Introduction

mcmodule is a framework for building modular Monte Carlo risk analysis models. It extends the capabilities of mc2d to make working with multiple risk pathways, variates, and scenarios easier. It was developed by epidemiologists, for the farmrisk project, and now it aims to help epidemiologists and other risk modellers save time and evaluate ambitious, complex risk pathways in R.

The mc2d R package (Pouillot and Delignette-Muller 2010) provides tools to build and analyse models involving multiple variables and uncertainties based on Monte Carlo simulations. The mcmodule package includes additional tools to:

Organize risk analysis in independent flexible modules
Perform multivariate mcnode operations
Automate the creation of mcnodes
visualise risk analysis models

Multivariate Monte-Carlo simulations

Quantitative risk analysis is the numerical assessment of risk to facilitate decision making in the face of uncertainty. Monte Carlo simulation is a technique used to model and analyse uncertainty (Vose 2008).

In mc2d, parameters are stored as mcnode class objects. These objects are arrays of numbers that represent random variables and have three dimensions: variability × uncertainty × variates. For more information, see the mc2d package vignette.

In the mcmodule framework an mcnode is an array of dimensions (u × 1 × i):

We combine variability and uncertainty into one dimension¹ (u)
We use variates (representing different groups or categories) in the other dimension (i)
The variables that define and distinguish these variates are called keys and are stored as metadata

In this document we will use the term “uncertainty dimension” to refer to the combined dimension of variability and uncertainty.

Example

If we are buying a number of cows and heifers from a specific region, an mcnode for herd prevalence would have:

Multiple variates (rows), defined by two keys: “animal category” and “pathogen”

Probability values generated through a PERT distribution, using minimum, mode, and maximum parameters across u iterations (columns) to model the uncertainty in these estimates

An mcnode representing herd prevalence of three pathogens, tuberculosis (TB), infectious bovine rhinotracheitis (IBR), and bovine viral diarrhoea (BVD), and two animal categories (cows and heifers). The herd prevalence is the same for both animal categories, but values differ due to the stochastic nature of random sampling from the PERT distribution.

Risk assessment

This section provides a brief introduction to risk assessment in R. Although this package is not intended for beginners in risk assessment, it can help you understand the logic behind mcmodule and its purpose.

A simple risk assessment

Consider a scenario where we need to purchase a cow from a farm that we know is infected with a disease our farm is free from. To reduce the risk of introducing the disease to our farm, we plan to perform a diagnostic test on the cow before bringing it to our farm. We want to calculate the probability of introducing the disease by purchasing one cow that tests negative.

We have an estimation (with some uncertainty) of both the probability of animal infection within a herd and the test sensitivity, so we want to conduct a stochastic risk assessment that properly accounts for this uncertainty.

The risk assessment for our cattle purchase can be performed using base R (2024) random sampling functions, or mc2d (Pouillot and Delignette-Muller 2010), a package provides additional probability distributions (such as rpert) and other useful tools for analysing stochastic (Monte-Carlo) simulations.

library(mc2d)
set.seed(123)
n_iterations <- 10000

# Within-herd prevalence
w_prev <- mcstoc(runif,
  min = 0.15, max = 0.2,
  nsu = n_iterations, type = "U"
)
# Test sensitivity
test_sensi <- mcstoc(rpert,
  min = 0.89, mode = 0.9, max = 0.91,
  nsu = n_iterations, type = "U"
)
# Probability an animal is tested in origin
test_origin <- mcdata(1, type = "0") # Yes


# EXPRESSIONS
# Probability that an animal in an infected herd is infected (a = animal)
infected <- w_prev
# Probability an animal is tested and is a false negative
# (test specificity assumed to be 100%)
false_neg <- infected * test_origin * (1 - test_sensi)
# Probability an animal is not tested
no_test <- infected * (1 - test_origin)
# Probability an animal is not detected
no_detect <- false_neg + no_test

mc_model <- mc(
  w_prev, infected, test_origin, test_sensi,
  false_neg, no_test, no_detect
)

# RESULT
hist(mc_model)

no_detect
#>   node    mode nsv   nsu nva variate    min   mean median    max Nas type outm
#> 1    x numeric   1 10000   1       1 0.0138 0.0175 0.0174 0.0218   0    U each

Multiple risk assessments at once

In the previous example, we calculated the risk for one specific case. However, we know that this farm is also positive for pathogen B, so it would be also interesting to calculate the risk of introducing it as well. Pathogen B has different within-herd prevalence and test sensitivity than Pathogen A.

