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\title{{\textbf{\texttt{microbiomeDASim}: Tools for Simulationg Longitudinal
Differential Abundance}}}
\author{Justin Williams, Hector Corrada Bravo, Jennifer Tom, Joseph Nathaniel
    Paulson}
\date{Modified: September 9, 2019. Compiled: \today}
\maketitle
\tableofcontents

\newpage

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\section{Introduction}
With an increasing emphasis on understanding the dynamics of microbial
communities in various settings, longitudinal sampling studies are underway.
Whole metagenomic shotgun sequencing and marker-gene survey studies have unique
technical artifacts that drive novel statistical methodological development for
estimating time intervals of differential abundance. In designing a study and
the frequency of collection prior to a study, one may wish to model the ability
to detect an effect, e.g., there may be issues with respect to cost, ease of
access, etc. Additionally, while every study is unique, it is possible that in
certain scenarios one statistical framework may be more appropriate than
another.

Here, we present a simulation paradigm in a R software package termed
\texttt{microbiomeDASim}. This package allows investigators to simulate
longitudinal differential abundance for single features, \texttt{mvrnorm\_sim},
or a community with multiple features, \texttt{gen\_norm\_microbiome}. The
functions allow the user to specify a variety of known functional forms
(e.g, linear, quadratic, oscillating, hockey stick) along with
flexible parameters to control desired signal to noise ratio. Different
longitudinal correlation structures are available including AR(1), compound,
and independent to account for expected within individual correlation. Comparing
estimation methods or designing a potential longidutinal investigation for
microbiome sequencing data using \texttt{microbiomeDASim} provides accurate and
reproducible results.

\section{Installation}
To install \texttt{microbiomeDASim} from \textit{Bioconductor} use the code
below:
<<pkg_install,eval=FALSE>>=
if(!requireNamespace("BiocManager", quietly = TRUE)){
    install.packages("BiocManager")
}
BiocManager::install("microbiomeDASim")
@

\section{Motivation}
In our analyses we want to try and simulate longitudinal differential abundance
for microbiome normalized counts generated from 16S rRNA sequencing in order to
compare different methodologies for estimating this differential abundance.
Simulating microbiome data presents a variety of different challenges based both
on biological and technical challenges.

These challenges include:

\begin{itemize}
    \item Non-negative restriction
    \item Presence of Missing Data/High Number of Zero Reads
    \item Low Number of Repeated Measurements
    \item Small Number of Subjects
\end{itemize}

For our initial simulation design, we are attempting to address many of these
challanges by simulating data across a variety of different parameter scenarios
where each element of the challenges listed above can be investigated. Since
many of the methods when analyzing microbiome data involve normalization
procedures and the central limit theory allows us to think about the normal
distribution as an asymptotic ideal, we will treat the Multivariate Normal
Distribution as our best case scenario for simulation purposes.

\section{Statistical Methodology}
Assume that we have data generated from the following distribution,
$$\mathbf{Y}\sim N(\mathbf{\mu}, \mathbf{\Sigma})$$

where
$$
\mathbf{Y}=\begin{pmatrix}
\mathbf{Y}_{1}^{T} \\
\mathbf{Y}_{2}^{T} \\
\vdots \\
\mathbf{Y}_{n}^{T}
\end{pmatrix}=\begin{pmatrix}
Y_{11} \\
Y_{12} \\
\vdots \\
Y_{1q_{1}}\\
Y_{21} \\
\vdots \\
Y_{2q_{2}}\\
\vdots \\
Y_{nq_{n}}
\end{pmatrix}
$$
with $Y_{ij}$ representing the $i^{th}$ patient at the $j^{th}$
timepoint where each patient has $q_{i}$ repeated measurements with
$i\in\{1,\dots,n\}$ and $j\in\{1,\dots,q_{i}\}$. We define the total number of
observations as $N=\sum_{i=1}^{n}q_{i}$. Therefore in our original assumption
defined above we can explicitly define the dimension of our objects as

$$
\mathbf{Y}_{N\times 1}\sim N_{N}(\mathbf{\mu}_{N\times 1},
\mathbf{\Sigma}_{N\times N})
$$

In our current simulations we choose to keep the number of repeated measurements
constant, i.e., $q_{i}=q \ \forall \ i\in\{1,\dots,n\}$. This means that the
total number of obseravtions simplifies to the expression $N=qn$. However, we
will vary the value of $q$ across the simulations to simulate data generated
from a study with a small ($q=3$), medium ($q=6$), or large ($q=12$) number of
repeated observations. Currently most studies with microbiome data collected
fall closest to the small case, but there are a few publically available
datasets that contain a large number of repeated observations for each
individual. For simplicity in the remaining description of the methodology
we will assume constant number of repeated observations.

