Analysing sparse ecological percent cover
data using gllvm
Pekka Korhonen
2026-06-22
This short document showcases how to use the gllvm
package to analyse multivariate percent cover data. Namely, we show how
to apply the hurdle beta GLLVM (with logistic link), as detailed in
Korhonen et al. (2024), to analyse the
kelp forest dataset from the Santa Barbara Coastal Long-Term Ecological
Research project, available from https://doi.org/10.6073/pasta/0af1a5b0d9dde5b4e5915c0012ccf99c.
A multivariate percent cover dataset often comes in the form of a
\(n \times m\) matrix \(\bf{Y}\), with \(n\) being the number of observational
units/sites, and \(m\) being the number
of species of plants, macroalgae, sessile invertebrates, et cetera. The
response \(y_{ij}\) is then the
recorded percentage of the relative area covered by species \(j\) on unit/site \(i\). Typically such datasets contain
considerable proportion of zero observations, as a given obs. unit is
often populated by only a small subset of the \(m\) species in total. More rarely, it might
even be the case that one of the species covers some obs. unit
completely.
Hurdle beta GLLVM
Traditionally, beta regression has been used to model responses that
take the form of percentages. If \(y^*_{ij}\) is in the (open) interval \((0,1)\), and is distributed according to
beta distribution, \(y^*_{ij} \sim
\text{Beta}(\mu_{ij}, \phi_j)\), then it has the following
density: \[ f_{\text{beta}}(y_{ij}^*;
\mu_{ij}, \phi_j) =
\frac{\Gamma(\phi_j)}{\Gamma(\mu_{ij}\phi_j)\Gamma(\phi_j-\phi_j\mu_{ij})}
(y_{ij}^*)^{\mu_{ij}\phi_j-1}(1-y_{ij}^*)^{\phi_j(1-\mu_{ij}-1)}.\]
The mean parameter \(\mu_{ij}\) can
be connected to a set of covariates and latent variables through some
link function, by the equation \(g(\mu_{ij})=\eta_{ij} = \beta_{0j} +
\boldsymbol{\beta}_j^\top\bf{x_i} +
\boldsymbol{\lambda}_j^\top\bf{u_i}\).
However, such a model is ill-suited for most percent cover datasets,
due to the fact that the beta distribution isn’t capable of handling
zero (or \(100\%\)) responses. To make
use of the beta regression model in such scenario, one needs to use some
transformation in order to ‘push’ the responses away from the
boundaries. This procedure might provide reasonable results when the
numbers of zeros or ones in the data are relatively low. On the other
hand, by transforming the zeros and ones, we might lose some important
information.
A more sophisticated way of tackling such issue is by considering a
zero-accommodating model, couple of which have been proposed recently.
One such model is the hurdle beta model, which models the two (can be
extended to three if \(100\%\) covers
are recorded, see Korhonen et al. (2024))
classes the response \(y_{ij}\) can
take, i.e., \(\{0\}\) and \((0,1)\), by separate processes. Namely, the
zeros are assumed to be generated by a Bernoulli process. Conditional on
\(y_{ij}\in(0,1)\), the response is
modeled using the standard beta distribution as presented above. The
likelihood function for a response \(Y_{ij}\) following the hurdle beta
distribution is of the form: \[
P(Y_{ij};\mu_{ij}, \mu_{ij}^0, \phi_j) = \begin{cases}
1-\mu_{ij}^0, & Y_{ij} = 0,\\
\mu_{ij}^0 \cdot f_{\text{beta}}(Y_{ij};\mu_{ij},\phi_j), &
Y_{ij} \in (0,1). \\
\end{cases}
\] where \(g(\mu_{ij}^0) = \eta_{ij}^0
= \beta_{0j}^0 + \bf{x_i}^\top\boldsymbol{\beta}_j^0 +
\bf{u_i}^\top\boldsymbol{\lambda}_j^0\) for and \(g(\mu_{ij}) = \eta_{ij} = \beta_{0j} +
\bf{x_i}^\top\boldsymbol{\beta}_j +
\bf{u_i}^\top\boldsymbol{\lambda}_j\) and \(g(\cdot)\) can be either probit or logistic
link function. Note, that here, the separate linear predictors share the
same environmental covariates \(\bf{x}_i\) and latent variable scores \(\bf{u}_i\), while the coefficients and
loadings are allowed to differ.
