Detection Data

The detection data frame contains one row for each detected target, whether that target is a single individual or a group of individuals. Columns in the detection data frame contain information like group size and perpendicular off-transect distance to the group. At a minimum, Rdistance requires the following information in the detection data frame:

• Detection Distance = The perpendicular distance (also known as off-transect distance) from the transect to the detected group or individual. Rdistance requires that this column of the detection data frame have measurement units (e.g., “m” or “ft”). Measurement units can be assigned using the units()<- or units::set_units functions. For example, units(sparrowDetectionData$dist) <- "m" makes all the measurements in column dist into meters. If column dist already has units, assigning meters in this way converts from the previous units into meters. Measurement units are required on all distances in Rdistance (e.g., off-transect distances, left and right cut-offs, scale locations, transect lengths, and study area size). Additional details are in help(dfuncEstim).
• Site ID = The ID of the transect on which the detection was made. Transect IDs are required to estimate abundance (in function abundEstim), but not to estimate a distance function (in function dfuncEstim).
• Group Sizes = The number of individuals in the group associated with each detection. Group sizes are required to estimate abundance (function abundEstim), but not to estimate a distance function (function dfuncEstim). If all groups are size 1, group sizes can be omitted.

The specific columns that contain distances and group sizes can be named anything and are specified in the formula argument of function dfuncEstim. Likewise, the siteID column(s) can be named anything and is specified in the transectID argument to function dfuncEstim. If siteID is not specified (or is NULL), Rdistance constructs the siteIDs from the set of common columns contained in the detection and sites data frames. See the Input data frames section of help(dfuncEstim) for additional details.

Line-transect distance-sampling analysis is performed on perpendicular off-transect distances, where off-transect distances are measured from the detected group to the closest point on the transect. In the field, observers commonly record straight-line sighting distance (from observer straight to the target) and sighting angle instead of perpendicular distance. Rdistance provides a utility function named perpDists that will compute perpendicular distances from sighting distance and angle. See help(perpDists) for details.

For reference, the first six rows of the sparrow detection data set are:

head(sparrowDetectionData)

##   siteID groupsize sightdist sightangle     dist
## 1     A1         1        65         15 16.8 [m]
## 2     A1         1        70         10 12.2 [m]
## 3     A1         1        25         75 24.1 [m]
## 4     A1         1        40          5  3.5 [m]
## 5     A1         1        70         85 69.7 [m]
## 6     A1         1        10         90 10.0 [m]

We will use siteID, groupsize, and dist in this tutorial. Details on the study and other columns are in help(sparrowDetectionData).

Site Data

The sites data frame contains one row for each surveyed site regardless of whether the site was “positive” (one or more targets detected) or “zero” (no targets detected). We use the term ‘site’ because it encompasses both continuous transects and point transects. In this tutorial, a ‘site’ is one transect. Rdistance views the study area as containing one or more surveyed ‘sites’, where each ‘site’ has a particular surveyed length. Along the surveyed length, zero or more groups were detected. Stratum and sub-units of the study area are not automatically accommodated in Rdistance. We view stratum as separate study areas, estimate abundance in each separately, and then sum over stratum to arrive as the study area estimate.

At a minimum, Rdistance requires the following information in the sites data frame:

• Site IDs = The ID of every surveyed site (here, transect) in the study area.
• Length = The length of transect surveyed in each site. The length column in the sites data frame must have measurement units assigned (e.g., “m” or “ft”). Like other distance measurements, units can be assigned using units()<- or units::set_units.

Note that a site data frame is not required to estimate a distance function. In addition to site IDs and length, the sites data set can contain covariates that are constant for detections on the same transect. We call such covariates site-level covariate to distinguish them from detection-level covariates that vary by detection.

