# 1 Abstract

unmarked aims to be a complete environment for the statistical analysis of data from surveys of unmarked animals. Currently, the focus is on hierarchical models that separately model a latent state (or states) and an observation process. This vignette provides a brief overview of the package - for a more thorough treatment see Fiske and Chandler (2011).

# 2 Overview of unmarked

Unmarked provides methods to estimate site occupancy, abundance, and density of animals (or possibly other organisms/objects) that cannot be detected with certainty. Numerous models are available that correspond to specialized survey methods such as temporally replicated surveys, distance sampling, removal sampling, and double observer sampling. These data are often associated with metadata related to the design of the study. For example, in distance sampling, the study design (line- or point-transect), distance class break points, transect lengths, and units of measurement need to be accounted for in the analysis. Unmarked uses S4 classes to store data and metadata in a way that allows for easy data manipulation, summarization, and model specification. Table 1 lists the currently implemented models and their associated fitting functions and data classes.

Table 1. Models handled by unmarked.
Model Fitting Function Data Citation
Occupancy occu unmarkedFrameOccu MacKenzie et al. (2002)
Royle-Nichols occuRN unmarkedFrameOccu Royle and Nichols (2003)
Point Count pcount unmarkedFramePCount Royle (2004a)
Distance-sampling distsamp unmarkedFrameDS Royle et al. (2004)
Generalized distance-sampling gdistsamp unmarkedFrameGDS Chandler et al. (2011)
Arbitrary multinomial-Poisson multinomPois unmarkedFrameMPois Royle (2004b)
Colonization-extinction colext unmarkedMultFrame MacKenzie et al. (2003)
Generalized multinomial-mixture gmultmix unmarkedFrameGMM Royle (2004b)

Each data class can be created with a call to the constructor function of the same name as described in the examples below.

# 3 Typical unmarked session

The first step is to import the data into R, which we do below using the read.csv function. Next, the data need to be formatted for use with a specific model fitting function. This can be accomplished with a call to the appropriate type of unmarkedFrame. For example, to prepare the data for a single-season site-occupancy analysis, the function unmarkedFrameOccu is used.

## 3.1 Importing and formatting data

library(unmarked)
y <- wt[,2:4]
siteCovs <-  wt[,c("elev", "forest", "length")]
obsCovs <- list(date=wt[,c("date.1", "date.2", "date.3")],
ivel=wt[,c("ivel.1",  "ivel.2", "ivel.3")])
wt <- unmarkedFrameOccu(y = y, siteCovs = siteCovs, obsCovs = obsCovs)
summary(wt)
## unmarkedFrame Object
##
## 237 sites
## Maximum number of observations per site: 3
## Mean number of observations per site: 2.81
## Sites with at least one detection: 79
##
## Tabulation of y observations:
##    0    1 <NA>
##  483  182   46
##
## Site-level covariates:
##       elev               forest              length
##  Min.   :-1.436125   Min.   :-1.265352   Min.   :0.1823
##  1st Qu.:-0.940726   1st Qu.:-0.974355   1st Qu.:1.4351
##  Median :-0.166666   Median :-0.064987   Median :1.6094
##  Mean   : 0.007612   Mean   : 0.000088   Mean   :1.5924
##  3rd Qu.: 0.994425   3rd Qu.: 0.808005   3rd Qu.:1.7750
##  Max.   : 2.434177   Max.   : 2.299367   Max.   :2.2407
##
## Observation-level covariates:
##       date               ivel
##  Min.   :-2.90434   Min.   :-1.7533
##  1st Qu.:-1.11862   1st Qu.:-0.6660
##  Median :-0.11862   Median :-0.1395
##  Mean   :-0.00022   Mean   : 0.0000
##  3rd Qu.: 1.30995   3rd Qu.: 0.5493
##  Max.   : 3.80995   Max.   : 5.9795
##  NA's   :42         NA's   :46

Alternatively, the convenience function csvToUMF can be used

wt <- csvToUMF(system.file("csv","widewt.csv", package="unmarked"),
long = FALSE, type = "unmarkedFrameOccu")

If not all sites have the same numbers of observations, then manual importation of data in long format can be tricky. csvToUMF seamlessly handles this situation.

pcru <- csvToUMF(system.file("csv","frog2001pcru.csv", package="unmarked"),
long = TRUE, type = "unmarkedFrameOccu")

To help stabilize the numerical optimization algorithm, we recommend standardizing the covariates.

