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Vis-MLM.tex
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pdftitle={Visualizing Multivariate Data and Models in R},
pdfauthor={Michael Friendly},
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\title{Visualizing Multivariate Data and Models in R}
\author{Michael Friendly}
\date{2024-01-08}
\begin{document}
\maketitle
% you may need to leave a few empty pages before the dedication page
%\cleardoublepage\newpage\thispagestyle{empty}\null
%\cleardoublepage\newpage\thispagestyle{empty}\null
%\cleardoublepage\newpage
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\begin{center}
To all the mistakes I've made (data, and otherwise) and those who tolerated my making them.
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\textbf{TODO}: Make this a more general introduction
This book is about graphical methods developed recently for multivariate
data, and their uses in understanding relationships when there are
several aspects to be considered together. Data visualization methods
for statistical analysis are well-developed for simple linear models
with a single outcome variable. However, with applied research in the
social and behavioral sciences, it is often the case that the phenomena
of interest (e.g., depression, job satisfaction, academic achievement,
childhood ADHD disorders, etc.) can be measured in several different
ways or related aspects.
For example, if academic achievement can be measured for adolescents by
reading, mathematics, science and history scores, how do predictors such
as parental encouragement, school environment and socioeconomic status
affect all these outcomes? In a similar way? In different ways? In such
cases, much more can be understood from a multivariate approach that
considers the correlations among the outcomes. Yet, sadly, researchers
typically examine the outcomes one by one which often only tells part of
the data story.
However, to do this it is useful to set the stage for multivariate
thinking, with a grand scheme for statistics and data visualization, a
parable, and an example of multivariate discovery.
\hypertarget{one-two-many}{%
\section*{ONE, TWO, MANY}\label{one-two-many}}
\addcontentsline{toc}{section}{ONE, TWO, MANY}
\markright{ONE, TWO, MANY}
There is an old and helpful idea I learned from John Hartigan in my
graduate days at Princeton:
\begin{quote}
In statistics and data visualization \emph{all} methods can be
classified by the number of dimensions contemplated, on a scale of
\textbf{ONE}, \textbf{TWO}, \textbf{MANY}.
\end{quote}
By this, he meant that, at a global level, all data, statistical
summaries, and graphical displays could be classified as:
\begin{itemize}
\tightlist
\item
\textbf{univariate}: a single variable, considered in isolation (age,
COVID cases, pizzas ordered). Univariate numerical summaries are
means, medians, measures of variablilty, and so forth. Univariate
displays include dot plots, boxplots, histograms and density
estimates.
\item
\textbf{bivariate}: two variables, considered jointly. Numerical
summaries include correlations, covariances and two-way tables of
frequencies or measures of association for categorical variables.
Bivariate displays include scatterplots and mosaic plots.
\item
\textbf{multivariate}: three or more variables, considered jointly.
Numerical summaries include correlation and covariance matrices,
consisting of all pairwise values, but also derived measures from the
analysis of these matrices (eigenvalues, eigenvectors). Graphical
displays of multivariate data can sometimes be shown in 3D, but often
involve multiple views of the data projected into 2D plots.
\end{itemize}
As a quasi-numerical scale, I refer to these as \textbf{1D}, \textbf{2D}
and \textbf{nD}. This admits the possibility of half-integer cases, such
as \textbf{1.5D}, where the main focus is on a single variable, but that
is classified by a simple factor (e.g., gender), or \textbf{2.5D} where
a 2D scatterplot can show other variables using color, shape or other
visual attributes His point in this classification was that once you've
reached three variables, all higher dimensions involve similar summaries
and data displays.
Univariate and bivariate methods and displays are well-known. This book
is about how these ideas can be extended to an \(n\)-dimensional world.
Three-dimensional data displays are now fairly easy to produce, even if
they are sometimes difficult to understand. But how can we even think
about four or more dimensions? The difficulty can be appreciated by
considering the tale of \emph{Flatland}.
