How to Avoid Errors in Your Error Analyses and Gain Confidence in Your Confidence Intervals.

Calculations of forest biomass and elemental content require many measurements and models, each contributing uncertainty to the final estimates.  While sampling error is commonly reported, based on replicate plots, error due to uncertainty in the regression used to estimate biomass from tree diameter is usually not quantified.  Some published estimates of uncertainty due to the regression models have used the uncertainty in the prediction of individuals, ignoring uncertainty in the mean, while others have propagated uncertainty in the mean while ignoring individual variation.  Using the simple case of the calcium concentration of sugar maple leaves, we compare the variation among individuals (the standard deviation) to the uncertainty in the mean (the standard error) and illustrate the declining importance in the prediction of individual concentrations as the number of individuals increases.  For allometric models, the analogous statistics are the prediction interval (or the residual variation in the model fit) and the confidence interval (describing the uncertainty in the best fit model).  The effect of propagating these two sources of error is illustrated using the mass of sugar maple foliage.  The uncertainty in individual tree predictions was large for plots with few trees; for plots with 30 trees or more, the uncertainty in individuals was less important than the uncertainty in the mean.  Authors of previously published analyses have reanalyzed their data to show the magnitude of these two sources of uncertainty in scales ranging from experimental plots to entire countries. The most correct analysis will take both sources of uncertainty into account, but for practical purposes, country-level reports of uncertainty in carbon stocks, as required by the IPCC, can ignore the uncertainty in individuals.  Ignoring the uncertainty in the mean will lead to exaggerated estimates of confidence in estimates of forest biomass and carbon and nutrient contents.