Generalized Linear Latent Variable Models for Multivariate Count and Biomass Data in Ecology

Abstract

In this paper we consider generalized linear latent variable models that can handle overdispersed counts and continuous but non-negative data. Such data are common in ecological studies when modelling multivariate abundances or biomass. By extending the standard generalized linear modelling framework to include latent variables, we can account for any covariation between species not accounted for by the predictors, notably species interactions and correlations driven by missing covariates. We show how estimation and inference for the considered models can be performed efficiently using the Laplace approximation method and use simulations to study the finite-sample properties of the resulting estimates. In the overdispersed count data case, the Laplace-approximated estimates perform similarly to the estimates based on variational approximation method, which is another method that provides a closed form approximation of the likelihood. In the biomass data case, we show that ignoring the correlation between taxa affects the regression estimates unfavourably. To illustrate how our methods can be used in unconstrained ordination and in making inference on environmental variables, we apply them to two ecological datasets: abundances of bacterial species in three arctic locations in Europe and abundances of coral reef species in Indonesia.

Supplementary materials accompanying this paper appear on-line.

Appendices

A Proofs

1.1 A.1 Laplace Approximations for the General Exponential Family

Assume that the responses $y_{ij}$ come from the exponential family of distributions with mean $\mu _{ij}=E(y_{ij})$, and write $f(y_{ij}|\varvec{u}_i,\varvec{\Psi }) = \exp \left\{ y_{ij}a_j(\mu _{ij})-b_j(\mu _{ij}) + c_j(y_{ij})\right\}$, where $a_j(\cdot )$, $b_j(\cdot )$ and $c_j(\cdot )$ are known functions, and $\varvec{\Psi }$ includes all model parameters. The log-likelihood function (5) for parameter vector $\varvec{\Psi }$ now equals

$$\begin{aligned} l(\varvec{ \Psi }) =&\sum \limits _{i=1}^n\log \displaystyle \int \left[ \prod \limits _{j=1}^m \exp \Big \{y_{ij}\, a_j(\mu _{ij}) - b_j(\mu _{ij}) + c_j(y_{ij})\Big \} \right] \times (2\pi )^{-\frac{d}{2}}\exp \left( -\frac{1}{2}\varvec{u}_i'\varvec{u}_i\right) \,d\varvec{u}_i, \end{aligned}$$

and the Laplace approximation of the log-likelihood function is

$$\begin{aligned} \tilde{l}(\varvec{ \Psi },\varvec{\hat{u}}_{i})&= \sum \limits _{i=1}^n\Bigg (-\frac{1}{2}\log \det \left\{ \varvec{\Gamma }(\varvec{\Psi },\varvec{\hat{u}}_i)\right\} + \sum \limits _{j=1}^m\left\{ y_{ij}\, a_j(\mu _{ij}) - b_j(\mu _{ij}) + c_j(y_{ij})\right\} - \frac{\varvec{\hat{u}}_i'\varvec{\hat{u}}_i}{2}\Bigg ), \end{aligned}$$

where

$$\begin{aligned} \varvec{\Gamma }(\varvec{\Psi },\varvec{\hat{u}}_i) = \sum \limits _{j=1}^m\frac{\partial ^2 \left\{ -y_{ij}\, a_j(\mu _{ij}) + b_j(\mu _{ij})\right\} }{\partial \varvec{u}_i'\partial \varvec{u}_i}\Bigg |_{\varvec{u}_i=\varvec{\hat{u}}_i} + \varvec{I}_d, \end{aligned}$$

and $\varvec{\hat{u}}_i$ is the maximum of $Q(\varvec{\Psi },\varvec{u}_{i}) = (1/m)\left( \sum \limits _{j=1}^m\log f(y_{ij}|\varvec{u}_i;\varvec{\Psi }) - \varvec{u}_i'\varvec{u}_i/2\right)$ with respect to $\varvec{u}_i$. The result has been proven in Huber et al. (2004).

