A permutation-based combination of sign tests for assessing habitat selection

Abstract

The analysis of habitat selection in radio-tagged animals is approached by comparing the portions of use against the portions of availability observed for each habitat type. Since data are linearly dependent with singular variance-covariance matrices, standard multivariate statistical tests cannot be applied. To bypass the problem, compositional data analysis is customarily performed via log-ratio transform of sample observations. The procedure is criticized in this paper, emphasizing the several drawbacks which may arise from the use of compositional analysis. An alternative nonparametric solution is proposed in the framework of multiple testing. The habitat use is assessed separately for each habitat type by means of the sign test performed on the original observations. The resulting p values are combined in an overall test statistic whose significance is determined permuting sample observations. The theoretical findings of the paper are checked by simulation studies. Applications to case studies previously considered in literature are discussed.

Appendices

Appendix 1: Different expressions for the hypothesis of random habitat use

The RHU hypothesis (2) actually constitutes a multivariate hypothesis which can be rewritten as

$$\begin{aligned} \text{ H }_{X0} :\bigcap _{j=1}^K \left\{ \text{ E }(X_{Uj} -X_{Aj})=0\right\} \end{aligned}$$

(12)

where $\text{ E }(X_{Uj} -X_{Aj})=0$ is the univariate hypothesis that the expected use of habitat $j$ coincides with its expected (or constant) availability. The obvious sense of (12) is that $\text{ H }_{X0}$ is true if all the univariate hypotheses are true. In turn, chosen a reference habitat $h$, (12) is equivalent to

$$\begin{aligned} \text{ H }_{X0}:\bigcap _{j\ne h=1}^K \left\{ \frac{\text{ E }(X_{Uj})}{\text{ E }(X_{Uh})}= \frac{\text{ E }(X_{Aj})}{\text{ E }(X_{Ah})}\right\} \end{aligned}$$

(13)

Indeed, if (2) is true, than for any habitat $j$ it follows from (12) that $\text{ E }(X_{Uj})=\text{ E }(X_{Aj})$ from which $\text{ E }(X_{Uj})/\text{ E }(X_{Aj})=1$. Accordingly, for the reference habitat $h$ and for each $j\ne h$, it follows that $\text{ E }(X_{Uj})/\text{ E }(X_{Aj})=\text{ E }(X_{Uh})/\text{ E }(X_{Ah})$ from which $\text{ E }(X_{Uj})/\text{ E }(X_{Uh})=\text{ E }(X_{Aj})/\text{ E }(X_{Ah})$. As to the reverse, if (13) is true, then for the reference habitat $h$ and for each $j\ne h$ it holds that $\text{ E }(X_{Uj})/\text{ E }(X_{Uh})=\text{ E }(X_{Aj})/\text{ E }(X_{Ah})$ or equivalently $\text{ E }(X_{Uj})/\text{ E }(X_{Aj})=\text{ E }(X_{Uh})/\text{ E }(X_{Ah})$, i.e. for each $j=1, \ldots , K$ it holds that $\text{ E }(X_{Uj})/\text{ E }(X_{Aj})=c$ or equivalently $\text{ E }(X_{Uj})=c\text{ E }(X_{Aj})$.

But since $\sum _{j=1}^K {\text{ E }(X_{Uj})} =\sum _{j=1}^K {\text{ E }(X_{Aj})} =1$, then $c=1$, which obviously implies (2).

In a similar way, chosen a reference habitat $h$, (3) constitutes a multivariate hypothesis which is equivalent to

$$\begin{aligned} \text{ H }_{Y0} :\bigcap _{j\ne h=1}^K {\left\{ \text{ E }(Y_{Uj}-Y_{Aj}) =0\right\} } \end{aligned}$$

or, more explicitly, to

$$\begin{aligned} \text{ H }_{Y0} :\bigcap _{j\ne h=1}^K \left\{ \text{ E }\left( \ln \frac{X_{Uj}}{X_{Uh}}\right) =\text{ E }\left( \ln \frac{X_{Aj}}{X_{Ah}}\right) \right\} \end{aligned}$$

(14)

From (13) and (14), it is at once apparent that (3) is equivalent to (2) if

$$\begin{aligned} \ln \left\{ \frac{\text{ E }(X_{Uj})}{\text{ E }(X_{Uh})}\right\} -\ln \left\{ \frac{\text{ E }(X_{Aj})}{\text{ E }(X_{Ah})} \right\} =\text{ E }\left\{ \ln \left( \frac{X_{Uj}}{X_{Uh}} \right) \right\} -\text{ E }\left\{ \ln \left( \frac{X_{Aj}}{X_{Ah}} \right) \right\} =0 \end{aligned}$$

(15)

for each $j\ne h=1,\ldots , K$. Since $\text{ E }\left\{ \ln (X)\right\}$ generally differs from $\ln \text{ E }(X)$, relation (15) does not generally hold.

