engen2008II: Measuring Habitat Selection Using the Method of Engen et al. (2008)

Description

These functions implements the method described by Engen et al. to measure the preference of animals for habitat variables in habitat selection studies.

Usage

engen2008II(us, av, id, nsim = 500, nsimra = 500)
engen2008I(us, av, nsimra=500)
# S3 method for engenetalI
print(x, …)
# S3 method for engenetalII
print(x, …)

Arguments

a data frame containing the value of numeric habitat variables (columns) in each site (rows) used by the animals.

a data frame containing the value of numeric habitat variables (columns) in each site (rows) available to the animals.

a factor with as many elements as there are rows in us, indicating the ID of the animal that used the corresponding rows in us.

nsim

the number of randomizations used in the calculation of the total variance.

nsimra

the number of random allocation of ranks used in the calculation of the normal score (see details).

an object of class engenetalI or engenetalII

…

additional arguments to be passed to other functions (currently unused)

Value

engen2008I returns a list of class engenetalI, and engen2008II returns a list of class engenetalII. Both types of list contain two elements:

raw

this is a list containing one data frame per habitat variable, containing the value of the correlation rho (for engenetalII objects), mean preferences and standard error of these preferences (columns) for each randomization performed (rows);

results

a data frame containing the mean values over all the randomizations, of these parameters (columns) for each habitat variable (rows).

Details

Engen et al. (2008) proposed an original approach to measure the preference of animals for values of each particular variable of a multivariate set of environmental variables. Their approach was originally developed for the case where there is a sample of used site is for each animal in a sample of identified animals (e.g. using radiotelemetry or GPS), with several sites per animal (i.e., design II according to the classification of Thomas and Taylor, 1990). However, we extended this approach to also include the case where habitat use is described by a sample of used site, with one site per unidentified animal (i.e., design I).

The original approach is the following: first, a normal score transformation of each habitat variable is performed: for each variable, the empirical cumulative distribution is computed, by dividing the rank of the value of each available site by the number of observations. Note that the ties are ranked randomly. Then, the inverse of the standard normal integral (see ?qnorm) of the cumulative distribution function is computed for all available sites: this results into a perfectly normal distribution of the habitat variables for the available sites. Then, the value of the cumulative distribution -- estimated from the available sites -- is computed for each used site. Then, the inverse of the standard normal integral is computed for each one.

Engen et al. (2008) suppose the following model describing how habitat use results from habitat availability. Let $$Z_{ij}$$ be the value of a given habitat variable (transformed according to the normal score) for the j-th site used by the i-th animal. Then this value can be described by the model:

$$Z_{ij} = \mu + U_i + V_{ij}$$

where $$\mu$$ is the preference for the habitat variable (0 indicates a non-preference), $$U_i$$ and $$V_{ij}$$ are normal distributions with means equal to zero and variances equal to $$\sigma^2$$ and $$\tau^2$$ respectively. Engen et al. give fomula for the estimation of these parameters. Their estimation is done by first estimating the total variance $$\sigma^2+ \tau^2$$ (this variance is estimated by sampling randomly one observation per animal -- the parameter nsim controls the number of samples used in this computation; see Engen et al. 2008). Note that the correlation between the value observed for two used units sampled from all the units used by a given animal is $$\rho = \sigma^2/(\sigma^2 + \tau^2)$$. A large value of rho indicates a large variation in the habitat used between animals (or a small within-animal variation). The main parameter of concern is here the preference. The function engen2008II allows to estimate these parameters.

The function engen2008I extends this model for design I studies (a sample of used sites and a sample of available sites, animals not identified), by considering the following model for these studies:

$$Z_{ij} = \mu + V_{ij}$$

where $$\mu$$ is the preference for the habitat variable (0 indicates a non-preference), and $$V_{ij}$$ are normal distributions with means equal to zero and variances equal to $$\sigma^2$$.

Note that the habitat variables may be correlated on the study area. In this case, observed preference for a given variable may be an artefact of other variables prefered by animals. Supposing that the data.frame containing the $$Z_{ij}$$'s is a realization of a multivariate normal distribution, we can compute, for each habitat variable and each used site, the *conditional* mean $$m_{ij}$$ and *conditional* standard deviation $$s_{ij}$$ of this variable at this site, *given* the values of the other habitat variables at this site (this is done using the algorithm described by Ripley, 1987, p.98). We then compute the standardized values $$P_{ij} = (Z_{ij} - m_{ij})/s_{ij}$$. The preference is then computed using these standardized values.

