Composite likelihood approach to the regression analysis of spatial multivariate ordinal data and spatial compositional data with exact zero values

Abstract

In many environmental and ecological studies, it is of interest to model compositional data. One approach is to consider positive random vectors that are subject to a unit-sum constraint. In landscape ecological studies, it is common that compositional data are also sampled in space with some elements of the composition absent at certain sampling sites. In this paper, we first propose a practical spatial multivariate ordered probit model for multivariate ordinal data, where the response variables can be viewed as the discretized non-negative compositions without the unit-sum constraint. We then propose a novel two-stage spatial mixture Dirichlet regression model. The first stage models the spatial dependence and the presence of exact zero values, and the second stage models all the non-zero compositional data. A maximum composite likelihood approach is developed for parameter estimation and inference in both the spatial multivariate ordered probit model and the two-stage spatial mixture Dirichlet regression model. The standard errors of the parameter estimates are computed by an estimate of the Godambe information matrix. A simulation study is conducted to evaluate the performance of the proposed models and methods. A land cover data example in landscape ecology further illustrates that accounting for spatial dependence can improve the accuracy in the prediction of presence/absence of different land covers as well as the magnitude of land cover compositions.

Appendices

Appendix 1: Technical details for Sect. 2.2

To efficiently obtain the first-order derivatives, for each response type we define a new vector of parameters that contains the cutoff parameters and regression coefficients: $\tilde{{\varvec{\beta }}}_i = (\alpha _{i0}, \alpha _{i1}, {\varvec{\alpha }}_i', \alpha _{iK}, {\varvec{\beta }}_i')'$. Let $\tilde{{\varvec{\beta }}} = (\tilde{{\varvec{\beta }}}_1', \ldots , \tilde{{\varvec{\beta }}}_I')'$, and we also let $\alpha _{i0} \equiv -\infty , \alpha _{i1} \equiv 0$, and $\alpha _{iK} \equiv +\infty$ only to simplify the notation, while the corresponding derivatives will not be considered in the score function. For response type i, we define new design matrices for the upper and lower limits in the integrals of (7) as

respectively, where ${\varvec{e}}_n$ is a $(p+1)$-dimensional basis vector with the nth entry being 1 and others being 0. For example, if $y_{ij} = 1$, then ${\varvec{e}}_{y_{ij}+1} = (0,1,0, \ldots ,0)'$. We then define ${\varvec{X}}_{\text {up}}$ and ${\varvec{X}}_{\text {lo}}$ as the matrices that include design matrices for all response types, that is,

$$\begin{aligned} {\varvec{X}}_{\text {up}} = ({\varvec{X}}_{\text {up}}^{(1)'}, \ldots , {\varvec{X}}_{\text {up}}^{(i)'}, \ldots , {\varvec{X}}_{\text {up}}^{(I)'})', {\varvec{X}}_{\text {lo}} = ({\varvec{X}}_{\text {lo}}^{(1)'}, \ldots , {\varvec{X}}_{\text {lo}}^{(i)'}, \ldots , {\varvec{X}}_{\text {lo}}^{(I)'})'. \end{aligned}$$

It follows that the bivariate density function $P(Y_{ij} = y_{ij}, Y_{i'j'}=y_{i'j'})$ can be rewritten as