For reference, the first six rows of the sparrow site data set are:

head(sparrowSiteData)

##   siteID  length observer bare herb shrub height shrubclass
## 1     A1 500 [m]     obs4 36.7 15.9  20.1   26.4       High
## 2     A2 500 [m]     obs4 38.7 16.1  19.3   25.0       High
## 3     A3 500 [m]     obs5 37.7 18.8  19.8   27.0       High
## 4     A4 500 [m]     obs5 37.7 17.9  19.9   27.1       High
## 5     B1 500 [m]     obs3 58.5 17.6   5.2   19.6        Low
## 6     B2 500 [m]     obs3 56.6 18.1   5.2   19.0        Low

We will use siteID and length in this tutorial.

Group Sizes

If group sizes are not specified, Rdistance assumes all detected groups consisted of one individual. The team collecting our example data occasionally detected two or three sparrows together. Group sizes should be entered in the detection data frame, and in our example data are contained in the groupsize column. We specify group sizes during distance function estimation as an groupsize() term in the formula for the distance function. Group sizes are not used in dfuncEstim to estimate parameters of the distance function, but will be used later in function abundEstim to estimate density and abundance.

Many distance analysts advocate dropping a small proportion of large distances to improve distance function estimation and reduce variance. Large but rare distances add variance to final estimates and can unduly influence (i.e., bias) distance functions, especially in small (e.g., n < 100) data sets. Dropping large observations establishes a maximum and is called right-truncation. Analysts typically drop between one and five percent of the largest distances depending on histogram shape and personal preference. The 95-th quantile of the sparrow detection distances is 104.18, while the 99-th quantile is 143.2. We will right-truncate the sparrow detection distances at 150 meters, which is a nice round number just above the 99-th quantile. In Rdistance, we right-truncate by specifying a value for parameter w.hi. Parameter w.hi must have measurement units because it is a distance.

Distance function estimation

We are ready to estimate a half-normal distance function with right-truncation at 150 m and non-unity group sizes on the sparrow data, i.e.,

rightTruncDistance <- units::set_units(150, "m")
dfuncSparrow<- dfuncEstim(formula = dist ~ 1 + offset(groupsize)
                          , detectionData = sparrowDetectionData
                          , likelihood = "halfnorm"
                          , w.hi = rightTruncDistance)
dfuncSparrow

## Call: dfuncEstim(formula = dist ~ 1 + offset(groupsize), detectionData
##    = sparrowDetectionData, likelihood = "halfnorm", w.hi =
##    rightTruncDistance)
## Coefficients:
##        Estimate  SE        z         p(>|z|)      
## Sigma  49.87369  2.014173  24.76138  2.338191e-135

The printout of a distance function object provides parameter estimates, convergence messages, fit information, and effective sampling distance. The estimated parameter of this half-normal distance function is 49.87. As an aside, when a half-normal distance function is estimated, the estimated parameter is the standard deviation of a normal distribution if it were fitted only to positive data and if we required the mean to be exactly zero. Approximately 68 percent of distances will fall between 0 and this parameter. Approximately 95 percent of distances will fall between 0 and twice this parameter. In the sparrow detection data, 69.4 percent of the observed distances are between 0 and 49.87 meters, while 93.5 percent are between 0 and 2(49.87) = 99.75 meters. The closeness of these observed proportions to their theoretical value provides information on fit of the detection function. Further comments on fit are outside the scope of this vignette.

The observation strip comprising the left-, and right-truncation levels appears in the line labeled ‘Strip:’. The Effective Strip Width (ESW) is a key piece of information needed to estimate abundance and it appears on the line labeled ‘Effective strip width (ESW):’. ESW appears in default Rdistance printouts and can be calculated separately using the ESW() function. ESW is the distance at which the same number of targets are missed nearer to the observer as were sighted farther from the observer. Missed targets and farther away targets balance themselves out at the ESW. Another way to say think of ESW is that a survey with imperfect detection and ESW equal to X will effectively cover the same area as a study with perfect detection out to a distance of X. See help(ESW) for details.

