1. Power analysis on simulated data
In this section we report the outcomes obtained for a simulated decrease
in population size over time, as increasing and decreasing trends showed
very similar results. The results for the case of simulated population
increases are presented in supplementary materials.
Sample counts were able to correctly identify the direction of a
population trend if it was as strong as -40% or more over 10 or 20
years (fig.1), or for a change in abundance of -30% if the yearly trend
variability was 0.1 or lower. If the population showed a change in
abundance of -20% in 10 or 20 years, sample counts were able to detect
the trend only under low trend variability (cvy = 0.05)
and at least 15 sampled sectors. Under a 10% overall trend sample
counts never reached a sufficient statistical power in identifying the
direction of the population trend, even sampling the entire area. When
the direction of the trend was detected, its magnitude was also
correctly identified with an error lower than 5% on the yearly trend.
However, sample counts were never able to reach a sufficient statistical
power in estimating the 10-year trend with an annual error lower than
2%, even when sampling the entire area. Conversely, on a time span of
20 years, trends of -30% or higher were detected within the 2% error
in at least 80% of the cases if the yearly trend variability was 0.05
(fig.2). With a higher annual variability in trends, the correct
magnitude over 20 years was never detected with sufficient power. A
number of sectors between 15 and 20 (out of 38 total) was sufficient to
estimate correctly both the direction and the magnitude of the trend,
corresponding to 23.4 - 60% of the total GPNP area (as the extent of
the sectors is highly variable). However, while estimating actual
abundance instead of the population trend, sample counts in 15-20
sectors were biased and produced a mean error of around +16% compared
to total counts.
The model selection for the effect of all the parameters on the
statistical power of the sample counts allowed us to choose a single
best predictive model (tab.2). The strongest effects in the model were
the number of sectors surveyed and the strength of the overall assigned
trend (tab.3). In particular a higher number of surveyed sectors
increased the statistical power of the analysis, while a higher r (thus
a lower decline over time) decreased the proportion of sample counts
that correctly detected the real population trend. The abundance trend
variability between years had also a strong effect on the statistical
power, with statistical power decreasing as the variability increases.
Other parameters, such as variability between sectors and variability of
detection probability, also influenced the outcome with a lower inverse
effect on statistical power, increasing the number of sectors needed to
be sampled. For example, with cvy = 0.05 and r = 0.982,
the number of sectors required to quantify the 20-years trend passed
from 12 with a low cvs to 22 for a high
cvs and from 13 with a low cvd to 19
with a high cvd (see supplementary materials).
Population size and detection probability did not have any effect in the
models. All the graphs for the effect of the parameters are provided in
supplementary materials.
When selecting the sectors to be sampled, choosing those with the
highest ibex abundance in the first year of the monitoring resulted in a
slightly higher statistical power, while a random selection in the first
year only or in each year decreased the reliability of the analysis
(tab.3), lowering the average statistical power respectively by 1.7%
and 2.6% for the 10-year trend and by 0.9% and 1.2% for the 20-years
trend. This difference was relevant in terms of the number of sectors to
be sampled to reach the 80% power threshold, as for instance, with
cvy = 0.05 and r = 0.978 the correct magnitude of a
20-year trend was detected with the 11 most abundant sectors, with 16
sectors selected at random in the first year or with 20 sectors selected
at random each year (see supplementary materials). However, if abundance
was estimated instead of the population trend, choosing the sectors with
the highest ibex abundance in the first year produced an overestimation
of population size, that was 41.7% higher on average than with a random
sector selection.
To monitor the population trend of Alpine ibex in GPNP (yearly
variability of 0.05 and sector variability of 0.05-0.1), sampling 15 to
20 sectors was in general sufficient (fig.3): the direction of the trend
was detected in at least 80% of the cases for trends equal to -20%
both in 10 and 20 years (fig.3a and fig.3b). The direction of trends of
-30% or more was correctly estimated even when sampling less than 5
sectors, while a trend of -10% was never detected even performing
censuses in all sectors. On a period of 20 years, surveying 10 sectors
was sufficient to correctly detect the magnitude of a trend of -30% or
more, while lower trends were not correctly quantified even sampling the
entire area (fig.3d). However, the magnitude of a 10-years trend was
impossible to detect with an error within 2% and the desired threshold
of statistical power of 80% whatever the real trend was (fig.3c).
The proportion of opposite (i.e., in the wrong direction) and
non-significant trends estimated with sample counts was low but
increased for weak trends (fig.4). Counting animals in half of the
sectors led, in the simulations, to 2.6% of non-significant and 1.2%
of opposite trends estimated when the population was declining by 30%
in 10 years (r = 0.965), while with the same yearly trend over 20 years
(r = 0.966) the population trend was always estimated in the right
direction and in 90% of the cases with the correct magnitude.
Monitoring low changes in abundance, such as a 10% decline in 10 or 20
years, led in 34.4% (in 10 years) and 36.4% (in 20 years) of the cases
to estimating incorrectly the direction of the population trend.