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R4PDE.tex
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\title{R4PDE}
\author{Emerson M. Del Ponte}
\date{}
\begin{document}
\frontmatter
\maketitle
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\mainmatter
\bookmarksetup{startatroot}
\hypertarget{welcome}{%
\chapter*{Welcome}\label{welcome}}
\addcontentsline{toc}{chapter}{Welcome}
\markboth{Welcome}{Welcome}
\textbf{R for Plant Disease Epidemiology} (R4PDE) is a dynamic book
project rooted in the teachings of the annual graduate course, FIP 602 -
Plant Disease Epidemiology, a key part of the curriculum in the
\href{http://www.dfp.ufv.br/grad/}{Graduate Program in Plant Pathology}
at Universidade Federal de Viçosa.
Designed for those passionate about studying and modeling plant disease
epidemics with \href{https://www.r-project.org/}{R}, the book offers an
exploration of diverse methods for describing, visualizing, and
analyzing epidemic data collected over time or space. Readers should
ideally have a foundational knowledge of R to best utilize the examples.
However, R4PDE is not a resource for learning data science through R, as
there are already well-established books such as
\href{https://r4ds.had.co.nz/}{R for data science} for that purpose.
Portuguese-speaking readers are recommended
\href{https://analises-ecologicas.netlify.app/}{Análises Ecológicas no
R} and \href{https://tiagoolivoto.github.io/e-bookr/index.html}{Software
R para avaliação de dados experimentais} as excellent R learning
resources, with an added focus on statistics using R.
The book often draws upon data and replicates analyses from \emph{The
Study of Plant Disease Epidemics} (Madden et al. 2007d). A mix of
general and specific R packages are utilized to conduct common plant
disease epidemiology data analysis, notably
\href{https://alvesks.github.io/epifitter/}{\{epifitter\}} and
\href{https://chgigot.github.io/epiphy/}{\{epiphy\}}, both designed by
plant pathologists. In conjunction with this book, a new R package
\href{https://github.com/emdelponte/r4pde}{\{r4pde\}} has been developed
and can be installed from GitHub using:
\begin{verbatim}
#install.packages("remotes")
remotes::install_github("emdelponte/r4pde")
\end{verbatim}
As a work in progress, the book is frequently updated and edited. The
website is free to use, licensed under a
\href{https://creativecommons.org/licenses/by-nc/4.0/}{Creative Commons
licence}, and the code for all analyses can be found on
\href{https://github.com/emdelponte/epidemiology-R}{GitHub}. While there
are no immediate plans for a printed version, it is under consideration
as the book further develops. The website is hosted by
https://www.netlify.com/.
Contributions to R4PDE are subject to a
\href{https://contributor-covenant.org/version/2/0/CODE_OF_CONDUCT.html}{Contributor
Code of Conduct}, and by contributing, you agree to adhere to its terms.
\hypertarget{acknowledgements}{%
\section*{Acknowledgements}\label{acknowledgements}}
\addcontentsline{toc}{section}{Acknowledgements}
\markright{Acknowledgements}
To \href{https://www.hgpenningtonart.co.uk/}{Helen Pennington} for
allowing me to use her painting of coffee leaf rust as book cover. Also
to those who have contributed fixes and improvements via pull request or
other form of contact: Adam Sparks
(\href{https://github.com/adamhsparks}{@adamhsparks}), Remco Stam
(\href{https://github.com/remco-stam}{@remco-stam}), Tiago Olivoto
(\href{https://github.com/TiagoOlivoto}{@TiagoOlivoto}), and Monalisa De
Cól (\href{https://github.com/Monalisacdc}{@Monalisacdc})
\bookmarksetup{startatroot}
\hypertarget{about-the-author}{%
\chapter*{About the author}\label{about-the-author}}
\addcontentsline{toc}{chapter}{About the author}
\markboth{About the author}{About the author}
\href{https://emersondelponte.netlify.app/}{Emerson M. Del Ponte} is a
Professor at the
\href{http://www.dfp.ufv.br/graduate/faculty/}{Departamento de
Fitopatologia}, Universidade Federal de Viçosa in Brazil. His academic
journey includes a DSc in Plant Pathology, obtained from Universidade
Federal de Pelotas in 2004, and a year-long visit to Cornell University
in the Bergstrom Lab. Following this, he spent nearly two years as a
postdoctoral associate working on a project related to disease risk
assessment and prediction at the Yang Lab, Iowa State University. This
experience led him to the Universidade Federal do Rio Grande do Sul,
Brazil, where he joined as an assistant professor of plant pathology.
Emerson Del Ponte is a fervent advocate for an open and reproducible
research model and culture, which he believes can lead to more
accessible, transparent, and reliable scientific knowledge. This belief
inspired him co-founding the
\href{http://www.openplantpathology.org/}{Open Plant Pathology}
initiative alongside \href{http://adamhsparks.netlify.app/}{Adam
Sparks}. In his Lab, students use the R language for all statistics and
data science-related activities. All data and computational codes
generated during the research are made accessible before the peer-review
process. The code can be located on
\href{https://github.com/emdelponte}{GitHub}.
