---
title: "Transforming BRAID kappa"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Transforming BRAID kappa}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)
```

```{r setup, include=FALSE}
library(braidReports)
set.seed(20240901)
```

## Introduction

One of the more subtle recurring issues we have encountered with communicating
about the BRAID model is the asymmetric nature of antagonism and synergy as
they are represented by the BRAID interaction parameter $\kappa$.  On the most
basic level, the parameter is quite straightforward: surfaces with negative
$\kappa$ values are antagonistic, and surfaces with positive $\kappa$ values are
synergistic.  But the mathematical form of the BRAID model means that $\kappa$
values can only go as low as $-2$, but they can stretch as far into the positive
domain as they like.  This means that values like $-1.96$ and $50$ can reflect
similarly extreme deviations from additivity, something that can be very 
confusing if values are plotted on a standard linear scale.

A solution to this issue when plotting is to transform $\kappa$ into a space
where antagonistic and synergistic values can both stretch indefinitely, much
as using a log-transformations allows positive values to be plotted further
and further out as they get closer to zero.  One such transformation is:

$$
T(\kappa) = \log\left(\frac{\kappa+2}{2}\right)
$$

This transformed value can extend to both negative and positive infinity, maps
BRAID additivity to zero (that is, $T(0)=0$), and gives similar shifts in
potency from antagonism and synergy similar magnitudes.  Unfortunately, writing
this transformation in every time one wants to plot a set of $\kappa$ values 
can get extremely tedious; so the `braidReports` package includes a bespoke
`transform` object to perform and label the transformation, as well as 
functions for plotting x- and y-coordinates in this $\kappa$-transformed space
in the `ggplot` plotting system.

## The Merck OPPS Dataset

The enclosed datasets `merckValues_unstable` and `merckValues_stable` contain
the results of running a version 1.0.0 BRAID model fit on the roughly twenty-two
thousand combinations in the Merck oncopolypharmacology screen (or OPPS).  The
only difference between the datasets is that `merckValues_unstable` contains
the results of performing BRAID fitting *without* Bayesian stabilization, while
`merckValues_stable` contains the results of fitting with the default "moderate"
Bayesian stabilization.  The two datasets are therefore ideal for examining the
effect of Bayesian stabilization on the broad behavior of $\kappa$.  Let's first
try plotting the unstabilized $\kappa$ values using a traditional linear scale:

```{r}
ggplot(merckValues_unstable,aes(x=kappa,y=factor(1)))+
	geom_jitter()+
	geom_violin(fill="#f3aaa9")+
	scale_x_continuous("BRAID kappa")
```

The result is not very informative.  It's clear that there is a pocket of 
$\kappa$ values which have been raised to a maximum value of 100, while the
vast majority of values are down near zero, but it is impossible to discern
any structure to the values closest to zero.  Is the value biased towards
synergy or antagonism?  Is there a similar pocket of extremal negative kappa values? 
A linearly scaled kappa makes these questions nearly impossible to answer.  
A properly transformed and symmetrized kappa, on the other hand, is much easier
to grasp:

```{r}
ggplot(merckValues_unstable,aes(x=kappa,y=factor(1)))+
	geom_jitter()+
	geom_violin(fill="#f3aaa9")+
	scale_x_kappa("BRAID kappa")
```

Now the patterns are much more clear.  There is indeed a pocket of values at 
the minimum allowed fit for $\kappa$ (in this case -1.96). Nevertheless, the
overwhelming majority of values lie in a smooth, nearly normal distribution
centered near zero, albeit with a slight bias towards antagonism.  But the
number of response surfaces with extreme $\kappa$ values is disheartening;
while it is possible that all of those 2000 combinations at either end truly
exhibit extremely pronounced interactions, the more mundane (and hence more
likely) explanation is that most or all of these fits result from over-fit
noise in under-determined surfaces.  To test this, let's see what the
distribution looks like *with* Bayesian stabilization:

```{r}
ggplot(merckValues_stable,aes(x=kappa,y=factor(1)))+
	geom_jitter()+
	geom_violin(fill="#9db2cb")+
	scale_x_kappa("BRAID kappa")
```

Sure enough, our pockets of extreme values have been almost completely
eliminated.  Yet importantly, the shape of the central distribution is nearly
identical, indicating that the introduction of Bayesian stabilization did not
serve to suppress the value of $\kappa$ altogether.

## Plotting with Other Values

The transformed kappa space is also ideal for plotting alongside other relevant
values and making comparisons between different sets of results.  The following
plot, for example, depicts the joint distribution of $\kappa$ values and IAE
(index of achievable efficacy) values for all combinations involving each of the
38 drugs in the OPPS:

```{r, fig.height=7, fig.width=8}
merckValues_full <- merckValues_stable
merckValues_full[,c("drugA","drugB")] <- merckValues_stable[,c("drugB","drugA")]
merckValues_full <- rbind(merckValues_stable,merckValues_full)

ggplot(merckValues_full,aes(x=kappa,y=IAE))+
	geom_density2d()+
	geom_vline(xintercept = 0,colour="black",linetype=2)+
	scale_x_kappa("BRAID kappa",labels=as.character)+
	scale_y_log10("IAE")+
	facet_wrap(vars(drugB),ncol=6)
```

While it should come as no surprise that combinations involving certain drugs
would be, on average, more potent than those involving others, this plot
makes such comparisons extremely clear.  Furthermore, it reveals the much less
obvious fact that some drugs exhibit noticeably different patterns of 
*interaction* than others, with some drugs, such as bortezomib, dinaciclib,
and methotrexate, exhibiting a clear bias towards antagonism, and others,
including the experimental MK-2206 and BEZ-235, showing a much more pronounced
tendency towards synergy.