Some researchers prefer to report overall probability of detection in the strip. Probability of detection is ESW divided by the observation strip width. Probability of detection is reported on the line below ESW and in this case equals 62.34 / (150 - 0) = 0.4156. We leave numbers reported in the remainder of the output for later tutorials.

A visual picture of the distance function is obtained using the plot method,

# Figure 2
plot(dfuncSparrow, nbins =40, col="grey")

The half-normal distance function fitted to sparrow off-transect detection distances.

In Figure 2, the histogram of distances is grey and the estimated distance function is red. ESW is area under the red curve. Rdistances’s plot methods are extensive. See Examples in help(plot.dfunc) examples for ways to modify bins, colors, line types, labels, etc.

Bootstrap Distribution

Rdistance stores bootstrap parameter values, density, and ESW values in the $B component of abundance objects. Storing bootstrap iterations provides flexibility for more advanced analyses. For example,

• Users can combine bootstrap iterations from multiple abundance objects by appending $B components together. New confidence intervals from the new $B component will be computed when the abundance object is printed.
• Users can compute confidence intervals using different methods (e.g., percentile instead of bia-corrected).
• Users can compute the variance of other quantities that depend on parameters, density, or ESW; for example, density in a particular strip.
  
• Users can plot the full bootstrap distribution of density (or ESW) to inspect statistical properties.

The first six lines of the bootstrap iterations are:

# Convert to hectares for readability
units(fit$B$density) <- "1/ha"

head(fit$B)

##            density  effDistance
## 1 0.7411862 [1/ha] 65.96033 [m]
## 2 0.8697759 [1/ha] 64.51214 [m]
## 3 0.8209953 [1/ha] 62.59340 [m]
## 4 0.8248241 [1/ha] 60.78737 [m]
## 5 0.9833776 [1/ha] 57.05958 [m]
## 6 0.9260318 [1/ha] 58.79328 [m]

The bootstrap distribution of sparrow density is,

# Figure 3
hist(fit$B$density
     , n = 30
     , xlab = "Density"
     , main = NULL)
# Show final density estimates, after converting to 1/ha
d <- fit$density
d.ci <- fit$density.ci
units(d) <- "1/ha"
units(d.ci) <- "1/ha"
abline(v = c(d, d.ci), col="blue")

Bootstrap distribution of sparrow density computed in the abundEstim routine.

In Figure 3, the density point estimate (0.8265237 [1/ha]) and confidence limits (0.6544368 [1/ha] to 1.0038680 [1/ha]) visually plot where we expect and encompass the center of the bootstrap distribution. The same figure for other data may show final estimates ‘shifted’ relative to the bootstrap distribution because the bias-corrected CI method attempts to remove estimation bias (i.e., removes distributional ‘shift’).

Output Measurment Units

It is possible to change the output measurement units on ESW and density by setting the outputUnits = parameter in the original call to dfuncEstim. Setting outputUnits only effects reporting units (output). There is no need to change the units of inputs because conversion is handled internally and automatically. For example, to output linear distances in kilometers, and density in square kilometers, issue the following:

dfuncSparrow <- dfuncEstim(formula = dist~1
                          , detectionData = sparrowDetectionData
                          , likelihood = "halfnorm"
                          , w.hi = rightTruncDistance
                          , outputUnits = "km")

fit <- abundEstim(dfuncSparrow
                , detectionData = sparrowDetectionData
                , siteData = sparrowSiteData
                , area = saSize
                , ci = 0.95
                )

fit

## Call: dfuncEstim(formula = dist ~ 1, detectionData =
##    sparrowDetectionData, likelihood = "halfnorm", w.hi =
##    rightTruncDistance, outputUnits = "km")
## Coefficients:
##        Estimate    SE           z         p(>|z|)      
## Sigma  0.04987416  0.002014229  24.76092  2.365001e-135
## 
## Density in sampled area: 78.64156 [1/km^2]
## Abundance in 4105 [km^2] study area: 322823.6