\bookmarksetup{startatroot}
\hypertarget{introduction}{%
\chapter{Introduction}\label{introduction}}
\hypertarget{defining-plant-disease}{%
\section{Defining plant disease}\label{defining-plant-disease}}
Disease in plants can be defined as \emph{any malfunctioning of host
cells and tissues that results from continuous irritation by a
pathogenic agent or environmental factor and leads to development of
symptoms} (Agrios 2005a). When caused by pathogenic agent, the disease
results from the combination of three elements: susceptible host plant,
a virulent pathogen, and favorable environmental conditions - the famous
disease triangle. When a pathogen population establishes and causes
\emph{disease} in a host population, the phenomenon is called an
\emph{epidemic}, or the disease in populations. Among several
definitions of epidemic, a comprehensive one is the \emph{change in
disease intensity in a host population over time and space} (Madden et
al. 2007d).
There exist numerous iterations of the disease triangle, incorporating
additional elements (e.g., human intervention and time) as points and/or
dimensions to provide a more comprehensive representation of an epidemic
(Agrios 2005b). We find the disease prism particularly illustrative,
where a sequence of stacked triangles represent the evolution of a plant
disease through time (Francl 2001).
\begin{figure}
{\centering \includegraphics[width=3.92708in,height=\textheight]{imgs/prism.png}
}
\caption{\label{fig-prism}The plant disease prism as a model of plant
disease epidemics}
\end{figure}
\hypertarget{importance-of-epidemics}{%
\section{Importance of epidemics}\label{importance-of-epidemics}}
Epidemics bear significant economic importance due to their potential to
decrease crop yields, diminish product quality, and escalate control
costs, contingent on their intensity level. Numerous historical examples
of widespread epidemics, reaching pandemic levels and resulting in
catastrophic effects on crops, have been documented (Agrios 2005b). The
Irish potato famine of 1845--1847, caused by the late blight pathogen
(\emph{Phytophthora infestans}), is a famous example of a
well-documented pandemic. This disease notably altered the course of
history in Europe and the United States, and was pivotal in the
evolution of the science of plant pathology. During the 1840s, the
pathogen ravaged potato crops, which were a dietary staple for the
Irish. The disease outbreak was triggered by the introduction of a
novel, virulent pathogen population that found suitable environmental
conditions (cool and wet weather) for infection and development within a
dense population of susceptible hosts.
However, there are several reasons why devastating epidemics may
continue to unfold. Recent history has seen severe epidemics reaching
pandemic levels due to the incursion of pathogens into regions where
they had previously been absent (refer to Box 1). Alternatively, new
pathogenic strains might emerge as a result of factors driving genetic
diversity within the local pathogen population. A case in point is the
Ug99 strain of the wheat stem rust, which poses a significant threat to
global wheat production. First identified in Uganda in 1998, an asexual
lineage has propagated through Africa and the Middle East, causing
catastrophic epidemics. Research suggests that Ug99 emerged via somatic
hybridization and nuclear exchange between isolates from different
lineages (Li et al. 2019). Finally, disease emergence or re-emergence
can be influenced by shifts in climatic patterns. For instance, the
Fusarium head blight of wheat caused by the fungus \emph{Fusarium
graminearum}. In Southern Brazil, the increased frequency of severe
epidemics resulting in greater yield loss since the early 1990s has been
linked to alterations in rainfall patterns across decades (Duffeck et
al. 2020).
\begin{tcolorbox}[enhanced jigsaw, titlerule=0mm, rightrule=.15mm, colbacktitle=quarto-callout-note-color!10!white, opacitybacktitle=0.6, toptitle=1mm, leftrule=.75mm, colback=white, colframe=quarto-callout-note-color-frame, bottomrule=.15mm, toprule=.15mm, breakable, bottomtitle=1mm, coltitle=black, title=\textcolor{quarto-callout-note-color}{\faInfo}\hspace{0.5em}{Box 1: Diseases on the move}, arc=.35mm, opacityback=0, left=2mm]
In Brazil, the soybean rust pathogen (\emph{Phakopsora pachyrhizi})
first reached southern Brazil in 2002 (Yorinori et al. 2005). The
disease spread to all production regions of the country in the following
few years, severely reducing yields. To overcome the problem, farmers
have relied on massive applications of fungicides on soybeans, which
dramatically increased the production costs with the need for sequential
fungicide sprays to combat the disease. Total economic loss have been
estimated at around US\$ 2 billion yearly (Godoy et al. 2016). More
recently, wheat blast, a disease that originated in the south of Brazil
in 1984, and have been restricted to South America, was firstly spotted
in South Asia, Bangladesh, in 2016. Blast epidemics in that occasion
devastated more than 15,000 ha of wheat and reduced yield of wheat in
the affected field up to 100\% (Malaker et al. 2016; Islam et al. 2019).