To estimate the risk for both pathogens with our previous models, we could:

Copy and paste the code twice with different parameters (against all good coding practices)
Wrap the code in a function and call it twice using each pathogen’s parameters as arguments
Create a loop

While these options work, they become messy or computationally intensive when the number of parameters or different situations to simulate increases.

The package mc2d offers a clever solution to this scalability problem: variates. In the previous example, our stochastic nodes only had uncertainty dimension. However, we can now use the variates dimension to calculate the risk of introduction of both pathogens at the same time.

set.seed(123)
n_iterations <- 10000

# Within-herd prevalence
w_prev_min <- mcdata(c(a = 0.15, b = 0.45), nvariates = 2, type = "0")
w_prev_max <- mcdata(c(a = 0.2, b = 0.6), nvariates = 2, type = "0")

w_prev <- mcstoc(runif,
  min = w_prev_min, max = w_prev_max,
  nsu = n_iterations, nvariates = 2, type = "U"
)

# Test sensitivity
test_sensi_min <- mcdata(c(a = 0.89, b = 0.80), nvariates = 2, type = "0")
test_sensi_mode <- mcdata(c(a = 0.9, b = 0.85), nvariates = 2, type = "0")
test_sensi_max <- mcdata(c(a = 0.91, b = 0.90), nvariates = 2, type = "0")

test_sensi <- mcstoc(rpert,
  min = test_sensi_min,
  mode = test_sensi_mode, max = test_sensi_max,
  nsu = n_iterations, nvariates = 2, type = "U"
)

# Probability an animal is tested in origin
test_origin <- mcdata(c(a = 1, b = 1), nvariates = 2, type = "0")


# EXPRESSIONS
# Probability that an animal in an infected herd is infected (a = animal)
infected <- w_prev
# Probability an animal is tested and is a false negative
# (test specificity assumed to be 100%)
false_neg <- infected * test_origin * (1 - test_sensi)
# Probability an animal is not tested
no_test <- infected * (1 - test_origin)
# Probability an animal is not detected
no_detect <- false_neg + no_test

mc_model <- mc(
  w_prev, infected, test_origin, test_sensi,
  false_neg, no_test, no_detect
)

# RESULT
no_detect
#>   node    mode nsv   nsu nva variate    min   mean median    max Nas type outm
#> 1    x numeric   1 10000   2       1 0.0139 0.0175 0.0174 0.0217   0    U each
#> 2    x numeric   1 10000   2       2 0.0477 0.0787 0.0783 0.1178   0    U each

When to use mcmodule?

The mc2d multivariate approach works well for basic multivariate risk analysis. However, if instead of purchasing one cow, you’re dealing with multiple cattle purchases, from different farms, across different pathogens, scenarios, and age categories, or modelling multiple risk pathways with different what-if scenarios, this approach becomes unwieldy.

mcmodule addresses these challenges by providing functions for multivariate operations and modular management of the risk model. It automates the process of creating mcnodes and assigns metadata to them (making it easy to identify which variate corresponds to which data row). Thanks to this mcnode metadata, it enables row-matching between nodes with different variates, combines probabilities across variates, and calculates multilevel trials. As your risk analysis grows, you can create separate modules for different pathways, each with independent parameters, expressions, and scenarios that can later be connected into a complete model.

This package is particularly useful for:

Working with complex models that involve multiple pathways, pathogens, or scenarios simultaneously
Dealing with large parameter sets (hundreds or thousands of parameters)
Handling different numbers of variates across different parts of your model that need to be combined
Creating modular risk assessments where different components need to be developed independently but later integrated (for example in collaborative projects)
Performing sophisticated sensitivity analyses across multiple model components

However, for simpler analyses, such as single pathway models, exploratory work, small models with few parameters, one-off analyses or learning risk assessment mcmodule’s additional structure may be unnecessary.

Installing mcmodule

Now, let’s explore the package! We can install it from CRAN:

install.packages("mcmodule")
library("mcmodule")

Or install latest development version from GitHub (requires devtool package):

# install.packages("devtools")
devtools::install_github("NataliaCiria/mcmodule")
library("mcmodule")

Other recommended packages to load along with mcmodule are:

# install.packages(c("dplyr","ggplot2","igraph","visNetwork"))
library(dplyr) # Data manipulation
library(igraph) # Network analysis
library(visNetwork) # Interactive network visualization

Building an mcmodule

To quickly understand the key components of an mcmodule, we’ll start by building one using the animal imports example included in the package.