$$\begin{aligned}&\varPhi _2({\varvec{x}}_{\text {up},(i-1)N+j}' \tilde{{\varvec{\beta }}},{\varvec{x}}_{\text {up},(i'-1)N+j'}'\tilde{{\varvec{\beta }}};\tilde{\rho }_{ii'jj'})\nonumber \\&\quad +\, \varPhi _2({\varvec{x}}_{\text {lo},(i-1)N+j}'\tilde{{\varvec{\beta }}},{\varvec{x}}_{\text {lo},(i'-1)N+j'}'\tilde{{\varvec{\beta }}};\tilde{\rho }_{ii'jj'}) \nonumber \\&\quad -\, \varPhi _2({\varvec{x}}_{\text {up},(i-1)N+j}'\tilde{{\varvec{\beta }}},{\varvec{x}}_{\text {lo},(i'-1)N+j'}'\tilde{{\varvec{\beta }}};\tilde{\rho }_{ii'jj'})\nonumber \\&\quad - \, \varPhi _2({\varvec{x}}_{\text {lo},(i-1)N+j}'\tilde{{\varvec{\beta }}},{\varvec{x}}_{\text {up},(i'-1)N+j'}'\tilde{{\varvec{\beta }}};\tilde{\rho }_{ii'jj'}), \end{aligned}$$

(14)

where $\tilde{{\varvec{\theta }}} = (\tilde{{\varvec{\beta }}}', {\varvec{\gamma }}', \sigma ^2, \rho )'$, and ${\varvec{x}}_{\text {up},(i-1)N+j}$ and ${\varvec{x}}_{\text {lo},(i-1)N+j}$ are the $\{(i-1)N+j\}$th row of ${\varvec{X}}_{\text {up}}$ and ${\varvec{X}}_{\text {lo}}$, respectively.

Further, let $\tilde{{\varvec{x}}}_{\text {up},ij} = {\varvec{x}}_{\text {up},(i-1)N+j}$ and $\tilde{{\varvec{x}}}_{\text {lo},ij} = {\varvec{x}}_{\text {lo},(i-1)N+j}$. The derivatives for any of the bivariate CDFs in (14) can be obtained as

$$\begin{aligned} \frac{\partial \varPhi _2(\tilde{{\varvec{x}}}_{ij}'\tilde{{\varvec{\beta }}},\tilde{{\varvec{x}}}_{i'j'}'\tilde{{\varvec{\beta }}};\tilde{\rho }_{ii'jj'})}{\partial {\tilde{{\varvec{\beta }}}}}= & {} \phi _1(\tilde{{\varvec{x}}}_{ij}'\tilde{{\varvec{\beta }}})\varPhi _1(\xi (ij,i'j'))\tilde{{\varvec{x}}}_{ij}'\nonumber \\&\quad +\, \phi _1(\tilde{{\varvec{x}}}_{i'j'}'\tilde{{\varvec{\beta }}})\varPhi _1(\xi (i'j',ij))\tilde{{\varvec{x}}}_{i'j'}', \nonumber \\ \frac{\partial \varPhi _2(\tilde{{\varvec{x}}}_{ij}'\tilde{{\varvec{\beta }}},\tilde{{\varvec{x}}}_{i'j'}'\tilde{{\varvec{\beta }}};\tilde{\rho }_{ii'jj'})}{\partial {\tilde{\rho }_{ii'jj'}}}= & {} \phi _2(\tilde{{\varvec{x}}}_{ij}'\tilde{{\varvec{\beta }}},\tilde{{\varvec{x}}}_{i'j'}'\tilde{{\varvec{\beta }}};\tilde{\rho }_{ii'jj'}), \end{aligned}$$

(15)

where $\phi _1(t) = (2\pi )^{-1/2}\exp \left( -t^2/2\right)$, $\xi (ij,i'j') = \big (\tilde{{\varvec{x}}}_{i'j'}'\tilde{{\varvec{\beta }}}- \tilde{\rho }_{ii'jj'}\tilde{{\varvec{x}}}_{ij}'\tilde{{\varvec{\beta }}}\big )(1-\tilde{\rho }_{ii'jj'}^2)^{-1/2}$, and $\tilde{{\varvec{x}}}_{ij}$ is either $\tilde{{\varvec{x}}}_{\text {up},ij}$ or $\tilde{{\varvec{x}}}_{\text {lo},ij}$. The partial derivatives with respect to $\gamma _{i_1i_2}$, $\sigma ^2$, and $\rho$ are obtained by the chain rule, that is,