The gllvm package implements the hurdle beta GLLVM
with two different estimation methods available. First, accessed by the
argument method="VA" when calling uses a hybrid approach,
where the method of variational approximations, or VA, is applied to the
Bernoulli-process part of the data (only probit link allowed), while the
method of extended variational approximations, see
Korhonen et al. (2023), is applied to the
beta distributed part. By instead specifying method="EVA",
the EVA method is applied to both parts of the
likelihood.
Example
In the following, we show how to fit the logistic hurdle beta GLLVM
using EVA on the SBC LTER marine macroalgae (i.e.,
seaweed) percent cover dataset. The data has been collected in 2000-2020
along 44 permanent transect lines along coastal southern California. We
will specify a model with two latent variables (for ordination) and will
include the rockiness of the seabed and the average number of stripes of
giant kelp as environmental covariates in the model.
library(gllvm)
data("kelpforest")
Yabund <- kelpforest$Y
Xenv <- kelpforest$X
SPinfo <- kelpforest$SPinfo
# Data contains both algae and sessile invertebrates
table(SPinfo$GROUP)
##
## ALGAE INVERT PLANT
## 61 69 2
# Select only the macroalgae:
Yalg <- Yabund[,SPinfo$GROUP=="ALGAE"]
# To demonstrate the models, use only the data from the year 2016:
Yalg <- Yalg[Xenv$YEAR==2016,]
Xenv <- Xenv[Xenv$YEAR==2016,]
# Remove species which have no observations or just one
Yalg <- Yalg[,-which(colSums(Yalg>0)<2)]
# Number of obs. and species:
dim(Yalg)
## [1] 44 42
# Specify the covariates in the linear predictor
Xformulai = ~KELP_FRONDS + PERCENT_ROCKY
After setting up the data, LV design and the covariates, the model is
estimated by
fit <- gllvm(Yalg, X=Xenv, formula = Xformulai, family = "betaH", method="EVA",
num.lv = 2, link="logit", control=list(reltol=1e-12))
To inspect e.g., the covariate effects, use
## KELP_FRONDS PERCENT_ROCKY
## AU -1.013880e-01 -4.724495e-04
## BF -2.487885e+00 1.099410e-01
## BO 2.098018e-02 -2.460852e-03
## BR -4.591168e-02 2.245169e-02
## BRA 3.250781e-01 -8.188835e-02
## CAL -1.739428e-01 -1.514354e-01
## CC 1.223957e-01 7.898618e-04
## CF -8.803738e-02 9.909499e-03
## CG -5.110920e-03 -3.599310e-02
## CO -5.379647e-02 8.438689e-03
## COF -3.677566e-01 5.539502e-02
## CP -1.422141e-02 -1.056876e-01
## CRYP 1.513781e-03 1.126785e-05
## CYOS 3.735364e-02 -8.421308e-03
## DL 6.822257e-02 -4.420682e-04
## DMH -1.434229e-01 -5.133406e-03
## DP -5.380764e-02 7.893054e-03
## DU -4.290070e-02 -7.821515e-03
## EAH 7.382062e+00 -6.536593e-02
## EC 3.950257e-02 1.790603e-02
## EH 6.156132e-01 -1.146798e-01
## ER 1.359653e-01 2.790188e-02
## FB -4.664432e-01 -8.944906e-04
## FTHR -2.513756e-01 -1.867772e-02
## GR 8.114077e-02 1.569813e-02
## GS -4.314578e-01 -6.898069e-03
## GYSP -1.702809e-01 8.491179e-03
## MH 9.455312e-02 -2.211462e-03
## NIE -3.137399e-02 -6.160912e-03
## PH 1.433539e+00 -2.049199e-01
## PHSE -5.433307e-01 8.347886e-03
## PL 8.135429e-02 8.546121e-03
## POLA -7.307152e-01 -1.160537e-02
## PRSP -4.404353e-06 1.117286e-07
## R 5.937542e-02 9.097785e-03
## RAT 5.659865e-02 2.412178e-03
## SAFU -8.986772e-01 2.385311e-02
## SAHO 3.787715e+00 -3.293743e-02
## SAMU -4.959294e-02 -7.051954e-04
## SCCA -1.329191e-04 2.275465e-02
## TALE 2.302233e-02 3.499769e-02
## UV -8.243220e-02 -1.195282e-02
## H01_AU -2.083531e-01 -2.726309e-02
## H01_BF 5.507560e-02 -5.187252e-02
## H01_BO 2.284767e-01 -1.700030e-02
## H01_BR -1.146060e-01 -5.203770e-04
## H01_BRA -4.398328e-01 -1.951335e-02
## H01_CAL -2.231247e-01 2.158039e-01
## H01_CC 2.031363e-01 -1.031977e-01
## H01_CF -7.217803e-03 -1.128072e-02
## H01_CG 3.901961e-02 3.085968e-01
## H01_CO 3.660236e-02 -9.952391e-03
## H01_COF -1.126976e+00 2.947424e-01
## H01_CP -1.415997e+00 2.812103e-01
## H01_CRYP 6.537794e-02 -2.201999e-02
## H01_CYOS 2.537823e-01 -3.834405e-02
## H01_DL 5.944980e-02 -1.413654e-02
## H01_DMH -2.651595e-01 -2.401626e-02
## H01_DP -1.305892e-01 2.194531e-02
## H01_DU -3.832930e-01 2.355605e-01
## H01_EAH -1.163850e+01 1.587828e-01
## H01_EC 2.057204e-01 5.111600e-02
## H01_EH -8.105573e-02 -2.296213e-02
## H01_ER -7.035114e-02 4.330515e-02
## H01_FB -3.341276e-01 -1.717165e-02
## H01_FTHR -8.632758e-01 -3.259739e-02
## H01_GR -1.706945e-01 3.685095e-01
## H01_GS -4.612208e-01 -2.297949e-01
## H01_GYSP 5.242699e-02 4.591198e-03
## H01_MH 2.330174e+00 -1.143238e-02
## H01_NIE 2.448747e-01 -6.436792e-02
## H01_PH 1.662762e-01 -9.340331e-02
## H01_PHSE -1.504011e+00 -8.769848e-02
## H01_PL 5.435793e-02 -1.490611e-03
## H01_POLA -2.645024e-01 -4.416796e-02
## H01_PRSP -2.011027e-01 -4.193284e-03
## H01_R -9.003936e-02 -4.408076e-03
## H01_RAT -1.265648e-01 2.084117e-02
## H01_SAFU -3.713454e+00 -1.745355e-01
## H01_SAHO -3.261900e+00 2.347616e-01
## H01_SAMU -1.456076e-01 3.791985e-03
## H01_SCCA -8.238921e-02 3.180287e-02
## H01_TALE -4.878582e-01 2.337806e-01
## H01_UV -5.544205e-01 -1.662595e-02
In the above, the prefix indicates that the coefficient relates to
the Bernoulli part of the hurdle model.
Ordination plot can then be generated as per usual:
ordiplot(fit, jitter = TRUE, s.cex = .8)
Ordered beta GLLVM
Another solution for modeling percentage cover data in gllvm is to
use ordered beta response model. It handles zeros and ones slightly
differently compared to hurdle beta model. Instead of assuming that the
zeros and ones comes from separate process from the percent cover, the
model assumes that there is an underlying process \(z_{ij}\) where all observations comes
from.
For species \(j = 1, . . . , m\),
let \(z_{ij}\) denote an underlying
continuous variable, and define two cutoff parameters \(\zeta_{j0} < \zeta_{j1}\) such that
\(Y_{ij} = 0\) occurs when \(z_{ij} < \zeta_{j0}\), \(Y_{ij}=1\) occurs when \(z_{ij}>\zeta_{j1}\), and \(Y_{ij} \in (0,1)\) occurs when \(\zeta_{j0} < z_{ij} < \zeta_{j1}\).