The disease was later found in Zambia, thus also becoming a threat to
wheat production in Africa (Tembo et al. 2020). In Brazil, the wheat
blast disease is a current threat to expansion of wheat cultivation in
the tropics(Cruz and Valent 2017).
\end{tcolorbox}
\hypertarget{history-of-epidemiology}{%
\section{History of Epidemiology}\label{history-of-epidemiology}}
Botanical epidemiology, or the study of plant disease epidemics, is a
discipline with roots tracing back to the early 1960s. However, its
origins can be linked to events from centuries and decades prior. For
instance, in 1728, Duhamel de Monceau presented the earliest known
epidemiological work on a disease, referred to as `Death,' that
afflicted saffron crocus (\emph{Rhizoctonia violacea}). Fast forward to
1858, a textbook detailing plant diseases, written by Julius Kuhn, made
its debut, introducing the concept of an epidemic as illustrated by the
Irish late blight epidemics of 1845-46. Subsequently, in 1901, H.M. Ward
adopted an ecological perspective to the study of plant diseases in his
seminal book, Disease in Plants. By 1946, Gäumann penned the first book
exclusively devoted to plant disease epidemiology.
Further evolution of this field was marked by the publication of a
chapter titled ``Analysis of Epidemics'' by J.E. Vanderplank in Plant
Pathology, vol.~3, edited by Horsfall and Dimond, in 1960. Vanderplank
elaborated on his pioneering ideas in his 1963 book, ``Plant Diseases:
Epidemics and Control''(Vanderplank 1963). He is universally recognized
as the foundational figure of plant disease epidemiology (Zadoks and
Schein 1988; Thresh 1998), his landmark book being the first to
comprehensively describe and quantify plant disease epidemics, and
offering a theoretical framework for epidemic analysis.
In the same year, the first International Epidemiology Workshop was
convened in Pau, France. This event constitutes an important milestone
in the historical narrative, significantly contributing to the molding
of this emergent discipline.
\begin{figure}
{\centering \includegraphics[width=5.38542in,height=\textheight]{imgs/iew1.png}
}
\caption{\label{fig-iew1}Group photo of the First International
Epidemiology Workshop}
\end{figure}
The \textbf{International Epidemiology Workshop (IEW)} is the principal
working group of plant disease epidemiology. This is an organization
with a rich history whose members have met approximately every 5 years
since 1963. Thus far, \href{https://iew13.net/about.html}{13 meetings}
have been organized/planned:
1963 - Pau, France\\
1971 - Wageningen, The Netherlands\\
1979 - Penn State, United States\\
1983 - NC State, Raleigh, United States\\
1986 - Jerusalem, Israel\\
1990 - Giessen, Germany\\
1994 - Papendal, The Netherlands\\
2001 - Ouro Preto, Brazil\\
2005 - Landerneau, France\\
2009 - Cornell, Geneva, United States\\
2013 - Beijing, China\\
2018 - Lillehammer, Norway\\
2024 - Iguassu Falls, Brazil
\hypertarget{other-resources}{%
\section{Other resources}\label{other-resources}}
\hypertarget{books}{%
\subsection{Books}\label{books}}
2006 - \href{https://link.springer.com/book/10.1007/1-4020-4581-6}{The
Epidemiology of Plant Diseases}\\
2007 -
\href{https://apsjournals.apsnet.org/doi/book/10.1094/9780890545058}{The
Study of Plant Disease Epidemics}\\
2017 -
\href{https://apsjournals.apsnet.org/doi/book/10.1094/9780890544426}{Exercises
in Plant Disease Epidemiology}\\
2017 -
\href{https://apsjournals.apsnet.org/doi/book/10.1094/9780890544877}{Application
of Information Theory to Epidemiology}\\
2020 -
\href{https://apsjournals.apsnet.org/doi/book/10.1094/9780890546383}{Emerging
Plant Diseases and Global Security}\\
\hypertarget{online-tutorials}{%
\subsection{Online tutorials}\label{online-tutorials}}
\href{https://www.apsnet.org/edcenter/disimpactmngmnt/topc/EcologyAndEpidemiologyInR/Pages/default.aspx}{Ecology
and Epidemiology in R}\\
\href{https://www.apsnet.org/edcenter/disimpactmngmnt/topc/EpidemiologyTemporal/Pages/default.aspx}{Plant
Disease Epidemiology - Temporal aspects}\\
\href{https://www.apsnet.org/edcenter/disimpactmngmnt/topc/BotanicalEpidemiology/Pages/default.aspx}{Simulation
Modeling in Plant Disease Epidemiology and Crop Loss Analysis}
\hypertarget{software}{%
\subsection{Software}\label{software}}
\href{http://adamhsparks.github.io/epicrop/}{Epicrop - Simulation
Modeling of Crop Diseases using a SEIR model}
\part{Epidemic data}
\hypertarget{disease-variables}{%
\chapter{Disease variables}\label{disease-variables}}
\hypertarget{disease-quantification}{%
\section{Disease quantification}\label{disease-quantification}}
Studies on the temporal progression or spatial spread of epidemics
cannot be conducted without field-collected data, or, in some cases,
simulated data. The study of plant disease quantification, termed
Phytopathometry, is a subdivision of plant pathology concerned with the
science of disease measurement. It has strong ties to the field of
epidemiology (Bock et al. 2021b).