$$\begin{aligned} \partial \tilde{\rho }_{ii'jj'}/\partial \gamma _{i_1i_2}= & {} (\sigma ^2\rho ^{d_{ii'jj'}})^{\mathcal {I}(d_{jj'}> 0)} \mathcal {I}(i=i_1,i'= i_2), \text { for } i_1 = 1, \cdots I-1, i_2 = i_1+1, \cdots ,I, \nonumber \\ \partial \tilde{\rho }_{ii'jj'}/\partial \sigma ^2= & {} \rho ^{d_{ii'jj'}} {\mathcal {I}(d_{ii'jj'}>0)} \gamma _{12}^{\mathcal {I}(i = 1, i'=2)} \gamma _{13}^{\mathcal {I}(i = 1, i'=3)} \cdots \gamma _{(I-1)I}^{\mathcal {I}(i = I-1, i'=I)}, \nonumber \\ \partial \tilde{\rho }_{ii'jj'}/\partial \rho= & {} d_{ii'jj'} \sigma ^2 \rho ^{d_{ii'jj'}-1} \gamma _{12}^{\mathcal {I}(i = 1, i'=2)} \gamma _{13}^{\mathcal {I}(i = 1, i'=3)} \cdots \gamma _{(I-1)I}^{\mathcal {I}(i = I-1, i'=I)}. \end{aligned}$$

(16)

Appendix 2: Technical details for Sect. 3.1

The conditional density is given by

$$\begin{aligned} f_{\mathrm{D}|\mathrm{B}}(\tilde{{\varvec{y}}}^{(j)}|{\varvec{y}}^{(j)}) = \varGamma \left( \sum _{i=1}^I\phi y_{ij} \mu _{ij}\right) \left\{ \prod _{i =1}^I \varGamma (\phi \mu _{ij})^{y_{ij}} \right\} ^{-1} \prod _{i =1}^I \tilde{y}_{ij}^{y_{ij}(\phi \mu _{ij} - 1)}, \end{aligned}$$

(17)

Thus, the conditional expectations can be expressed as

$$\begin{aligned} E(\tilde{Y}_{ij}|{\varvec{Y}}^{(j)})= & {} {Y_{ij}\exp ({\varvec{x}}_{j}'{\varvec{\eta }}_{i})}\{\sum _{i=1}^{I}Y_{ij}\exp ({\varvec{x}}_{j}'{\varvec{\eta }}_{i})\}^{-1} \nonumber \\ E(\tilde{Y}^2_{ij}|{\varvec{Y}}^{(j)})= & {} \{Y_{ij}\exp ({\varvec{x}}_{j}'{\varvec{\eta }}_{i})\}\{Y_{ij}\exp ({\varvec{x}}_{j}'{\varvec{\eta }}_{i})+1\}\\&\times \left[ \left\{ \sum _{i=1}^{I}Y_{ij}\exp ({\varvec{x}}_{j}'{\varvec{\eta }}_{i})\}\{1+\sum _{i=1}^{I}Y_{ij}\exp ({\varvec{x}}_{j}'{\varvec{\eta }}_{i})\right\} \right] ^{-1} \\ E(\tilde{Y}_{ij} \tilde{Y}_{i'j} |{\varvec{Y}}^{(j)})= & {} \{Y_{ij}\exp ({\varvec{x}}_{j}'{\varvec{\eta }}_{i})\}\{Y_{i'j}\exp ({\varvec{x}}_{j}'{\varvec{\eta }}_{i'})\}\\&\times \left[ \left\{ \sum _{i=1}^{I}Y_{ij}\exp ({\varvec{x}}_{j}'{\varvec{\eta }}_{i})\}\{1+\sum _{i=1}^{I}Y_{ij}\exp ({\varvec{x}}_{j}'{\varvec{\eta }}_{i}) \right\} \right] ^{-1} \end{aligned}$$

for $i \ne i'$, where ${\varvec{Y}}^{(j)} = (Y_{1j}, \ldots , Y_{Ij})'$ with $Y_{Ij} \equiv 1$.

