10
Mark Burgman et al.
recently, irrespective of any evidence of population decline through the observation period. Grimson’s equation is sensitive to patterns of observations other than
the time since the last observation.
Simulations
Power tests for these equations involve setting up a scenario in which the “true”
circumstances concerning a species’ decline or loss are known. Then, the scenario
is sampled repeatedly, and the equation to be tested is applied to the resulting data.
Power is measured by the probability that an equation results in a significant test
whenever there is a real difference.
There is some probability of observing an extant species in each time step. The
applications described above assume implicitly that this probability is a function
of the underlying population size of the species. If the underlying scenario involves no real change in population size, the equations should produce, on average, a significant result fewer than once in 20 replications (i.e., the type I error rate
is less than 5%). If the underlying scenario involves a real change in the population size, the best method will be the one that is most likely to produce a significant result. The type II error rate is the probability of concluding the patterns are
random, when, in fact, there is an underlying change. A more powerful method is
more likely to detect a true change. In these examples, we assumed that survey
intensity remained constant throughout the period, so that McCarthy’s (1998)
equation reduced to Solow’s (1993) equation, and the results from the two equations were identical.
Three scenarios were developed. In the first, population size did not change for
the duration of the observation period. The mean number of observations per time
step was set at one of five levels (i.e., 0.05, 0.1, 0.2, 0.5, and 1.0 observations per
unit time). The frequency of observations in each cell (time step) was sampled
from a Poisson distribution, and each level of the mean number of observations
per time step was replicated 1,000 times. This first scenario was used as a control
to test the α level (the type I error rate) of the formulas.
In the second scenario, the underlying population size declined linearly. The
rate of decline was set at one of four levels: 0.5, 1, 2, and 5%. The mean number of
observations per time step declined from a maximum of one in each case (Fig.
2.1). The third scenario involved a stepped decline at the midpoint of the time
period, the 50th time cell. The magnitude of the decline was set at one of five
levels: 20, 50, 70, 90, and 100%. As in the first two scenarios, the mean number of
observations per time step declined from a maximum of one per time step in each
case (Fig. 2.2). Each combination of decline and frequency of observation for
each of the scenarios was replicated 1,000 times.
Analyses of the Acacia Data
The information associated with collections of all Western Australian conservation Acacia taxa toward the end of 1995 was extracted from the specimen database
Mark Burgman et al.
recently, irrespective of any evidence of population decline through the observation period. Grimson’s equation is sensitive to patterns of observations other than
the time since the last observation.
Simulations
Power tests for these equations involve setting up a scenario in which the “true”
circumstances concerning a species’ decline or loss are known. Then, the scenario
is sampled repeatedly, and the equation to be tested is applied to the resulting data.
Power is measured by the probability that an equation results in a significant test
whenever there is a real difference.
There is some probability of observing an extant species in each time step. The
applications described above assume implicitly that this probability is a function
of the underlying population size of the species. If the underlying scenario involves no real change in population size, the equations should produce, on average, a significant result fewer than once in 20 replications (i.e., the type I error rate
is less than 5%). If the underlying scenario involves a real change in the population size, the best method will be the one that is most likely to produce a significant result. The type II error rate is the probability of concluding the patterns are
random, when, in fact, there is an underlying change. A more powerful method is
more likely to detect a true change. In these examples, we assumed that survey
intensity remained constant throughout the period, so that McCarthy’s (1998)
equation reduced to Solow’s (1993) equation, and the results from the two equations were identical.
Three scenarios were developed. In the first, population size did not change for
the duration of the observation period. The mean number of observations per time
step was set at one of five levels (i.e., 0.05, 0.1, 0.2, 0.5, and 1.0 observations per
unit time). The frequency of observations in each cell (time step) was sampled
from a Poisson distribution, and each level of the mean number of observations
per time step was replicated 1,000 times. This first scenario was used as a control
to test the α level (the type I error rate) of the formulas.
In the second scenario, the underlying population size declined linearly. The
rate of decline was set at one of four levels: 0.5, 1, 2, and 5%. The mean number of
observations per time step declined from a maximum of one in each case (Fig.
2.1). The third scenario involved a stepped decline at the midpoint of the time
period, the 50th time cell. The magnitude of the decline was set at one of five
levels: 20, 50, 70, 90, and 100%. As in the first two scenarios, the mean number of
observations per time step declined from a maximum of one per time step in each
case (Fig. 2.2). Each combination of decline and frequency of observation for
each of the scenarios was replicated 1,000 times.
Analyses of the Acacia Data
The information associated with collections of all Western Australian conservation Acacia taxa toward the end of 1995 was extracted from the specimen database