12
Mark Burgman et al.
species collected in each year as a covariate. The results of the analyses were
compared with current listings of conservation taxa and were interpreted by one of
us (BM) familiar with their abundance, distribution, and taxonomic status.
Results
Type I Error Rates
The type I error rate for a formula (the α value) is the probability of rejecting the
null hypothesis that the observations are randomly distributed when the null
hypothesis is, in fact, true. Convention has set the acceptance criterion for the type
I error rate at 0.05, or the probability that one case in 20 or fewer will be
considered significant when there is no true significance. The formulas are well
specified for the convention of α = 0.05. For the scenario in which there was no
change in the underlying population, the equations returned a type I error rate that
was equal to or less than 0.05, for all levels of the mean number of observations
per time period.
Statistical Power
The power of a statistic is the probability of detecting a change when there is a
change present (i.e., the probability of rejecting the null hypothesis when it is
false). Statistical power equals (1 − β), where β is the type II error rate, the
probability of accepting the null hypothesis when it is false (i.e., the probability of
failing to detect a change when a change has occurred). The statistical power of a
formula in this study was equivalent to the proportion of replications detected as
being significantly different from random in each of the scenarios that involve a
continuous or a stepped decline. Power will generally increase with increasing
sample size (Siegal and Castellan 1988).
Neither of the equations was effective at detecting continuous declines for rates
of 0.5% per time step (Fig. 2.3). There was no corresponding increase in power
with an increase in mean number of observations per time step. When the underlying population declined by 1% per time step, the equations were likely to detect a
change if the mean number of observations per time step was greater than about
0.25 for the period of 100 time steps. At decline rates of 1% and higher, an
increase in mean number of observations per time step corresponded with an
increase in power. The likelihood that a true change would be detected approached
100% when the rate of decline was 2% or higher and the mean number of
observations per time step was greater than about 0.5.
Solow’s equation was the most powerful in circumstances in which there was a
continuous decline in the underlying population size. This conclusion held for all
rates of decline and all levels of the mean number of observations per time step. In
most cases, most of the data generated by random sampling resulted in single
observations within time cells. The relative power of Grimson’s equation may
Mark Burgman et al.
species collected in each year as a covariate. The results of the analyses were
compared with current listings of conservation taxa and were interpreted by one of
us (BM) familiar with their abundance, distribution, and taxonomic status.
Results
Type I Error Rates
The type I error rate for a formula (the α value) is the probability of rejecting the
null hypothesis that the observations are randomly distributed when the null
hypothesis is, in fact, true. Convention has set the acceptance criterion for the type
I error rate at 0.05, or the probability that one case in 20 or fewer will be
considered significant when there is no true significance. The formulas are well
specified for the convention of α = 0.05. For the scenario in which there was no
change in the underlying population, the equations returned a type I error rate that
was equal to or less than 0.05, for all levels of the mean number of observations
per time period.
Statistical Power
The power of a statistic is the probability of detecting a change when there is a
change present (i.e., the probability of rejecting the null hypothesis when it is
false). Statistical power equals (1 − β), where β is the type II error rate, the
probability of accepting the null hypothesis when it is false (i.e., the probability of
failing to detect a change when a change has occurred). The statistical power of a
formula in this study was equivalent to the proportion of replications detected as
being significantly different from random in each of the scenarios that involve a
continuous or a stepped decline. Power will generally increase with increasing
sample size (Siegal and Castellan 1988).
Neither of the equations was effective at detecting continuous declines for rates
of 0.5% per time step (Fig. 2.3). There was no corresponding increase in power
with an increase in mean number of observations per time step. When the underlying population declined by 1% per time step, the equations were likely to detect a
change if the mean number of observations per time step was greater than about
0.25 for the period of 100 time steps. At decline rates of 1% and higher, an
increase in mean number of observations per time step corresponded with an
increase in power. The likelihood that a true change would be detected approached
100% when the rate of decline was 2% or higher and the mean number of
observations per time step was greater than about 0.5.
Solow’s equation was the most powerful in circumstances in which there was a
continuous decline in the underlying population size. This conclusion held for all
rates of decline and all levels of the mean number of observations per time step. In
most cases, most of the data generated by random sampling resulted in single
observations within time cells. The relative power of Grimson’s equation may