This vignette illustrates the use of the `PLN`

function and the methods accompaning the R6 class `PLNfit`

.

From the statistical point of view, the function `PLN`

adjusts a multivariate Poisson lognormal model to a table of counts, possibly after correcting for effects of offsets and covariates. `PLN`

is the building block for all the multivariate models found in the `PLNmodels`

package: having a basic understanding of both the mathematical background and the associated set of `R`

functions is a good place to start.

The packages required for the analysis are **PLNmodels** plus some others for data manipulation and representation:

We illustrate our point with the trichoptera data set, a full description of which can be found in the corresponding vignette. Data preparation is also detailed in the specific vignette.

The `trichoptera`

data frame stores a matrix of counts (`trichoptera$Abundance`

), a matrix of offsets (`trichoptera$Offset`

) and some vectors of covariates (`trichoptera$Wind`

, `trichoptera$Temperature`

, etc.)

The multivariate Poisson lognormal model (in short PLN, see Aitchison and Ho (1989)) relates some \(p\)-dimensional observation vectors \(\mathbf{Y}_i\) to some \(p\)-dimensional vectors of Gaussian latent variables \(\mathbf{Z}_i\) as follows

\[\begin{equation} \begin{array}{rcl} \text{latent space } & \mathbf{Z}_i \sim \mathcal{N}({\boldsymbol\mu},\boldsymbol\Sigma), \\ \text{observation space } & Y_{ij} | Z_{ij} \quad \text{indep.} & \mathbf{Y}_i | \mathbf{Z}_i\sim\mathcal{P}\left(\exp\{\mathbf{Z}_i\}\right). \end{array} \end{equation}\]

The parameter \({\boldsymbol\mu}\) corresponds to the main effects and the latent covariance matrix \(\boldsymbol\Sigma\) describes the underlying residual structure of dependence between the \(p\) variables. The following figure provides insights about the role played by the different layers

This model generalizes naturally to a formulation closer to a multivariate generalized linear model, where the main effect is due to a linear combination of \(d\) covariates \(\mathbf{x}_i\) (including a vector of intercepts). We also let the possibility to add some offsets for the \(p\) variables in in each sample, that is \(\mathbf{o}_i\). Hence, the previous model generalizes to

\[\begin{equation} \mathbf{Y}_i | \mathbf{Z}_i \sim \mathcal{P}\left(\exp\{\mathbf{Z}_i\}\right), \qquad \mathbf{Z}_i \sim \mathcal{N}({\mathbf{o}_i + \mathbf{x}_i^\top\boldsymbol\Theta},\boldsymbol\Sigma), \\ \end{equation}\] where \(\boldsymbol\Theta\) is a \(d\times p\) matrix of regression parameters. When all individuals \(i=1,\dots,n\) are stacked together, the data matrices available to feed the model are

- the \(n\times p\) matrix of counts \(\mathbf{Y}\)
- the \(n\times d\) matrix of design \(\mathbf{X}\)
- the \(n\times p\) matrix of offsets \(\mathbf{O}\)

Inference in PLN then focuses on the regression parameters \(\boldsymbol\Theta\) and on the covariance matrix \(\boldsymbol\Sigma\).

Technically speaking, we adopt in **PLNmodels** a variational strategy to approximate the log-likelihood function and optimize the consecutive variational surrogate of the log-likelihood with a gradient-ascent-based approach. To this end, we rely on the CCSA algorithm of Svanberg (2002) implemented in the C++ library (Johnson 2011), which we link to the package.

The standard PLN model described above is adjusted with the function `PLN`

. We now review its usage on a the trichoptera data set.

In order to become familiar with the function `PLN`

and its outputs, let us first fit a simple PLN model with just an intercept for each species:

```
##
## Initialization...
## Adjusting a PLN model with full covariance model
## Post-treatments...
## DONE!
```

Note the use of the `formula`

object to specify the model: the vector \(\boldsymbol\mu\) of main effects in the mathematical formulation (one per column species) is specified in the call with the term `~ 1`

in the right-hand-side of the formula. `Abundance`

is a variable in the data frame `trichoptera`

correspondings to a matrix of 17 columns and the *response* in the model, occurring on the left-hand-side of the formula.