Appendix 2: Dirichlet distributions and log-ratio transforms

The Dirichlet distribution is probably the most familiar model adopted for positive random vectors $\mathbf{X}=\left[ X_1, \ldots , X_K\right] ^{\mathrm{T}}$ subject to the constraint $\mathbf{1}^{\mathrm{T}}\mathbf{X}=1$. A $K$-variate random vector X is said to have a Dirichlet distribution with parameters $\delta >0$ and ${\varvec{\uptheta }}=\left[ \theta _1, \ldots , \theta _K\right] ^{\mathrm{T}}$ with $\theta _{j}>0$ for each $j=1,\ldots , K$ if the joint probability density function at $\mathbf{x}=\left[ x_1, \ldots , x_K \right] ^{\mathrm{T}}$ with $\mathbf{1}^{\mathrm{T}}\mathbf{x}=1$ is given by

$$\begin{aligned} f(\mathbf{x})=\frac{\Gamma (\delta \theta )}{\prod \nolimits _{j=1}^{K} {\Gamma (\delta \theta _j)}}\prod _{j=1}^K {x_j^{\delta \theta _j -1}} \end{aligned}$$

where $\theta =\mathbf{1}^{\mathrm{T}}{\varvec{\uptheta }}$. As is well known (e.g. Fang et al. 1990), each marginal variable $X_j$ has a beta distribution on [0,1] with shape parameters $\delta \theta _j$ and $\delta (\theta -\theta _j)$ in such a way that

$$\begin{aligned} \text{ E }(X_j)=\frac{\theta _j}{\theta } \end{aligned}$$

and

$$\begin{aligned} \text{ V }(X_j)=\frac{\theta _j (\theta -\theta _j)}{\theta ^{2}(\delta \theta +1)} \end{aligned}$$

Accordingly, marginal expectations do not depend on $\delta$ and marginal variances increase as $\delta$ decreases. In the framework of habitat selection analysis, $\delta$ obviously accounts for the variability of portions of animal trajectories or home ranges within habitat types. However, when these quantities are estimated on the field by means of animal’s radio locations, $\delta$ also accounts for the number of radio locations adopted in the study, since marginal variances decrease as the $r_i\text{ s }$ increase and estimates become close to the real values.

If X has a Dirichlet distribution with parameters $\delta$ and ${\varvec{\uptheta }}$, the log-ratio transform $\mathbf{Y}=lr_h (\mathbf{X})$ is a random vector on $R^{K-1}$ whose $j$-th marginal random variable $Y_j =\ln (X_j/X_h)$ has a generalized logistic distribution of type IV with expectation

$$\begin{aligned} \text{ E }(Y_j)=\varphi (\delta \theta _j)-\varphi (\delta \theta _h) \end{aligned}$$

(16)

where $\varphi (x)=\partial \ln \Gamma (x)/\partial x$ denotes the digamma function (e.g. Johnson et al. 1995, p. 142, Fang et al. 1990, Problem 1.5).

In the case of Johnson’s second order selection, denote by a the vector of portions of habitat types in the study area and suppose that $\mathbf{X}_U$ has a Dirichlet distribution with parameters $\delta _U$ and a, in such a way that $\text{ H }_{X0}$ is true. Thus, in accordance with (16), the squared value of the unreliability measure of CODA-based procedure turns out to be

$$\begin{aligned} \Delta ^{2}=\frac{1}{K}\sum _{h=1}^K {\sum _{j\ne h} {\left\{ \phi (\delta _U a_j)-\phi (\delta _U a_h)-\ln (a_j/a_h)\right\} ^{2}}} \end{aligned}$$

(17)

