Geoscience Reference
In-Depth Information
4.4 Statistics
Our rather lengthy, somewhat philosophical, discussions of uncertainty and emer-
gence have set the stage now for our discussion of statistical analysis in den-
drochronology. Dendrochronology is predominantly an empirical science that
is deeply rooted in statistical modeling and its attendant probabilistic nature.
Therefore, biological uncertainty and emergence are generally still expressed
through statistical descriptions of the data. This approach places an added burden
on the results and interpretations of dendrochronological analyses because, as we all
know, statistical analysis can lead to the incorrect acceptance of apparently strong,
yet utterly false, statistical associations. Hence the famous quote fromMark Twain's
and statistics.' Clearly, Mark Twain did not have much trust in statistics!
In statistical jargon, the generic way in which we test empirical associations is
through
hypothesis testing
. We start out by assuming that no experimental treatment
effect or association exists between the statistical samples or variables being com-
pared. This is our
null hypothesis
.The
alternate hypothesis
is simply the converse
of the null hypothesis; i.e., that the experimental treatment effect or association is
detectable with some level of confidence. In standard statistical notation, the null
and alternate hypotheses are expressed as H
0
:
ρ
=
0 and H
a
:
ρ
=
0, respectively,
where
is the statistic being tested, such as the Pearson correlation coefficient
r
.
We then test the statistical significance of the treatment effect or association via
an appropriate statistical test. When the null hypothesis is falsely rejected, this is
known as a Type-1 error in statistics. In other words, we have concluded that the
tested effect or association is true, when in fact it is not true. This is a serious mis-
take that can lead us down the wrong scientific path. To counterbalance Type-1 error,
there is Type-2 error, which is the false rejection of the alternate hypothesis when
it is true. This mistake is also serious, but in a different way, because now we may
have missed an important scientific discovery.
The balancing act between Type-1 and Type-2 errors is generally based on
the chosen
ρ
α
-level probability used for rejecting the null hypothesis. Commonly,
α =
0.05 is selected (a 1-in-20 chance of being wrong), which is equivalent to the
95% significance level (1 -
α
), but there is no theoretical reason for picking any
particular
-level. It all depends on how willing the analyst being wrong is in not
accepting the null hypothesis as true. The '1-in-20 chance of being wrong' example
also illustrates the way in which statistical hypothesis testing is generally weighted
towards accepting the null hypothesis. What also matters is the choice of a one-tailed
or two-tailed hypothesis test. If the sign of the outcome has no a priori expectation
(e.g., the correlation
r
may be either positive or negative to be statistically signifi-
cant), a two-tailed hypothesis test is used. Conversely, if the sign of the outcome can
only be positive or negative to be meaningfully significant, a one-tailed hypothesis
test is used. In this case, the alternate hypothesis notation changes from H
a
:
α
ρ
=
0
to either H
a
:
> 0 to account for the sign of the outcome being impor-
tant. In our later examples of modeling the climate signals in tree rings, we do not
assume that we know the signs of the statistical associations a priori and, thus, we
ρ
<0orH
a
:
ρ