1.2 A.2 Poisson Responses

Species counts can be modelled as Poisson distributed responses, $y_{ij}\sim Poisson(\mu _{ij})$, and log link function. Then $a_j(\mu _{ij}) = \log (\mu _{ij}), b_j(\mu _{ij})=\mu _{ij}$, and $c_j(y_{ij})=-\log (y_{ij}!)$. Then the following Laplace approximation $\tilde{l}$ for the log-likelihood function is obtained

$$\begin{aligned} \tilde{l} (\varvec{ \Psi },\varvec{\hat{u}}_{i})&= \sum \limits _{i=1}^n\Bigg (-\frac{1}{2}\log \,\det \left( \varvec{\Gamma }(\varvec{\Psi },\varvec{\hat{u}}_i) \right) + \sum \limits _{j=1}^m\big [ y_{ij} \hat{\eta }_{ij} - \exp (\hat{\eta }_{ij}) - \log (y_{ij}!) \big ] - \frac{\varvec{\hat{u}}_i'\varvec{\hat{u}}_i}{2}\Bigg ), \end{aligned}$$

where $\varvec{\Gamma }(\varvec{\Psi },\varvec{\hat{u}}_i)= \sum \nolimits _{j=1}^m \exp (\hat{\eta }_{ij})\varvec{\gamma }_j\varvec{\gamma }_j' + \varvec{I}_d$, with $\hat{\eta }_{ij}=\alpha _i + \beta _{0j} + \varvec{x}'_i \varvec{\beta }_j + \hat{\varvec{u}_i}'\varvec{\gamma }_j$, and $\varvec{\hat{u}}_i$ is the maximum of

$$\begin{aligned} Q(\varvec{\Psi },\varvec{u}_{i}) =&\frac{1}{m}\Bigg [\sum \limits _{j=1}^m\big [ y_{ij}\eta _{ij} - \exp (\eta _{ij}) - \log (y_{ij}!) \big ] - \frac{\varvec{u}_i'\varvec{u}_i}{2} - \frac{d}{2}\log (2\pi )\Bigg ]. \end{aligned}$$

1.3 A.3 Proof of Theorem 2

Assume that the responses $y_{ij}$ come from the zero-inflated Poisson distribution with mean $E(y_{ij})=(1-p_j)\mu _{ij}$ and density of the form (3). The log-likelihood function (5) then equals

$$\begin{aligned} l(\varvec{ \Psi })= & {} \sum \limits _{i=1}^n\log \bigg (\int \prod \limits _{j=1}^m \exp \left( \log \left[ p_j + (1-p_j)\exp \{-\exp (\eta _{ij})\}\right] I_{(y_{ij} = 0)} \right. \\&+ \left. \left\{ \log (1-p_j) - \exp (\eta _{ij})+y_{ij}\eta _{ij} -\log (y_{ij}!)\right\} I_{(y_{ij} > 0)} \right) \bigg )\\&\times (2\pi )^{-\frac{d}{2}}\exp \left( -\frac{1}{2}\varvec{u}_i'\varvec{u}_i\right) \,d\varvec{u}_i. \end{aligned}$$

Hence, the Laplace approximation of the log-likelihood function is

$$\begin{aligned} \tilde{l}(\varvec{ \Psi },\varvec{\hat{u}}_{i})= & {} \sum \limits _{i=1}^n\Bigg (-\frac{1}{2}\log \det \left\{ \varvec{\Gamma }(\varvec{\Psi },\varvec{\hat{u}}_i)\right\} + \sum \limits _{j=1}^m \log f(y_{ij}|\varvec{\hat{u}}_i;\varvec{\Psi }) - \frac{\varvec{\hat{u}}_i'\varvec{\hat{u}}_i}{2}\Bigg ) \\= & {} \sum \limits _{i=1}^n\Bigg (-\frac{1}{2}\log \det \left\{ \varvec{\Gamma }(\varvec{\Psi },\varvec{\hat{u}}_i)\right\} + \sum \limits _{j=1}^m\Big (\log \big (p_j + (1-p_j) \hat{A}_{ij}\big )I_{(y_{ij} = 0)} \\&+ \left\{ \log (1-p_j) - \exp (\hat{\eta }_{ij})+y_{ij}\hat{\eta }_{ij} -\log (y_{ij}!)\right\} I_{(y_{ij} > 0)}\Big ) - \frac{\varvec{\hat{u}}_i'\varvec{\hat{u}}_i}{2}\Bigg ), \end{aligned}$$

