Calculating unbiased tree-ring indices for the study of climatic and environmental change

In dendroclimatology, tree-ring indices are traditionally calculated as part of the tree-ring chronology development process. This is accomplished by fitting a growth curve to the ring-width series and using it as a series of expectations for more or less well specified null conditions (uniform climate perhaps) of annual radial growth. The ratio of the actual ring widths to these expectations produces a set of dimensionless indices that can be averaged arithmetically with cross-dated indices from other trees into a mean chronology suitable for studies of climatic and environmental change. We show that tree-ring indices calculated in this manner can be systematically biased. The shape of this bias is defined by the reciprocal of the growth curve used to calculate the indices, and its magnitude depends on the proximity of the growth curve to the time axis and its intercept. The underlying cause, however, is lack of fit. To avoid this bias, residuals from the growth curve, rather than ratios, can be computed. If this is done, in conjunction with appropriate transformations to stabilize the variance, the resulting tree-ring chronology will not be biased in the way that ratios can be. This bias problem is demonstrated in an annual tree-ring chronology of bristlecone pine from Campito Mountain, which has been used previously in global change studies. We show that persistent growth increase since AD 1900 in that series is over-estimated by 23.6% on average when ratios are used instead of residuals, depending on how the ring widths are transformed. Such bias in ratios is not always serious, as it depends on the joint behaviour of the growth curve and data, particularly near the ends of the data interval. Consequently, ratios can still be used safely in many situations. However, to avoid the possibility of ratio bias problems, we recommend that variance-stabilized residuals be used.