Author: PLN team
Date: 2022-01-31

# Analyzing multivariate count data with the Poisson log-normal model

#### PLN team

#### 2022-01-31

## Preliminaries

This vignette illustrates the use of the `PLN` function and the methods accompanying 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.

### Requirements

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

    library(PLNmodels)
    library(ggplot2)
    library(corrplot)

### Data set

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.

    data(trichoptera)
    trichoptera <- prepare_data(trichoptera$Abundance, trichoptera$Covariate)

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.)

### Mathematical background

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

PLN: geometrical view

#### Covariates and offsets

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 \(\boldsymbol\Sigma\).

#### Optimization by Variational inference

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.

## Analysis of trichoptera data with a PLN model

The standard PLN model described above is adjusted with the function `PLN`. We now review its usage on a the trichoptera data set.

### A PLN model with latent main effects

#### Adjusting a fit

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:

    myPLN <- PLN(Abundance ~ 1, trichoptera)

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

#### The `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:

    myPLN

    ## A multivariate Poisson Lognormal fit with full covariance model.
    ## ==================================================================
    ##  nb_param    loglik      BIC       ICL
    ##       170 -1130.055 -1460.86 -2259.872
    ## ==================================================================
    ## * Useful fields
    ##     $model_par, $latent, $latent_pos, $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.

#### Field access

Accessing public fields of a `PLNfit` object can be done just like with a traditional list, e.g.,

    c(myPLN$loglik, myPLN$BIC, myPLN$ICL)

    ## [1] -1130.055 -1460.860 -2259.872

    myPLN$criteria

    ##   nb_param    loglik      BIC       ICL
    ## 1      170 -1130.055 -1460.86 -2259.872

#### GLM-like interface

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 learned correctly from the data1:

    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()

fitted value vs. observation

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 coefficients, 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 | -2.8946988 | -3.8933488 | 1.0475508 | -2.2502452 | 3.2843832
stderr | 0.5772689 | 0.5766974 | 0.0739221 | 0.4079502 | 0.0129229

The residual covariance matrix better displays as an image matrix:

    corrplot(sigma(myPLN), is.corr = FALSE)

#### Observation weights

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()

### Accounting for covariates and offsets

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 parameters2:

    rbind(
      myPLN$criteria,
      myPLN_offsets$criteria
    ) %>% knitr::kable()

nb_param | loglik | BIC | ICL
--- | --- | --- | ---
170 | -1130.055 | -1460.860 | -2259.872
170 | -1052.488 | -1383.292 | -2226.842

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

    myPLN_wind <- PLN(Abundance ~ 1 + Wind + offset(log(Offset)), data = trichoptera)

    ##
    ##  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 chooses not to include this variable:

    rbind(
      myPLN_offsets$criteria,
      myPLN_wind$criteria
    ) %>% knitr::kable()

nb_param | loglik | BIC | ICL
--- | --- | --- | ---
170 | -1052.488 | -1383.292 | -2226.842
187 | -1028.432 | -1392.317 | -2096.695

### Covariance models (full, diagonal, spherical)

It is possible to change a bit the parametrization used for modeling 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_spherical <-
      PLN(
        Abundance ~ 1 + offset(log(Offset)),
        data = trichoptera, control = list(covariance = "spherical", trace = 0)
      )

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

Note that, by default, the model chosen is `covariance = "spherical"`, so that the two following calls are equivalents:

    myPLN_default <-
      PLN(Abundance ~ 1, data = trichoptera, )

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

    myPLN_full <-
      PLN(Abundance ~ 1, data = trichoptera, control = list(covariance = "full"))

    ##
    ##  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
--- | --- | --- | --- | ---
full | 170 | -1052.488 | -1383.292 | -2226.842
diagonal | 34 | -1110.282 | -1176.443 | -2102.677
spherical | 18 | -1158.871 | -1193.898 | -2144.225

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
--- | --- | --- | ---
187 | -1028.432 | -1392.317 | -2096.695
34 | -1110.282 | -1176.443 | -2102.677
51 | -1074.021 | -1173.262 | -1838.669

## References

Aitchison, J., and C. H. Ho. 1989. “The Multivariate Poisson-Log Normal Distribution.” Biometrika 76 (4): 643–53.
Johnson, Steven G. 2011. The NLopt Nonlinear-Optimization Package. https://nlopt.readthedocs.io/en/latest/.
Svanberg, Krister. 2002. “A Class of Globally Convergent Optimization Methods Based on Conservative Convex Separable Approximations.” SIAM Journal on Optimization 12 (2): 555–73.

1. We use a log-log scale in our plot in order not to give an excessive importance to the higher counts in the fit↩︎

2. In PLNmodels the R-squared is a pseudo-R-squared that can only be trusted between model where the same offsets term was used↩︎