Traditionally, disease quantification has been executed through visual
evaluation. However, the past few decades have witnessed significant
advancements in imaging and remote sensing technologies (which don't
necessitate contact with the object), leaving a profound impact on this
field. As such, disease quantity can now be gauged through estimation
(visually, by the human eye) or measurement (through remote sensing
technologies such as RGB, MSI, and HSI)
Figure~\ref{fig-disease_measure}.
While the utilization of digital or remote sensing technology for
disease measurement or estimation provides a more objective approach,
visual assessment is largely subjective. It is known to vary among human
raters, as these raters differ in their innate abilities, training, and
how they are influenced by the chosen method (e.g., scales). Disease is
estimated or measured on a specimen within a population, or on a sample
of specimens drawn from that population. The specimen in question can be
a plant organ, an individual plant, a group of plants, a field, or a
farm. The specific specimen type also determines the terminology used to
describe disease quantity.
\begin{figure}
{\centering \includegraphics[width=5.10417in,height=\textheight]{imgs/disease_measure.png}
}
\caption{\label{fig-disease_measure}Different approaches used to obtain
estimates or measures of plant disease. RGB = red, green, blue; MSI =
multispectral imaging; HSI = hyperspectral imaging.}
\end{figure}
inally, while developing new or refining existing disease assessment
methods, it is crucial to evaluate the reliability of the assessments
made by different raters or instruments, as well as their
accuracy---specifically, how close the estimations or measurements are
to the reference (or gold standard) values. Several methods are
available for assessing the reliability, precision, and accuracy of
these estimates or measurements (see
\href{data-accuracy.html}{definitions}). The choice of methods depends
on the objective of the work, but largely on the type or nature of the
data. These considerations will be further discussed.
\hypertarget{disease-variables-1}{%
\section{Disease variables}\label{disease-variables-1}}
A common term used to reference the quantity of disease, irrespective of
how it is expressed, is `disease intensity'. This term, however, has
minimal practical value as it only implies that the disease is more or
less ``intense''. We require more specific terminology to standardize
the reference to disease quantity and methodology. One of the primary
tasks in disease assessment is classifying each specimen, often in a
sample or within a population, as diseased or not diseased. This binary
(yes/no or 1/0) evaluation may sufficiently express disease intensity if
the goal is to ascertain the number or proportion of diseased specimens
in a sample or a population.
This discussion brings us to two terms: disease incidence and
prevalence. Incidence is typically used to denote the proportion or
number (count) of plants (or their organs) deemed as observational units
at the field scale or below. On the other hand, prevalence refers to the
proportion or number of fields or farms with diseased plants within a
larger production area or region (Nutter et al. 2006)
Figure~\ref{fig-prevalence_incidence}. Therefore, prevalence is
analogous to incidence, with the only difference being the spatial scale
of the sampling unit.
\begin{figure}
{\centering \includegraphics[width=5in,height=\textheight]{imgs/prevalence_incidence.png}
}
\caption{\label{fig-prevalence_incidence}Schematic representation of how
prevalence and incidence of plant diseases are calculated depending on
the spatial scale of the assessment}
\end{figure}
In many instances, it's necessary to determine the degree to which a
specimen is diseased, a concept defined as disease severity. In certain
contexts, severity is narrowly defined as the proportion of the unit
that exhibits symptoms (Nutter et al. 2006). However, a more expansive
view of severity includes additional metrics such as nominal or ordinal
scores, lesion count, and percent area affected (ratio scale). Ordinal
scales are broken down into rank-ordered classes (see specific
\href{data-ordinal.html}{section}), defined based on either a percentage
scale or descriptions of symptoms (Bock et al. 2021b). Occasionally,
disease is expressed in terms of (average) lesion size or area, which
could be regarded as a measure of severity. These variables represent
different levels of measurements that provide varying degrees of
information about the disease quantity - from low (nominal scale) to
high (ratio scale) Figure~\ref{fig-severity}.
\begin{figure}
{\centering \includegraphics[width=5.32292in,height=\textheight]{imgs/severity.png}
}
\caption{\label{fig-severity}Scales and associated levels of measurement
used to describe severity of plant diseases}
\end{figure}
\hypertarget{data-types}{%
\section{Data types}\label{data-types}}
The data used to express disease as incidence or any form of severity
measurements can be discrete or continuous in nature.
