Size Biased Distributions.

sbd

Size Biased Distributions

An R package for fitting and plotting size-biased non-negative distributions. Size-biased distributions occur when the probability of observing something is proportional to its size. The example motivating this package is observations of animal speeds made by camera traps, where faster moving animals are more likely to encounter camera traps and hence be recorded, with probability of observation expected to be proportional to speed (Rowcliffe et al. 2016).

The core function is sbm, which performs model fitting and estimates the underlying average of a set of size biased data, given a model formula and a dataframe containing the variables named in the formula. Default usage is non-parametric, providing the harmonic mean as the estimator of underlying average:

library(sbd)
data("BCI_speed_data")
agoutiData <- subset(BCI_speed_data, species=="agouti")
mod <- sbm(speed~1, agoutiData)
mod

## Call:
## speed ~ 1
## 
## Probability distribution:
## none
## 
## Estimates:
##         est          se       lcl       ucl
## 1 0.1322636 0.004164446 0.1241013 0.1404259

The distribution can be plotted using the hist method:

hist(mod)

Three parametric distributions are also available - log-normal, gamma, and Weibull. These can be selected using the pdf argument, and model support can be evaluated using the AIC method:

lmod <- sbm(speed~1, agoutiData, pdf = "lnorm")
gmod <- sbm(speed~1, agoutiData, pdf = "gamma")
wmod <- sbm(speed~1, agoutiData, pdf = "weibull")
AIC(lmod, gmod, wmod)

##   model     PDF       AIC     dAIC
## 1  lmod   lnorm -919.6315  0.00000
## 3  wmod weibull -872.9454 46.68610
## 2  gmod   gamma -854.1374 65.49404

For parametric models without covariates, a fitted probability density function is added to the data distribution when it is plotted:

hist(lmod)

Parametric models can accept covariates, for example, testing the effect of species body mass on average speed:

lmod_null <- sbm(speed~1, BCI_speed_data, "lnorm")
lmod_mass <- sbm(speed~mass, BCI_speed_data, "lnorm")
AIC(lmod_null, lmod_mass)

##       model   PDF       AIC     dAIC
## 2 lmod_mass lnorm -1789.306  0.00000
## 1 lmod_null lnorm -1776.837 12.46826

The parameters of the underlying linear model can be inspected using the summary method, where lmean is the natural log of the underlying mean, and lsig is the natural log of the underlying standard deviation:

summary(lmod_mass)

##                       estimate    stdError    tValue       pValue
## lmean.(Intercept) -2.000214738 0.025891134 77.254814 0.000000e+00
## lmean.mass         0.007705719 0.002022373  3.810235 1.427234e-04
## lsig              -0.219988199 0.015221563 14.452405 2.948531e-45

The predict method can be used to calculate estimated averages and their standard errors for any desired predictor value, for example:

nd <- data.frame(mass = c(1,10,100))
predict(lmod_mass, newdata = nd)

##   mass       est          se       lcl       ucl
## 1    1 0.1363529 0.003415827 0.1298537 0.1434708
## 2   10 0.1461448 0.002977502 0.1402656 0.1520938
## 3  100 0.2923970 0.055006174 0.2011866 0.4161327