where

$$\begin{aligned} \varvec{\Gamma }(\varvec{\Psi },\varvec{\hat{u}}_i)= & {} \frac{\partial ^2}{\partial \varvec{u}_i'\partial \varvec{u}_i} \Bigg [ - \sum \limits _{j=1}^m\log f(y_{ij}|\varvec{u}_i;\varvec{\Psi }) + \frac{\varvec{u}_i'\varvec{u}_i}{2}\Bigg ] \Bigg |_{\varvec{u}_i=\varvec{\hat{u}}_i}\\= & {} \sum \limits _{j=1}^m \frac{\partial ^2 \left\{ \exp (\eta _{ij})I_{(y_{ij}> 0)} - \log (p_j + (1-p_j)A_{ij})I_{(y_{ij} = 0)}\right\} }{\partial \varvec{u}_i'\partial \varvec{u}_i}\Bigg |_{\varvec{u}_i=\varvec{\hat{u}}_i}+ \varvec{I}_d \\= & {} \sum \limits _{j=1}^m\Bigg [\exp (\hat{\eta }_{ij})I_{(y_{ij} > 0)} - \Bigg (\frac{(1-p_j)\hat{A}_{ij}\exp (\hat{\eta }_{ij})(\exp (\hat{\eta }_{ij})-1)}{p_j + (1-p_j)\hat{A}_{ij}} \\&- \frac{(1-p_j)^2\hat{A}_{ij}^2\exp (2\hat{\eta }_{ij})}{(p_j + (1-p_j)\hat{A}_{ij})^2}\Bigg )I_{(y_{ij} = 0)}\Bigg ]\varvec{\gamma }_j\varvec{\gamma }_j' + \varvec{I}_d, \end{aligned}$$

with $\hat{\eta }_{ij}=\alpha _i + \beta _{0j} + \varvec{x}'_i \varvec{\beta }_j + \hat{\varvec{u}_i}'\varvec{\gamma }_j$ and $\hat{A}_{ij}=\exp \{-\exp (\hat{\eta }_{ij})\}$, and $\varvec{\hat{u}}_i$ is the maximum of $Q(\varvec{\Psi },\varvec{u}_{i}) = (1/m)\left( \sum \nolimits _{j=1}^m\log f(y_{ij}|\varvec{u}_i;\varvec{\Psi }) - \varvec{u}_i'\varvec{u}_i/2\right)$.

1.4 A.4 Proof of Theorem 3

Assume that the responses $y_{ij}$ come from the Tweedie distribution with mean $E(y_{ij})=\mu _{ij}$ and density of the form (4). The log-likelihood function (5) then equals

$$\begin{aligned} l(\varvec{ \Psi })= & {} \sum \limits _{i=1}^n\log \Bigg (\int \prod \limits _{j=1}^m \exp \left( - \frac{\mu _{ij}^{2-\nu }}{\phi _j(2-\nu )}\right) I_{(y_{ij} = 0)} \\&+\, \frac{1}{y_{ij}}\tilde{W}(y_{ij},\phi _j,\nu )\exp \left\{ \frac{1}{\phi _j}\left( \frac{y_{ij}\mu _{ij}^{1-\nu }}{1-\nu } - \frac{\mu _{ij}^{2-\nu }}{2-\nu }\right) \right\} I_{(y_{ij} > 0)} \Bigg )\\&\times (2\pi )^{-\frac{d}{2}}\exp \left( -\frac{1}{2}\varvec{u}_i'\varvec{u}_i\right) \,d\varvec{u}_i. \end{aligned}$$

