In DistanceDevelopment/Distance: Distance Sampling Detection Function and Abundance Estimation

knitr::opts_chunk$set(eval=TRUE, echo=TRUE, message=FALSE, warnings=FALSE)

In this exercise, we use R [@r_core_team_r_2019] and the Distance package [@miller_distance_2019] to fit different detection function models to point transect survey data of savanna sparrows (Passerculus sandwichensis) density and abundance. These data were part of a study examining the effect of livestock grazing upon vegetation structure and consequently upon the avian community described by Knopf et al. [-@knopf_guild_1988]. This dataset was also used to demonstrate point transect analysis

Objectives

Fit a detection function pooling data across pastures,
Fit pasture-specific detection functions,
Choose most appropriate analysis using model selection.

Survey design

A total of 373 point transects were placed in three pastures in the Arapaho National Wildlife Refuge in Colorado (Figure \@ref(fig:fig)). Elevation of these pastures was ~2500m. In this example, we will perform pasture-level analysis of these data.

knitr::include_graphics("arapaho.jpg")

The fields of the Savannah_sparrow_1980 data set are:

Region.Label - three pastures that constituted sections of the study area.
Area - size of the study region. A place holder, because pasture sizes are not known. Estimates of density and abundance will be equivalent.
Sample.Label - point transect identifier (total of 273)
Effort - number of visits to each point
object - unique identifier for each detected savanna sparrow
distance - radial distance (metres) to each detection
Study.Area - only data for savanna sparrow (SASP) are included in this data set

Make the data available for R session

This command assumes that the dsdata package has been installed on your computer. The R workspace Savannah_sparrow_1980 contains detections of savanna sparrows from point transect surveys of Knopf et al. [-@knopf_guild_1988].

library(Distance)
data(Savannah_sparrow_1980)
conversion.factor <- convert_units("meter", NULL, "hectare")

Separate data into pasture-specific data sets

The simplest way to fit pasture-specific detection functions is to subset the data. This could be done at the time the ds() function is called, but we perform the step here as a data preparation step.

sasp.past1 <- subset(Savannah_sparrow_1980, Region.Label == "PASTURE 1")
sasp.past2 <- subset(Savannah_sparrow_1980, Region.Label == "PASTURE 2")
sasp.past3 <- subset(Savannah_sparrow_1980, Region.Label == "PASTURE 3")

Pasture (stratum)-specific detection functions

Fit half-normal key functions without adjustments to each pasture separately after performing 5\% right truncation.

past1.hn <- ds(data=sasp.past1, key="hn", adjustment=NULL,
              transect="point", convert_units=conversion.factor, truncation="5%")
past2.hn <- ds(data=sasp.past2, key="hn", adjustment=NULL,
              transect="point", convert_units=conversion.factor, truncation="5%")
past3.hn <- ds(data=sasp.past3, key="hn", adjustment=NULL,
              transect="point", convert_units=conversion.factor, truncation="5%")

The total AIC for the model that fits separate detection functions to each pasture is the sum of the AICs for the individual pastures.

model.separate.AIC <- sum(AIC(past1.hn, past2.hn, past3.hn)$AIC)

Common detection function across pastures

This model is much simpler to fit because there is only a single call to ds() using the original data.

model.pooled <- ds(data=Savannah_sparrow_1980, key="hn", adjustment=NULL,
                   transect="point", convert_units = conversion.factor, truncation = "5%")
model.pooled.AIC <- AIC(model.pooled)

Comparison of AIC scores

cat(paste("Stratum-specific detection AIC", round(model.separate.AIC),
      "\nCommon detection function AIC", round(model.pooled.AIC$AIC)), sep=" ")

Because the AIC for model with stratum-specific detection functions (r round(model.separate.AIC)) is less than AIC for model with pooled detection function (r round(model.pooled.AIC$AIC)), we base our inference upon the stratum-specific detection function model (depicted in Figure \@ref(fig:threeplot)).

cutpoints <- c(0,5,10,15,20,30,40,53)
par(mfrow=c(1,3))
plot(past1.hn, breaks=cutpoints, pdf=TRUE, main="Pasture 1")
plot(past2.hn, breaks=cutpoints, pdf=TRUE, main="Pasture 2")
plot(past3.hn, breaks=cutpoints, pdf=TRUE, main="Pasture 3")

Absolute goodness of fit

Always best to check the fit of the preferred model to the data.

gof_ds(past1.hn, plot = FALSE)
gof_ds(past2.hn, plot = FALSE)
gof_ds(past3.hn, plot = FALSE)

Further exploration of analyses involving stratification can be found in the example of dung survey analysis.

Comments

Note there is a difference of r round(model.pooled.AIC$AIC - model.separate.AIC) AIC units between the model using stratum-specific detection functions and the model using a pooled detection function, with the stratum-specific detection function model being preferrable. To be thorough, absolute goodness of fit for the three stratum-specific detection functions is checked, and all models fit the data adequately.

This vignette focuses upon use of stratum-specific detection functions as a model selection exercise. Consequently, the vignette does not examine stratum-specific abundance or density estimates. That output is not included in this example analysis, but can easily be produced by continuing the analysis begun in this example.