Uncertainty, Emergence, and Statistics in Dendrochronology - Dendroclimatology - page 87

will use a two-tailed hypothesis test. For examples of using a one-tailed hypothesis
test in modeling the effects of temperature on tree growth in dendrochronology, see
Pederson et al. ( 2004 ) .
The interpretation of any given -level probability is complicated by the fact that

we often mine tree-ring data for statistically significant associations with climate
through multiple tests of the same statistic. This is because we do not know what
to expect more often than we would like. For example, if we wish to calculate the
monthly correlations between tree rings and climate over, say, a 16-month dendro-
climatic year (Fritts 1976 ; Blasing et al. 1984 ) beginning in the May of the previous
year and ending in August of the current year of growth, and we do this for both tem-
perature and precipitation, we are making 32 tests simultaneously and then looking
for any statistically significant correlations to tell us how to interpret the tree-ring
series as a function of climate. Unfortunately, this multiplicity of tests weakens the
apriori

α

-level probability for any given correlation out of the 32 candidates based
on a standard t -test for r , and one can therefore no longer legitimately claim that any
given correlation is truly significant at the a priori 1 -

α

level. Most reported associa-
tions between tree rings and climate in the literature ignore this multiplicity problem
in describing statistically significant months in climate correlation functions.
A simple correction for the effects of multiplicity on significance levels can
be made as P

α

p ) m , where p is the a priori probability (same as our

= 1 − (1 −

α -level described above), m is the number of tests being made, and P is the resulting

a posteriori probability (Mitchell et al. 1966 ; Yamaguchi 1994 ) . In our hypothetical
correlation function example, if p

0.806, which does
not instill much confidence in our statistical interpretations if the correlations barely
exceed the a priori 95% significance level to begin with. To achieve an a posteri-
ori probability P

= 0.05 and m = 32, then P =

0.05 requires an approximate a priori probability p ~0.001 in
our hypothetical case. This is equivalent to an a priori 95% significance level for r
of ~0.25 and an a posteriori 95% significance level of ~0.46 for all 32 correlations
considered jointly. This correction for multiplicity would render many correlations
between tree rings and climate reported in the literature (including ours!) not statis-
tically significant. As will be shown by example below, we should not err on the side
of excessive statistical rigor at this stage, because those same climate correlations
may yet have significant dendroclimatic meaning even if most of them do not exceed
the a priori 95% significance level, let alone the stringent correction for multiplic-
ity. So what do we mean here? An example of correlation and response function
analysis applied to an annual tree-ring chronology will serve as an illustration.

=

4.5 Correlation and Response Function Analysis

Consider the case where few, if any, simple correlations are statistically significant
( p < 0.05) after correction for multiplicity. Figure 4.3 shows such an example for
an eastern hemlock ( Tsuga canadensis ) tree-ring chronology from a xeric, quartzite
conglomerate, talus slope site located near Mohonk Lake, Ulster County, New York
(41.76 ◦ N, 74.15 ◦ W; elev. 379 m). Correlation and response functions (Fritts 1976 ;