### WWRC 86-08

Estimating Uncertainty in Population Growth Rates: Jackknife vs. Bootstrap Techniques

Abstract

Although per capita rates of increase (*r*) have been calculated by population biologists
for decades, the inability to estimate uncertainty (variance) associated with *r* values has until recently
precluded statistical comparisons of population growth rates. In this study, we used two computer-
intensive techniques, Jackknifing and Bootstrapping, to estimate bias, standard errors, and sampling
distributions of *r* for real and hypothetical populations of cladocerans. Results generated using the
two techniques, using data on laboratory cohorts of *Daphnia pulex*. were almost identical, as were
results for a hypothetical *D. pulex* population whose sampling distribution was approximately normal.
However, for another hypothetical population whose sampling distribution was negatively skewed
due to high juvenile mortality. Bootstrap and full-sample estimates of *r* were negatively biased by 3.3
and 1.8%, respectively. A bias adjustment reduced the bias in the Bootstrap estimate and produced
estimates of *r* and SE(*r*) almost identical to those of the Jackknife technique. In general, our simulations
show that the Jackknife will provide more cost-effective point and interval estimates of *r* for cladoceran
populations, except when juvenile mortality is high (at least > 25%). Coefficients of variation in the
mean of *r* within laboratory cohorts of *D. pulex* were one-half to one-third the magnitude of the
corresponding coefficients of variation in the mean of total reproduction and in the mean day to death
(range of values of CV[*r*] = 1.6 to 3.8%). This suggests that extremes in reproductive output and survival
of individuals tend to be dampened at the population level, and that within-cohort variability in *r* is
not explosive. Moreover, between-cohort variability in *r* can be much greater than within-cohort
variability, as indicated by a statistically significant difference of 30% (P << .01) between the high and
low *r* values that were computed for four cohorts of *D. pulex* born during a 1-mo period from the
same laboratory stock population. Based on variability in per capita rates of increase that have been
estimated for several cladoceran species, we suggest that the precision for reporting *r* values should
in most cases be limited to two significant figures.
*Key words: Bootstrap;* Daphnia pulex; *Jackknife; per capita rate of increase; statistical comparisons,
temporal variability.*

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