Discrete variables are countable (involving integers) at a particular
point in time. In other words, only a finite number of values (nominal
or ordinal) is possible, and these cannot be subdivided. For instance, a
plant or plant part can be either diseased or not diseased (nominal
data). It's not possible to count 1.5 diseased plants. Furthermore, a
plant classified as diseased may exhibit a certain number of lesions
(count data), or be categorized into a specific severity class (ordinal
data, common in ordinal scales, e.g., 1-9). Disease data in the form of
counts often relates to the number of infections per sampling units.
Most commonly, these counts refer to the assessed pathogen population,
such as the number of airborne or soilborne propagules.
In contrast to discrete variables, continuous variables can be measured
on a scale and can assume any numeric value between two points. For
example, the size of a lesion on a plant can be measured at a very
precise scale (cm or mm). An estimate of severity on a percent scale (\%
diseased area) can take any value between non-zero and 100\%. Although
incidence at the individual level is discrete, at the sample level it
can be treated as continuous, as it can assume any value in proportion
or percentage.
Disease variables can also be characterized by a statistical
distribution, which are models that provide the probability of a
specific value (or a range of values) being drawn from a particular
distribution. Understanding statistical or mathematical distributions is
a crucial step in improving our grasp of data collection methods,
experiment design, and data analysis processes such as data
summarization or hypothesis testing.
\hypertarget{statistical-distributions-and-simulation}{%
\section{Statistical distributions and
simulation}\label{statistical-distributions-and-simulation}}
\hypertarget{binomial-distribution}{%
\subsection{Binomial distribution}\label{binomial-distribution}}
For incidence (and prevalence), the data is binary at the individual
level, as there are only two possible outcomes in a \emph{trial}: the
plant or plant part is disease or not diseased. The statistical
distribution that best describe the incidence data at the individual
level is the \emph{binomial distribution}.
Let's simulate the binomial outcomes for a range of probabilities in a
sample of 100 units, using the \texttt{rbinom()} function in R. For a
single trial (e.g., status of plants in a single plant row), the
\texttt{size} argument is set to 1.
\begin{Shaded}
\begin{Highlighting}[]
\FunctionTok{library}\NormalTok{(tidyverse)}
\FunctionTok{library}\NormalTok{(r4pde)}
\FunctionTok{set.seed}\NormalTok{(}\DecValTok{123}\NormalTok{) }\CommentTok{\# for reproducibility}
\NormalTok{P}\FloatTok{.1} \OtherTok{\textless{}{-}} \FunctionTok{rbinom}\NormalTok{(}\DecValTok{100}\NormalTok{, }\AttributeTok{size =} \DecValTok{1}\NormalTok{, }\AttributeTok{prob =} \FloatTok{0.1}\NormalTok{)}
\NormalTok{P}\FloatTok{.3} \OtherTok{\textless{}{-}} \FunctionTok{rbinom}\NormalTok{(}\DecValTok{100}\NormalTok{, }\AttributeTok{size =} \DecValTok{1}\NormalTok{, }\AttributeTok{prob =} \FloatTok{0.3}\NormalTok{)}
\NormalTok{P}\FloatTok{.7} \OtherTok{\textless{}{-}} \FunctionTok{rbinom}\NormalTok{(}\DecValTok{100}\NormalTok{, }\AttributeTok{size =} \DecValTok{1}\NormalTok{, }\AttributeTok{prob =} \FloatTok{0.7}\NormalTok{)}
\NormalTok{P}\FloatTok{.9} \OtherTok{\textless{}{-}} \FunctionTok{rbinom}\NormalTok{(}\DecValTok{100}\NormalTok{, }\AttributeTok{size =} \DecValTok{1}\NormalTok{, }\AttributeTok{prob =} \FloatTok{0.9}\NormalTok{)}
\NormalTok{binomial\_data }\OtherTok{\textless{}{-}} \FunctionTok{data.frame}\NormalTok{(P}\FloatTok{.1}\NormalTok{, P}\FloatTok{.3}\NormalTok{, P}\FloatTok{.7}\NormalTok{, P}\FloatTok{.9}\NormalTok{)}
\end{Highlighting}
\end{Shaded}
We can then visualize the plots.
