In this example we apply generalized linear latent variable model (GLLVMs) on the bacterial species data discussed in Nissinen, Mannisto, and van Elsas (2012). The sequence data is published in European Nucleotide Archive with the project number PRJEB17695. The subset of the data used in our analyses is included in the gllvm package. This example follows the one provided in Niku et al. (2017), except that here we use variational approximation method to fit GLLVMs instead of the Laplace approximation method. Altogether eight different sampling sites were selected from three locations. Three of the sites were in Kilpisjarvi, Finland, three in Ny-Alesund, Svalbard, Norway, and two in Mayrhofen, Austria. From each sampling site, several soil samples were taken and their bacterial species were recorded. The data consist of \(m = 985\) bacterial species counts measured from \(n = 56\) sites. The sites can be considered as independent from each other since bacterial communities are known to be very location specific. In addition to bacteria counts, three continuous environmental variables (pH, available phosphorous and soil organic matter) were measured from each soil sample. The data set is available in object Ysoil and the environmental variables and information regarding the sampling sites are given in object Xenv. In addition to environmental variables, Xenv contains information from the sampling location (Region), sampling site at each region (Site) and soil sample type (Soiltype, top soil (T) or bottom soil (B)). Using GLLVMs we try to find out if soils physico-chemical properties or region affect the structure of bacterial communities. The package and the dataset can be loaded along the following lines:

## Loading required package: TMB
## Loading required package: mvabund
## 
## Attaching package: 'gllvm'
## The following object is masked from 'package:mvabund':
## 
##     coefplot
## The following objects are masked from 'package:stats':
## 
##     nobs, predict, simulate, vcov
data("microbialdata")
Ysoil <- microbialdata$Y
Xenv <- microbialdata$Xenv
dim(Ysoil)
## [1]  56 985
head(Xenv, 3)
##      SOM   pH Phosp Region Site Soiltype
## AB2 0.01 6.70  2.63    Aus    A        B
## AB3 0.01 6.67  2.16    Aus    A        B
## AB4 0.00 8.19  0.62    Aus    A        B

Let’s take a look at the means and variances of the species counts:

par(mfrow = c(1, 2))
plot(apply(Ysoil,2, mean), main = "Species means")
plot(apply(Ysoil,2, var), log = "y", main = "Species variances")

As it can be seen there is a lot of variation in species’ means and variances.

In order to study if the effect of environmental variables is seen in an unconstrained ordination plot, we first consider a GLLVM with two latent variables and no predictors, and constructed an ordination plot based on the predicted latent variables. For count data we consider here the negative binomial (NB) distribution. We also include random effects for sites in order to account for the differences in site totals. This can be done by defining the structure for random row effects in an argument row.eff and including a data frame with ‘Site’ variable to an argument ‘X’. Note that in the examples we consider below we don’t necessarily need standard errors for the parameters for this specific model so we define sd.errors = FALSE in the function call. When needed, standard errors can also be calculated afterwards using function se.

Site<-data.frame(Site=Xenv$Site)
ftNULL <- gllvm(Ysoil, family = "negative.binomial", row.eff = ~(1|Site), num.lv = 2, sd.errors = FALSE)

The print of the model object:

ftNULL
## Call: 
## gllvm(y = Ysoil, X = data.frame(Site = Xenv$Site), num.lv = 2, 
##     family = "negative.binomial", row.eff = ~(1 | Site), sd.errors = FALSE)
## family: 
## [1] "negative.binomial"
## method: 
## [1] "VA"
## 
## log-likelihood:  -59822.73 
## Residual degrees of freedom:  51220 
## AIC:  127525.5 
## AICc:  128131.8 
## BIC:  135505.3

Using the residual diagnostic plot we check that the chosen NB distribution is suitable for the data at hand. The Dunn-Smyth residuals given by the NB model are normally distributed around zero, thus supporting the choice. Using argument n.plot we can choose a number of randomly selected species to be plotted in order to make any patterns in the residual plots more apparent. This can be useful argument when data is high-dimensional.

par(mfrow = c(1, 2))
plot(ftNULL, which = 1:2, var.colors = 1, n.plot = 100)

The ordination of sites based on the fitted negative binomial GLLVM can be plotted using function ordiplot. The sites can be colored according to their environmental variable values, pH, SOM and phosp using an argument s.colors. In addition, the ordination points are labeled according to the sampling location (Kilpisjarvi, Ny-Alesund and Innsbruck). This can be done by setting symbols = TRUE and defining the symbols for each site using argument pch, see below:

# Define colors according to the values of pH, SOM and phosp
library(grDevices)
ph <- Xenv$pH
rbPal <- colorRampPalette(c('mediumspringgreen', 'blue'))
Colorsph <- rbPal(20)[as.numeric(cut(ph, breaks = 20))]
breaks <- seq(min(ph), max(ph), length.out = 30)
som <- Xenv$SOM
Colorssom <- rbPal(20)[as.numeric(cut(som, breaks = 20))]
breaks <- seq(min(som), max(som), length.out = 30)
phosp <- Xenv$Phosp
Colorsphosp <- rbPal(20)[as.numeric(cut(phosp, breaks = 20))]
breaks <- seq(min(phosp), max(phosp), length.out = 30)
# Define symbols for different sampling locations:
pchr = NULL
pchr[Xenv$Region == "Kil"] = 1
pchr[Xenv$Region == "NyA"] = 2
pchr[Xenv$Region == "Aus"] = 3

# Ordination plots. Dark color indicates high environmental covariate value.
ordiplot(ftNULL, main = "Ordination of sites, color: pH",
         symbols = TRUE, pch = pchr, s.colors = Colorsph)
legend("topleft", legend = c("Kil", "NyA", "Mayr"), pch = c(1, 2, 3), bty = "n")

ordiplot(ftNULL, main = "Ordination of sites, color: SOM", 
         symbols = TRUE, pch = pchr, s.colors = Colorssom)
legend("topleft", legend = c("Kil", "NyA", "Mayr"), pch = c(1, 2, 3), bty = "n")

ordiplot(ftNULL, main = "Ordination of sites, color: phosphorous",
         symbols = TRUE, pch = pchr, s.colors = Colorsphosp)
legend("topleft", legend = c("Kil", "NyA", "Mayr"), pch = c(1, 2, 3), bty = "n")

A clear gradient in the pH values of sites is observed, whereas there is less evidence of such pattern with the two other soil variables. It is also clear that the three sampling locations differ in terms of species composition. Standard deviation for the random site effects can be extracted by ftNULL$params$sigma. By plotting the predicted random site effects, we can possibly see differences in sampling intensity of the eight sites.

# Sampling locations of the eight sampling sites:
locaSites<-c(3,3,1,1,2,2,2,1)
plot(ftNULL$params$row.params, xlab = "site", col = locaSites, pch = locaSites, 
     main = "Site effects", ylab = "Site effect", xaxt = 'n', ylim = c(-1,1))
axis(1, at=1:8, labels=levels(Site$Site))
legend("topleft", legend = c("Kil", "NyA", "Mayr"), pch = c(1, 2, 3), 
       col = c(1, 2, 3), bty = "n")

Next we produce a biplot based on GLLVM. Below, column indices of the 15 species with largest factor loadings are added in the (rotated) ordination plot. The biplot suggests a small set of indicator species which prefer sites with low pH values and a larger set of indicator species for high pH sites.

# Plot the species using column indices of the species:
rownames(ftNULL$params$theta) <- 1:ncol(Ysoil)
ordiplot(ftNULL, main = "Ordination of sites and species", xlim = c(-6, 5), 
         ylim = c(-4, 4), symbols = TRUE, pch = pchr, s.colors = Colorsph, 
         biplot = TRUE, ind.spp = 15, cex.spp = 0.9)
legend("topleft", legend = c("Kil", "NyA", "Mayr"), pch=c(1, 2, 3), bty = "n")

In order to study if pH value alone is capable of explaining the variation in species composition across sites, we included it as explanatory variable in the GLLVM. When the Poisson distribution performed so poorly on the null model, we consider only NB GLLVMs in the following examples.