\hypertarget{flatland}{%
\section*{Flatland}\label{flatland}}
\addcontentsline{toc}{section}{Flatland}
\markright{Flatland}
\begin{quote}
To comport oneself with perfect propriety in Polygonal society, one
ought to be a Polygon oneself. --- Edwin A. Abbott, \emph{Flatland}
\end{quote}
In 1884, an English schoolmaster, Edwin Abbott Abbott, shook the world
of Victorian culture with a slim volume, \emph{Flatland: A Romance of
Many Dimensions} (\protect\hyperlink{ref-Abbott:1884}{Abbott, 1884}). He
described a two-dimensional world, \emph{Flatland}, inhabited entirely
by geometric figures in the plane. His purpose was satirical, to poke
fun at the social and gender class system at the time: Women were mere
line segments, while men were represented as polygons with varying
numbers of sides--- a triangle was a working man, but acute isosceles
were soldiers or criminals of very small angle; gentlemen and
professionals had more sides. Abbot published this under the pseudonym,
``A Square'', suggesting his place in the hierarchy.
\begin{quote}
True, said the Sphere; it appears to you a Plane, because you are not
accustomed to light and shade and perspective; just as in Flatland a
Hexagon would appear a Straight Line to one who has not the Art of Sight
Recognition. But in reality it is a Solid, as you shall learn by the
sense of Feeling. --- Edwin A. Abbott, \emph{Flatland}
\end{quote}
But how did it feel to be a member of a flatland society? How could a
point (a newborn child?) understand a line (a woman)? How does a
Triangle ``see'' a Hexagon or even a infinitely-sided Circle? Abbott
introduces the very idea of different dimensions of existence through
dreams and visions:
\begin{itemize}
\item
A Square dreams of visiting a one-dimensional \emph{Lineland} where
men appear as lines, and women are merely ``illustrious points'', but
the inhabitants can only see the Square as lines.
\item
In a vision, the Square is visited by a Sphere, to illustrate what a
2D Flatlander could understand from a 3D sphere
(Figure~\ref{fig-flatland-spheres}) that passes through the plane he
inhabits. It is a large circle when seen at the moment of its'
greatest extent. As the Spehere rises, it becomes progressively
smaller, until it becomes a point, and then vanishes.
\end{itemize}
\begin{figure}
{\centering \includegraphics[width=0.9\textwidth,height=\textheight]{images/flatland-spheres.jpg}
}
\caption{\label{fig-flatland-spheres}A 2D Flatlander seeing a sphere as
it passes through Flatland. The line, labeled `My Eye' indicates what
the Flatlander would see. Source: Abbott
(\protect\hyperlink{ref-Abbott:1884}{1884})}
\end{figure}
Abbott goes on to state what could be considered as a demonstration (or
proof) by induction of the difficulties of seeing in 1, 2, 3 dimensions,
and how the idea motion over time (one more dimension) could allow
citizens of any 1D, 2D, 3D world to contemplate one more dimension.
\begin{quote}
In One Dimensions, did not a moving Point produce a Line with two
terminal points? In two Dimensions, did not a moving Line produce a
Square with four terminal points? In Three Dimensions, did not a moving
Square produce - did not the eyes of mine behold it - that blessed
being, a Cube, with eight terminal points? And in Four Dimensions, shall
not a moving Cube - alas, for Analogy, and alas for the Progress of
Truth if it be not so - shall not, I say the motion of a divine Cube
result in a still more divine organization with sixteen terminal points?
--- Edwin A. Abbott
\end{quote}
For Abbot, the way for a citizen of any world to imagine one more
dimension was to consider how a higher-dimensional object would change
over time.\footnote{In his famous TV series, \emph{Cosmos}, Carl Sagan
provides \href{https://youtu.be/UnURElCzGc0}{an intriguing video
presentation} Flatland and the 4th dimension. However, as far back as
1754 (\protect\hyperlink{ref-Cajori:1926}{Cajori, 1926}), the idea of
adding a fourth dimension appears in Jean le Rond d'Alembert's
``Dimensions'', and one realization of a four-dimensional object is a
\emph{tesseract}, shown in Figure~\ref{fig-1D-4D}.} A line moved over
time could produce a rectangle as shown in Figure~\ref{fig-1D-4D}; that
rectangle moving in another direction over time would produce a 3D
figure, and so forth.