Because there may be ties in the distribution of values of habitat variables, the results may vary depending on the random order chosen for ties when computing the normal scores. Engen et al. recommended to repeat the function a large number of times, and to use the mean values as estimates of the parameters. This is what the function does, and the number of randomization is controlled by the parameter nsimra.

Note that all these methods rely on the following hypotheses: (i) independence between animals, (ii) independence between sites, and (iii) all animals are selecting habitat in the same way (in addition to "traditional" hypotheses in these kinds of studies: no territoriality, all animals having equal access to all available resource units, etc., see Manly et al. 2002 for further details).

That the examples below provide an illustration and discussion of interesting and (at first sight) surprising properties of this method.

References

Engen, S., Grotan, V., Halley, D. and Nygard, T. (2008) An efficient multivariate approach for estimating preference when individual observations are dependent. Journal of Animal Ecology, 77: 958--965.

Thomas, D. and Taylor, E. (1990) Study designs and tests for comparing resource use and availability. Journal of Wildlife Management, 54, 322--330.

Ripley, B. (1987) Stochastic Simulation. John Wiley and Sons.

Examples

Run this code

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## Practical use of engen2008II

data(puechabon)
kasc <- puechabon$kasc

## Removes the aspect (no factor allowed in the function)
kasc$Aspect <- NULL

## engen2008II:
avail <- kasc2df(kasc)$tab
use <- join.kasc(puechabon$locs[,c("X","Y")], kasc)
id <- puechabon$locs$Name

## This function can be very long:
engen2008II(use, avail, id, nsimra=10)




## Practical use of engen2008I

data(lynxjura)
ka <- lynxjura$map
lo <- lynxjura$locs[,1:2]
av <- kasc2df(ka)$tab
us <- join.kasc(lo, ka)

## Idem, be patient here:
engen2008I(us, av, nsimra=10)




#################################
#################################
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##
##  For a deeper discussion on
##  this method... a simulation:



#################################
##
## First, simulation of a dataset
## copy and paste this part into R,
## but skip the reading of the
## comments if you are not interested
## into this simulation

## simulate the available points

set.seed(235)
av <- cbind(rnorm(1000, mean=0, sd=3), rnorm(1000, mean=0, sd=0.5))
tt <- cbind(c(cos(-pi/4),sin(-pi/4)),c(cos(pi/4), sin(pi/4)))
av <- as.data.frame(as.matrix(av)%*%tt)

## simulate the used points: we simulate a selection on the first
## principal component of the PCA of the data.frame describing the
## availability. In other words, we simulate the case where the
## habitat selection occurs on the "common part" of the two habitat
## variable (no preference for one particular variable).

us <- do.call("rbind", lapply(1:5, function(i) {
    us1 <- cbind(rnorm(30, mean=rnorm(1, -4, 1), sd=0.5),
                 rnorm(30, mean=rnorm(1, 0, 0.5), sd=0.2))

    return(us1%*%tt)
}))
colnames(us) <- colnames(av) <- c("var1", "var2")
id <- gl(5,30)


#################################
##
## Study of the habitat selection on these data
## The data are:
## - us: a matrix containing the used sites for two
##       habitat variables
## - av: a matrix containing the available sites for two
##       habitat variables
## - id: a vector containing the id of 5 animals

## First illustrate the use and availability of the two variables:

plot(av, xlab="Habitat variable 1", ylab="Habitat variable 2",
     col="grey", pch=16)
lius <- split(as.data.frame(us), id)
junk <- lapply(1:5, function(i) points(lius[[i]], pch=16, col=i))

## -----> ***It is very clear that there is a selection***:
## the animals select the low values of both habitat variables.
## (this is what we actually simulated


## Now perform the method of Engen et al. (2008):
engen2008II(us, av, id)


## Surprisingly, the method seems to fail to identify the clear
## habitat selection identified graphically...
##
## In fact, it does not fail:
## this method identifies the part of habitat selection that is clearly
## attributable to a given variable.  Here the animals select the
## the common factor expressed in the two variables, and it is impossible
## to identify whether the selection is due only to the variable 1 or to
## the variable 2: it is caused by both variable simultaneously.
## Once the selection on the variable 2 (including the common part)
## has been removed, there is no longer appearant selection on
## variable 1.  Once the selection caused by the variable 1
## (including the common part) has been removed, there is no
## longer selection on variable 2...
##
## For this reason, Engen et al. recommended to use this method
## concurrently with other factor analyses of the habitat selection
## such as madifa, kselect, niche (in ade4 package), etc.
##
## Note also the strong correlation between the value of two random
## points used by a given animal. This indicates a strong variability
## among animals...



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