Data

Let’s consider a scenario where we want to evaluate the risk of introducing pathogen A and pathogen B into our region from animal imports from different regions (north, south, east, and west). We have gathered the following data:

animal_imports: number of animal imports with their mean and standard deviation values per region, and the number of exporting farms in each region.

animal_imports
#>   origin farms_n animals_n_mean animals_n_sd
#> 1   nord       5            100            6
#> 2  south      10            130           10
#> 3   east       7            140           12

prevalence_region: estimates for both herd and within-herd prevalence ranges for pathogens A and B, as well as an indicator of how often tests are performed in origin

prevalence_region
#>   pathogen origin h_prev_min h_prev_max w_prev_min w_prev_max test_origin
#> 1        a   nord       0.08       0.10       0.15        0.2   sometimes
#> 2        a  south       0.02       0.05       0.15        0.2   sometimes
#> 3        a   east       0.10       0.15       0.15        0.2       never
#> 4        b   nord       0.50       0.70       0.45        0.6      always
#> 5        b  south       0.25       0.30       0.37        0.4   sometimes
#> 6        b   east       0.30       0.50       0.45        0.6     unknown

test_sensitivity: estimates of test sensitivity values for pathogen A and B

test_sensitivity
#>   pathogen test_sensi_min test_sensi_mode test_sensi_max
#> 1        a           0.89            0.90           0.91
#> 2        b           0.80            0.85           0.90

Now we will use dplyr::left_join() to create our imports module data:

imports_data <- prevalence_region %>%
  left_join(animal_imports) %>%
  left_join(test_sensitivity) %>%
  relocate(pathogen, origin, test_origin)
#> Joining with `by = join_by(origin)`
#> Joining with `by = join_by(pathogen)`

Data keys

From now on we will use only the merged imports_data table. However, it is useful to understand which input dataset each parameter comes from, as each dataset provides information for different keys. In this context, keys are fields that (combined) uniquely identify each row in a table. In our example:

animal_imports provided information by region of "origin"
prevalence_region provided information by "pathogen" and region of "origin"
test_sensitivity provided information by "pathogen" only

The resulting merged table, imports_data, will therefore have two keys: "pathogen" and "origin". However, not all parameters will use both keys, for example, "test_sensi" only has information by "pathogen". Knowing the keys for each parameter is crucial when performing multivariate operations, such as calculating totals.

To make these relationships explicit in the model, we need to provide the data keys. These are defined in a list with one element for each input dataset, specifying both the columns and the keys for each dataset.

imports_data_keys <- list(
  animal_imports = list(
    cols = names(animal_imports),
    keys = "origin"
  ),
  prevalence_region = list(
    cols = names(prevalence_region),
    keys = c("pathogen", "origin")
  ),
  test_sensitivity = list(
    cols = names(test_sensitivity),
    keys = "pathogen"
  )
)

mcnodes table

With values and keys established, we still need some information to build our stochastic parameters. The mcnode table specifies how to build mcnodes from the data table. It specifies which parameters are included in the model, the type of parameters (those with an mc_func are stochastic), and what columns to look for in the data table to build these mcnodes (the name of the mcnode, or another variable in the data columns), as well as transformations that are useful to encode categorical data values into mcnodes that must always be numeric.

mcnode: Name of the Monte Carlo node (required)
description: Description of the parameter
mc_func: Probability distribution
from_variable: Column name, if it comes from a column with a name different to the mcnode
transformation: Transformation to be applied to the original column values
sensi_baseline: Parameters for baseline mock distribution ²
sensi_variation: OAT variation expression using value placeholder³

Here we have the imports_mctable for our example. While the mctable can be hard-coded in R, it’s more efficient to prepare it in a CSV or other external file. This approach also allows the table to be included as part of the model documentation.