obsCovs(pcru) <- scale(obsCovs(pcru))

## 3.2 Fitting models

Occupancy models can then be fit with the occu() function:

fm1 <- occu(~1 ~1, pcru)
## Warning in truncateToBinary(designMats$y): Some observations were > 1. These ## were truncated to 1. fm2 <- occu(~ MinAfterSunset + Temperature ~ 1, pcru) ## Warning in truncateToBinary(designMats$y): Some observations were > 1. These
## were truncated to 1.
fm2
##
## Call:
## occu(formula = ~MinAfterSunset + Temperature ~ 1, data = pcru)
##
## Occupancy:
##  Estimate    SE    z  P(>|z|)
##      1.54 0.292 5.26 1.42e-07
##
## Detection:
##                Estimate    SE      z  P(>|z|)
## (Intercept)      0.2098 0.206  1.017 3.09e-01
## MinAfterSunset  -0.0855 0.160 -0.536 5.92e-01
## Temperature     -1.8936 0.291 -6.508 7.60e-11
##
## AIC: 356.7591

Here, we have specified that the detection process is modeled with the MinAfterSunset and Temperature covariates. No covariates are specified for occupancy here. See ?occu for more details.

## 3.3 Back-transforming parameter estimates

unmarked fitting functions return unmarkedFit objects which can be queried to investigate the model fit. Variables can be back-transformed to the unconstrained scale using backTransform. Standard errors are computed using the delta method.

backTransform(fm2, 'state')
## Backtransformed linear combination(s) of Occupancy estimate(s)
##
##  Estimate     SE LinComb (Intercept)
##     0.823 0.0425    1.54           1
##
## Transformation: logistic

The expected probability that a site was occupied is 0.823. This estimate applies to the hypothetical population of all possible sites, not the sites found in our sample. For a good discussion of population-level vs finite-sample inference, see Royle and Dorazio (2008) page 117. Note also that finite-sample quantities can be computed in unmarked using empirical Bayes methods as demonstrated at the end of this document.

Back-transforming the estimate of $$\psi$$ was easy because there were no covariates. Because the detection component was modeled with covariates, $$p$$ is a function, not just a scalar quantity, and so we need to be provide values of our covariates to obtain an estimate of $$p$$. Here, we request the probability of detection given a site is occupied and all covariates are set to 0.

backTransform(linearComb(fm2, coefficients = c(1,0,0), type = 'det'))
## Backtransformed linear combination(s) of Detection estimate(s)
##
##  Estimate    SE LinComb (Intercept) MinAfterSunset Temperature
##     0.552 0.051    0.21           1              0           0
##
## Transformation: logistic

Thus, we can say that the expected probability of detection was 0.552 when time of day and temperature are fixed at their mean value. A predict method also exists, which can be used to obtain estimates of parameters at specific covariate values.

newData <- data.frame(MinAfterSunset = 0, Temperature = -2:2)
round(predict(fm2, type = 'det', newdata = newData, appendData=TRUE), 2)
##   Predicted   SE lower upper MinAfterSunset Temperature
## 1      0.98 0.01  0.93  1.00              0          -2
## 2      0.89 0.04  0.78  0.95              0          -1
## 3      0.55 0.05  0.45  0.65              0           0
## 4      0.16 0.03  0.10  0.23              0           1
## 5      0.03 0.01  0.01  0.07              0           2

Confidence intervals are requested with confint, using either the asymptotic normal approximation or profiling.

confint(fm2, type='det')
confint(fm2, type='det', method = "profile")
##                        0.025      0.975
## p(Int)            -0.1946871  0.6142292
## p(MinAfterSunset) -0.3985642  0.2274722
## p(Temperature)    -2.4638797 -1.3233511
##                        0.025      0.975
## p(Int)            -0.1929210  0.6208837
## p(MinAfterSunset) -0.4044794  0.2244221
## p(Temperature)    -2.5189984 -1.3789261

## 3.4 Model selection and model fit

Model selection and multi-model inference can be implemented after organizing models using the fitList function.

fms <- fitList('psi(.)p(.)' = fm1, 'psi(.)p(Time+Temp)' = fm2)
modSel(fms)
##                    nPars    AIC  delta   AICwt cumltvWt
## psi(.)p(Time+Temp)     4 356.76   0.00 1.0e+00     1.00
## psi(.)p(.)             2 461.00 104.25 2.3e-23     1.00
predict(fms, type='det', newdata = newData)
##    Predicted         SE     lower      upper
## 1 0.98196076 0.01266193 0.9306044 0.99549474
## 2 0.89123189 0.04248804 0.7763166 0.95084836
## 3 0.55225129 0.05102660 0.4514814 0.64890493
## 4 0.15658708 0.03298276 0.1021713 0.23248007
## 5 0.02718682 0.01326263 0.0103505 0.06948653

