Geoscience Reference
In-Depth Information
Dennis and Taper (1994) offered a procedure known as the parametric
bootstrap likelihood ratio test that was similar to that of Pollard et al. (1987).
Instead of permuting the data they generated 1,000 simulated data sets under
the null model (equation 6.2) of density independence with randomly chosen
error terms. As Pollard et al. (1987) had done, they then calculated a distribu-
tion of slopes with which they compared the actual slope generated from the
data by ordinary linear regression. After applying both of these techniques to
the data given in figure 6.1, I found only equivocal support for density
dependence (Elkinton et al. 1996). Three of eight population time series were
identified as density dependent (at
p
< 0.05) with the test of Dennis and Taper
(1994) and two out of eight with the test of Pollard et al. (1987).
However, neither of these techniques is without problems; first they all lack
statistical power. One needs 20-30 generations of data to reliably find density
dependence when it exits (Solow and Steele 1990; Dennis and Taper 1994).
Data sets that long are rare in ecology. Second, Shenk et al. (1998) simulated
the effect of measurement error on both of these tests. They concluded that
both tests were highly prone to type I error (concluding density dependence
when it did not exist) and hence of little use when measurement error is sig-
nificant, as it usually is in most data sets. Dennis and Taper (1994) conducted
analogous simulations and found that their test was robust against measure-
ment error. The difference in the two studies was in how measurement error
was modeled and it is too early to tell which view will prevail. Finally, correla-
tions between error terms from one year to the next can also lead to spurious
conclusions of density dependence when it does not exist (Solow 1990; Red-
dingius 1990). It would not be very surprising to have autocorrelated errors
because they include the effects of all other variables on population growth
other than density. Suppose a population was determined largely by the action
of a generalist predator whose density was not linked to that of its prey and
that might or might not cause density-dependent mortality. Variation in pre-
dation rates caused by fluctuation in predator density would be embodied in
the error term and would probably be influenced by predator densities at pre-
vious time steps. The error term would thus be autocorrelated (Williams and
Liebhold 1995).
ANALYSIS OF DATA ON MORTALITY OR SURVIVAL
Many investigators collect data not just on density but on mortality or fraction
surviving in particular age categories or life stages. They may be interested
in particular agents of mortality and want to know whether these are capable
