If you have not already done so, I would read through the
*pibble* vignette before this one.

fido can be used for jointly modeling multivariate count data and
multivariate Gaussian data. For example, this would be a reasonable
model to jointly model 16S microbiome data and metabolomics data
jointly. Because of the “two-headed” nature of this model, e.g., two
observed data-sets, I named this model *orthus*, a two-headed dog and
brother of Cerberus in Greek Mythology. The *orthus* model
can be written as

\[ \begin{align} Y_j & \sim \text{Multinomial}\left(\pi_j \right) \\ \pi_j & = \phi^{-1}(\eta_j) \\ \begin{bmatrix}\eta_j \\ Z_j \end{bmatrix} &\sim N(\Lambda X, \Sigma) \\ \Lambda &\sim N(\Theta, \Sigma, \Gamma) \\ \Sigma &\sim W^{-1}(\Xi, \upsilon) \end{align} \]

Note this looks nearly identical to the *pibble* model but we
have appended the second (Gaussian) dataset (\(Z\)) onto \(\eta\). In doing this, the definition of
\(\Lambda\) changes (it is now larger
with the bottom rows dictating how the covariates \(X\) influence the second dataset).
Similarly, \(\Sigma\) now is much
larger and can be though of as \[
\Sigma = \begin{bmatrix} \Sigma_{(\eta, \eta)} & \Sigma_{(\eta, Z)}
\\
\Sigma_{(Z, \eta)} & \Sigma_{(Z,
Z)}\end{bmatrix}
\] where \(\Sigma_{(\eta,
\eta)}\) describes the covariance between log-ratios (e.g., the
covariance among the multinomial categories in log-ratio space), \(\Sigma_{(Z, Z)}\) describes the covariance
between the dimensions of \(Z\) (e.g.,
between metabolites if Z is metabolomics data), and \(\Sigma_{(\eta, Z)} = \Sigma_{(Z, \eta)}^T\)
represents the covariance between log-ratios and dimensions of \(Z\) (e.g., between microbial taxa and
metabolites). Similar to \(\Sigma\) and
\(\Lambda\), the parameters \(\Xi\) and \(\Theta\) undergo a similar expansion to
accommodate the second dataset.

To demonstrate *orthus* I will perform a toy analysis on data
from Kashyap et al. (2013) and made
available by Callahan et al. (2016) as
part of their recently published microbiome data analysis workflow (Callahan et al. 2016). I follow the data
preprocessing of Callahan et al. (2016) we
just don’t drop taxa but instead amalgamate those that don’t pass
filtering to a category called “other”. I do this to maintain the proper
variance in the multinomial model.

```
metab_path <- system.file("extdata/Kashyap2013", "metabolites.csv", package="fido")
microbe_path <- system.file("extdata/Kashyap2013", "microbe.rda", package="fido")
metab <- read.csv(metab_path, row.names = 1)
metab <- as.matrix(metab)
microbe <- get(load(microbe_path))
## Preprocessing ##
# Metabolite Preprocessing
keep_ix <- rowSums(metab == 0) <= 3
metab <- metab[keep_ix, ]
# 16S Preprocesing - plus some weirdness to rename amalgamated category to "other"
keep_ix <- taxa_sums(microbe) > 4
keep_ix <- keep_ix & (rowSums(otu_table(microbe)>2)>3)
microbe <- merge_taxa(microbe, taxa_names(microbe)[!keep_ix])
nms <- taxa_names(microbe)
rnm <- which(taxa_names(microbe)==taxa_names(microbe)[!keep_ix][1])
nms[rnm] <- "other"
taxa_names(microbe) <- nms
rm(nms, rnm)
# bit of preprocessing
metab <- log10(1 + metab)
```

Now I am going to do just a bit of processing to get data into a format for orthus. Note I have no extra metadata so we are just going to use an intercept in our model at this time.

```
Y <- otu_table(microbe, taxa_are_rows=TRUE)
Z <- metab #(metabolites are rows)
X <- matrix(1, 1, phyloseq::nsamples(microbe))
# save dims for easy reference
N <- ncol(Y)
P <- nrow(Z)
Q <- nrow(X)
D <- nrow(Y)
```

Now I am going to set up the priors. My priors are going to be
similar to that of *pibble* but now we need to think about a
prior for the covariance among the metabolites and between the
metabolites and the log-ratios of the taxa. Remember, that priors must
be defined in the \(ALR_D\) (e.g., ALR
with the reference being the D-th taxa; this may be changed in the
future to make specifying priors more user friendly).

I am going to form our prior for \(\Sigma\) by specifying \(\upsilon\) and \(\Xi\). I will specify that I have weak prior belief that the taxa are independent in terms of their log absolute abundance. We can translate this statement about covariance of log absolute abundance into a statement about log-ratio covariance by pre- and post-multiplying by the \(ALR_D\) contrast matrix (which I refer to as \(GG\) below). Additionally, I believe that there is likely no substantial covariance between the taxa and the metabolites and I assume the metabolites are likely independent.

```
upsilon <- (D-1+P)+10 # weak-ish prior on covariance over joint taxa and metabolites
Xi <- diag(D-1+P)
GG <- cbind(diag(D-1), -1)
Xi[1:(D-1), 1:(D-1)] <- GG%*%diag(D) %*% t(GG)
Xi <- Xi * (upsilon-D-P) # this scales Xi to have the proper mean we wanted
image(Xi)
```

Note the structure of this prior, everything is independent but there is a moderate positive covariance between the log-ratios based on their shared definition in terms of the \(D\)-th taxa.

The other parts of the prior are less interesting. We are going to state that our mean for \(\Lambda\) is centered about \(\mathbf{0}\) and that the signal-to-noise ratio in the data is approximately 1 (this later part is specified by \(\Gamma=I\)).