The parametric bootstrap can be used to check the adequacy of model fit. Here we use a $$\chi^2$$ statistic appropriate for binary data.

chisq <- function(fm) {
umf <- fm@data
y <- umf@y
y[y>1] <- 1
sr <- fm@sitesRemoved
if(length(sr)>0)
y <- y[-sr,,drop=FALSE]
fv <- fitted(fm, na.rm=TRUE)
y[is.na(fv)] <- NA
sum((y-fv)^2/(fv*(1-fv)), na.rm=TRUE)
}

(pb <- parboot(fm2, statistic=chisq, nsim=100, parallel=FALSE))
##
## Call: parboot(object = fm2, statistic = chisq, nsim = 100, parallel = FALSE)
##
## Parametric Bootstrap Statistics:
##    t0 mean(t0 - t_B) StdDev(t0 - t_B) Pr(t_B > t0)
## 1 356           17.5             17.6        0.149
##
## t_B quantiles:
##      0% 2.5% 25% 50% 75% 97.5% 100%
## t*1 302  312 327 334 347   379  391
##
## t0 = Original statistic computed from data
## t_B = Vector of bootstrap samples

We fail to reject the null hypothesis, and conclude that the model fit is adequate.

## 3.5 Derived parameters and empirical Bayes methods

The parboot function can be also be used to compute confidence intervals for estimates of derived parameters, such as the proportion of $$N$$ sites occupied $$\mbox{PAO} = \frac{\sum_i z_i}{N}$$ where $$z_i$$ is the true occurrence state at site $$i$$, which is unknown at sites where no individuals were detected. The colext vignette shows examples of using parboot to obtain confidence intervals for such derived quantities. An alternative way achieving this goal is to use empirical Bayes methods, which were introduced in unmarked version 0.9-5. These methods estimate the posterior distribution of the latent variable given the data and the estimates of the fixed effects (the MLEs). The mean or the mode of the estimated posterior distibution is referred to as the empirical best unbiased predictor (EBUP), which in unmarked can be obtained by applying the bup function to the estimates of the posterior distributions returned by the ranef function. The following code returns an estimate of PAO using EBUP.

re <- ranef(fm2)
EBUP <- bup(re, stat="mode")
sum(EBUP) / numSites(pcru)
## [1] 0.8076923

Note that this is similar, but slightly lower than the population-level estimate of $$\psi$$ obtained above.

A plot method also exists for objects returned by ranef, but distributions of binary variables are not so pretty. Try it out on a fitted abundance model instead.

# References

Chandler, R. B., J. A. Royle, and D. I. King. 2011. Inference about density and temporary emigration in unmarked populations. Ecology 92:1429–1435.

Fiske, I., and R. Chandler. 2011. Unmarked: An R package for fitting hierarchical models of wildlife occurrence and abundance. Journal of Statistical Software 43:1–23.

MacKenzie, D. I., J. D. Nichols, J. E. Hines, M. G. Knutson, and A. B. Franklin. 2003. Estimating site occupancy, colonization, and local extinction when a species is detected imperfectly. Ecology 84:2200–2207.

MacKenzie, D. I., J. D. Nichols, G. B. Lachman, S. Droege, J. A. Royle, and C. A. Langtimm. 2002. Estimating site occupancy rates when detection probabilities are less than one. Ecology 83:2248–2255.

Royle, J. A. 2004a. N-mixture models for estimating population size from spatially replicated counts. Biometrics 60:108–115.

Royle, J. A. 2004b. Generalized estimators of avian abundance from count survey data. Animal Biodiversity and Conservation 27:375–386.

Royle, J. A., D. K. Dawson, and S. Bates. 2004. Modeling abundance effects in distance sampling. Ecology 85:1591–1597.

Royle, J. A., and R. M. Dorazio. 2008. Hierarchical modeling and inference in ecology: The analysis of data from populations, metapopulations and communities. Academic Press.

Royle, J. A., and J. D. Nichols. 2003. Estimating abundance from repeated presence-absence data or point counts. Ecology 84:777–790.