Hence, the Laplace approximation of the log-likelihood function is

$$\begin{aligned} \tilde{l}(\varvec{ \Psi },\varvec{\hat{u}}_{i})= & {} \sum \limits _{i=1}^n\Bigg (-\frac{1}{2}\log \det \left\{ \varvec{\Gamma }(\varvec{\Psi },\varvec{\hat{u}}_i)\right\} + \sum \limits _{j=1}^m \log f(y_{ij}|\varvec{\hat{u}}_i;\varvec{\Psi }) - \frac{\varvec{\hat{u}}_i'\varvec{\hat{u}}_i}{2}\Bigg ) \\= & {} \sum \limits _{i=1}^n\Bigg (-\frac{1}{2}\log \det \left\{ \varvec{\Gamma }(\varvec{\Psi },\varvec{\hat{u}}_i) \right\} + \sum \limits _{j=1}^m\bigg [\left\{ \log \tilde{W}(y_{ij},\phi _j,\nu ) - \log (y_{ij})\right\} I_{(y_{ij}>0)} \\&+ \frac{1}{\phi _j}\left( \frac{y_{ij}\exp \{(1-\nu )\hat{\eta }_{ij}\}}{1-\nu } - \frac{\exp \{(2-\nu )\hat{\eta }_{ij}\}}{2-\nu }\right) \bigg ] - \frac{\varvec{\hat{u}}_i'\varvec{\hat{u}}_i}{2}\Bigg ), \end{aligned}$$

where

$$\begin{aligned} \varvec{\Gamma }(\varvec{\Psi },\varvec{\hat{u}}_i)= & {} \frac{\partial ^2}{\partial \varvec{u}_i'\partial \varvec{u}_i} \Bigg [ - \sum \limits _{j=1}^m\log f(y_{ij}|\varvec{u}_i;\varvec{\Psi }) + \frac{\varvec{u}_i'\varvec{u}_i}{2}\Bigg ] \Bigg |_{\varvec{u}_i=\varvec{\hat{u}}_i}\\= & {} \sum \limits _{j=1}^m \frac{\partial ^2}{\partial \varvec{u}_i'\partial \varvec{u}_i} \frac{1}{\phi _j}\left( - \frac{y_{ij}\exp \{(1-\nu )\eta _{ij}\}}{1-\nu } + \frac{\exp \{(2-\nu )\eta _{ij}\}}{2-\nu }\right) \Bigg |_{\varvec{u}_i=\varvec{\hat{u}}_i}+ \varvec{I}_d \\= & {} \sum \limits _{j=1}^m\frac{1}{\phi _j}\left[ (2-\nu )\exp \{(2-\nu )\hat{\eta }_{ij}\} - y_{ij}(1-\nu )\exp \{(1-\nu )\hat{\eta }_{ij}\} \right] \varvec{\gamma }_j\varvec{\gamma }_j' + \varvec{I}_d, \end{aligned}$$

with $\hat{\eta }_{ij}=\alpha _i + \beta _{0j} + \varvec{x}'_i \varvec{\beta }_j + \hat{\varvec{u}_i}'\varvec{\gamma }_j$ and $\hat{A}_{ij}=\exp \{-\exp (\hat{\eta }_{ij})\}$, and $\varvec{\hat{u}}_i$ is the maximum of $Q(\varvec{\Psi },\varvec{u}_{i}) = (1/m)\left( \sum \nolimits _{j=1}^m\log f(y_{ij}|\varvec{u}_i;\varvec{\Psi }) - \varvec{u}_i'\varvec{u}_i/2\right)$.

B Additional Application Results

See Figs. 5, 6, 7, 8 and 9.

Fig. 5

The ordination of $n=56$ sites based on generalized linear latent variable model without any covariates assuming negative binomial distributed responses. The sites in ordination are coloured according to their a soil organic matter (SOM) values and b phosphorous (P) values, and labelled according to the sampling site.

Fig. 6

Ranked point estimates with $95\%$ confidence intervals for the three environmental variables based on negative binomial GLLVM. Grey confidence intervals include the zero value.

Fig. 7

The ordination of $n=56$ sites based on generalized linear latent variable model with pH, soil organic matter and phosphorous as covariates, and assuming negative binomial distributed responses. The sites in ordination are coloured according to their a pH values, b soil organic matter (SOM) values and c phosphorous (P) values, and labelled according to the sampling site. The effect of environmental variables vanishes, but the ordination is affected by the sampling location few Kilpisjärvi sites being different from the others what comes to species composition.

Fig. 8

Dunn–Smyth residuals against linear predictors for the a Poisson, b zero-inflated Poisson and c negative binomial GLLVM models with pH, soil organic matter, phosphorous and categorical site as covariates. Lowess curves are included in the plots.

Fig. 9

Dunn–Smyth residuals against linear predictors for the Tweedie models a without site effect and b with site effect.