\begin{Shaded}
\begin{Highlighting}[]
\NormalTok{binomial\_data }\SpecialCharTok{|\textgreater{}}
\FunctionTok{pivot\_longer}\NormalTok{(}\DecValTok{1}\SpecialCharTok{:}\DecValTok{4}\NormalTok{, }\AttributeTok{names\_to =} \StringTok{"P"}\NormalTok{,}
\AttributeTok{values\_to =} \StringTok{"value"}\NormalTok{) }\SpecialCharTok{|\textgreater{}}
\FunctionTok{ggplot}\NormalTok{(}\FunctionTok{aes}\NormalTok{(value)) }\SpecialCharTok{+}
\FunctionTok{geom\_histogram}\NormalTok{(}\AttributeTok{fill =} \StringTok{"\#339966"}\NormalTok{,}
\AttributeTok{bins =} \DecValTok{10}\NormalTok{) }\SpecialCharTok{+}
\FunctionTok{facet\_wrap}\NormalTok{( }\SpecialCharTok{\textasciitilde{}}\NormalTok{ P) }\SpecialCharTok{+}
\FunctionTok{theme\_r4pde}\NormalTok{()}
\end{Highlighting}
\end{Shaded}
\begin{figure}
\includegraphics{data-terminology_files/figure-latex/fig-binomial-1.pdf} \hfill{}
\caption{\label{fig-binomial}Binomial distribution to describe binary
data}
\end{figure}
\hypertarget{beta-distribution}{%
\subsection{Beta distribution}\label{beta-distribution}}
In many studies, it's often useful to express these quantities as a
proportion of the total population or sample size, rather than absolute
numbers. This helps standardize the data, making it easier to compare
between different populations or different time periods.
For example, if we're studying a plant disease, we could express disease
incidence as the proportion of plants that are newly diseased during a
given time period. Similarly, disease severity could be expressed as the
proportion of each plant's organ area that is affected by the disease.
These proportions are ratio variables, as they can take on any value
between 0 and 1, and ratios of these variables are meaningful.
The Beta distribution is a probability distribution that is defined
between 0 and 1, which makes it ideal for modeling data that represents
proportions. It's a flexible distribution, as its shape can take many
forms depending on the values of its two parameters, often denoted as
alpha and beta.
Let's simulate some data using the \texttt{rbeta()} function.
\begin{Shaded}
\begin{Highlighting}[]
\NormalTok{beta1}\FloatTok{.5} \OtherTok{\textless{}{-}} \FunctionTok{rbeta}\NormalTok{(}\AttributeTok{n =} \DecValTok{1000}\NormalTok{, }\AttributeTok{shape1 =} \DecValTok{1}\NormalTok{, }\AttributeTok{shape2 =} \DecValTok{5}\NormalTok{)}
\NormalTok{beta5}\FloatTok{.5} \OtherTok{\textless{}{-}} \FunctionTok{rbeta}\NormalTok{(}\AttributeTok{n =} \DecValTok{1000}\NormalTok{, }\AttributeTok{shape1 =} \DecValTok{5}\NormalTok{, }\AttributeTok{shape2 =} \DecValTok{5}\NormalTok{)}
\NormalTok{beta\_data }\OtherTok{\textless{}{-}} \FunctionTok{data.frame}\NormalTok{(beta1}\FloatTok{.5}\NormalTok{, beta5}\FloatTok{.5}\NormalTok{)}
\end{Highlighting}
\end{Shaded}
Notice that there are two shape parameters in the beta distribution:
\texttt{shape1} and \texttt{shape2} to be defined. This makes the
distribution very flexible and with different potential shapes as we can
see below.
\begin{Shaded}
\begin{Highlighting}[]
\NormalTok{beta\_data }\SpecialCharTok{|\textgreater{}}
\FunctionTok{pivot\_longer}\NormalTok{(}\DecValTok{1}\SpecialCharTok{:}\DecValTok{2}\NormalTok{, }\AttributeTok{names\_to =} \StringTok{"P"}\NormalTok{,}
\AttributeTok{values\_to =} \StringTok{"value"}\NormalTok{) }\SpecialCharTok{|\textgreater{}}
\FunctionTok{ggplot}\NormalTok{(}\FunctionTok{aes}\NormalTok{(value)) }\SpecialCharTok{+}
\FunctionTok{geom\_histogram}\NormalTok{(}\AttributeTok{fill =} \StringTok{"\#339966"}\NormalTok{,}
\AttributeTok{color =} \StringTok{"white"}\NormalTok{,}
\AttributeTok{bins =} \DecValTok{15}\NormalTok{) }\SpecialCharTok{+}
\FunctionTok{scale\_x\_continuous}\NormalTok{(}\AttributeTok{limits =} \FunctionTok{c}\NormalTok{(}\DecValTok{0}\NormalTok{, }\DecValTok{1}\NormalTok{)) }\SpecialCharTok{+}
\FunctionTok{facet\_wrap}\NormalTok{( }\SpecialCharTok{\textasciitilde{}}\NormalTok{ P)}\SpecialCharTok{+}
\FunctionTok{theme\_r4pde}\NormalTok{()}
\end{Highlighting}
\end{Shaded}
\begin{figure}
\includegraphics{data-terminology_files/figure-latex/fig-betabin-1.pdf} \hfill{}
\caption{\label{fig-betabin}Binomial distribution to describe proportion
data}
\end{figure}
\hypertarget{beta-binomial-distribution}{%
\subsection{Beta-binomial
distribution}\label{beta-binomial-distribution}}
The Beta-Binomial distribution is a mixture of the Binomial distribution
with the Beta distribution acting as a prior on the probability
parameter of the binomial. Disease probabilities can vary across trials
due to a number of unobserved or unmeasured factors. This variability
can result in overdispersion, a phenomenon where the observed variance
in the data is greater than what the binomial distribution expects.
