Uncertainty, Emergence, and Statistics in Dendrochronology - Dendroclimatology - page 90

1991 ; Biondi and Waikul 2004 ) and (2) by using a t -test derived from principal com-
ponents regression theory (Fekedulegn et al. 2002 ) . We use the t -test described by
Fekedulegn et al. ( 2002 ) here.
The number of climate eigenvectors included in the response function (four in
our example) also affects the number of resulting 'significant' variables. This effect
is illustrated in the four steps used in calculating the final response function. The
Step-1 response function (Fig. 4.4a ) is based on only one eigenvector that explains
19.5% of the variance. Yet, all 32 monthly variables are 'significant' based on the
calculated 95% confidence limits. Step-2 cumulatively explains 25% of the variance,
but now has only 26 'significant' variables. Step-3 cumulatively explains 30.3% of
the variance and has 27 'significant' variables. Final Step-4 cumulatively explains
34.2% of the variance and now has only 24 significant variables. So, from Step-1 to
Step-4, the explained variance increases by 75%, but the number of significant vari-
ables decreases by 25%. A similar result can be found in Fritts ( 1976 , Fig. 7.13 ,p.
367). This tendency for an inverse dependence between the number of eigenvectors
in a response function model and the resulting number of 'significant' monthly vari-
ables means that one must be careful about interpreting response functions based
on few climate eigenvectors because the number of 'significant' months may be
inflated; this will be especially the case when only one climate eigenvector is used.
Using simulation methods, Cropper ( 1982b ) illustrated how the number of 'sig-
nificant' months can be inflated in response functions. This inflation occurs in part
because the monthly patterns of the climate eigenvectors are determined by the
intercorrelations between the monthly variables themselves. Some of these climate
intercorrelations will be based on true physical associations (e.g., temperature may
be inversely correlated with precipitation for a given month or season). However,
some of the intercorrelations will also be unique to the analysis period or occur by
chance alone, and thus have little or no true physical meaning. Yet, they will show
up in the monthly patterns of the climate eigenvectors and may be carried into the
response function when they are part of the climate eigenvectors that best explain
tree growth. We must always keep in mind that the climate eigenvectors are math-
ematically defined orthogonal modes that are not constrained to have any physical
meaning whatsoever (although they will often have some in practice) and certainly
do not have any inherent biological meaning. So the number of statistically signif-
icant months in a response function is not a good diagnostic for determining the
successful application of the method. Response function monthly confidence inter-
vals also do not take into account the multiplicity problem described earlier. Gray
et al. ( 1981 ) tackled this problem by using the binomial distribution to determine the
minimum number of significant ( p < 0.05) months needed for a response function
to have overall significance ( P < 0.05). For response functions based on 32 monthly
coefficients (Fig. 4.3b ) , one needs a minimum of four to five significant coefficients
( p < 0.05) to claim that the response function is significant (overall response func-
tion P < 0.05), which in our example is the case. Our correlation function would
barely pass this test, however.
This finding brings up another issue that can greatly affect the estimation of
response functions. Fekedulegn et al. ( 2002 ) note that response function analysis