# Scale environmental variables
Xsoils <- cbind(scale(Xenv[, 1:3]),Site)
ftXph <- gllvm(Ysoil, Xsoils, formula = ~pH, family = "negative.binomial", 
               row.eff = ~(1|Site), num.lv = 2)
ftXph
## Call: 
## gllvm(y = Ysoil, X = Xsoils, formula = ~pH, num.lv = 2, family = "negative.binomial", 
##     row.eff = ~(1 | Site))
## family: 
## [1] "negative.binomial"
## method: 
## [1] "VA"
## 
## log-likelihood:  -58976.08 
## Residual degrees of freedom:  50235 
## AIC:  127802.2 
## AICc:  128768.1 
## BIC:  137777

Ranked point estimates with 95% confidence intervals are plotted below using function coefplot and indicate that pH value strongly affects to the species composition as so many of the confidence intervals do not contain zero value (black). The species names are not informative in the coefficient plot when the number of species is so large and can be removed using an argument y.label.

coefplot(ftXph, cex.ylab = 0.5, y.label = FALSE)

The corresponding ordination plot given below indicates that the gradient in the pH values of the sites vanishes, but the ordination still exhibits a sampling location effect. In particular, several Kilpisjarvi sites seem to differ what comes to the bacterial species composition and all Mayrhofen sites are located at the top of the ordination.

ordiplot(ftXph, main = "Ordination of sites", 
         symbols = TRUE, pch = pchr, s.colors = Colorsph)
legend("topleft", legend = c("Kil", "NyA", "Mayr"), pch = c(1, 2, 3), bty = "n")

Next we include all environmental variables as explanatory variables in the GLLVM.

ftX <- gllvm(Ysoil, Xsoils, family = "negative.binomial", row.eff = ~(1|Site), num.lv = 2)
ftX
## Call: 
## gllvm(y = Ysoil, X = Xsoils, num.lv = 2, family = "negative.binomial", 
##     row.eff = ~(1 | Site))
## family: 
## [1] "negative.binomial"
## method: 
## [1] "VA"
## 
## log-likelihood:  -57178.88 
## Residual degrees of freedom:  48265 
## AIC:  128147.8 
## AICc:  130118.1 
## BIC:  142112.6

The information criterias for the model with all covariates were worse than for the model with only pH as a covariate. Below, point estimates with 95% confidence intervals below indicate that pH value is the main covariate affecting the species composition.

coefplot(ftX, cex.ylab = 0.5, y.label = FALSE, mar = c(4, 2, 2, 1))

The corresponding ordination plot given below is very similar to the ordination plot of the model which includes only pH value as a covariate, and the ordination still exhibits a sampling location effect.

ordiplot(ftX, main = "Ordination of sites", 
         symbols = TRUE, pch = pchr, s.colors = Colorsph)
legend("topleft", legend = c("Kil", "NyA", "Mayr"), pch = c(1, 2, 3), bty = "n")

To account for this we add the sampling location as a categorical covariate into the model.

Xenv <- data.frame(Xsoils, Region = factor(Xenv$Region),
                   Soiltype = factor(Xenv$Soiltype))
ftXi <- gllvm(Ysoil, Xenv, formula = ~ SOM + pH + Phosp + Region, 
              family = "negative.binomial", row.eff = ~(1|Site), num.lv = 2,
              sd.errors = FALSE)
ftXi
## Call: 
## gllvm(y = Ysoil, X = Xenv, formula = ~SOM + pH + Phosp + Region, 
##     num.lv = 2, family = "negative.binomial", row.eff = ~(1 | 
##         Site), sd.errors = FALSE)
## family: 
## [1] "negative.binomial"
## method: 
## [1] "VA"
## 
## log-likelihood:  -54759.57 
## Residual degrees of freedom:  46295 
## AIC:  127249.1 
## AICc:  130644.7 
## BIC:  145203.9

The resulting ordination plot shows that there is no more visible pattern in sampling location.

ordiplot(ftXi, main = "Ordination of sites",  
         symbols = TRUE, pch = pchr, s.colors = Colorsph)
legend("topleft", legend = c("Kil", "NyA", "Mayr"), pch = c(1, 2, 3), bty = "n")

When comparing nested models, in particular, the model with environmental covariates to the null model, variance explained can be quantified by using methods like extensions of pseudo R2. For example, below we compare the total covariation in the ftNULL and ftX based on the traces of the residual covariance matrices. This suggests that environmental variables explain 43% of the total covariation.

1 - getResidualCov(ftX)$trace/getResidualCov(ftNULL)$trace
## [1] 0.4391057

References

Niku, J., D. I. Warton, F. K. C. Hui, and S. Taskinen. 2017. “Generalized Linear Latent Variable Models for Multivariate Count and Biomass Data in Ecology.” 22: 498–522.

Nissinen, R., M. Mannisto, and J van Elsas. 2012. “Endophytic Bacterial Communities in Three Arctic Plants from Low Arctic Fell Tundra Are Cold-Adapted and Host-Plant Specific.” 82: 510–22.