2. Inferring Threat from Scientific Collections
19
tionally with fewer than 30 observations. Levels of accuracy may be unsatisfactory or unknown for asymptotic methods applied to sample sizes larger than 30.
Combinatorial methods provide “exact” rather than approximate or asymptotic
results, and their applications require relatively mild conditions or assumptions
about the data (Grimson 1993).
All the methods were poor at detecting relatively small changes in the underlying population. Changes such as a 20% step decline and a 0.5% continuous
decline were difficult to detect because of the low probability of there being empty
cells in the sequence. Grimson et al. (1992) anticipated this result, suggesting that
Grimson’s equation should have good power mostly in situations in which there is
a rationale for zero occurrences in certain periods of time.
Solow (1993) conducted power tests for Equation (2.1) and found that the test
had low power when the extinction time was near the end of the sequence, and
when prior to extinction, the population had declined to a level at which the
sighting rate was close to zero. The results here suggest that the formulas had low
power when the rate of decline was 1% or less or when the magnitude of a step
decline was less than about 50%.
Perhaps counter-intuitively, Solow’s equation was more likely to detect a
change in sequences with a 2% continuous decline than in sequences with a 5%
continuous decline, when the mean number of observations per time step was less
than 0.2 (Fig. 2.3). This is due to a relationship between the length of a run of
zeros at the end of the sequence and the number of observations made before the
population became extinct. The scenario involving a 2% continuous decline has a
minimum of 50 empty cells at the end of the sequence, and the 5% continuous
decline has a minimum of 80 empty cells at the end of the sequence. The number
of time cells from the first observation to the last observation is divided by the
total sequence length (100 time cells) and is then raised to the power of the
number of observations (Equation [2.1]). As it takes longer for the population
experiencing a 2% decline to reach extinction than it does for the population
experiencing a 5% decline, the value of n will be higher in the scenario involving a
2% decline. The effect of the increased sample size results in it being easier to
detect situations with more observations and a shorter run of empty cells.
The power of the formulas to detect the step declines (Fig. 2.4) was slightly
higher than their ability to detect the corresponding continuous declines (Fig. 2.3).
For example, a stepped decline of 100% may be considered equivalent to a
continuous decline of 2% because both result in the loss of the population in the
50th time step. The stepped decline scenario results in more observations before
the loss of the population. As a result, all equations are more likely to detect a
change in a population that experiences a step decline than in a population that
experiences a continuous decline, even if the populations are lost at the same time.
Relevance of Taxonomy
The observation that 44 conservation taxa were not recognized prior to 1970
serves to make the somewhat obvious point that conservation research, and the
19
tionally with fewer than 30 observations. Levels of accuracy may be unsatisfactory or unknown for asymptotic methods applied to sample sizes larger than 30.
Combinatorial methods provide “exact” rather than approximate or asymptotic
results, and their applications require relatively mild conditions or assumptions
about the data (Grimson 1993).
All the methods were poor at detecting relatively small changes in the underlying population. Changes such as a 20% step decline and a 0.5% continuous
decline were difficult to detect because of the low probability of there being empty
cells in the sequence. Grimson et al. (1992) anticipated this result, suggesting that
Grimson’s equation should have good power mostly in situations in which there is
a rationale for zero occurrences in certain periods of time.
Solow (1993) conducted power tests for Equation (2.1) and found that the test
had low power when the extinction time was near the end of the sequence, and
when prior to extinction, the population had declined to a level at which the
sighting rate was close to zero. The results here suggest that the formulas had low
power when the rate of decline was 1% or less or when the magnitude of a step
decline was less than about 50%.
Perhaps counter-intuitively, Solow’s equation was more likely to detect a
change in sequences with a 2% continuous decline than in sequences with a 5%
continuous decline, when the mean number of observations per time step was less
than 0.2 (Fig. 2.3). This is due to a relationship between the length of a run of
zeros at the end of the sequence and the number of observations made before the
population became extinct. The scenario involving a 2% continuous decline has a
minimum of 50 empty cells at the end of the sequence, and the 5% continuous
decline has a minimum of 80 empty cells at the end of the sequence. The number
of time cells from the first observation to the last observation is divided by the
total sequence length (100 time cells) and is then raised to the power of the
number of observations (Equation [2.1]). As it takes longer for the population
experiencing a 2% decline to reach extinction than it does for the population
experiencing a 5% decline, the value of n will be higher in the scenario involving a
2% decline. The effect of the increased sample size results in it being easier to
detect situations with more observations and a shorter run of empty cells.
The power of the formulas to detect the step declines (Fig. 2.4) was slightly
higher than their ability to detect the corresponding continuous declines (Fig. 2.3).
For example, a stepped decline of 100% may be considered equivalent to a
continuous decline of 2% because both result in the loss of the population in the
50th time step. The stepped decline scenario results in more observations before
the loss of the population. As a result, all equations are more likely to detect a
change in a population that experiences a step decline than in a population that
experiences a continuous decline, even if the populations are lost at the same time.
Relevance of Taxonomy
The observation that 44 conservation taxa were not recognized prior to 1970
serves to make the somewhat obvious point that conservation research, and the