abn: Additive Bayesian Networks

The R package abn is a tool for Bayesian network analysis, a form of probabilistic graphical model. It derives a directed acyclic graph (DAG) from empirical data that describes the dependency structure between random variables. The package provides routines for structure learning and parameter estimation of additive Bayesian network models.

Installation

The most recent development version is available from Github and can be installed with:

devtools::install_github("furrer-lab/abn")

It is recommended to install abn within a virtual environment, e.g., using renv which can be done with:

renv::install("bioc::graph")
renv::install("bioc::Rgraphviz")
renv::install("abn", dependencies = c("Depends", "Imports", "LinkingTo", "Suggests"))

Please note that the abn package is currently unavailable on CRAN. We are dedicated to providing a robust and reliable package, and we appreciate your understanding as we work towards making abn available on CRAN soon. [^1]

[^1]: The abn package includes certain features, such as multiprocessing and integration with the INLA package, which are limited or available only on specific CRAN flavors. While it is possible to relax the testing process by, e.g., excluding tests of these functionalities, we believe that rigorous testing is important for reliable software development, especially for a package like abn that includes complex functionalities. We have implemented a rigorous testing framework similar to CRAN's to validate these functionalities in our development process. Our aim is to maximize the reliability of the abn package under various conditions, and we are dedicated to providing a robust and reliable package. We appreciate your understanding as we work towards making abn available on CRAN soon.

Additional libraries

The following additional libraries are recommended to best profit from the abn features.

• INLA, which is an R package used for model fitting. It is hosted separately from CRAN and is easy to install on common platforms (see instructions on the INLA website).

install.packages("INLA", repos = c(getOption("repos"), INLA = "https://inla.r-inla-download.org/R/stable"), dep = TRUE)

• Rgraphviz is used to produce plots of network graphs and is hosted on Bioconductor.

if (!requireNamespace("BiocManager", quietly = TRUE))
    install.packages("BiocManager")
BiocManager::install("Rgraphviz", version = "3.8")

• JAGS is a program for analyzing Bayesian hierarchical models using Markov Chain Monte Carlo (MCMC) simulation. Its installation is platform-dependent and is, therefore, not covered here.

Quickstart

Explore the basics of data analysis using additive Bayesian networks with the abn package through our simple example. The datasets required for these examples are included within the abn package.

For a deeper understanding, refer to the manual pages on the abn homepage, which include numerous examples. Key pages to visit are fitAbn(), buildScoreCache(), mostProbable(), and searchHillClimber(). Also, see the examples below for a quick overview of the package's capabilities.

Features

The R package abn provides routines for determining optimal additive Bayesian network models for a given data set. The core functionality is concerned with model selection - determining the most likely model of data from interdependent variables. The model selection process can incorporate expert knowledge by specifying structural constraints, such as which arcs are banned or retained.

The general workflow with abn follows a three-step process:

1. Determine the model search space: The function buildScoreCache() builds a cache of pre-computed scores for each possible DAG.

For this, it's required to specify the data types of the variables in the data set and the structural constraints of the model (e.g. which arcs are banned or retained and the maximum number of parents per node).

2. Structure learning: abn offers different structure learning algorithms:

   • The exact structure learning algorithm from Koivisto and Sood (2004) is implemented in C and can be called with the function mostProbable(), which finds the most probable DAG for a given data set.

   The function searchHeuristic() provides a set of heuristic search algorithms. These include the hill-climber, tabu search, and simulated annealing algorithms implemented in R. searchHillClimber() searches for high-scoring DAGs using a random re-start greedy hill-climber heuristic search and is implemented in C. It slightly deviates from the method initially presented by Heckerman et al. 1995 (for details consult the respective help page ?abn::searchHillClimber()).

3. Parameter estimation: The function fitAbn() estimates the model's parameters based on the DAG from the previous step.

abn allows for two different model formulations, specified with the argument method:

• method = "mle" fits a model under the frequentist paradigm using information-theoretic criteria to select the best model.