mcnode	description	mc_func	from_variable	transformation	sensi_baseline	sensi_variation
h_prev	Herd prevalence	runif	NA	NA	min = 0.1, max = 0.3	pmin(1, pmax(0, value * 1.5))
w_prev	Within herd prevalence	runif	NA	NA	min = 0.1, max = 0.3	pmin(1, pmax(0, value * 1.5))
test_sensi	Test sensitivity	rpert	NA	NA	min = 0.7, mode = 0.85, max = 0.95	pmin(1, pmax(0, value * 1.5))
farms_n	Number of farms exporting animals	NA	NA	NA	value = 10	value * 1.5
animals_n	Number of animals exported per farm	rnorm	NA	NA	mean = 50, sd = 10	value * 1.5
test_origin_unk	Unknown probability of the animals being tested in origin (true = unknown)	NA	test_origin	value==“unknown”	value = “unknown”	ifelse(value == “unknown”, “always”, value)
test_origin	Probability of the animals being tested in origin	NA	NA	ifelse(value == “always”, 1, ifelse(value == “sometimes”, 0.5, ifelse(value == “never”, 0, NA)))	value = 0.5	pmin(1, pmax(0, value * 1.5))

The data table and the mctable must complement each other:

mcnodes without an mc_func (like farms_n), needs the matching column name ("farms_n") in the data table
mcnodes with an mc_func, you need columns for each probability distribution argument in the data table. For example:
- h_prev with runif distribution requires "h_prev_min" and "h_prev_max"
- animals_n with rnorm distribution requires "animals_n_mean" and "animals_n_sd"

For encoding categorical variables as mcnodes (or any other data transformation), you can use any R code with value as a placeholder for the mcnode name or column name (specified in from_variable)

Expressions

Finally, we need to write the model’s mathematical expression. These expressions should ideally include only arithmetic operations, not R functions (with some exceptions that will be covered later in “tricks and tweaks”). We’ll wrap them using quote() so they aren’t executed immediately but stored for later evaluation with eval_model().

imports_exp <- quote({
  # Probability that an animal in an infected herd is infected (a = animal)
  infected <- w_prev
  # Probability an animal is tested and is a false negative
  # (test specificity assumed to be 100%)
  false_neg <- infected * test_origin * (1 - test_sensi)
  # Probability an animal is not tested
  no_test <- infected * (1 - test_origin)
  # Probability an animal is not detected
  no_detect <- false_neg + no_test
})

Creating mcnodes within expressions

Starting with version 1.2.0, you can create mcnodes directly inside expressions using mcstoc() or mcdata() from the mc2d package. However, these functions do not behave exactly as they do in mc2d when used within mcmodule. Keep the following points in mind:

Do not set nvariates. It is automatically set to match the number of rows in your data.
Use type = "V" (variability, the default) or type = "0" (deterministic).
All other mcstoc() and mcdata() arguments behave as documented in mc2d.

# Example expression with mcnodes created on-the-fly
imports_exp_inline <- quote({
  # Probability that an animal in an infected herd is infected
  infected <- w_prev
  
  # Create a clinic sensitivity parameter directly in the expression
  # (no need to specify nvariates, it's inferred from data rows)
  clinic_sensi <- mcstoc(runif, min = 0.6, max = 0.8)
  
  # Probability an animal is tested and is a false negative
  # Now accounting for both test sensitivity and clinic sensitivity
  false_neg <- infected * test_origin * (1 - test_sensi) * (1 - clinic_sensi)
  
  # Probability an animal is not tested but detected at clinic
  no_test <- infected * (1 - test_origin) * (1 - clinic_sensi)
  
  # Probability an animal is not detected
  no_detect <- false_neg + no_test
})

Evaluating an mcmodule

With all components in place, we’re now ready to create our first mcmodule using eval_module().

imports <- eval_module(
  exp = c(imports = imports_exp_inline),
  data = imports_data,
  mctable = imports_mctable,
  data_keys = imports_data_keys
)
#> imports evaluated
#> mcmodule created (expressions: imports)

class(imports)
#> [1] "mcmodule"

An mcmodule is an S3 object class, and it is essentially a list that contains all risk assessment components in a structured format.

names(imports)
#> [1] "data"      "exp"       "node_list"

