Revisiting the winter wren point transects with cue counts.

In this exercise, we use `R`

(R Core Team, 2019) and the
`Distance`

package (Miller, Rexstad, Thomas, Marshall, & Laake,
2019) to fit different detection function models to point
transect cue count survey data of winter wren *(Troglodytes
troglodytes)* density and abundance. These data were part of a study
described by Buckland (2006).

- Estimate density of cues from point transect data
- Convert cue density to animal density using rate of song production

Each of the 32 point count stations were visited twice. During each visit, the observer recorded distances to all songs detected during a 5-minute sampling period (Figure 1).

In addition, 43 male winter wrens were observed and their rate of song production was measured. The mean cue rate, along with its standard error (between individuals) was calculated and included in the data set to serve as a multiplier.

The fields of the `wren_cuecount`

data set are:

- Region.Label - identifier of regions: in this case there is only one region and set to ‘Montrave’
- Area - size of the study region (hectares): 33.2ha
- Sample.Label - point transect identifier (numbered 1-32)
- Cue.rate - production of cues (per minute)
- Cue.rate.SE - standard error of cue production rate (between individuals)
- object - unique identifier for each detected winter wren
- distance - radial distance (metres) to each detection
- Search.time - Duration of listening at each station (minutes)
- Study.Area - this is the name of the study, ‘Montrave 3’

`Distance`

package and cue count dataThis command assumes that the `dsdata`

package has been
installed on your computer. The R workspace `wren_cuecount`

contains detections of winter wrens from the line transect surveys of
Buckland (2006).

Examine the first few rows of `wren_cuecount`

using the
function `head()`

```
head(wren_cuecount)
```

```
Region.Label Area Sample.Label Cue.rate Cue.rate.SE object distance
1 Montrave 33.2 1 1.4558 0.2428 38 50
2 Montrave 33.2 1 1.4558 0.2428 39 55
3 Montrave 33.2 1 1.4558 0.2428 40 55
4 Montrave 33.2 1 1.4558 0.2428 41 55
5 Montrave 33.2 1 1.4558 0.2428 46 50
6 Montrave 33.2 1 1.4558 0.2428 47 50
Study.Area Search.time
1 montrave 3 10
2 montrave 3 10
3 montrave 3 10
4 montrave 3 10
5 montrave 3 10
6 montrave 3 10
```

Note there is no field in the data to indicate sampling effort. With
line transects, the lengths of each transect were provided to measure
effort. For point transects, the number of visits to each station was
specified. In this data set, all that is specified is
`Search.time`

the length of time each station was sampled.
*Note*, each station was visited twice and sampling was 5 minutes
in length on each visit. Hence `Search.time`

is recorded as
10. *Note also* the units of measure of `Search.time`

must be consistent with the units of measure of cue rate.

Gain familiarity with the perpendicular distance data using the
`hist()`

function (Figure 2).

```
hist(wren_cuecount$distance, xlab="Distance (m)", main="Song detection distances")
```

Note the long right tail we will cut off with the
`truncation`

argument to `ds()`

.

`ds`

As noted above, **Effort** is missing from the data.
With cue count surveys, effort is measured in time rather than length or
number of visits. Therefore we define a new field `Effort`

and set it equal to the `Search.time`

field.

*Note*: no `converstion.factor`

is specified in the
call to `ds()`

because it is only the detection function that
is of interest at this step of the analysis, nothing about density or
abundance.

```
conversion.factor <- convert_units("meter", NULL, "hectare")
wren_cuecount$Effort <- wren_cuecount$Search.time
wrensong.hr <- ds(wren_cuecount, transect="point", key="hr", adjustment=NULL,
truncation=100)
```

Visually inspect the fitted detection function with the
`plot()`

function, specifying the cutpoints histogram with
argument `breaks`

(Figure 3).

```
cutpoints <- c(0,5,10,15,20,30,40,50,65,80,100)
plot(wrensong.hr, breaks=cutpoints, pdf=TRUE, main="Hazard rate function fit to winter wren song counts.")
```

Do not examine the abundance or density estimates produced by
`summary(wrensong.hr)`

because as the results it contains are
*nonsense*. These summary values do not properly recognise that
the unit of effort is time rather than visits for the point count
survey. This additional component of the analysis is provided in the
next step.