• method = "bayes" estimates the posterior distribution of the model parameters based on two Laplace approximation methods, that is, a method for Bayesian inference and an alternative to Markov Chain Monte Carlo (MCMC): A standard Laplace approximation is implemented in the abn source code but switches in specific cases (see help page ?fitAbn) to the Integrated Nested Laplace Approximation from the INLA package requiring the installation thereof.

To generate new observations from a fitted ABN model, the function simulateAbn() simulates data based on the DAG and the estimated parameters from the previous step. simulateAbn() is available for both method = "mle" and method = "bayes" and requires the installation of the JAGS package.

Supported Data types

The abn package supports the following distributions for the variables in the network:

• Gaussian distribution for continuous variables.

• Binomial distribution for binary variables.

• Poisson distribution for variables with count data.

• Multinomial distribution for categorical variables (only available with method = "mle").

Unlike other packages, abn does not restrict the combination of parent-child distributions.

Multilevel Models for Grouped Data Structures

The analysis of "hierarchical" or "grouped" data, in which observations are nested within higher-level units, requires statistical models with parameters that vary across groups (e.g. mixed-effect models).

abn allows to control for one-layer clustering, where observations are grouped into a single layer of clusters that are themself assumed to be independent, but observations within the clusters may be correlated (e.g. students nested within schools, measurements over time for each patient, etc). The argument group.var specifies the discrete variable that defines the group structure. The model is then fitted separately for each group, and the results are combined.

For example, studying student test scores across different schools, a varying intercept model would allow for the possibility that average test scores (the intercept) might be higher in one school than another due to factors specific to each school. This can be modeled in abn by setting the argument group.var to the variable containing the school names. The model is then fitted as a varying intercept model, where the intercept is allowed to vary across schools, but the slope is assumed to be the same for all schools.

Under the frequentist paradigm (method = "mle"), abn relies on the lme4 package to fit generalized linear mixed models (GLMMs) for Binomial, Poisson, and Gaussian distributed variables. For multinomial distributed variables, abn fits a multinomial baseline category logit model with random effects using the mclogit package. Currently, only one-layer clustering is supported (e.g., for method = "mle", this corresponds to a random intercept model).

With a Bayesian approach (method = "bayes"), abn relies on its own implementation of the Laplace approximation and the package INLA to fit a single-level hierarchical model for Binomial, Poisson, and Gaussian distributed variables. Multinomial distributed variables in general (see Section Supported Data Types) are not yet implemented with method = "bayes".

Basic Background

Bayesian network modeling is a data analysis technique ideally suited to messy, highly correlated and complex datasets. This methodology is rather distinct from other forms of statistical modeling in that its focus is on structure discovery—determining an optimal graphical model that describes the interrelationships in the underlying processes that generated the data. It is a multivariate technique and can be used for one or many dependent variables. This is a data-driven approach, as opposed to relying only on subjective expert opinion to determine how variables of interest are interrelated (for example, structural equation modeling).

Below and on the package's website, we provide some cookbook-type examples of how to perform Bayesian network structure discovery analyses with observational data. The particular type of Bayesian network models considered here are additive Bayesian networks. These are rather different, mathematically speaking, from the standard form of Bayesian network models (for binary or categorical data) presented in the academic literature, which typically use an analytically elegant but arguably interpretation-wise opaque contingency table parametrization. An additive Bayesian network model is simply a multidimensional regression model, e.g., directly analogous to generalized linear modeling but with all variables potentially dependent.

An example can be found in the American Journal of Epidemiology, where this approach was used to investigate risk factors for child diarrhea. A special issue of Preventive Veterinary Medicine on graphical modeling features several articles that use abn to fit epidemiological data. Introductions to this methodology can be found in Emerging Themes in Epidemiology and in Computers in Biology and Medicine where it is compared to other approaches.

What is an additive Bayesian network?

Additive Bayesian network (ABN) models are statistical models that use the principles of Bayesian statistics and graph theory. They provide a framework for representing data with multiple variables, known as multivariate data.

ABN models are a graphical representation of (Bayesian) multivariate regression. This form of statistical analysis enables the prediction of multiple outcomes from a given set of predictors while simultaneously accounting for the relationships between these outcomes.