The mcmodule contains the input data and mathematical expressions (exp) that ensure traceability. All input and calculated parameters are stored in node_list. Each node contains not only the mcnode itself but also important metadata: node type (input or output), source dataset and columns, keys, calculation method, and more. The specific metadata varies depending on the node’s characteristics. Here are a few examples:

imports$node_list$w_prev
#> $type
#> [1] "in_node"
#> 
#> $mc_func
#> [1] "runif"
#> 
#> $description
#> [1] "Within herd prevalence"
#> 
#> $inputs_col
#> [1] "w_prev_min" "w_prev_max"
#> 
#> $input_dataset
#> [1] "prevalence_region"
#> 
#> $keys
#> [1] "pathogen" "origin"  
#> 
#> $exp_name
#> [1] "imports"
#> 
#> $mc_name
#> [1] "w_prev"
#> 
#> $mcnode
#>   node    mode  nsv nsu nva variate  min  mean median max Nas type outm
#> 1    x numeric 1001   1   6       1 0.15 0.175  0.175 0.2   0    V each
#> 2    x numeric 1001   1   6       2 0.15 0.175  0.173 0.2   0    V each
#> 3    x numeric 1001   1   6       3 0.15 0.176  0.176 0.2   0    V each
#> 4    x numeric 1001   1   6       4 0.45 0.524  0.524 0.6   0    V each
#> 5    x numeric 1001   1   6       5 0.37 0.385  0.385 0.4   0    V each
#> 6    x numeric 1001   1   6       6 0.45 0.525  0.525 0.6   0    V each
#> 
#> $data_name
#> [1] "imports_data"
imports$node_list$no_detect
#> $function_call
#> [1] TRUE
#> 
#> $type
#> [1] "out_node"
#> 
#> $node_exp
#> [1] "false_neg + no_test"
#> 
#> $inputs
#> [1] "false_neg" "no_test"  
#> 
#> $exp_name
#> [1] "imports"
#> 
#> $mc_name
#> [1] "no_detect"
#> 
#> $keys
#> [1] "pathogen" "origin"  
#> 
#> $exp_param
#> [1] "false_neg" "no_test"  
#> 
#> $mcnode
#>   node    mode  nsv nsu nva variate    min   mean median    max Nas type outm
#> 1    x numeric 1001   1   6       1 0.0169 0.0288 0.0286 0.0435   0    V each
#> 2    x numeric 1001   1   6       2 0.0173 0.0288 0.0288 0.0433   0    V each
#> 3    x numeric 1001   1   6       3 0.0305 0.0522 0.0519 0.0795   0    V each
#> 4    x numeric 1001   1   6       4 0.0113 0.0237 0.0231 0.0415   0    V each
#> 5    x numeric 1001   1   6       5 0.0427 0.0665 0.0659 0.0936   0    V each
#> 6    x numeric 1001   1   6       6 0.0910 0.1566 0.1550 0.2357   0    V each
#> 
#> $data_name
#> [1] "imports_data"

And now that we have an mcmodule, we can begin exploring its possibilities!

Understanding mcnodes operations

When arithmetic operations are performed between nodes in mcmodule, they are applied on matching elements and keep the original dimensions, allowing uncertainties and variates to propagate through the calculations.

Working with an mcmodule

Visualizing

We can visualise an mc_module with the mc_network() function. For this, you will need to have igraph (Csardi and Nepusz 2006) and visNetwork (Almende B. V. and Benoit Thieurmel 2025) installed.

In these network visualizations, input datasets appear in blue, input data files, input columns and input mcnodes appear in different shades of dark-grey-blue, output mcnodes in green, and total mcnodes (as we will see later) in orange. The numbers displayed when clicked correspond to the median and the 95% confidence interval of the first variate of each mcnode.

mc_network(imports, legend = TRUE)

Summarizing

In the imports mcmodule, we can already see the raw mcnode results for the probability of an imported animal not being detected (no_detect). However, it’s difficult to determine which pathogen or region these results refer to. The mc_summary() function solves this problem by linking mcnode results with their key columns in the data.

Note that while the printed summary looks similar to the raw mcnode, it’s actually just a dataframe containing statistical measures, whereas the actual mcnode is a large array of numbers with dimensions (uncertainty × 1 × variates),