`dht2`

The function `dht2`

provides additional capacity for
providing density or abundance estimates in novel situations such as cue
counts where multipliers need to be incorporated.

The argument `multipliers`

in `dht2`

provides
the mechanism whereby the cue production rate and its uncertainty are
incorporated into the analysis.

To properly perform the calculations responsible for converting song
density to bird density, we enlist the aide of the function
`dht2`

. The additional information about cue rates and their
variability are provided in a `list`

. The multiplier in the
list is **required** to have the name `creation`

and it contains both the cue rate point estimate and its associated
measure of precision.

```
cuerate <- unique(wren_cuecount[ , c("Cue.rate","Cue.rate.SE")])
names(cuerate) <- c("rate", "SE")
(mult <- list(creation=cuerate))
```

```
$creation
rate SE
1 1.4558 0.2428
```

Additional arguments are also passed to `dht2`

.
`flatfile`

is the name of the data set and
`strat_formula`

contains information about stratification
that might exist in the survey design. The Montrave study had no
stratification, inference was only for the 33 hectare woodland, so
`strat_formula`

here is simply constant `~1`

.

Results of the overall winter wren density estimate is provided by a
`print`

method, specifying `report="density"`

. The
alternative for the `report`

argument is
`report="abundance"`

.

```
wren.estimate <- dht2(wrensong.hr, flatfile=wren_cuecount, strat_formula=~1,
multipliers=mult, convert_units=conversion.factor)
print(wren.estimate, 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 1005.31 320 771 32 2.409 0.236 0.098
Density estimates:
.Label Estimate se cv LCI UCI df
Total 1.2018 0.238 0.198 0.8173 1.7674 520.679
Component percentages of variance:
.Label Detection ER Multipliers
Total 4.83 24.38 70.79
```

We assess the goodness of fit of the hazard rate model to the winter wren cue count data (Figure 4).

```
gof_ds(wrensong.hr)
```

```
Goodness of fit results for ddf object
Distance sampling Cramer-von Mises test (unweighted)
Test statistic = 1.69439 p-value = 6.24744e-05
```

Note the distinct lack of fit to the song data. This is because of
many detections at the identical distances from birds being stationary
and singing. This induces a phenomenon known as *over
dispersion*.

This vignette uses the function `dht2`

because that
function knows how to incorporate multipliers such as cue rates and
propogate the uncertainty in cue rate into overall uncertainty in
density and abundance. Because there is uncertainty coming not only from
encounter rate variability and uncertainty in detection function
parameters, but also from cue rate variability, the relative
contribution of each source of uncertainty is tablated. This is the last
table produced by printing the `wren.estimate`

object. For
the Montrave winter wren data, only 4% of the uncertainty in the density
estimate is attributable to the detection function, 24% attributable to
encounter rate variability and 71% attributable to between-individual
variability in call rate.

This insight suggests that if this survey was to be repeated, exerting more effort in measuring between-individual variation in call rate would likely yield the most benefits in tightening the precision in density estimates.

Also note the poor fit of the model to the data; the P-value for the Cramer von-Mises test is <<0.05. This is caused by over-dispersion in the distribution of detected call distances. A single individual may sit on a tree branch and emit many song bursts, leading to a jagged distribution of call distances that is not well fitted by a smooth detection function. That over-dispersion will not bias the density estimates.

Buckland, S. T. (2006). Point transect surveys for songbirds: Robust
methodologies. *The Auk*, *123*(2), 345–345. https://doi.org/10.1642/0004-8038(2006)123[345:psfsrm]2.0.co;2

Miller, D. L., Rexstad, E., Thomas, L., Marshall, L., & Laake, J. L.
(2019). Distance sampling in r. *Journal of Statistical
Software*, *89*(1), 1–28. https://doi.org/10.18637/jss.v089.i01

R Core Team. (2019). *R: A language and environment for statistical
computing*. Vienna Austria: R Foundation for
Statistical Computing.