Without loss of generality we can split the total patients ($n$) into two
groups, control ($n_{0}$) vs ($n_{1}$), with the first $n_{0}$ patients
representing the control patients and the remaining $n-n_{0}$ representing the
treatment. Partitioning our observations in this way allows us to also partition
the mean vector as $\mathbf{\mu}=(\mathbf{\mu}_{0}, \mathbf{\mu}_{1})$. In our
current simulation shown below we assume that the mean for the control group is
constant $\mu_{0}\mathbf{1}_{n_{0}\times 1}$, but we allow our mean vector for
the treatment group to vary depending on time $\mu_{1ij}(t)=\mu_{0} + f(t_{j})$
for $i=1,\dots, n_{1}$ and $j=1,\dots,q$. In our simulation we will choose a
parametric form for the function $f(t_{j})$ primarly from the polynomial family
where
$$
f(t_{j})=\beta_{0}+\beta_{1}t_{j}+\beta_{2}t_{j}^{2}+\dots+\beta_{p}t_{j}^{p}
$$
for a $p$ dimensional polynomial. For instance, to define a linear polynomial we
would have
$$f(t_{j})=\beta_{0}+\beta_{1}t_{j}$$
where $\beta_{1}>0$ for an increasing linear trend and $\beta_{1}<0$ for a
decreasing linear trend. For a list of available functional forms currently
implemented and the expected form of the input see documentation for
`microbiomeDASim::mean\_trend`.

For the covariance matrix, we want to encode our longitudinal dependencies so
that observations within an individual are correlated, i.e.,
$\text{Cor}(Y_{ij}, Y_{ij'})\ne0$, but that observations between individuals
are independent, i.e.,
$\text{Cor}(Y_{ij}, Y_{i'j})=0 \ \forall i\ne i' \ \text{and} \ j$.
To accomplish this we define the matrix $\mathbf{\Sigma}_{N\times N}$ as
$\mathbf{\Sigma}=\text{bdiag}(\mathbf{\Sigma}_{1},\dots,\mathbf{\Sigma}_{n})$,
where each $\mathbf{\Sigma}_{i}$ is a $q\times q$ matrix a specific longitudinal
structure.

For instance if we want to specify an autoregressive correlation structure for
individual $i$ we could define their covariance matrix as
$$\mathbf{\Sigma}_{i}=\sigma^{2}\begin{bmatrix}
1 & \rho & \rho^{2} & \cdots & \rho^{|1 - q|} \\
\rho & 1 & \rho & \cdots & \rho^{|2-q|} \\
\rho^{2} & \rho & 1 & \cdots & \\
\vdots &  & & \ddots & \vdots \\
\rho^{|q-1|} & \rho^{|q-2|} & \cdots & \cdots & 1
\end{bmatrix}$$
In this case we are using the first order autoregressive definition and
therefore will refer to this as "ar1".

Alternatively, for the compound correlation structure for an individual $i'$ we
could define the covariance matrix as
$$\mathbf{\Sigma}_{i'}=\sigma^{2}\begin{bmatrix}
1 & \rho & \rho & \cdots & \rho \\
\rho & 1 & \rho & \cdots & \rho \\
\rho & \rho & 1 & \cdots & \\
\vdots &  & & \ddots & \vdots \\
\rho & \rho & \cdots & \cdots & 1
\end{bmatrix}$$

Finally, the independent correlation structure for an individual $i''$ would be
defined as
$$
\mathbf{\Sigma}_{i''}=\sigma^{2}\begin{bmatrix}
1 & 0 & 0 & \cdots & 0 \\
0 & 1 & 0 & \cdots & 0 \\
0 & 0 & 1 & \cdots & \\
\vdots &  & & \ddots & \vdots \\
0 & 0 & \cdots & \cdots & 1
\end{bmatrix}
$$