mc_summary(mcmodule = imports, mc_name = "no_detect")
#>     mc_name pathogen origin       mean          sd        Min       2.5%
#> 1 no_detect        a   nord 0.02876295 0.005883480 0.01687456 0.01875305
#> 2 no_detect        a  south 0.02876181 0.005994587 0.01726271 0.01870397
#> 3 no_detect        a   east 0.05221667 0.011010899 0.03050241 0.03392336
#> 4 no_detect        b   nord 0.02368331 0.005669560 0.01132022 0.01435750
#> 5 no_detect        b  south 0.06651522 0.012734194 0.04266336 0.04540683
#> 6 no_detect        b   east 0.15664447 0.033952337 0.09097242 0.09980820
#>          25%        50%        75%      97.5%        Max  nsv Na's
#> 1 0.02387016 0.02863349 0.03333063 0.03959466 0.04351491 1001    0
#> 2 0.02375014 0.02877089 0.03307491 0.04063392 0.04326582 1001    0
#> 3 0.04312578 0.05186702 0.06064605 0.07355557 0.07948500 1001    0
#> 4 0.01930997 0.02309890 0.02751195 0.03579437 0.04150531 1001    0
#> 5 0.05585011 0.06591137 0.07797919 0.08748444 0.09355962 1001    0
#> 6 0.12933887 0.15496802 0.18097518 0.22205690 0.23574153 1001    0

Filtering

Sometimes you may want to focus your analysis on specific subsets of your data, such as a particular pathogen or region. The mc_filter() function allows you to filter mcnodes based on conditions, similar to dplyr::filter().

Filtering a single condition

Let’s filter the no_detect node to only include results for pathogen “a”:

# Filter for pathogen a only
imports <- mc_filter(
  imports,
  "no_detect",
  pathogen == "a",
  name = "no_detect_pathogen_a"
)

# View the filtered results
imports$node_list$no_detect_pathogen_a$summary
#>                         mc_name pathogen origin       mean          sd
#> 1 no_detect_pathogen_a_filtered        a   nord 0.02876295 0.005883480
#> 2 no_detect_pathogen_a_filtered        a  south 0.02876181 0.005994587
#> 3 no_detect_pathogen_a_filtered        a   east 0.05221667 0.011010899
#>          Min       2.5%        25%        50%        75%      97.5%        Max
#> 1 0.01687456 0.01875305 0.02387016 0.02863349 0.03333063 0.03959466 0.04351491
#> 2 0.01726271 0.01870397 0.02375014 0.02877089 0.03307491 0.04063392 0.04326582
#> 3 0.03050241 0.03392336 0.04312578 0.05186702 0.06064605 0.07355557 0.07948500
#>    nsv Na's
#> 1 1001    0
#> 2 1001    0
#> 3 1001    0

Filtering with multiple conditions

You can also apply multiple filter conditions at once. For example, to analyze only pathogen “b” from the “nord” region:

# Filter for pathogen b from nord region
imports <- mc_filter(
  imports,
  "no_detect",
  pathogen == "b",
  origin == "nord",
  name = "no_detect_b_nord"
)

# View the filtered results
imports$node_list$no_detect_b_nord$summary
#>                     mc_name pathogen origin       mean         sd        Min
#> 4 no_detect_b_nord_filtered        b   nord 0.02368331 0.00566956 0.01132022
#>        2.5%        25%       50%        75%      97.5%        Max  nsv Na's
#> 4 0.0143575 0.01930997 0.0230989 0.02751195 0.03579437 0.04150531 1001    0

The filtered nodes maintain all the metadata from the original node and can be used in subsequent calculations just like any other node in the mcmodule.

Calculating totals

Most of the following probability calculations are based on Chapter 5 of the Handbook on Import Risk Analysis for Animals and Animal Products Volume 2. Quantitative risk assessment (Murray 2004). More details can be found on the Multivariate operations vignette.

Single-level trials

In imports, we know the probability that an infected animal from an infected farm goes undetected ("no_detect"). We can use the total number of animals selected per farm ("animals_n") as the number of trials (trials_n) to determine the probability that at least one infected animal from an infected farm is not detected (no_detect_set).

In single-level trials, each trial is independent with the same probability of success (\(trial\_p\)). For a set of \(trials\_n\) trials, the probability of at least one success is:

\[ set\_p= 1-(1-trial\_p)^{trials\_n} \]

# Probability of at least one imported animal from an infected herd is not detected
imports <- trial_totals(
  mcmodule = imports,
  mc_names = "no_detect",
  trials_n = "animals_n",
  mctable = imports_mctable
)

The trial_totals() function returns the mcmodule with some additional nodes: the probability of at least one success and the expected number of successes. These total nodes have special metadata fields, and always include a summary by default.