`PLNfit`

object`myPLN`

is an `R6`

object with class `PLNfit`

, which comes with a couple of methods, as recalled when printing/showing such an object in the `R`

console:

```
## A multivariate Poisson Lognormal fit with full covariance model.
## ==================================================================
## nb_param loglik BIC ICL R_squared
## 170 -1129.429 -1460.234 -2305.272 0.99
## ==================================================================
## * Useful fields
## $model_par, $latent, $var_par, $optim_par
## $loglik, $BIC, $ICL, $loglik_vec, $nb_param, $criteria
## * Useful S3 methods
## print(), coef(), sigma(), vcov(), fitted(), predict(), standard_error()
```

See also `?PLNfit`

for more comprehensive information.

Accessing public fields of a `PLNfit`

object can be done just like with a traditional list, *e.g.*,

`## [1] -1129.4293389 -1460.2340642 -2305.2717607 0.9898361`

```
## nb_param loglik BIC ICL R_squared
## 1 170 -1129.429 -1460.234 -2305.272 0.9898361
```

We provide a set of S3-methods for `PLNfit`

that mimic the standard (G)LM-like interface of `R::stats`

, which we present now.

One can access the fitted value of the counts (`Abundance`

– \(\hat{\mathbf{Y}}\)) and check that the algorithm basically learnt correctly from the data^{1}:

```
data.frame(
fitted = as.vector(fitted(myPLN)),
observed = as.vector(trichoptera$Abundance)
) %>%
ggplot(aes(x = observed, y = fitted)) +
geom_point(size = .5, alpha =.25 ) +
scale_x_log10() +
scale_y_log10() +
theme_bw() + annotation_logticks()
```

We can also reach the matrix of regression parameters \(\mathbf{\Theta}\) and the residual variance/covariance matrix \(\boldsymbol{\Sigma}\) of the latent variable with the traditional functions found in `R`

for (G)LM manipulation: for the regression coefficents, we can use the `coef`

(or `coefficients`

) method. Approximated standard errors of the coefficients are also accessible via `standard_error`

:

```
data.frame(
rbind(t(coef(myPLN)), t(standard_error(myPLN))),
row.names = c("effect", "stderr")
) %>% select(1:5) %>% knitr::kable()
```

Che | Hyc | Hym | Hys | Psy | |
---|---|---|---|---|---|

effect | -3.1794155 | -4.0691522 | 1.0442124 | -2.3197069 | 3.2551465 |

stderr | 0.5772879 | 0.5772882 | 0.0739204 | 0.4082495 | 0.0129228 |

The residual covariance matrix better displays as an image matrix:

It is also possible to use observation weights like in standard (G)LMs:

```
myPLN_weighted <-
PLN(
Abundance ~ 1,
data = trichoptera,
weights = runif(nrow(trichoptera)),
control = list(trace = 0)
)
data.frame(
unweighted = as.vector(fitted(myPLN)),
weighted = as.vector(fitted(myPLN_weighted))
) %>%
ggplot(aes(x = unweighted, y = weighted)) +
geom_point(size = .5, alpha =.25 ) +
scale_x_log10() +
scale_y_log10() +
theme_bw() + annotation_logticks()
```

For ecological count data, it is generally a good advice to include the sampling effort via an offset term whenever available, otherwise samples are not necessarily comparable:

```
myPLN_offsets <-
PLN(Abundance ~ 1 + offset(log(Offset)),
data = trichoptera, control = list(trace = 0))
```

Note that we use the function `offset`

with a log-transform of the total counts since it acts in the latent layer of the model. Obviously the model with offsets is better since the log-likelihood is higher with the same number of parameters^{2}:

nb_param | loglik | BIC | ICL | R_squared |
---|---|---|---|---|

170 | -1129.429 | -1460.234 | -2305.272 | 0.9898361 |

170 | -1051.468 | -1382.273 | -2192.545 | 0.9222185 |

Let us try to correct for the wind effect in our model:

```
##
## Initialization...
## Adjusting a PLN model with full covariance model
## Post-treatments...
## DONE!
```