\begin{figure}
{\centering \includegraphics[width=0.9\textwidth,height=\textheight]{images/1D-4D.png}
}
\caption{\label{fig-1D-4D}Geometrical objects in 1 to 4 dimensions. One
more dimension can be thought of as the trace of movement over time.}
\end{figure}
But wait! Where does that 4D thing (a \emph{tesseract}) come from? To
really see a tesseract it helps to view it in an animation over time
(\textbf{?@fig-tesseract}). But like the Square, contemplating 3D from a
2D world, it takes some imagination.
Yet the deep mathematics of more than three dimensions only emerged in
the 19th century. In Newtonian mechanics, space and time were always
considered independent of each other. Our familiar three-dimensional
space, of length, width, and height had formed the backbone of Euclidean
geometry for millenea. However, the idea that space and time are indeed
interwoven was first proposed by German mathematician Hermann Minkowski
(1864--1909) in 1908. This was a powerful idea. It bore fruit when
Albert Einstein revolutionized the Newtonian conceptions of gravity in
1915 when he presented a theory of general relativity which was based
primarily on the fact that mass and energy warp the fabric of
four-dimensional spacetime.
The parable of \emph{Flatland} can provide inspiration for statistical
thinking and data visualization. Once we go beyond bivariate statistics
and 2D plots, we are in a multivariate world of possibly MANY
dimensions. It takes only some imagination and suitable methods to get
there.
Like Abbott's \emph{Flatland}, this book is a romance, in many
dimensions, of what we can learn from modern methods of data
visualization.
\hypertarget{eureka}{%
\section*{EUREKA!}\label{eureka}}
\addcontentsline{toc}{section}{EUREKA!}
\markright{EUREKA!}
Even modest sized multivariate data can have secrets that can be
revealed in the right view. As an example, David Coleman at RCA
Laboratories in Princeton, N.J. generated a dataset of five (fictitious)
measurements of grains of pollen for the 1986 Data Exposition at the
Joint statistical Meetings. The first three variables are the lengths of
geometric features 3848 observed sampled pollen grains -- in the x, y,
and z dimensions: a \texttt{ridge} along x, a \texttt{nub} in the y
direction, and a \texttt{crack} in along the z dimension. The fourth
variable is pollen grain \texttt{weight}, and the fifth is
\texttt{density}. The challenge was to ``find something interesting'' in
this dataset.
Those who solved the puzzle were able to find an orientation of this
5-dimensional dataset, such that zooming in revealed a magic word,
``EUREKA'' spelled in points, as in the following figure.
\begin{figure}
\begin{minipage}[t]{0.50\linewidth}
{\centering
\raisebox{-\height}{
\includegraphics{images/pollen-eureka1.png}
}
}
\end{minipage}%
%
\begin{minipage}[t]{0.50\linewidth}
{\centering
\raisebox{-\height}{
\includegraphics{images/pollen-eureka2.png}
}
}
\end{minipage}%
\newline
\begin{minipage}[t]{0.50\linewidth}
{\centering
\raisebox{-\height}{
\includegraphics{images/pollen-eureka4.png}
}
}
\end{minipage}%
%
\begin{minipage}[t]{0.50\linewidth}
{\centering
\raisebox{-\height}{
\includegraphics{images/pollen-eureka3.png}
}
}
\end{minipage}%
\caption{\label{fig-pollen-eureka}Four views of the \texttt{pollen}
data, zooming in, clockwise from the upper left to discover the word
``EUREKA''.}
\end{figure}
This can be seen better in a 3D animation. \texttt{rgl}
(\protect\hyperlink{ref-R-rgl}{Adler \& Murdoch, 2023}) is used to
create a 3D scatterplot of the first three variables. Then the
\texttt{animation} package (\protect\hyperlink{ref-R-animation}{Xie,
2021}) is use to record a sequence of images, adjusting the
\texttt{rgl::par3d(zoom)} value.