In a similar way, in the case of Johnson’s third order selection, suppose that $\mathbf{X}_U$ and $\mathbf{X}_A$ have Dirichlet distributions with the same parameter a and variability parameters $\delta _U$ and $\delta _A$, respectively, in such a way that $\text{ H }_{X0}$ is true. From (16), the squared value of unreliability measure is

$$\begin{aligned} \Delta ^{2}=\frac{1}{K}\sum _{h=1}^K {\sum _{j\ne h}{\left\{ \phi (\delta _U a_j)-\phi (\delta _U a_h)-\phi (\delta _A a_j)-\phi (\delta _A a_h)\right\} ^{2}}} \end{aligned}$$

(18)

Appendix 3: Generating dependent compositional data

It is worth noting that $\mathbf{X}_U$ and $\mathbf{X}_A$ arise from the choice of the same animal and as such they should be realistically presumed as dependent random vectors. However, the general problem of constructing dependent random vectors $\mathbf{X}_1 =\left[ X_{11}, \ldots , X_{1K}\right] ^{\mathrm{T}}$ and $\mathbf{X}_2 =\left[ X_{21}, \ldots , X_{2K}\right] ^{\mathrm{T}}$ subject to the constraint $\mathbf{1}^{\mathrm{T}}\mathbf{X}_1 =\mathbf{1}^{\mathrm{T}}\mathbf{X}_2 =1$ is difficult to solve in the framework of Dirichlet model since any couple of subvectors $\mathbf{X}_1,\,\mathbf{X}_2$ partitioning a vector X with a Dirichlet distribution turn out to be independent with marginal Dirichlet distributions (see Fang et al. 1990, Theorem 1.4).

For this purpose, it is convenient to consider one vector, say $\mathbf{X}_1$, distributed as a Dirichlet random vector with parameters $\delta >0$ and ${\varvec{\uptheta }}$ in such a way that $\mathbf{1}^{\mathrm{T}}\mathbf{X}_1 =1$, and then obtaining $\mathbf{X}_2$ by means of $\mathbf{X}_1 +\mathbf{U}$, where U is a random vector in which $K-1$ components, say $U_1, \ldots , U_{K-1}$, are random variables in the range $(-W,W)$ with

$$\begin{aligned} W=\text{ min }\left( X_{11}, \ldots , X_{1K-1}, \frac{X_{1K}}{K-1}\right) \end{aligned}$$

and $U_K =-(U_1 +\cdots +U_{K-1})$. Indeed, after a straightforward algebra it can be proven that $0<X_{2j} <1$ for each $j=1,\ldots , K$ while $\mathbf{1}^{\mathrm{T}}\mathbf{X}_2 =1$ by construction. Obviously $\text{ E }(X_{2j})=\text{ E }(X_{1j})+\text{ E }(U_j)$, while $\text{ V }(X_{2j})=\text{ V }(X_{1j})+\text{ V }(U_j)$, providing that $\mathbf{X}_1$ and U are independent. If $\text{ E }(\mathbf{U})=\mathbf 0$, then $\mathbf{X}_1$ and $\mathbf{X}_2$ are dependent with the same mean vector. Moreover, if the $U_j\text{ s }$ are symmetrically distributed around 0, than $\text{ Pr }(X_{2j}>X_{1j})=0.5$ for each $j=1,\ldots , K$. These two last features can be readily achieved if the $U_j\text{ s }$ are independent beta variables on $(-W,W)$ with shape parameters both equal to $\beta >0$ in such a way that they turn out to be symmetric around 0, with variance

$$\begin{aligned} \text{ V }(U_j)=\frac{1}{4(\beta ^{2}+1)} \end{aligned}$$

Accordingly the $U_j\text{ s }$ inflate the variances of the $X_{1j}$ by a term which increases as $\beta$ approaches 0.

If $\mathbf{X}_1$ coincides with the vector of constants a, then if $\text{ E }(\mathbf{U})=\mathbf 0$ and the $U_j\text{ s }$ are symmetrically distributed around 0, $\text{ E }(\mathbf{X}_2)=\mathbf{a},\,\text{ Pr }(X_{2j} >a_j)=0.5$ and $\text{ V }(X_{2j})=\text{ V }(U_j)$ for each $j=1,\ldots , K$. Obviously, in this case the $U_j\text{ s }$ varies on $(-w,w)$ with $w=\min (a_1, \ldots , a_{K-1}, \frac{a_K}{K-1})$.