Simple detection function analysis of duck nests

The Distance package written by Dave Miller was intended to permit analysis of simple datasets in R.

Using the “flatfile” feature of Distance, loading data is simple. We show as examples, the analysis of a simulated survey of duck nests, consisting of 20 transects, each 128.8km in length. These data exist as a comma-delimited text file in the current directory (one can find the current working directory in R using getwd() and change it with setwd()).

library(Distance)
tidy.opts <- list(width.cutoff=40)

ducknests <- read.csv("ducknests.csv")
head(ducknests)

     Region.Label Area Sample.Label Effort distance
1 Monte_Vista_NWR    0            1  128.8     0.06
2 Monte_Vista_NWR    0            1  128.8     0.07
3 Monte_Vista_NWR    0            1  128.8     0.04
4 Monte_Vista_NWR    0            1  128.8     0.01
5 Monte_Vista_NWR    0            1  128.8     0.37
6 Monte_Vista_NWR    0            1  128.8     0.36

Note the following:

• transect length is measured in kilometres but perpendicular distance is measured in metres, therefore a conversion factor of 0.001 converts perpendicular distance metres into kilometres. Hence the reported nest density will be in nests kilometres^-2.
• we have not provided the area of the study area, therefore no estimate of abundance will be produced, but density can be estimated
• sample label simply refers to the transect number (of which there are 20).

Detection function fitting

The below code fits three detection functions: half-normal with cosine adjustments, uniform with cosine adjustments and hazard-rate with simple polynomial adjustments. In each case the number of adjustment terms is selected by AIC. The selection process can be seen in the output.

# Convert.units adjusts for distance measured in metres and effort in km
halfnorm.ducks <- ds(ducknests, key="hn", adjustment="cos", convert.units = 0.001)

## Starting AIC adjustment term selection.
## Fitting half-normal key function
## AIC= 928.134
## Fitting half-normal key function with cosine(2) adjustments
## Warning: Detection function is not weakly monotonic!
## Warning: Detection function is not strictly monotonic!
## Warning: Detection function is greater than 1 at some distances
## AIC= 929.834
## 
## 
## half-normal key function selected!

unifcos.ducks <- ds(ducknests, key="unif", adjustment="cos", mono="strict", convert.units = 0.001)

## Starting AIC adjustment term selection.
## Fitting uniform key function with cosine(1) adjustments
## AIC= 928.48
## Fitting uniform key function with cosine(1,2) adjustments
## AIC= 929.383
## 
## 
## uniform key function with cosine(1) adjustments selected!

hazard.ducks <- ds(ducknests, key="hr",  adjustment="poly", convert.units = 0.001)

## Starting AIC adjustment term selection.
## Fitting hazard-rate key function
## AIC= 929.799
## Fitting hazard-rate key function with simple polynomial(2) adjustments
## Warning: Detection function is not weakly monotonic!
## Warning: Detection function is not strictly monotonic!
## Warning: Detection function is greater than 1 at some distances
## AIC= 931.673
## 
## 
## hazard-rate key function selected!

For both the half-normal and hazard-rate functions, ds complains that the detection function is non-monotone (that detection probability increases with distance at some points in the fitted function), however these are for the models with adjustments which are not selected.

Model selection

AIC scores for the three fitted models:

Model	AIC
Half normal	928.1338
Uniform cosine	928.4797
Hazard rate	929.799

Model fit

Having completed the model selection phase, produce depiction of detection function, show its goodness of fit.

par(mfrow=c(1,2))
plot(halfnorm.ducks, main="Duck nests, half-normal detection function")
fit.test <- ddf.gof(halfnorm.ducks$ddf)

par(mfrow=c(1,1))

My preference is to pluck out the Cramer-von Mises goodness of fit test. For these data, this test has a test statistic value of 0.0354 with an associated p-value of 0.9554

Content with adequate fit of the model to these data, we want to present our density estimate.

Population inference

From the first table below, we see that 534 nests were encountered along 2574km of transects. The encounter rate (ER) is the ratio of these values, implying that roughly 5 nests were encountered per kilometre of transect. We also see the variablity of the encounter rate.

kable(halfnorm.ducks$dht$individuals$summary,format="markdown")

Region	Area	CoveredArea	Effort	n	k	ER	se.ER	cv.ER
Monte_Vista_NWR	12.36	12.36	2575	534	20	0.2074	0.008	0.0384

This second table produces our estimate of duck nest density within the region covered by the survey transects. The estimate of 49.7 is in units of nests kilometre^-2, with associated measures of precision in the form of standard error, coefficient of variation and 95% confidence interval bounds.

kable(halfnorm.ducks$dht$individuals$D,format="markdown")

Label	Estimate	se	cv	lcl	ucl	df
Total	49.7	2.937	0.0591	44.2	55.87	99.56

Finally at the end of the analysis, reset the R environment in preparation for another analysis.

rm(list=ls())