# Probability of at least one
imports$node_list$no_detect_set$summary
#>         mc_name pathogen origin      mean           sd       Min      2.5%
#> 1 no_detect_set        a   nord 0.9353649 3.883557e-02 0.7861311 0.8430134
#> 2 no_detect_set        a  south 0.9688685 2.434001e-02 0.8521708 0.9053670
#> 3 no_detect_set        a   east 0.9982705 2.713104e-03 0.9779484 0.9896240
#> 4 no_detect_set        b   nord 0.8925258 5.809912e-02 0.6801729 0.7620536
#> 5 no_detect_set        b  south 0.9994921 8.394029e-04 0.9937705 0.9971897
#> 6 no_detect_set        b   east 0.9999999 1.007295e-06 0.9999716 0.9999996
#>         25%       50%       75%     97.5%       Max  nsv Na's
#> 1 0.9110563 0.9449214 0.9664750 0.9839338 0.9903200 1001    0
#> 2 0.9556170 0.9767071 0.9875921 0.9960537 0.9987422 1001    0
#> 3 0.9978473 0.9994127 0.9998526 0.9999866 0.9999976 1001    0
#> 4 0.8568207 0.9028213 0.9383441 0.9732375 0.9913663 1001    0
#> 5 0.9993476 0.9998565 0.9999701 0.9999948 0.9999994 1001    0
#> 6 1.0000000 1.0000000 1.0000000 1.0000000 1.0000000 1001    0

# Expected number of animals
imports$node_list$no_detect_set_n$summary
#>           mc_name pathogen origin      mean        sd      Min      2.5%
#> 1 no_detect_set_n        a   nord  2.881746 0.6124382 1.529033  1.833821
#> 2 no_detect_set_n        a  south  3.740006 0.8357107 1.894541  2.335889
#> 3 no_detect_set_n        a   east  7.330762 1.6711952 3.754963  4.480383
#> 4 no_detect_set_n        b   nord  2.361640 0.5839372 1.132675  1.423825
#> 5 no_detect_set_n        b  south  8.613970 1.7498539 4.958773  5.733913
#> 6 no_detect_set_n        b   east 22.012662 5.1394951 9.971617 13.895154
#>         25%       50%       75%     97.5%       Max  nsv Na's
#> 1  2.391553  2.857134  3.339671  4.051653  4.536046 1001    0
#> 2  3.079032  3.707894  4.320097  5.424437  6.535555 1001    0
#> 3  6.003530  7.242773  8.544794 10.827406 12.409943 1001    0
#> 4  1.923674  2.303978  2.749540  3.555514  4.654709 1001    0
#> 5  7.123986  8.552104 10.002242 11.658403 13.687274 1001    0
#> 6 18.048291 21.529684 25.817968 32.375069 36.016324 1001    0

Multilevel trials

Simple multilevel

We can also calculate the probability that at least one infected animal from at least one infected farm is not detected, but here, we need to consider two levels: animals and farms.

Selection (in blue) of 4 animals from 3 farms, with a 20% regional herd prevalence and 50% within-herd prevalence.

We import animals from "farms_n" farms. Each farm has a probability "h_prev" (regional herd prevalence) of being infected. From each farm, we import "animals_n" animals. In an infected farm, each animal has a probability "w_prev" (within-herd prevalence) of being infected. We’ve already used this to calculate "no_detect", which is the probability that an infected animal is not detected.

The probability of at least one success in this hierarchical structure is given by:

\[ set\_p= 1-(1-subset\_p \cdot (1-(1-trial\_p)^{trial\_n}))^{subset\_n} \]

Where:

trials_p represents the probability of a trial in a subset being a success
trials_n represents the number of trials in subset
subset_p represents the probability of a subset being selected
subset_n represents the number of subsets
set_p represents the probability of a at least one trial of at least one subset being a success

# Probability of at least one animal from at least one herd being is not detected (probability of a herd being infected: h_prev)
imports <- trial_totals(
  mcmodule = imports,
  mc_names = "no_detect",
  trials_n = "animals_n",
  subsets_n = "farms_n",
  subsets_p = "h_prev",
  mctable = imports_mctable,
)

