Analysis with multipliers solution
Analysis with multipliers
Dung survey of deer
Returning to the data described in Marques et al. (2001), the following code loads the relevant packages and data. The perpendicular distances are measured in centimetres, effort along the transects measured in kilometres and areas in square kilometres.
Here we did not perform a comprehensive examination of fitting a detection function to the detected pellet groups, however, as a general guideline, we truncated the longest 10% perpendicular distances.
deer.df <- ds(sikadeer, key="hn", truncation="10%", convert_units = conversion.factor)
plot(deer.df)
Region Area CoveredArea Effort n k ER se.ER cv.ER
1 A 13.9 0.005950 1.70 1217 13 715.88234 119.918872 0.1675120
2 B 10.3 0.003850 1.10 396 10 359.99999 86.859289 0.2412758
3 C 8.6 0.001575 0.45 17 3 37.77778 8.521202 0.2255612
4 E 8.0 0.002975 0.85 30 5 35.29412 16.568939 0.4694533
5 F 14.0 0.000700 0.20 29 1 145.00000 0.000000 0.0000000
6 G 15.2 0.001400 0.40 32 3 80.00000 39.686269 0.4960784
7 H 11.3 0.000700 0.20 3 1 15.00000 0.000000 0.0000000
8 J 9.6 0.000350 0.10 7 1 70.00000 0.000000 0.0000000
9 Total 90.9 0.017500 5.00 1731 37 201.90876 0.000000 0.0000000The summary above shows that in blocks F, H and J there was only one transect and, as a consequence, it is not possible to calculate a variance empirically for the encounter rate in those blocks.
Estimating decay rate from data
A paper by Laing et al. (2003) describes field protocol for collecting data to estimate the mean persistence time of dung or nests to be used as multipliers. The code segment shown earlier analyses a file of such data via logistic regression to produce an estimate of mean persistence time and its associated uncertainty.
Mean persistence time SE %CV
163.396748 14.226998 8.707026 Using the output from calling the MIKE.persistence function, the multipliers can be specified:
# Create list of multipliers
mult <- list(creation = data.frame(rate=25, SE=0),
decay = data.frame(rate=163, SE=14))
print(mult)$creation
rate SE
1 25 0
$decay
rate SE
1 163 14deer_ests <- dht2(deer.df, flatfile=sikadeer, strat_formula=~Region.Label,
convert_units=conversion.factor, multipliers=mult,
stratification="effort_sum", total_area = 100)
print(deer_ests, report="abundance")Abundance estimates from distance sampling
Stratification : effort_sum
Variance : R2, n/L
Multipliers : creation, decay
Sample fraction : 1
Summary statistics:
Region.Label Area CoveredArea Effort n k ER se.ER cv.ER
A 13.9 0.005950 1.70 1217 13 715.882 119.919 0.168
B 10.3 0.003850 1.10 396 10 360.000 86.859 0.241
C 8.6 0.001575 0.45 17 3 37.778 8.521 0.226
E 8.0 0.002975 0.85 30 5 35.294 16.569 0.469
F 14.0 0.000700 0.20 29 1 145.000 0.000 0.000
G 15.2 0.001400 0.40 32 3 80.000 39.686 0.496
H 11.3 0.000700 0.20 3 1 15.000 0.000 0.000
J 9.6 0.000350 0.10 7 1 70.000 0.000 0.000
Total 100.0 0.017500 5.00 1731 37 346.200 45.234 0.131
Abundance estimates:
Region.Label Estimate se cv LCI UCI df
A 1027 197.466 0.192 691 1527 20.797
B 383 99.167 0.259 220 667 11.955
C 34 8.200 0.244 15 75 2.759
E 29 13.959 0.479 9 99 4.329
F 210 19.752 0.094 174 252 60310.077
G 126 63.396 0.505 18 858 2.147
H 18 1.649 0.094 15 21 60310.077
J 69 6.538 0.094 58 83 60310.077
Total 3574 575.837 0.161 2560 4990 20.215
Component percentages of variance:
Region.Label Detection ER Multipliers
A 4.07 75.96 19.97
B 2.24 86.76 10.99
C 2.52 85.14 12.34
E 0.66 96.13 3.22
F 16.93 0.00 83.07
G 0.59 96.52 2.89
H 16.93 0.00 83.07
J 16.93 0.00 83.07
Total 8.09 91.91 0.00There are a few things to notice:
- overall estimate of density
- most effort took place in woodland A where deer density was high. Therefore, the overall estimate is between the estimated density in woodland A and the lower densities in the other woodlands.
- components of variance
- we now have uncertainty associated with the encounter rate, detection function and decay rate (note there was no uncertainty associated with the production rate) and so the components of variation for all three components are provided.
In woodland A, there were 13 transects on which over 1,200 pellet groups were detected: uncertainty in the estimated density was 19% and the variance components were apportioned as detection probability 4%, encounter rate 76% and multipliers 20%.
In woodland E, there were 5 transects and 30 pellet groups resulting in a coefficient of variation (CV) of 48%: the variance components were apportioned as detection probability 0.7%, encounter rate 96% and multipliers 3%.
In woodland F only a single transect was placed and the CV of density of 9% was apportioned as detection probability 17% and multipliers 83%. Do you trust this assessment of uncertainty in the density of deer in this woodland? We are missing a component of variation because we were negligent in placing only a single transect in this woodland and so are left to ‘assume’ there is no variability in encounter rate in this woodland.
By the same token, we are left to assume there is no variability in production rates between deer because we have not included a measure of uncertainty in this facet of our analysis.
Cue counting survey of songbirds
Analysis of the cue count data of winter wrens described by Buckland (2006).
data(wren_cuecount)
cuerate <- unique(wren_cuecount[ , c("Cue.rate","Cue.rate.SE")])
names(cuerate) <- c("rate", "SE")
mult <- list(creation=cuerate)
print(mult)$creation
rate SE
1 1.4558 0.2428The sampling fraction for these data will be 1 because the full circle around the observer was searched.
conversion.factor <- convert_units("meter", NULL, "hectare")
w3.est <- dht2(w3.hr, flatfile=wren_cuecount, strat_formula=~1,
multipliers=mult, convert_units=conversion.factor)
# NB "Effort" here is sum(Search.time) in minutes
# NB "CoveredArea" here is pi * w^2 * sum(Search.time)
print(w3.est, report="density")Density estimates from distance sampling
Stratification : geographical
Variance : P2, n/L
Multipliers : creation
Sample fraction : 1
Summary statistics:
.Label Area CoveredArea Effort n k ER se.ER cv.ER
Total 33.2 860.1681 320 765 32 2.391 0.236 0.099
Density estimates:
.Label Estimate se cv LCI UCI df
Total 1.2092 0.242 0.2 0.8195 1.7843 522.541
Component percentages of variance:
.Label Detection ER Multipliers
Total 6.14 24.33 69.54Note the large proportion of the uncertainty in winter wren density stems from variability in cue (song) rate. Analyses of the cue count data are necessarily rather subjective as the data show substantial over-dispersion (a single bird may give many song bursts all from the same location during a five minute count). In this circumstance, goodness-of-fit tests are misleading and care must be taken not to over-fit the data (i.e. fit a complicated detection function).
Goodness of fit results for ddf object
Distance sampling Cramer-von Mises test (unweighted)
Test statistic = 1.70062 p-value = 6.04794e-05