This is where the Beta-Binomial distribution comes in handy. By
combining the Beta distribution's flexibility in modeling probabilities
with the Binomial distribution's discrete event modeling, it provides an
extra layer of variability to account for overdispersion. The
Beta-Binomial distribution treats the probability of success (disease
occurrence in this context) as a random variable itself, following a
Beta distribution. This means the probability can vary from trial to
trial.
Therefore, when we observe data that shows more variance than the Beta
distribution can account for, or when we believe there are underlying
factors causing variability in the probability of disease occurrence,
the Beta-Binomial distribution is a more appropriate model. It captures
both the variability in success probability as well as the occurrence of
the discrete event (disease incidence).
When combined with the Binomial distribution, which handles discrete
events (e.g.~whether an individual is diseased or not), the
Beta-Binomial distribution allows us to make probabilistic predictions
about these events. For example, based on prior data (the Beta
distribution), we can estimate the likelihood of a particular individual
being diseased (the Binomial distribution).
In R, the \texttt{rBetaBin} function of the \emph{FlexReg} package
generates random values from the beta-binomial distribution. The
arguments of the function are \texttt{n}, or the number of values to
generate; if length(\texttt{n}) \textgreater{} 1, the length is taken to
be the number required. \texttt{size} is he total number of trials.
\texttt{mu} is the mean parameter. It must lie in (0, 1). \texttt{theta}
is the overdispersion parameter. It must lie in (0, 1). \texttt{phi} the
precision parameter. It is an alternative way to specify the theta
parameter. It must be a positive real value.
\begin{Shaded}
\begin{Highlighting}[]
\FunctionTok{library}\NormalTok{(FlexReg) }
\NormalTok{betabin3}\FloatTok{.6} \OtherTok{\textless{}{-}} \FunctionTok{rBetaBin}\NormalTok{(}\AttributeTok{n =} \DecValTok{100}\NormalTok{, }\AttributeTok{size =} \DecValTok{40}\NormalTok{, }\AttributeTok{mu =}\NormalTok{ .}\DecValTok{3}\NormalTok{, }\AttributeTok{theta =}\NormalTok{ .}\DecValTok{6}\NormalTok{)}
\NormalTok{betabin7}\FloatTok{.3} \OtherTok{\textless{}{-}} \FunctionTok{rBetaBin}\NormalTok{(}\AttributeTok{n =} \DecValTok{100}\NormalTok{, }\AttributeTok{size =} \DecValTok{40}\NormalTok{, }\AttributeTok{mu =}\NormalTok{ .}\DecValTok{7}\NormalTok{, }\AttributeTok{theta =}\NormalTok{ .}\DecValTok{3}\NormalTok{)}
\NormalTok{betabin\_data }\OtherTok{\textless{}{-}} \FunctionTok{data.frame}\NormalTok{(betabin3}\FloatTok{.6}\NormalTok{, betabin7}\FloatTok{.3}\NormalTok{)}
\end{Highlighting}
\end{Shaded}
\begin{Shaded}
\begin{Highlighting}[]
\NormalTok{betabin\_data }\SpecialCharTok{|\textgreater{}}
\FunctionTok{pivot\_longer}\NormalTok{(}\DecValTok{1}\SpecialCharTok{:}\DecValTok{2}\NormalTok{, }\AttributeTok{names\_to =} \StringTok{"P"}\NormalTok{,}
\AttributeTok{values\_to =} \StringTok{"value"}\NormalTok{) }\SpecialCharTok{|\textgreater{}}
\FunctionTok{ggplot}\NormalTok{(}\FunctionTok{aes}\NormalTok{(value)) }\SpecialCharTok{+}
\FunctionTok{geom\_histogram}\NormalTok{(}\AttributeTok{fill =} \StringTok{"\#339966"}\NormalTok{,}
\AttributeTok{color =} \StringTok{"white"}\NormalTok{,}
\AttributeTok{bins =} \DecValTok{15}\NormalTok{) }\SpecialCharTok{+}
\FunctionTok{facet\_wrap}\NormalTok{( }\SpecialCharTok{\textasciitilde{}}\NormalTok{ P) }\SpecialCharTok{+}
\FunctionTok{theme\_r4pde}\NormalTok{()}
\end{Highlighting}
\end{Shaded}
\begin{figure}
\includegraphics{data-terminology_files/figure-latex/fig-beta-1.pdf} \hfill{}
\caption{\label{fig-beta}Beta-binomial distribution to describe
proportion data}
\end{figure}
\hypertarget{poisson-distribution}{%
\subsection{Poisson distribution}\label{poisson-distribution}}
When conducting studies in epidemiology, specifically plant diseases,
researchers often collect data on the number of diseased plants,
infected plant parts, or individual symptoms, such as lesions. These
variables are counted in whole numbers - 1, 2, 3, etc., making them
discrete variables. Discrete variables contrast with continuous
variables that can take any value within a defined range and can include
fractions or decimals. In addition to being discrete, these variables
are also non-negative, meaning they cannot take negative values. After
all, you can't have a negative number of diseased plants or lesions.