\subsection{Trivial Example}
We can construct a trivial example where $n=2$ ($n_0=1$ and $n_{1}=1$)
and $q=2$, an ar1 correlation structure where $\rho=0.8$, $\sigma=1$, a linear
functional form with $\mathbf{\beta}=(0, 1)^{T}$, control mean is constant at
$\mu_{0}=2$ and $t=(1, 2)$.
$$Y\sim\text{N}_{4}\begin{pmatrix}
\begin{pmatrix}
\mu_{0} \\
\mu_{0} \\
\mu_{0} +  \beta_0 + \beta_1\times t_{1}\\
\mu_{0} +  \beta_0 + \beta_1\times t_{2}
\end{pmatrix},\sigma^{2}
\begin{pmatrix}
1 & \rho & 0 & 0 \\
\rho & 1 & 0 & 0 \\
0 & 0 & 1 & \rho \\
0 & 0 & \rho & 1
\end{pmatrix}
\end{pmatrix}=
\text{N}_{4}\begin{pmatrix}
\begin{pmatrix}
2 \\
2 \\
3 \\
4
\end{pmatrix},
\begin{pmatrix}
1 & 0.8 & 0 & 0 \\
0.8 & 1 & 0 & 0 \\
0 & 0 & 1 & 0.8 \\
0 & 0 & 0.8 & 1
\end{pmatrix}
\end{pmatrix}
$$

Next, we can now look at sample data generated for our trivial example,
<<tr_ex,fig.align='center',fig.height=4,fig.width=6,warning=FALSE,tidy=FALSE>>=
require(microbiomeDASim)
triv_ex <- mvrnorm_sim(n_control=10, n_treat=10, control_mean=2,
                        sigma=1, num_timepoints=2, t_interval=c(1, 2), rho=0.8,
                        corr_str="ar1", func_form="linear",
                        beta= c(1, 2), missing_pct=0,
                        missing_per_subject=0, dis_plot=TRUE)
head(triv_ex$df)
triv_ex$Sigma[seq_len(2), seq_len(2)]
@

Note that there are a variety of flexible choices for the the functional
form of the trend:

\begin{itemize}
    \item Linear
    \item Quadratic
    \item Cubic
    \item M/W (Oscillating Trends)
    \item L\_up/L\_down (Hockey Stick Trends)
\end{itemize}

Below we show an example using a Hockey Stick trend where we graph the true
functional form, and then simulate data with this trend.
<<hockey_stick, fig.align='center', fig.height=4, fig.width=6, tidy=FALSE>>=
true_mean <- mean_trend(timepoints=1:10, form="L_up", beta=0.5, IP=5,
                            plot_trend=TRUE)
hockey_sim <- mvrnorm_sim(n_control=10, n_treat=10, control_mean=2, sigma=1,
                            num_timepoints=10, t_interval=c(0, 9), rho=0.8,
                            corr_str="ar1", func_form="L_up", beta= 0.5, IP=5,
                            missing_pct=0, missing_per_subject=0, dis_plot=TRUE,
                            asynch_time=TRUE)
@

The \texttt{mvrnorm\_sim} method generates a single feature with a specified
longitudinal differential abundance pattern. However, we may want to simulate a
microboime environment with multiple features where certain features have
differential abundance while others do not.

To address this we can use the function \texttt{gen\_microbiome}, which
specifies thenumber of features and the number of differentiall abundant
features. All features selected to have differential abundance will have the
same type of functional form.
<<bug_gen, tidy=FALSE>>=
bug_gen <- gen_norm_microbiome(features=6, diff_abun_features=3, n_control=30,
                                n_treat=20, control_mean=2, sigma=2,
                                num_timepoints=4, t_interval=c(0, 3),
                                rho=0.9, corr_str="compound",
                                func_form="M", beta=c(4, 3), IP=c(2, 3.3, 6),
                                missing_pct=0.2, missing_per_subject=2,
                                miss_val=0)
head(bug_gen$bug_feat)
bug_gen$Y[, 1:5]
names(bug_gen)
@

Note that we now have two objects returned in this function.

\begin{itemize}
    \item \texttt{Y} is our observed feature matrix with rows representing
    features and columns indicating our repeated samples
    \item \texttt{bug\_feat} identifies the subject ID, time, group
    (Control vs. Treatment) and corresponding Sample ID from the columns of
    \texttt{Y}.
\end{itemize}

\section{Session Info}
<<session_info>>=
sessionInfo()
@
\end{document}
