2. Inferring Threat from Scientific Collections
9
p =
C e
C T
n
(2.1)
and it gives the likelihood that a species has not become extinct since the last
observation.
The time period (0, T) is partitioned into C T equally sized units of time (months
or years). C e is the number of time intervals between the start of observations and
the last collection. The total number of observations for the species is n.
Grimson et al. (1992) developed a test for the longest run of empty cells in a
time series composed of counts. It is a generalization of Bradley’s (1968) runs test,
which calculates the probability of the longest run of empty cells (time cells with
no observations) in the sequence of cells, C T . Grimson’s test assumes that the
occurrences of observations (counts) are independent. The probability of a run of
empty cells as long or longer than the longest run of observed consecutive absences is
p = C T
−n ͚
i ≥ 1
k ≥ 1
(−1)
k +1
j + 1
k (C T − rk) i S(n, j)
(2.2)
where ( ) j is a falling factorial (for an integer, a, the falling factorial (a) j = a(a−1)
. . . (a − j+1), and S( ) is a Stirling number of the second kind for which Grimson et
al. (1992) provide a table of values for small values of n and j. The term r is the
maximum number of consecutive empty units of time within the sequence of C T
units of time, and n is the total number of cases arising in the C T units.
Detecting systematic changes in natural populations is a type of trend analysis.
McCarthy (1998) suggested that changes in abundance (represented by the number of collections of a species per unit time) through time may be detected more
efficiently by incorporating an index of collection effort such as the total number
of records of, say, all plant species in the relevant database. He rewrote Solow’s
(1993) equation as
p =
C e
͚
i =1
C T
͚
i =1
e i
e i
n
(2.3)
where e i is the index of collection effort each time step, C e , and C T and n are
defined above. When the collection effort is constant, the equation reduces to
Solow’s (1993) equation, and McCarthy (1998) called it the partial Solow equation, because collection effort is treated as a covariate. Solow’s equation determines the probability of the run of empty cells occurring at the end of a sequence,
giving the probability that a taxon still exists. The equation is a special case of
Grimson’s equation, with the run of empty cells restricted to the end of the
sequence. Solow’s equation results in relatively large p values (i.e., small probabilities of extinction) if a single observation of a species was made relatively
9
p =
C e
C T
n
(2.1)
and it gives the likelihood that a species has not become extinct since the last
observation.
The time period (0, T) is partitioned into C T equally sized units of time (months
or years). C e is the number of time intervals between the start of observations and
the last collection. The total number of observations for the species is n.
Grimson et al. (1992) developed a test for the longest run of empty cells in a
time series composed of counts. It is a generalization of Bradley’s (1968) runs test,
which calculates the probability of the longest run of empty cells (time cells with
no observations) in the sequence of cells, C T . Grimson’s test assumes that the
occurrences of observations (counts) are independent. The probability of a run of
empty cells as long or longer than the longest run of observed consecutive absences is
p = C T
−n ͚
i ≥ 1
k ≥ 1
(−1)
k +1
j + 1
k (C T − rk) i S(n, j)
(2.2)
where ( ) j is a falling factorial (for an integer, a, the falling factorial (a) j = a(a−1)
. . . (a − j+1), and S( ) is a Stirling number of the second kind for which Grimson et
al. (1992) provide a table of values for small values of n and j. The term r is the
maximum number of consecutive empty units of time within the sequence of C T
units of time, and n is the total number of cases arising in the C T units.
Detecting systematic changes in natural populations is a type of trend analysis.
McCarthy (1998) suggested that changes in abundance (represented by the number of collections of a species per unit time) through time may be detected more
efficiently by incorporating an index of collection effort such as the total number
of records of, say, all plant species in the relevant database. He rewrote Solow’s
(1993) equation as
p =
C e
͚
i =1
C T
͚
i =1
e i
e i
n
(2.3)
where e i is the index of collection effort each time step, C e , and C T and n are
defined above. When the collection effort is constant, the equation reduces to
Solow’s (1993) equation, and McCarthy (1998) called it the partial Solow equation, because collection effort is treated as a covariate. Solow’s equation determines the probability of the run of empty cells occurring at the end of a sequence,
giving the probability that a taxon still exists. The equation is a special case of
Grimson’s equation, with the run of empty cells restricted to the end of the
sequence. Solow’s equation results in relatively large p values (i.e., small probabilities of extinction) if a single observation of a species was made relatively