In this post I’m going to show you a few simple steps to illustrate continuous distributions. As an example, we consider reaction time data, which are typically positively skewed and can differ in different ways. Reaction time distributions are also a rich source of information to constrain cognitive theories and models. So unless the distributions are at least illustrated, this information is lost (which is typically the case when distributions are summarised using a single value like the mean). Other approaches not covered here include explicit mathematical models of decision making and fitting functions to model the shape of the distributions (Balota & Yap, 2011).
For our current example, I made up data for 2 independent groups with four patterns of differences:
- no clear differences;
- uniform shift between distributions;
mostly late differences;
mostly early differences.
The R code is on GitHub.
For our first visualisation, we use
ggplot2. The 1D scatterplots give us a good idea of how the groups differ but they’re not the easiest to read. The main reason is probably that we need to estimate local densities of points in different regions and compare them between groups.
For the purpose of this exercise, each group (g1 and g2) is composed of 1,000 observations, so the differences in shapes are quite striking. With smaller sample sizes the evaluation of these graphs could be much more challenging.
Kernel density plots
Relative to scatterplots, I find that kernel density plots make the comparisons between groups much easier.
Scatterplots and kernel density plots can be combined by using beeswarm plots. Here we create scatterplots shaped by local density using the
geom_quasirandom() function from the
ggbeeswarm package. Essentially, the function creates violin plots in which the constituent points are visible.
To make the plots even more informative, I’ve superimposed quantiles – here deciles computed using the Harrell-Davis quantile estimator. The deciles are represented by vertical black lines, with medians shown with thicker lines. Medians are informative about the location of the bulk of the observations and comparing the lower to upper quantiles let us appreciate the amount of asymmetry within distributions. Comparing quantiles between groups give us a sense of the amount of relative compression/expansion on each side of the distributions. This information would be lost if we only compared the medians.
If we remove the scatterplots and only show the quantiles, we obtain quantile plots, which provide a compact description of how distributions differ (please post a comment if you know of older references using quantile plots). Because the quantiles are superimposed, they are easier to compare than in the previous scatterplots. To help with the group comparisons, I’ve also added plots of the quantile differences, which emphasise the different patterns of group differences.
An alternative to quantiles are Vincentiles, which are computed by sorting the data and splitting them in equi-populated bins (there is the same number of observations in each bin). Then the mean is computed for each bin (Balota et al. 2008; Jiang et al. 2004). Below means were computed for 9 equi-populated bins. As expected from the way they are computed, quantile plots and Vincentile plots look very similar for our large samples from continuous variables.
Group quantile and Vincentile plots can be created by averaging quantiles and Vincentiles across participants (Balota & Yap, 2011; Ratcliff, 1979). This will be the topic of another post.
Related to quantile plots and Vincentile plots, delta plots show the difference between conditions, bin by bin (for each Vincentile) along the y-axis, as a function of the mean across conditions for each bin along the x-axis (De Jong et al., 1994). Not surprisingly, these plots have very similar shapes to the quantile difference plots we considered earlier.
Negative delta plots (nDP, delta plots with a negative slope) have received particular attention because of their theoretical importance (Ellinghaus & Miller, 2018; Schwarz & Miller, 2012).
Delta plots are related to the shift function, a powerful tool introduced in the 1970s: it consists in plotting the difference between the quantiles of two groups as a function of the quantiles in one group, with some measure of uncertainty around the difference (Doksum, 1974; Doksum & Sievers, 1976; Doksum, 1977). It was later refined by Rand Wilcox (Rousselet et al. 2017). This modern version is shown below, with deciles estimated using the Harrell-Davis quantile estimator, and percentile bootstrap confidence intervals of the quantile differences. The sign of the difference is colour-coded (purple for negative, orange for positive).
Unlike other graphical quantile techniques presented here, the shift function affords statistical inferences because of it’s use of confidence intervals (the shift function also comes in a few Bayesian flavours). It is probably one of the easiest ways to compare entire distributions, without resorting to explicit models of the distributions. But the shift function and the other graphical methods demonstrated in this post are not meant to compete with hierarchical models. Instead, they can be used to better understand data patterns within and between participants, before modelling attempts. They also provide powerful alternatives to the mindless application of t-tests and bar graphs, helping to nudge researchers away from the unique use of the mean (or the median) and towards considering the rich information available in continuous distributions.
Balota, D.A. & Yap, M.J. (2011) Moving Beyond the Mean in Studies of Mental Chronometry: The Power of Response Time Distributional Analyses. Curr Dir Psychol Sci, 20, 160-166.
Balota, D.A., Yap, M.J., Cortese, M.J. & Watson, J.M. (2008) Beyond mean response latency: Response time distributional analyses of semantic priming. J Mem Lang, 59, 495-523.
Clarke, E. & Sherrill-Mix, S. (2016) ggbeeswarm: Categorical Scatter (Violin Point) Plots.
De Jong, R., Liang, C.C. & Lauber, E. (1994) Conditional and Unconditional Automaticity – a Dual-Process Model of Effects of Spatial Stimulus – Response Correspondence. J Exp Psychol Human, 20, 731-750.
Doksum, K. (1974) Empirical Probability Plots and Statistical Inference for Nonlinear Models in the two-Sample Case. Ann Stat, 2, 267-277.
Doksum, K.A. (1977) Some graphical methods in statistics. A review and some extensions. Statistica Neerlandica, 31, 53-68.
Doksum, K.A. & Sievers, G.L. (1976) Plotting with Confidence – Graphical Comparisons of 2 Populations. Biometrika, 63, 421-434.
Ellinghaus, R. & Miller, J. (2018) Delta plots with negative-going slopes as a potential marker of decreasing response activation in masked semantic priming. Psychol Res, 82, 590-599.
Jiang, Y., Rouder, J.N. & Speckman, P.L. (2004) A note on the sampling properties of the Vincentizing (quantile averaging) procedure. J Math Psychol, 48, 186-195.
Ratcliff, R. (1979) Group Reaction-Time Distributions and an Analysis of Distribution Statistics. Psychol Bull, 86, 446-461.
Rousselet, G.A., Pernet, C.R. & Wilcox, R.R. (2017) Beyond differences in means: robust graphical methods to compare two groups in neuroscience. The European journal of neuroscience, 46, 1738-1748.
Schwarz, W. & Miller, J. (2012) Response time models of delta plots with negative-going slopes. Psychon B Rev, 19, 555-574.