When we compare the models, the gain is clear in terms of log-likelihood. However, the BIC choses not to include this variable:

nb_param | loglik | BIC | ICL | R_squared |
---|---|---|---|---|

170 | -1051.468 | -1382.273 | -2192.545 | 0.9222185 |

187 | -1027.783 | -1391.668 | -2084.740 | 0.8903289 |

It is possible to change a bit the parametrization used for modelling the residual covariance matrix \(\boldsymbol\Sigma\), and thus reduce the total number of parameters used in the model. By default, the residual covariance is fully parameterized (hence \(p \times (p+1)/2\) parameters). However, we can chose to only model the variances of the species and not the covariances, by means of a diagonal matrix \(\boldsymbol\Sigma_D\) with only \(p\) parameters. In an extreme situation, we may also chose a single variance parameter for the whole matrix \(\boldsymbol\Sigma = \sigma \mathbf{I}_p\). This can be tuned in `PLN`

with the `control`

argument, a list controlling various aspects of the underlying optimization process:

```
myPLN_diagonal <-
PLN(
Abundance ~ 1 + offset(log(Offset)),
data = trichoptera, control = list(covariance = "diagonal", trace = 0)
)
myPLN_spherical <-
PLN(
Abundance ~ 1 + offset(log(Offset)),
data = trichoptera, control = list(covariance = "spherical", trace = 0)
)
```

Note that, by default, the model chosen is , so that the two following calls are equivalents:

```
##
## Initialization...
## Adjusting a PLN model with full covariance model
## Post-treatments...
## DONE!
```

```
##
## Initialization...
## Adjusting a PLN model with full covariance model
## Post-treatments...
## DONE!
```

Different covariance models can then be compared with the usual criteria: it seems that the gain brought by passing from a diagonal matrix to a fully parameterized covariance is not worth having so many additional parameters:

```
rbind(
myPLN_offsets$criteria,
myPLN_diagonal$criteria,
myPLN_spherical$criteria
) %>%
as.data.frame(row.names = c("full", "diagonal", "spherical")) %>%
knitr::kable()
```

nb_param | loglik | BIC | ICL | R_squared | |
---|---|---|---|---|---|

full | 170 | -1051.468 | -1382.273 | -2192.545 | 0.9222185 |

diagonal | 34 | -1109.309 | -1175.470 | -2041.994 | 0.9165038 |

spherical | 18 | -1792.270 | -1827.297 | -2691.099 | 0.8634665 |

A final model that we can try is the diagonal one with the wind as a covariate, which gives a slight improvement.

```
myPLN_final <-
PLN(
Abundance ~ 1 + Wind + offset(log(Offset)),
data = trichoptera, control = list(covariance = "diagonal", trace = 0)
)
rbind(
myPLN_wind$criteria,
myPLN_diagonal$criteria,
myPLN_final$criteria
) %>% knitr::kable()
```

nb_param | loglik | BIC | ICL | R_squared |
---|---|---|---|---|

187 | -1027.783 | -1391.668 | -2084.740 | 0.8903289 |

34 | -1109.309 | -1175.470 | -2041.994 | 0.9165038 |

51 | -1072.813 | -1172.054 | -1755.988 | 0.8762200 |

Aitchison, J., and C.H. Ho. 1989. “The Multivariate Poisson-Log Normal Distribution.” *Biometrika* 76 (4). Oxford University Press: 643–53.

Johnson, Steven G. 2011. *The Nlopt Nonlinear-Optimization Package*. http://ab-initio.mit.edu/nlopt.

Svanberg, Krister. 2002. “A Class of Globally Convergent Optimization Methods Based on Conservative Convex Separable Approximations.” *SIAM Journal on Optimization* 12 (2). SIAM: 555–73.