Two distributions can differ in many ways, yet the standard approach in neuroscience & psychology is to assume differences in means. That’s why the first step in exploratory data analysis should always be detailed graphical representations (Rousselet et al. 2016, 2017). To help quantify how two distributions differ, a fantastic tool is the shift function – an example is provided below. It consists in plotting the difference between group quantiles, as a function of the quantiles in one group. The technique was first described in the 1970s by Doksum (1974; Doksum & Sievers, 1976), and later refined by Wilcox using the Harrell-Davis quantile estimator (Harrell & Davis, 1982) in conjunction with two percentile bootstrap methods (Wilcox 1995; Wilcox et al. 2014). The technique is related to delta plots and relative distribution methods (see details in Rousselet et al. 2017).

The goal of this post is to get feedback on my first attempt to make a Bayesian version of the shift function. Below I describe three potential strategies. My main motivation for making a Bayesian version is that the shift function comes with frequentist confidence intervals and p values. Although I still use confidence intervals to describe sampling variability, they are inherently linked to p values and tend to be associated with major flaws in interpretation (Morey et al. 2016). And my experience is that if p values are available, most researchers will embark on lazy and irrational decision making (Wagenmakers 2007). P values would not be a problem if they were just used as one of many pieces of evidence, without any special status (McShane et al. 2017).

Let’s consider a toy example: two independent exGaussian distributions, with n = 100 in each group.

The two groups clearly differ, as expected from the generative process (see online code). A t-test on means suggests a large uncertainty about the group difference:

difference = – 65 ms [-166, 37]

t = -1.26

p = 0.21

A shift function provides much more information about how the two groups differ.

The x-axis shows the quantiles of group 1 (here only the 9 deciles, which is a good default). The y-axis shows the difference between deciles of group 1 and group 2. Intuitively, the difference shows by how much group 2 would need to be shifted to match group 1, for each decile. The coloured labels show the quantile differences. I let you go back and forth between the density estimates and the shift function to understand what’s going on. Another detailed description is provided here if needed.

# Strategy 1: Bayesian bootstrap

A simple strategy to make a Bayesian shift function is to use the Bayesian bootstrap instead of a percentile bootstrap. The percentile bootstrap can already be considered a very cheap way to create Bayesian posterior distributions, if we make the strong (and wrong) assumption that our observations are the only possible ones. Rasmus Bååth provides a detailed introduction to the Bayesian bootstrap, and it’s R implementation on his blog. There is also a video.

Using the Bayesian bootstrap, and 95% highest density intervals (HDI) of the posterior distributions, the shift function looks very similar to the original version, as expected.

Except now we’re dealing with credible intervals, and there are no p values, so users have to focus on quantification!

# Strategy 2: Bayesian quantile regression

Another strategy is to use quantile regression, which comes in a Bayesian flavour using the asymmetric Laplacian likelihood family. To do that, I’m using the amazing `brms`

R package by Paul Bürkner.

To get started with Bayesian statistics and the `brms`

package, I recommend this excellent blog post by Matti Vuorre. Many other great posts are available here.

Using the default priors from `brms`

, we can fit a quantile regression line for each group and for each decile. The medians (or means, or modes) and 95% HDIs of the posterior distributions can then be used to create a shift function.

Again, the results are quite similar to the original ones, although this time the quantiles do differ a bit from those in the previous versions because we use the medians of the posterior distributions to estimate them. Also, I haven’t looked in much detail at how well the model fits the data. The posterior predictive samples suggest the fits could be improved, but I have too little experience to make a call.

# Strategy 3: Bayesian model with exGaussians

The third strategy is to fit a descriptive model to the distributions, generate samples from the posterior distributions, and compute quantiles from these predicted values. Here, since our toy model simulates reaction time data using exGaussian distributions, it makes sense to fit an exGaussian family to the data. More generally, exGaussian distributions are very good at capturing the shape of RT data (Matzke & Wagenmakers, 2009).

Again, the results look similar to the original ones. This strategy has the advantage to require fitting only one model, instead of a new model for each quantile in strategy 2. In addition, we get an exGaussian fit of the data, which is very useful in itself. So that would probably be my favourite strategy.

# Questions to you, reader

Does this make even remotely sense?

Which strategy seems more promising?

The clear benefit of strategies 2 and 3 is that they can easily be extended to create hierarchical shift functions, by fitting simultaneously observations from multiple participants. Whereas for the original shift function using the percentile bootstrap, I don’t see any obvious way to make a hierarchical version.

You might want to ask, why not simply focus on modelling the distributions instead of looking at the quantiles? Modelling the shape of the distributions is great of course, but I don’t think it can achieve the same fine-grained quantification achieved by the shift function. Another approach of course would be to fit a more psychologically motivated model, such as a diffusion model. I think these three approaches are complementary.

# References

Doksum, K. (1974) Empirical Probability Plots and Statistical Inference for Nonlinear Models in the two-Sample Case. Annals of Statistics, 2, 267-277.

Doksum, K.A. & Sievers, G.L. (1976) Plotting with Confidence – Graphical Comparisons of 2 Populations. Biometrika, 63, 421-434.

Harrell, F.E. & Davis, C.E. (1982) A new distribution-free quantile estimator. Biometrika, 69, 635-640.

Blakeley B. McShane, David Gal, Andrew Gelman, Christian Robert, Jennifer L. Tackett (2017) Abandon Statistical Significance. arXiv:1709.07588

Matzke, D. & Wagenmakers, E.J. (2009) Psychological interpretation of the ex-Gaussian and shifted Wald parameters: a diffusion model analysis. Psychon Bull Rev, 16, 798-817.

Morey, R.D., Hoekstra, R., Rouder, J.N., Lee, M.D. & Wagenmakers, E.J. (2016) The fallacy of placing confidence in confidence intervals. Psychon Bull Rev, 23, 103-123.

Rousselet, G.A., Foxe, J.J. & Bolam, J.P. (2016) A few simple steps to improve the description of group results in neuroscience. The European journal of neuroscience, 44, 2647-2651.

Rousselet, G., Pernet, C. & Wilcox, R. (2017) Beyond differences in means: robust graphical methods to compare two groups in neuroscience, figshare.

Wagenmakers, E.J. (2007) A practical solution to the pervasive problems of p values. Psychonomic bulletin & review, 14, 779-804.

Wilcox, R.R. (1995) Comparing Two Independent Groups Via Multiple Quantiles. Journal of the Royal Statistical Society. Series D (The Statistician), 44, 91-99.

Wilcox, R.R., Erceg-Hurn, D.M., Clark, F. & Carlson, M. (2014) Comparing two independent groups via the lower and upper quantiles. J Stat Comput Sim, 84, 1543-1551.