# Transforming BRAID kappa

## 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:

    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:

    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:

    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:

    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.