In other words, additive Bayesian network models extend the concept of generalized linear models (GLMs), which are typically used to predict a single outcome, to scenarios with multiple dependent variables. This makes them a powerful tool for understanding complex, multivariate datasets.

The term Bayesian network is interpreted differently across various fields.

Bayesian network models often involve binary nodes, arguably the most frequently used type of Bayesian network. These models typically use a contingency table instead of an additive parameter formulation. This approach allows for mathematical elegance and enables key metrics like model goodness of fit and marginal posterior parameters to be estimated analytically (i.e., from a formula) rather than numerically (an approximation). However, this parametrization may not be parsimonious, and the interpretation of the model parameters is less straightforward than the usual Generalized Linear Model (GLM) type models, which are prevalent across all scientific disciplines.

While this is a crucial practical distinction, it’s a relatively low-level technical one, as the primary aspect of BN modeling is that it’s a form of graphical modeling – a model of the data’s joint probability distribution. This joint – multidimensional – aspect makes this methodology highly attractive for complex data analysis and sets it apart from more standard regression techniques, such as GLMs, GLMMs, etc., which are only one-dimensional as they assume all covariates are independent. While this assumption is entirely reasonable in a classical experimental design scenario, it’s unrealistic for many observational studies in fields like medicine, veterinary science, ecology, and biology.

Examples

• Example 1: Basic usage
• Example 2: Restrict model search space
• Example 3: Grouped Data Structures
• Example 4: Using INLA vs internal Laplace approximation

Example 1: Basic Usage

This is a basic example which shows the basic workflow:

library(abn)

# Built-in toy dataset with two Gaussian variables G1 and G2, two Binomial variables B1 and B2, and one multinomial variable C
str(g2b2c_data)

# Define the distributions of the variables
dists <- list(G1 = "gaussian",
              B1 = "binomial",
              B2 = "binomial",
              C = "multinomial",
              G2 = "gaussian")


# Build the score cache
cacheMLE <- buildScoreCache(data.df = g2b2c_data,
                         data.dists = dists,
                         method = "mle",
                         max.parents = 2)

# Find the most probable DAG
dagMP <- mostProbable(score.cache = cacheMLE)

# Print the most probable DAG
print(dagMP)

# Plot the most probable DAG
plot(dagMP)

# Fit the most probable DAG
myfit <- fitAbn(object = dagMP,
                method = "mle")

# Print the fitted DAG
print(myfit)

Example 2: Restrict Model Search Space

Based on example 1, we may know that the arc G1->G2 is not possible and that the arc from C -> G2 must be present. This "expert knowledge" can be included in the model by banning the arc from G1 to G2 and retaining the arc from C to G2.

The retain and ban matrices are specified as an adjacency matrix of 0 and 1 entries, where 1 indicates that the arc is banned or retained, respectively. Row and column names must match the variable names in the data set. The corresponding column is a parent of the variable in the row. Each column represents the parents, and the row is the child. For example, the first row of the ban matrix indicates that G1 is banned as a parent of G2.

Further, we can restrict the maximum number of parents per node to 2.

# Ban the edge G1 -> G2
banmat <- matrix(0, nrow = 5, ncol = 5, dimnames = list(names(dists), names(dists)))
banmat[1, 5] <- 1

# retain always the edge C -> G2
retainmat <- matrix(0, nrow = 5, ncol = 5, dimnames = list(names(dists), names(dists)))
retainmat[5, 4] <- 1

# Limit the maximum number of parents to 2
max.par <- 2

# Build the score cache
cacheMLE_small <- buildScoreCache(data.df = g2b2c_data,
                            data.dists = dists,
                            method = "mle",
                            dag.banned = banmat,
                            dag.retained = retainmat,
                            max.parents = max.par)
print(paste("Without restrictions from example 1: ", nrow(cacheMLE$node.defn)))
print(paste("With restrictions as in example 2: ", nrow(cacheMLE_small$node.defn)))

Example 3: Grouped Data Structures

Depending on the data structure, we may want to control for one-layer clustering, where observations are grouped into a single layer of clusters that are themselves assumed to be independent, but observations within the clusters may be correlated (e.g., students nested within schools, measurements over time for each patient, etc.).