Conditional on \(Y_{ij} \in (0,1)\),
the response variable follows a beta distribution. By assuming \(z_{ij}\) follows a logistic distribution,
then marginalising over \(z_{ij}\) we
obtain the following distribution for the percent cover responses that
characterizes the ordered beta GLLVM,
\[\begin{align}
P(Y_{ij};\eta_{ij}, \phi_j) =
\begin{cases}
\rho^0_{ij}, & if Y_{ij} = 0 ,\\
\left(\rho^1 - \rho^0 \right) \cdot f_{\text{beta}}(Y_{ij};
\mu_{ij}, \phi_j), & if Y_{ij} \in (0,1) ,\\
1-\rho^1_{ij}, & if Y_{ij} = 1 ,\\
\end{cases}
\end{align}\]
Adjusting ordered beta to data without ones
Lets demonstrate the ordered beta response model for the previous
example. As the data has no ones:
## [1] 0
We can accommodate the model to better handle data that by fixing the
upper cutoff parameters to some large value, for example 20. With
gllvm this can be done by setting starting values for the
cutoff parameter, ´zetacutoff = c(0, 20)´ and fixing the upper cutoff
parameters with ´setMap = list(zeta = factor(rbind(1:m, rep(NA, m))))´.
We assume that the number of species/response variables in the data is
saved to object m. There are ´m´ (number of species) lower
cutoff parameters \(\zeta_{j0}\) we let
be freely estimated (that’s indexes ´1:m´ in ´setMap´) and ´m´ upper
cutoff parameters \(\zeta_{j1}\) that
we fix (that’s ´rep(NA, m)´ in ´setMap´).
Setting shape parameter to common across species
Some species are observed only a few times:
## AU BF BO BR BRA CAL CC CF CG CO COF CP CRYP CYOS DL DMH
## 23 3 11 5 2 5 30 13 5 24 5 3 3 33 26 8
## DP DU EAH EC EH ER FB FTHR GR GS GYSP MH NIE PH PHSE PL
## 14 4 3 36 2 6 6 6 11 20 7 36 13 2 5 4
## POLA PRSP R RAT SAFU SAHO SAMU SCCA TALE UV
## 6 4 31 17 6 2 7 7 5 6
so there is not much information to estimate the shape parameter of
the beta distribution for each species separately. Thus we can also set
the shape parameter to be common across species with ´disp.formula =
rep(1, m)´. This can be applied for all beta based models in
gllvm.
Model fit
Now we are ready to fit the model:
# save the number of species to object m
m <- ncol(Yalg)
fit_ob <- gllvm(Yalg, X=Xenv, formula = Xformulai, family = "orderedBeta",
method="EVA", num.lv = 2, link="logit",
disp.formula = rep(1, m), zetacutoff = c(0, 20),
setMap = list(zeta = factor(rbind(1:m, rep(NA, m)))) )
fit_ob
## Call:
## gllvm(y = Yalg, X = Xenv, formula = Xformulai, family = "orderedBeta",
## num.lv = 2, method = "EVA", link = "logit", disp.formula = rep(1,
## m), setMap = list(zeta = factor(rbind(1:m, rep(NA, m)))),
## zetacutoff = c(0, 20))
## family:
## [1] "orderedBeta"
## method:
## [1] "EVA"
##
## log-likelihood: 239.7871
## Residual degrees of freedom: 1554
## AIC: 108.4259
## AICc: 220.1194
## BIC: 1731.852
Now if we check the cutoff values we see that the upper bounds are
fixed to 20
## cutoff0 cutoff1
## AU -2.253214 20
## BF -1.251314 20
## BO -2.718055 20
## BR -1.361771 20
## BRA -1.806604 20
## CAL -2.030746 20
## CC -5.133260 20
## CF -2.881754 20
## CG -3.627948 20
## CO -2.939729 20
## COF -5.513601 20
## CP -3.693915 20
## CRYP -1.305949 20
## CYOS -3.771781 20
## DL -4.666261 20
## DMH -2.610885 20
## DP -2.349122 20
## DU -2.682536 20
## EAH -5.857012 20
## EC -4.137155 20
## EH -3.390855 20
## ER -2.625228 20
## FB -1.011205 20
## FTHR -2.597725 20
## GR -4.372955 20
## GS -3.898109 20
## GYSP -1.956803 20
## MH -4.344225 20
## NIE -3.129506 20
## PH -1.497350 20
## PHSE -2.489283 20
## PL -1.423868 20
## POLA -2.558219 20
## PRSP -2.436488 20
## R -4.645781 20
## RAT -3.181538 20
## SAFU -4.601453 20
## SAHO -8.748899 20
## SAMU -1.484021 20
## SCCA -2.419619 20
## TALE -3.101808 20
## UV -1.586696 20
Ordination plot:
ordiplot(fit_ob, jitter = TRUE, s.cex = .8)