\begin{Shaded}
\begin{Highlighting}[]
\FunctionTok{library}\NormalTok{(animation)}
\FunctionTok{library}\NormalTok{(rgl)}
\FunctionTok{data}\NormalTok{(pollen, }\AttributeTok{package =} \StringTok{"animation"}\NormalTok{)}
\NormalTok{oopt }\OtherTok{=} \FunctionTok{ani.options}\NormalTok{(}\AttributeTok{interval =} \FloatTok{0.05}\NormalTok{)}
\DocumentationTok{\#\# adjust the viewpoint}
\NormalTok{uM }\OtherTok{=}
\FunctionTok{matrix}\NormalTok{(}\FunctionTok{c}\NormalTok{(}\SpecialCharTok{{-}}\FloatTok{0.370919227600098}\NormalTok{, }\SpecialCharTok{{-}}\FloatTok{0.513357102870941}\NormalTok{,}
\SpecialCharTok{{-}}\FloatTok{0.773877620697021}\NormalTok{, }\DecValTok{0}\NormalTok{, }\SpecialCharTok{{-}}\FloatTok{0.73050606250763}\NormalTok{, }\FloatTok{0.675815105438232}\NormalTok{,}
\SpecialCharTok{{-}}\FloatTok{0.0981751680374146}\NormalTok{, }\DecValTok{0}\NormalTok{, }\FloatTok{0.573396027088165}\NormalTok{, }\FloatTok{0.528906404972076}\NormalTok{,}
\SpecialCharTok{{-}}\FloatTok{0.625681936740875}\NormalTok{, }\DecValTok{0}\NormalTok{, }\DecValTok{0}\NormalTok{, }\DecValTok{0}\NormalTok{, }\DecValTok{0}\NormalTok{, }\DecValTok{1}\NormalTok{), }\DecValTok{4}\NormalTok{, }\DecValTok{4}\NormalTok{)}
\FunctionTok{open3d}\NormalTok{(}\AttributeTok{userMatrix =}\NormalTok{ uM, }
\AttributeTok{windowRect =} \FunctionTok{c}\NormalTok{(}\DecValTok{10}\NormalTok{, }\DecValTok{10}\NormalTok{, }\DecValTok{510}\NormalTok{, }\DecValTok{510}\NormalTok{))}
\FunctionTok{plot3d}\NormalTok{(pollen[, }\DecValTok{1}\SpecialCharTok{:}\DecValTok{3}\NormalTok{])}
\CommentTok{\# zoom in}
\NormalTok{zm }\OtherTok{=} \FunctionTok{seq}\NormalTok{(}\DecValTok{1}\NormalTok{, }\FloatTok{0.045}\NormalTok{, }\AttributeTok{length =} \DecValTok{200}\NormalTok{)}
\FunctionTok{par3d}\NormalTok{(}\AttributeTok{zoom =} \DecValTok{1}\NormalTok{)}
\ControlFlowTok{for}\NormalTok{ (i }\ControlFlowTok{in} \DecValTok{1}\SpecialCharTok{:}\FunctionTok{length}\NormalTok{(zm)) \{}
\FunctionTok{par3d}\NormalTok{(}\AttributeTok{zoom =}\NormalTok{ zm[i])}
\FunctionTok{ani.pause}\NormalTok{()}
\NormalTok{\}}
\FunctionTok{ani.options}\NormalTok{(oopt)}
\end{Highlighting}
\end{Shaded}
\begin{figure}
{\centering
}
\caption{\label{fig-pollen-eureka-fig}Animation of zooming in on the
\texttt{pollen} data.}
\end{figure}
Lest this example seem contrived (which it admittedly is), multivariate
visualization has played an important role in quite a few scientific
discoveries. Among these \textbf{TODO} \ldots{} diabetes classification
\ldots.
\hypertarget{what-i-assume}{%
\section*{What I assume}\label{what-i-assume}}
\addcontentsline{toc}{section}{What I assume}
\markright{What I assume}
It is assumed that the reader has a background in intermediate
statistics including material on univariate linear models including
analysis of variance (ANOVA) and multiple regression. This means you
should be familiar with \ldots{}
There will also be some mathematics in the book where words and diagrams
are not enough. The mathematical level will be intermediate, mostly
consisting of simple algebra. No derivations, proofs, theorems here! For
multivariate methods, it will be useful to express ideas using matrix
notation to simplify presentation. I'm using math to express ideas, and
all you will need is a reading-level of understanding.
