# A clearer explanation of the shift function

The shift function is a power tool to compare two marginal distributions. It’s covered in detail in this previous post. Below is a new illustration which might help better understand the graphical representation of the shift function. The R code to generate the figure is available in the README of the `rogme` package.

Panel A illustrates two distributions, both n = 1000, that differ in spread. The observations in the scatterplots were jittered based on their local density, as implemented in `ggforce::geom_sina`.

Panel B illustrates the same data from panel A. The dark vertical lines mark the deciles of the distributions. The thicker vertical line in each distribution is the median. Between distributions, the matching deciles are joined by coloured lined. If the decile difference between group 1 and group 2 is positive, the line is orange; if it is negative, the line is purple. The values of the differences for deciles 1 and 9 are indicated in the superimposed labels.

Panel C focuses on the portion of the x-axis marked by the grey shaded area at the bottom of panel B. It shows the deciles of group 1 on the x-axis – the same values that are shown for group 1 in panel B. The y-axis shows the differences between deciles: the difference is large and positive for decile 1; it then progressively decreases to reach almost zero for decile 5 (the median); it becomes progressively more negative for higher deciles. Thus, for each decile the shift function illustrates by how much one distribution needs to be shifted to match another one. In our example, we illustrate by how much we need to shift deciles from group 2 to match deciles from group 1.

More generally, a shift function shows quantile differences as a function of quantiles in one group. It estimates how and by how much two distributions differ. It is thus a powerful alternative to the traditional t-test on means, which focuses on only one, non-robust, quantity. Quantiles are robust, intuitive and informative. # Problems with small sample sizes

In psychology and neuroscience, the typical sample size is too small. I’ve recently seen several neuroscience papers with n = 3-6 animals. For instance, this article uses n = 3 mice per group in a one-way ANOVA. This is a real problem because small sample size is associated with:

• low statistical power

• inflated false discovery rate

• inflated effect size estimation

• low reproducibility

Here is a list of excellent publications covering these points:

Button, K.S., Ioannidis, J.P., Mokrysz, C., Nosek, B.A., Flint, J., Robinson, E.S. & Munafo, M.R. (2013) Power failure: why small sample size undermines the reliability of neuroscience. Nature reviews. Neuroscience, 14, 365-376.

Colquhoun, D. (2014) An investigation of the false discovery rate and the misinterpretation of p-values. R Soc Open Sci, 1, 140216.

Forstmeier, W., Wagenmakers, E.J. & Parker, T.H. (2016) Detecting and avoiding likely false-positive findings – a practical guide. Biol Rev Camb Philos Soc.

Lakens, D., & Albers, C. J. (2017, September 10). When power analyses based on pilot data are biased: Inaccurate effect size estimators and follow-up bias. Retrieved from psyarxiv.com/b7z4q

This important aspect is illustrated in the figure below. Columns show distributions that differ in four different ways. The rows illustrate samples of different sizes. The scatterplots were jittered using `ggforce::geom_sina` in R. The vertical black bars indicate the mean of each sample. In row 1, examples 1, 3 and 4 have exactly the same mean. In example 2 the means of the two distributions differ by 2 arbitrary units. The remaining rows illustrate random subsamples of data from row 1. Above each plot, the t value, mean difference and its confidence interval are reported. Even with 100 observations we might struggle to approximate the shape of the parent population. Without additional information, it can be difficult to determine if an observation is an outlier, particularly for skewed distributions. In column 4, samples with n = 20 and n = 5 are very misleading. 