Given these characteristics, a suitable distribution to model such data
is the Poisson distribution. This distribution is particularly suitable
for counting the number of times an event occurs in a given time or
space.
In R, we can used the \texttt{rpois()} function to obtain 100 random
observations following a Poisson distribution. For such, we need to
inform the number of observation (n = 100) and \texttt{lambda}, the
vector of means.
\begin{Shaded}
\begin{Highlighting}[]
\NormalTok{poisson5 }\OtherTok{\textless{}{-}} \FunctionTok{rpois}\NormalTok{(}\DecValTok{100}\NormalTok{, }\AttributeTok{lambda =} \DecValTok{10}\NormalTok{)}
\NormalTok{poisson35 }\OtherTok{\textless{}{-}} \FunctionTok{rpois}\NormalTok{(}\DecValTok{100}\NormalTok{, }\AttributeTok{lambda =} \DecValTok{35}\NormalTok{)}
\NormalTok{poisson\_data }\OtherTok{\textless{}{-}} \FunctionTok{data.frame}\NormalTok{(poisson5, poisson35)}
\end{Highlighting}
\end{Shaded}
\begin{Shaded}
\begin{Highlighting}[]
\NormalTok{poisson\_data }\SpecialCharTok{|\textgreater{}}
\FunctionTok{pivot\_longer}\NormalTok{(}\DecValTok{1}\SpecialCharTok{:}\DecValTok{2}\NormalTok{, }\AttributeTok{names\_to =} \StringTok{"P"}\NormalTok{,}
\AttributeTok{values\_to =} \StringTok{"value"}\NormalTok{) }\SpecialCharTok{|\textgreater{}}
\FunctionTok{ggplot}\NormalTok{(}\FunctionTok{aes}\NormalTok{(value)) }\SpecialCharTok{+}
\FunctionTok{geom\_histogram}\NormalTok{(}\AttributeTok{fill =} \StringTok{"\#339966"}\NormalTok{,}
\AttributeTok{color =} \StringTok{"white"}\NormalTok{,}
\AttributeTok{bins =} \DecValTok{15}\NormalTok{) }\SpecialCharTok{+}
\FunctionTok{facet\_wrap}\NormalTok{( }\SpecialCharTok{\textasciitilde{}}\NormalTok{ P) }\SpecialCharTok{+}
\FunctionTok{theme\_r4pde}\NormalTok{()}
\end{Highlighting}
\end{Shaded}
\begin{figure}
\includegraphics{data-terminology_files/figure-latex/fig-poisson-1.pdf} \hfill{}
\caption{\label{fig-poisson}Poisson distribution to describe count data}
\end{figure}
\hypertarget{negative-binomial-distribution}{%
\subsection{Negative binomial
distribution}\label{negative-binomial-distribution}}
While the Poisson distribution is indeed suitable for modeling count
data, it assumes that the mean and variance of the data are equal.
However, in real-world scenarios, especially in epidemiology, it is
common to encounter overdispersed data - where the variance is greater
than the mean. This could occur, for instance, if there's greater
variability in disease incidence across different plant populations than
would be expected under the Poisson assumption.
In such cases, the Negative Binomial distribution is a better
alternative. The Negative Binomial distribution is a discrete
probability distribution that models the number of successes in a
sequence of independent and identically distributed Bernoulli trials
before a specified (non-random) number of failures occurs.
One of the key features of the Negative Binomial distribution is its
ability to handle overdispersion. Unlike the Poisson distribution, which
has one parameter (lambda, representing the mean and variance), the
Negative Binomial distribution has two parameters. One parameter is the
mean, but the other (often denoted as `size' or `shape') governs the
variance independently, allowing it to be larger than the mean if
necessary. Thus, it provides greater flexibility than the Poisson
distribution for modeling count data and can lead to more accurate
results when overdispersion is present.
In R, we can use the \texttt{rnbinom()} function to generate random
variates from a Negative Binomial distribution. This function requires
the number of observations (\texttt{n}), the target for the number of
successful trials (\texttt{size}), and the probability of each success
(\texttt{prob}).
Here's an example:
\begin{Shaded}