Finding confidence limits on population growth rates: Bootstrap and analytic methods

Abstract

When predicting population dynamics, the value of the prediction is not enough and should be accompanied by a confidence interval that integrates the whole chain of errors, from observations to predictions via the estimates of the parameters of the model. Matrix models are often used to predict the dynamics of age- or size-structured populations. Their parameters are vital rates. This study aims (1) at assessing the impact of the variability of observations on vital rates, and then on model's predictions, and (2) at comparing three methods for computing confidence intervals for values predicted from the models. The first method is the bootstrap. The second method is analytic and approximates the standard error of predictions by their asymptotic variance as the sample size tends to infinity. The third method combines use of the bootstrap to estimate the standard errors of vital rates with the analytical method to then estimate the errors of predictions from the model. Computations are done for an Usher matrix models that predicts the asymptotic (as time goes to infinity) stock recovery rate for three timber species in French Guiana. Little difference is found between the hybrid and the analytic method. Their estimates of bias and standard error converge towards the bootstrap estimates when the error on vital rates becomes small enough, which corresponds in the present case to a number of observations greater than 5000 trees.

Evaluating the Reliability of Counting Bacteria Using Epifluorescence Microscopy

“…However, it is highly challenging to obtain the variance of the true arithmetic mean value or any other measures of central tendencies, like the median or harmonic mean, in microscopic methods from samples that do not have a normal distribution [5]. To estimate the bias and variance of the arithmetic mean value of bacterial counts irrespective of the number of fields of view counted, a bootstrap method could be used to estimate these statistics because the assumption of normality is not necessary [19][20][21][22][23].…”

Stock recovery rates are not the panacea to assess timber yield sustainability: Evidence from managed Central African forests

“…This would counteract any adjustment of the stock recovery by parameter tuning. Techniques for computing the confidence interval of a stock recovery rate have been presented elsewhere (Picard et al, 2008a(Picard et al, , 2009a, but implementing them would require additional data on vital rates collected in permanent sample plots in logged-over and undisturbed forests. We also recommend that the interpretation of the stock recovery rates should take account of the ecological profiles of the species, in particular when grouping species.…”

“…The sensitivity of the stock recovery rate R to a parameter x is the partial derivative r x = @R/@x, whereas the elasticity of the stock recovery rate to this parameter is: e x = @(lnR)/@(lnx) = (x/R) r x . The quantity r x  D gives the amount by which R would change if parameter x was changed by a small additive perturbation D, whereas e x  D gives the proportional change of R that would be brought by a small proportional perturbation of parameter x in a proportion D. Analytic expressions for the sensitivity and the elasticity of R in the general case can be obtained using matrix differential calculus (Magnus and Neudecker, 2007;Chagneau et al, 2009; Picard et al, 2009a). The specific expressions in the present case are given in Appendix A.…”

“…This is desirable because the sensitivities of the stock recovery rate to its parameters determine how the estimation error on the parameters propagates to the estimation of the stock recovery rate (Picard et al, 2009a). Because the estimation error of a parameter is often proportional to its value, the elasticities better reflect the estimation uncertainties than the sensitivities.…”

“…Appendix A specifies an analytic condition for this to occur. Nevertheless, a full comparison of the two formulas would require integrating the estimation errors of the parameters, to get the estimation error of the stock recovery rate (Picard et al, 2009a). It is worth noting that this computation is not currently done in the forest management plans of Central Africa: stock recovery rates are given without a confidence interval or any assessment of their estimation error.…”