Currently, abn supports only one layer clustering.

# Built-in toy data set
str(g2pbcgrp)

# Define the distributions of the variables
dists <- list(G1 = "gaussian",
              P = "poisson",
              B = "binomial",
              C = "multinomial",
              G2 = "gaussian") # group is not among the list of variable distributions

# Ban arcs such that C has only B and P as parents
ban.mat <- matrix(0, nrow = 5, ncol = 5, dimnames = list(names(dists), names(dists)))
ban.mat[4, 1] <- 1
ban.mat[4, 4] <- 1
ban.mat[4, 5] <- 1

# Build the score cache
cache <- buildScoreCache(data.df = g2pbcgrp,
                         data.dists = dists,
                         group.var = "group",
                         dag.banned = ban.mat,
                         method = "mle",
                         max.parents = 2)

# Find the most probable DAG
dag <- mostProbable(score.cache = cache)

# Plot the most probable DAG
plot(dag)

# Fit the most probable DAG
fit <- fitAbn(object = dag,
              method = "mle")

# Plot the fitted DAG
plot(fit)

# Print the fitted DAG
print(fit)

Example 4: Using INLA vs internal Laplace approximation

Under a Bayesian approach, abn automatically switches to the Integrated Nested Laplace Approximation from the INLA package if the internal Laplace approximation fails to converge. However, we can also force the use of INLA by setting the argument control=list(max.mode.error=100).

The following example shows that the results are very similar. It also shows how to constrain arcs as formula objects and how to specify different parent limits for each node separately.

library(abn)

# Subset of the build-in dataset, see  ?ex0.dag.data
mydat <- ex0.dag.data[,c("b1","b2","g1","g2","b3","g3")] ## take a subset of cols

# setup distribution list for each node
mydists <- list(b1="binomial", b2="binomial", g1="gaussian",
                g2="gaussian", b3="binomial", g3="gaussian")

# Structural constraints
## ban arc from b2 to b1
## always retain arc from g2 to g1
## parent limits - can be specified for each node separately
max.par <- list("b1"=2, "b2"=2, "g1"=2, "g2"=2, "b3"=2, "g3"=2)

# now build the cache of pre-computed scores according to the structural constraints
res.c <- buildScoreCache(data.df=mydat, data.dists=mydists,
                         dag.banned= ~b1|b2, 
                         dag.retained= ~g1|g2, 
                         max.parents=max.par)


# repeat but using R-INLA. The mlik's should be virtually identical.
if(requireNamespace("INLA", quietly = TRUE)){
  res.inla <- buildScoreCache(data.df=mydat, data.dists=mydists,
                              dag.banned= ~b1|b2, # ban arc from b2 to b1
                              dag.retained= ~g1|g2, # always retain arc from g2 to g1
                              max.parents=max.par,
                              control=list(max.mode.error=100)) # force using of INLA
  
  ## comparison - very similar
  difference <- res.c$mlik - res.inla$mlik
  summary(difference)
}

Contributing

We greatly appreciate contributions from the community and are excited to welcome you to the development process of the abn package. Here are some guidelines to help you get started:

1. Seeking Support:

If you need help with using the abn package, you can seek support by creating a new issue on our GitHub repository. Please describe your problem in detail and include a minimal reproducible example if possible.

2. Reporting Issues or Problems:

If you encounter any issues or problems with the software, please report them by creating a new issue on our GitHub repository. When reporting an issue, try to include as much detail as possible, including steps to reproduce the issue, your operating system and R version, and any error messages you received.

3. Software Contributions:

We encourage contributions directly via pull requests on our GitHub repository. Before starting your work, please first create an issue describing the contribution you wish to make. This allows us to discuss and agree on the best way to integrate your contribution into the package.

By participating in this project, you agree to abide by our code of conduct. We are committed to making participation in this project a respectful and harassment-free experience for everyone.