Finally I fit the model.

Next we are going to transform the log-ratios from \(ALR_D\) to the \(CLR\). I have written all the
transformation functions, *e.g.*, `to_clr`

etc… to
work on `orthusfit`

objects in a similar manner to how they
work on `pibblefit`

objects. For `orthusfit`

objects they only transform the log-ratio components of parameters
leaving the other parts of inferred model parameters (*i.e.*, the
parts associated with the metabolites) untouched.

There are a ton of ways to visualize the inferred model. I could make network diagrams relating taxa to taxa, taxa to metabolites and metabolites to metabolites. I could look at a low dimensional representation of joint covariance to create something very much akin to canonical correlation analysis (CCA). I could look at how well the metabolites predict the taxa and vice-versa. But for the sake of simplicity I will do something much simpler. Here I am just going to find a list of taxa metabolite covariances that the model is very confident about.

```
# First just look ath the cross-covariances fit by the model
# (covariance between taxa in CLR coordinates and metabolites)
# This requires that we extract the corner of Sigma.
xcor <- fit$Sigma[1:D, D:(D-1+P),]
# Initial preprocessing to speed up computation of posterior intervals
# As there are a lot of cross-covariance terms we are going to first
# weed down the list of things we have to look at by first pass
# selecting only those taxa that have a large posterior mean for the covariance
xcor.mean <- apply(xcor, c(1,2), mean)
to.analyze <- fido::gather_array(xcor.mean, cov, taxa, metabolite) %>%
arrange(-abs(cov)) %>%
.[1:1000,] %>%
mutate(tm =paste0(taxa, "_", metabolite))
# Subset Covariance to those we are interested in and calculate posterior
# confidence intervals.
xcor.summary <- fido::gather_array(xcor, cov, taxa, metabolite, iter) %>%
mutate(tm=paste0(taxa, "_", metabolite)) %>%
filter(tm %in% to.analyze$tm) %>%
mutate(taxa = rownames(Y)[taxa], metabolite = rownames(Z)[metabolite]) %>%
group_by(taxa, metabolite) %>%
fido:::summarise_posterior(cov) %>%
arrange(mean) %>%
filter(taxa != 'other') # we don't care about these
# Select those covariances where the model has high certainty (95%) that
# the true covariance is not zero.
xcor.summary %>%
filter(sign(p2.5)==sign(p97.5)) %>%
filter(abs(mean) > 2)
#> # A tibble: 218 x 8
#> # Groups: taxa [17]
#> taxa metabolite p2.5 p25 p50 mean p75 p97.5
#> <chr> <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
#> 1 722 206.0445922 -6.55 -3.96 -3.08 -3.33 -2.32 -1.51
#> 2 7816 206.0445922 -5.63 -3.23 -2.45 -2.65 -1.85 -1.05
#> 3 722 290.9298419 -5.18 -3.09 -2.42 -2.62 -1.85 -1.23
#> 4 18182 380.1846197 -5.20 -3.07 -2.38 -2.55 -1.83 -1.03
#> 5 722 181.4504354 -5.19 -3.04 -2.36 -2.55 -1.80 -1.13
#> 6 722 177.0565368 -4.98 -3.01 -2.30 -2.50 -1.78 -1.08
#> 7 722 180.072273 -5.03 -3.06 -2.30 -2.49 -1.71 -0.986
#> 8 19517 380.1846197 -5.15 -3.06 -2.33 -2.49 -1.71 -0.952
#> 9 2943 380.1846197 -4.89 -2.97 -2.34 -2.49 -1.83 -1.03
#> 10 722 176.0343919 -4.93 -2.95 -2.25 -2.48 -1.79 -1.09
#> # ... with 208 more rows
```

So it looks there there are a few hundred covariances that we can be fairly confident about.

Please note, I performed this analysis to demonstrate the use of
*orthus* which is a model that I have been repeatedly asked for.
I think its a cool model and could be quite useful in the right
circumstances. But I would like to point out a few philosophical points
about the analysis I performed above.

First, I performed this analysis just to demonstrate *orthus*.
I really don’t know the data showcased here. What is metabolite
`206.0445922`

? I have no idea. For some reason this is how
the metabolites in that dataset were named. For the same reason I have
left the taxa indexed by sequence variant number.

Second (and more important), identifying relationships between taxa
and metabolites (or between any two high-dimensional multivariate
data-sets) is really difficult! Here we are looking at just 114 taxa and
405 but this leads to 46170 possible covariances and here we only have
12 samples! Yes *orthus* is a Bayesian model, and Yes, Bayesian
models can be quite useful when there are more parameters than samples,
but there is a limit of reasonability. Really, Bayesian models are great
when you can perfectly capture your prior beliefs with your prior. But
how often can that really be done perfectly? As such I would caution
users, to use *orthus* carefully. Consider which metabolites and
taxa you really care about and if you can, isolate your analyses to
those.

Alright, that’s probably enough philosophizing for an R package
Vignette. I hope you enjoy *orthus*.

Callahan, Ben J, Kris Sankaran, Julia A Fukuyama, Paul J McMurdie, and
Susan P Holmes. 2016. “Bioconductor Workflow for Microbiome Data
Analysis: From Raw Reads to Community Analyses.”
*F1000Research* 5.

Kashyap, Purna C, Angela Marcobal, Luke K Ursell, Samuel A Smits, Erica
D Sonnenburg, Elizabeth K Costello, Steven K Higginbottom, et al. 2013.
“Genetically Dictated Change in Host Mucus Carbohydrate Landscape
Exerts a Diet-Dependent Effect on the Gut Microbiota.”
*Proceedings of the National Academy of Sciences* 110 (42):
17059–64.