrap resampling approach can be
used to determine field significance of identified trends. Details of all these
methods along with their advantages and disadvantages can be found in Livezey
and Chen (1983), Yue et al. (2003), Elmore et al. (2006) and Khaliq et al.
(2009a).
In the above mentioned techniques of field significance analysis, the
number of sites with significant trends at a given level of local significance
(i.e., the significance level used to identify trends at each of the selected sites)
are counted and assessed if this number has arisen due to purely coincidence.
Because of the involved counting procedure, these methods of field significance
testing are generally categorized as counting techniques. These techniques
have been criticized because of the integer valued nature of the result of the
counting procedure and because of the binary view of the results of local
testing. Local null hypotheses that are very strongly rejected (i.e. local p -
values that are very much smaller than the local significance level) carry no
greater weight in the field significance test than do local tests for which the p -
values are only slightly smaller than the local significance level (Wilks, 2006).
In addition to this issue, the above tests only indicate whether the overall
results are field significant or not but they do not specify where and how the
results are field significant. These shortcomings of the counting procedures to
field significance assessment can in general be improved upon through the
use of test statistics that depend on the magnitudes of individual p -values of
the local tests. One of such kind of tests is the false discovery rate (FDR) test
proposed by Benjamini and Hochberg (1995), which is a relatively new
statistical procedure for simultaneous evaluation of multiple tests by recognizing
that a certain number of false rejections of the null hypothesis are to be
expected. Ventura et al. (2004) and Wilks (2006) demonstrated through
extensive simulation experiments that the FDR test is robust to spatial
correlations. This procedure works with any statistical test for which one can
generate a p -value. Thus, as long as the effects of serial structure of time series
is taken care of appropriately for evaluating at-site p -values in a hydrological
network/region, the FDR test could be applied for field significance analysis.
A step-by-step procedure for applying the FDR test can be found in Wilks
(2006) and an application of this method to hydrological time series in Khaliq
et al. (2009a).