# Result
imports$node_list$no_detect_set$summary
#>         mc_name pathogen origin      mean          sd       Min      2.5%
#> 1 no_detect_set        a   nord 0.3554524 0.022994131 0.2879914 0.3121972
#> 2 no_detect_set        a  south 0.2918725 0.062035936 0.1707131 0.1841581
#> 3 no_detect_set        a   east 0.6031702 0.044184854 0.5206358 0.5262034
#> 4 no_detect_set        b   nord 0.9746355 0.016235524 0.9080894 0.9353931
#> 5 no_detect_set        b  south 0.9593916 0.008324214 0.9435923 0.9445593
#> 6 no_detect_set        b   east 0.9662035 0.021220418 0.9176483 0.9211130
#>         25%       50%       75%     97.5%       Max  nsv Na's
#> 1 0.3379731 0.3553841 0.3732035 0.3966264 0.4037282 1001    0
#> 2 0.2363502 0.2957188 0.3458267 0.3884192 0.3984394 1001    0
#> 3 0.5662145 0.6055760 0.6393578 0.6753059 0.6787207 1001    0
#> 4 0.9653477 0.9783262 0.9873528 0.9947945 0.9963197 1001    0
#> 5 0.9523539 0.9602221 0.9666917 0.9714390 0.9717183 1001    0
#> 6 0.9508097 0.9717442 0.9848915 0.9915584 0.9921264 1001    0

It also provides the probability of at least one and the expected number of infected animals by subset (in this case a farm)

# Probability of at least one in a farm
imports$node_list$no_detect_subset$summary
#>            mc_name pathogen origin       mean          sd        Min       2.5%
#> 1 no_detect_subset        a   nord 0.08418689 0.006531174 0.06567697 0.07211792
#> 2 no_detect_subset        a  south 0.03425691 0.008450613 0.01854480 0.02014774
#> 3 no_detect_subset        a   east 0.12435837 0.013995100 0.09971337 0.10121463
#> 4 no_detect_subset        b   nord 0.53616683 0.061836633 0.37959809 0.42182969
#> 5 no_detect_subset        b  south 0.27549957 0.014904524 0.24987461 0.25117068
#> 6 no_detect_subset        b   east 0.39950577 0.057881309 0.30000310 0.30428818
#>         25%        50%       75%      97.5%        Max  nsv Na's
#> 1 0.0791792 0.08407440 0.0891952 0.09610651 0.09824438 1001    0
#> 2 0.0266043 0.03445036 0.0415504 0.04798151 0.04955292 1001    0
#> 3 0.1124718 0.12445096 0.1355792 0.14844909 0.14973428 1001    0
#> 4 0.4895580 0.53528475 0.5827474 0.65062396 0.67403048 1001    0
#> 5 0.2624308 0.27562373 0.2883684 0.29922709 0.29991533 1001    0
#> 6 0.3496821 0.39920048 0.4505998 0.49443719 0.49944352 1001    0

Multiple group multilevel trials

This trial_totals() application is beyond the scope of this vignette, but there are cases where you might have several variates from the same subset. For example we could deal with different animal categories (cow, calf, bull…) from the same farm. In this case, the infection probability of animals within the same farm is not independent, and this should be taken into account. For more information, see Multilevel trials in the Multivariate operations vignette.

Selection (in blue) of 2 cows and 5 calves from 3 farms, with a 20% regional herd prevalence and 50% within-herd prevalence for adult animals and 30% within-herd prevalence for young animals.

Aggregated totals

Until this point, all mcnode operations were element-wise, keeping the original dimensions, and allowing uncertainties and variates to propagate through the calculations. However, sometimes, we need to aggregate variates to calculate totals, for example, to total risk of introducing a pathogen across all regions. In this case, we want to preserve the uncertainty dimension but reduce the variates dimension. With agg_totals() we can calculate overall probabilities or sum quantities across groups.

imports <- agg_totals(
  mcmodule = imports,
  mc_name = "no_detect_set",
  agg_keys = "pathogen"
)
#> 3 variates per group for no_detect_set

# Result
imports$node_list$no_detect_set_agg$summary
#>             mc_name pathogen      mean           sd       Min      2.5%
#> 1 no_detect_set_agg        a 0.8188637 2.656678e-02 0.7330214 0.7635315
#> 4 no_detect_set_agg        b 0.9999646 3.709019e-05 0.9996568 0.9998676
#>         25%       50%       75%     97.5%       Max  nsv Na's
#> 1 0.8005996 0.8199115 0.8383779 0.8657543 0.8790562 1001    0
#> 4 0.9999557 0.9999769 0.9999892 0.9999968 0.9999988 1001    0

Now we can visualise our mcmodule again and see all these new nodes created by the totals functions.

mc_network(imports, legend = TRUE)

mcmodule

Modular Monte-Carlo Risk Analysis