I also assume the reader to have at least a basic familiarity with R.
While R fundamentals are outside the scope of the book, I believe that
this language provides a rich set of resources, far beyond that offered
by other statistical software packages, and is well worth learning.
For those not familiar with R, I recommend Matloff
(\protect\hyperlink{ref-Matloff-2011}{2011}), Wickham
(\protect\hyperlink{ref-Wickham2014}{2014}), and Cotton
(\protect\hyperlink{ref-Cotton-2013}{2013}) for introductions to
programming in the language, and Fox \& Weisberg
(\protect\hyperlink{ref-FoxWeisberg:2018}{2018}) and Teetor
(\protect\hyperlink{ref-Teetor2011}{2011}) for learning about how to
conduct basic statistical analyses. \textbf{TODO}: Revise this list.
\hypertarget{conventions-used-in-this-book}{%
\section*{Conventions used in this
book}\label{conventions-used-in-this-book}}
\addcontentsline{toc}{section}{Conventions used in this book}
\markright{Conventions used in this book}
The following typographic conventions are used in this book:
\begin{itemize}
\item
\emph{italic} : indicates terms to be \emph{emphasized} or defined in
the text, \ldots{}
\item
\textbf{bold} : is used for names of R packages
\item
\texttt{fixed-width} : is used in program listings as well as in text
to refer to variable and function names, R statement elements and
keywords.
\item
\emph{\texttt{fixed-width\ italic}} : isn't used yet, but probably
should be.
\end{itemize}
For R functions in packages, we use the notation
\texttt{package::function()}, for example: \texttt{car::Anova()} to
identify where those functions are defined
\mainmatter
\bookmarksetup{startatroot}
\hypertarget{sec-introduction}{%
\chapter{Introduction}\label{sec-introduction}}
This material may or may not survive; it was taken from an earlier
article.
\hypertarget{multivariate-vs.-multivariable-methods}{%
\section{Multivariate vs.~multivariable
methods}\label{multivariate-vs.-multivariable-methods}}
\begin{quote}
multivariate \(\ne\) multivariable
\end{quote}
In this era of multivitamins, multitools, multifactor authentication and
even the multiverse, it is well to understand the distinction between
\emph{multivariate} and \emph{multivariable} methods as these terms are
generally used and as I use them here in relation to statistical methods
and data visualization. The distinction is simple:
\begin{itemize}
\item
\textbf{Multivariate methods} for linear models such as multivariate
regression have more than one dependent, response or outcome variable.
Other multivariate methods such as principal components analysis or
factor analysis treat all variables on an equal footing.
\item
\textbf{Multivariable methods} have a single dependent variable and
more than one independent variables or covariates.
\end{itemize}
\hypertarget{why-use-a-multivariate-design}{%
\section{Why use a multivariate
design}\label{why-use-a-multivariate-design}}
A particular research outcome (e.g., depression, neuro-cognitive
functioning, academic achievement, self-concept, attention deficit
hyperactivity disorders) might take on a multivariate form if it has
several observed measurement scales or related aspects by which it is
quantified, or if there are multiple theoretically distinct outcomes
that should be assessed in conjunction with each other (e.g., using
depression, generalized anxiety, and stress inventories to model overall
happiness). In this situation, the primary concern of the researcher is
to ascertain the impact of potential predictors on two or more response
variables simultaneously.
For example, if academic achievement is measured for adolescents by
their reading, mathematics, science, and history scores, the following
questions are of interest:
\begin{itemize}
\item
Do predictors such as parent encouragement, socioeconomic status and
school environmental variables affect \emph{all} of these outcomes?
\item
Do they affect them in the \emph{same} or \emph{different} ways?
\item
How many different aspects of academic achievement can be
distinguished in the predictors? Equivalently, is academic achievement
\emph{unidimensional} or \emph{multidimensional} in relation to the
predictors?
