Stratification

E Rexstad

22 January 2020

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 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. (1988). This dataset was also used to demonstrate point transect analysis

Objectives

Survey design

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

Summer grazed pastures along Illinois River Arapaho National Wildlife Refuge, Colorado. Figure from [@knopf_guild_1988].

Summer grazed pastures along Illinois River Arapaho National Wildlife Refuge, Colorado. Figure from (Knopf et al., 1988).

The fields of the Savannah_sparrow_1980 data set are:

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. (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=" ")
## Stratum-specific detection AIC 2007 
## Common detection function AIC 2022

Because the AIC for model with stratum-specific detection functions (2007) is less than AIC for model with pooled detection function (2022), we base our inference upon the stratum-specific detection function model.

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.

## 
## Goodness of fit results for ddf object
## 
## Distance sampling Cramer-von Mises test (unweighted)
## Test statistic = 0.0939637 p-value = 0.615284
## 
## Goodness of fit results for ddf object
## 
## Distance sampling Cramer-von Mises test (unweighted)
## Test statistic = 0.0478569 p-value = 0.889167
## 
## Goodness of fit results for ddf object
## 
## Distance sampling Cramer-von Mises test (unweighted)
## Test statistic = 0.0402974 p-value = 0.931609

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

Comments

Note there is a difference of roughly 14 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 provided.

References

Knopf, F. L., Sedgwick, J. A., & Cannon, R. W. (1988). Guild structure of a riparian avifauna relative to seasonal cattle grazing. The Journal of Wildlife Management, 52(2), 280–290. https://doi.org/10.2307/3801235

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.