Quantifying the risk of mis-estimating correlation significance of climate–tree growth relationships

Abstract

In dendroecology, sampling effort has a strong influence of both regional chronology properties and climate–tree growth relationships assessment. Recent studies evidenced that decreasing sample size leads to a weakening of the bootstrapped correlation coefficients (${\text{BCC}}$). The present analysis focused on the risk of mis-estimating the significance of population ${\text{BCC}}\,\left( {{\text{BCC}}_{\text{POP}} } \right)$ from a sample of N trees, and then proposed an approach to detect and correct mis-estimations using the properties of the sample. The sample size effect and the limits of the correction were illustrated from 840 individual growth chronologies of Corsican pine (Pinus nigra Arnold ssp. laricio Poiret var. Corsicana) sampled in Western France. The 840 trees were used to assess the population characteristics, and the effect of sampling effort was investigated through a simulation approach based on a resampling procedure of N trees amongst 840 (N Є [5; 50]). Our results evidenced that the risk strongly varied amongst the climatic regressors. The highest risks were evidenced for significant ${\text{BCC}}_{\text{POP}}$, with a percentage of mis-estimation ranging from 25 to 80. On the contrary, small samples allowed providing an reliable estimation of the significance of non-significant ${\text{BCC}}_{\text{POP}}$. To a lesser extent, the risk slightly decreased with increasing N, according to a negative exponential trend. The detection and correction method was found relevant to detect mis-estimation only for significant ${\text{BCC}}_{\text{POP}}$; otherwise, the ${\text{BCC}}_{\text{POP}}$ significance was generally overestimated.

Appendix

Range of $\sigma_{\text{BIAS,CLM}}$

$BCC_{N}$ varies between −1 and 1; using the ratio $BCC_{N} /BCC_{\text{POP}}$, the range of $S_{N}$ is:

$$- \frac{1}{{{\text{BCC}}_{\text{POP}} }} \le S_{N} \le \frac{1}{{{\text{BCC}}_{\text{POP}} }} $$

(9)

Similarly, considering Eq. (7):

$$- \frac{1}{{{\text{BCC}}_{\text{POP}} \times B_{N} }} \le B_{sPOP} \le \frac{1}{{{\text{BCC}}_{\text{POP}} \times B_{N} }} $$

(10)

Equation (10) is formally equivalent to:

$$- \frac{{\sigma_{N} \times \sigma_{\text{CLM}} }}{{\sigma_{\text{POP,CLM}} }} \le B_{\text{sPOP}} \le \frac{{\sigma_{N} \times \sigma_{\text{CLM}} }}{{\sigma_{\text{POP,CLM}} }} $$

(11)

As $\sigma_{\text{BIAS,CLM}} = \left( {B_{\text{sPOP}} - 1} \right) \times \sigma_{\text{POP,CLM}}$ (cf. Eqs. (6) and (7)), it can be deduced that:

$$- \sigma_{N} \times \sigma_{\text{CLM}} - \sigma_{\text{POP,CLM}} \le \sigma_{\text{BIAS,CLM}} \le \sigma_{N} \times \sigma_{\text{CLM}} - \sigma_{\text{POP,CLM}} $$

(12)