\end{itemize}
Similarly, if psychiatric patients in various diagnostic categories are
measured on a battery of tests related to social skills and cognitive
functioning, we might want to know:
\begin{itemize}
\item
Which measures best discriminate among the diagnostic groups?
\item
Which measures are most predictive of positive outcomes?
\item
Further, how are the \emph{relationships} between the outcomes
affected by the predictors?
\end{itemize}
Such questions obviously concern more than just the separate univariate
relations of each response to the predictors. Equally, or perhaps more
importantly, are questions of how the response variables are predicted
\emph{jointly}.
\begin{tcolorbox}[enhanced jigsaw, opacitybacktitle=0.6, titlerule=0mm, breakable, rightrule=.15mm, leftrule=.75mm, colbacktitle=quarto-callout-note-color!10!white, left=2mm, toptitle=1mm, bottomtitle=1mm, opacityback=0, coltitle=black, title=\textcolor{quarto-callout-note-color}{\faInfo}\hspace{0.5em}{SEM}, arc=.35mm, toprule=.15mm, colframe=quarto-callout-note-color-frame, bottomrule=.15mm, colback=white]
Structural equation modeling (SEM) offers another route to explore and
analyze the relationships among multiple predictors and multiple
responses. They have the advantage of being able to test potentially
complex systems of linear equations in very flexible ways; however,
these methods are often far removed from data analysis \emph{per se} and
except for path diagrams offer little in the way of visualization
methods to aid in understanding and communicating the results. The
graphical methods we describe here can also be useful in a SEM context.
\end{tcolorbox}
\hypertarget{linear-models-univariate-to-multivariate}{%
\section{Linear models: Univariate to
multivariate}\label{linear-models-univariate-to-multivariate}}
For classical linear models for ANOVA and regression, the step from a
univariate model for a single response, \(y\), to a multivariate one for
a collection of \(p\) responses, \(\mathbf{y}\) is conceptually very
easy. That's because the univariate model,
\[y_i = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \dots + \beta_q x_q + u_i , \]
or, in matrix terms,
\[\mathbf{y} = \mathbf{X} \; \mathbf{\beta} + \mathbf{u}, \quad\mbox{ with }\quad \mathbf{u} \sim \mathcal{N} (0, \sigma^2 \mathbf{I}) ,\]
generalizes directly to an analogous multivariate linear model (MLM),
\[\mathbf{Y} = [\mathbf{y_1}, \mathbf{y_2}, \dots, \mathbf{y_p}] = \mathbf{X} \; \mathbf{B} + \mathbf{U} \quad\mbox{ with }\quad \mathbf{U} \sim \mathcal{N} (\mathbf{0}, \mathbf{\Sigma})\]
for multiple responses (as will be discussed in detail). The design
matrix, \(\mathbf{X}\) remains the same, and the vector \(\beta\) of
coefficients becomes a matrix \(\mathbf{B}\), with one column for each
of the \(p\) outcome variables.
Happily as well, hypothesis tests for the MLM are also straight-forward
generalizations of the familiar \(F\) and \(t\)-tests for univariate
response models. Moreover, there is a rich geometry underlying these
generalizations which we can exploit for understanding and
visualization.
\hypertarget{visualization-is-harder}{%
\section{Visualization is harder}\label{visualization-is-harder}}
However, with two or more response variables, visualizations for
multivariate models are not as simple as they are for their univariate
counterparts for understanding the effects of predictors, model
parameters, or model diagnostics. Consequently, the results of such
studies are often explored and discussed solely in terms of coefficients
and significance, and visualizations of the relationships are only
provided for one response variable at a time, if at all. This tradition
can mask important nuances, and lead researchers to draw erroneous
conclusions.