The covariance between two response types at any two different sites j and $j'$ is given by

$$\begin{aligned}&\text {Cov}(\tilde{Y}_{ij}, \tilde{Y}_{i'j'}) = E(\tilde{Y}_{ij} \tilde{Y}_{i'j'}) - E(\tilde{Y}_{ij}) E(\tilde{Y}_{i'j'}) \\&\quad = E\{ E(\tilde{Y}_{ij} \tilde{Y}_{i'j'} |{\varvec{Y}}) \} - E\{ E(\tilde{Y}_{ij} |{\varvec{Y}}) \} E\{ E(\tilde{Y}_{i'j'} |{\varvec{Y}}) \} \\&\quad = E\{ E(\tilde{Y}_{ij} \tilde{Y}_{i'j'} |{\varvec{Y}}^{(j)}, {\varvec{Y}}^{(j')}) \} - E\{ E(\tilde{Y}_{ij} |{\varvec{Y}}^{(j)}) \} E\{ E(\tilde{Y}_{i'j'} |{\varvec{Y}}^{(j')}) \} \\&\quad = \sum _{y_{1j} = 0}^1 \ldots \sum _{y_{(I-1)j}=0}^1 \sum _{y_{1j'} = 0}^1 \ldots \sum _{y_{(I-1)j'}=0}^1 E(\tilde{Y}_{ij} \tilde{Y}_{i'j'} | {\varvec{y}}^{(j)}, {\varvec{y}}^{(j')})P_\mathrm{B}({\varvec{Y}}^{(j)} = {\varvec{y}}^{(j)}, {\varvec{Y}}^{(j')} = {\varvec{y}}^{(j')}) \\&\qquad - \left\{ \sum _{y_{1j} = 0}^1 \cdots \sum _{y_{(I-1)j}=0}^1 E(\tilde{Y}_{ij}|{\varvec{y}}^{(j)})P_\mathrm{B}({\varvec{Y}}^{(j)} = {\varvec{y}}^{(j)}) \right\} \\&\qquad \quad \times \left\{ \sum _{y_{1j'} = 0}^1 \cdots \sum _{y_{(I-1)j'}=0}^1 E(\tilde{Y}_{i'j'}|{\varvec{y}}^{(j')})P_\mathrm{B}({\varvec{Y}}^{(j')} = {\varvec{y}}^{(j')}) \right\} \\&\quad = \sum _{y_{1j} = 0}^1 \cdots \sum _{y_{(I-1)j}=0}^1 \sum _{y_{1j'} = 0}^1 \cdots \sum _{y_{(I-1)j'}=0}^1 E(\tilde{Y}_{ij} | {\varvec{y}}^{(j)}) E(\tilde{Y}_{i'j'} |{\varvec{y}}^{(j')}) \\&\qquad \times \left\{ P_\mathrm{B}({\varvec{Y}}^{(j)} = {\varvec{y}}^{(j)}, {\varvec{Y}}^{(j')} = {\varvec{y}}^{(j')}) - P_\mathrm{B}({\varvec{Y}}^{(j)} = {\varvec{y}}^{(j)}) P_\mathrm{B}({\varvec{Y}}^{(j')} = {\varvec{y}}^{(j')}) \right\} , \end{aligned}$$

where $P_\mathrm{B}(\cdot )$ is the probability mass function of a multivariate Bernoulli distribution. The third equality in (18) is by the definition of expectation, and for the last equality, $E(\tilde{Y}_{ij} \tilde{Y}_{i'j'} | {\varvec{y}}^{(j)}, {\varvec{y}}^{(j')}) = E(\tilde{Y}_{ij} | {\varvec{y}}^{(j)}) E(\tilde{Y}_{i'j'} |{\varvec{y}}^{(j')})$ is due to the definition (9).