Citation

If you use abn in your research, please cite it as follows:

> citation("abn")
To cite the methodology of the R package 'abn' use:

  Kratzer G, Lewis F, Comin A, Pittavino M, Furrer R (2023). “Additive Bayesian Network Modeling with the R Package abn.” _Journal of Statistical Software_,
  *105*(8), 1-41. doi:10.18637/jss.v105.i08 <https://doi.org/10.18637/jss.v105.i08>.

To cite an example of a typical ABN analysis use:

  Kratzer, G., Lewis, F.I., Willi, B., Meli, M.L., Boretti, F.S., Hofmann-Lehmann, R., Torgerson, P., Furrer, R. and Hartnack, S. (2020). Bayesian Network
  Modeling Applied to Feline Calicivirus Infection Among Cats in Switzerland. Frontiers in Veterinary Science, 7, 73

To cite the software implementation of the R package 'abn' use:

  Furrer, R., Kratzer, G. and Lewis, F.I. (2023). abn: Modelling Multivariate Data with Additive Bayesian Networks. R package version 2.7-2.
  https://CRAN.R-project.org/package=abn

License

The abn package is licensed under the GNU General Public License v3.0.

Code of Conduct

Please note that the abn project is released with a Contributor Code of Conduct. By contributing to this project, you agree to abide by its terms.

Applications

The abn website provides a comprehensive set of documented case studies, numerical accuracy/quality assurance exercises, and additional documentation.

Technical articles

• Kratzer et al. (2023): Additive Bayesian Network Modeling with the R Package abn

• Kratzer et al. (2020) Bayesian Networks modeling applied to Feline Calicivirus infection among cats in Switzerland

• Kratzer et al. (2018): Comparison between Suitable Priors for Additive Bayesian Networks

• Koivisto et al. (2004): Exact Bayesian structure discovery in Bayesian networks

• Friedman et al. (2003): Being Bayesian about network structure. A Bayesian approach to structure discovery in Bayesian networks

• Friedman et al. (1999): Data analysis with Bayesian networks: A bootstrap approach

• Heckerman et al. (1995): Learning Bayesian Networks – The Combination of Knowledge And Statistical-Data

Application articles

• Delucchi et al. (2022): Bayesian network analysis reveals the interplay of intracranial aneurysm rupture risk factors

• Guinat et al. (2020) Biosecurity risk factors for highly pathogenic avian influenza (H5N8) virus infection in duck farms, France

• Hartnack et al. (2019) Additive Bayesian networks for antimicrobial resistance and potential risk factors in non-typhoidal Salmonella isolates from layer hens in Uganda

• Ruchti et al. (2019): Progression and risk factors of pododermatitis in part-time group housed rabbit does in Switzerland

• Comin et al. (2019) Revealing the structure of the associations between housing system, facilities, management and welfare of commercial laying hens using Additive Bayesian Networks

• Ruchti et al. (2018): Pododermatitis in group housed rabbit does in Switzerland – prevalence, severity and risk factors

• Pittavino et al. (2017): Comparison between generalised linear modelling and additive Bayesian network; identification of factors associated with the incidence of antibodies against Leptospira interrogans sv Pomona in meat workers in New Zealand

• Hartnack et al. (2017): Attitudes of Austrian veterinarians towards euthanasia in small animal practice: impacts of age and gender on views on euthanasia

• Lewis et al. (2012): Revealing the Complexity of Health Determinants in Resource-poor Settings

• Lewis et al. (2011): Structure discovery in Bayesian networks: An analytical tool for analysing complex animal health data

Workshops

Causality:

• 4 December 2018, Beate Sick & Gilles Kratzer of the 1st Causality workshop talk, Bayesian Networks meet Observational data. (UZH, Switzerland)

ABN modeling

• 07 July 2021, workshop at the UseR! Conference on Additive Bayesian Networks Modeling. (Online)

• 29 March 2019, workshop at the SVEPM conference on Multivariate analysis using Additive Bayesian Networks. (Utrecht, Netherland)

Presentations

• 4 October 2018, talk in Nutricia (Danone). Multivariable analysis: variable and model selection in system epidemiology. (Utrecht, Netherland)

• 30 May 2018. Brown Bag Seminar in ZHAW. Presentation: Bayesian Networks Learning in a Nutshell. (Winterthur, Switzerland)