The aim of this book is to describe and illustrate some central methods
that we have developed over the last ten years that aid in the
understanding and communication of the results of multivariate linear
models (\protect\hyperlink{ref-Friendly-07-manova}{Friendly, 2007};
\protect\hyperlink{ref-FriendlyMeyer:2016:DDAR}{Friendly \& Meyer,
2016}). These methods rely on \emph{data ellipsoids} as simple,
minimally sufficient visualizations of variance that can be shown in 2D
and 3D plots. As will be demonstrated, the \emph{Hypothesis-Error (HE)
plot} framework applies this idea to the results of multivariate tests
of linear hypotheses.
Further, in the case where there are more than just a few outcome
variables, the important nectar of their relationships to predictors can
often be distilled in a multivariate juicer--- a \textbf{projection} of
the multivariate relationships to the predictors in the low-D space that
captures most of the flavor. This idea can be applied using
\emph{canonical correlation plots} and with \emph{canonical discriminant
HE plots}.
\begin{figure}
{\centering \includegraphics{images/Cover-image-GBE.png}
}
\caption{\textbf{Projection}: The cover image from Hofstadter's
\emph{Goedel, Bach and Escher} illustrates projection of 3D solids onto
each 2D plane.}
\end{figure}
\hypertarget{sec-problems}{%
\section{Problems in understanding and communicating MLM
results}\label{sec-problems}}
In my consulting practice within the Statistical Consulting Service at
York University, I see hundreds of clients each year ranging from
advanced undergraduate thesis students, to graduate students and faculty
from a variety of fields. Over the last two decades, and across each of
these groups, I have noticed an increasing desire to utilize
multivariate methods. As researchers are exposed to the utility and
power of multivariate tests, they see them as an appealing alternative
to running many univariate ANOVAs or multiple regressions for each
response variable separately.
However, multivariate analyses are more complicated than such
approaches, especially when it comes to understanding and communicating
results. Output is typically voluminous, and researchers will often get
lost in the numbers. While software (SPSS, SAS and R) make tabular
summary displays easy, these often obscure the findings that researchers
are most interested in. The most common analytic oversights that we have
observed are:
\begin{itemize}
\item
\textbf{Atomistic data screening}: Researchers have mostly learned the
assumptions (the Holy Trinity of normality, constant variance and
independence) of univariate linear models, but then apply
\emph{univariate} tests (e.g., Shapiro-Wilk) and diagnostic plots
(normal QQ plots) to every predictor and every response.
\item
\textbf{Bonferroni everywhere}: Faced with the task of reporting the
results for multiple response measures and a collection of predictors
for each, a common tendency is to run (and sometimes report) each of
the separate univariate response models and then apply a correction
for multiple testing. Not only is this confusing and awkward to
report, but it is largely unnecessary because the multivariate tests
provide protection for multiple testing.
\item
\textbf{Reverting to univariate visualizations}: To display results,
SPSS and SAS make some visualization methods available through menu
choices or syntax, but usually these are the wrong (or at least
unhelpful) choices, in that they generate separate univariate graphs
for the individual responses.
\end{itemize}
This book to discusses a few essential procedures for multivariate
linear models, how their interpretation can be aided through the use of
well-crafted (though novel) visualizations, and provides replicable
sample code in R to showcase their use in applied behaviorial research.
A later section {[}ref?{]} provides some practical guidelines for
analyzing, visualizing and reporting such models to help avoid these and
other problems.
\begin{Shaded}
\begin{Highlighting}[]
\CommentTok{\#cat("Packages used here:\textbackslash{}n")}
\FunctionTok{write\_pkgs}\NormalTok{(}\AttributeTok{file =}\NormalTok{ .pkg\_file)}
\CommentTok{\#\textgreater{} 8 packages used here:}
\CommentTok{\#\textgreater{} base, datasets, graphics, grDevices, knitr, methods, stats, utils}
\end{Highlighting}
\end{Shaded}
\bookmarksetup{startatroot}
\hypertarget{sec-getting_started}{%
\chapter{Getting Started}\label{sec-getting_started}}
\hypertarget{sec-why_plot}{%
\section{Why plot your data?}\label{sec-why_plot}}
\begin{quote}
Getting information from a table is like extracting sunlight from a
cucumber. Farquhar \& Farquhar
(\protect\hyperlink{ref-FarquharFarquhar:91}{1891})
\end{quote}