# Comparing all quantiles of two distributions simultaneously

Summary: A new function in the WRS package compares many quantiles of two distributions simultaneously while controlling the overall alpha error.

When comparing data from two groups, approximately 99.6% of all psychological research compares the *central tendency* (that is a subjective estimate).

In some cases, however, it would be sensible to compare different parts of the distributions. For example, in reaction time (RT) experiments two groups may only differ in the fast RTs, but not in the long. Measures of central tendency might obscure or miss this pattern, as following example demonstrates.

Imagine RT distributions for two experimental conditions (âblackâ and âredâ). Participants in the red condition have some very fast RTs:

A naĂŻve (but common) approach would be to compare both distributions with a *t* test:

######################

data: RT1 and RT2

t = -0.3778, df = 168.715, p-value = 0.706

alternative hypothesis: true difference in means is not equal to 0

95 percent confidence interval:

-22.74478 15.43712

sample estimates:

mean of x mean of y

341.8484 345.5022

Results show that both groups do not differ in their central tendency.

**Now letâs do the same with a new method!**

The function

from the WRS package compares user-defined quantiles of both distributions using a HarrellâDavis estimator in conjunction with a percentile bootstrap. The method seems to improve over other methods: âCurrently, when there are tied values, no other method has been found that performs reasonably well. Even with no tied values, method HD can provide a substantial gain in power when q â€ .25 or q â„ .75 compared to other techniques that have been proposedâ. The method is described in the paper âComparing two independent groups via the upper and lower quantilesâ by Wilcox, Erceg-Hurn, Clark and Carlson (2013).

You can use the function as soon as you install the latest version of the WRS package following this installation instruction.

Letâs compare all percentiles from the 10th to the 90th:

The graphical output shows how groups differ in the requested quantiles, and the confidence intervals for each quantile:

The text output (see below) also shows that groups differ significantly in the 10th, the 50th, and the 60th percentile. The column labeled ââ.valueââshows the p value for a single quantile bootstrapping test. As we do multiple tests (one for each quantile), the overall Type 1 error (defaulting to .05) is controlled by the Hochberg method. Therefore, for each *p* value a critical *p* value is calculated that must be undercut (see column ‘_crit’. The column ‘signify’ marks all tests which fulfill this condition:

1 0.1 100 100 285.8276 218.4852 67.342399 41.04707 84.67980495 0.005555556 0.001 *

2 0.2 100 100 297.5061 264.7904 32.715724 -16.52601 68.80486452 0.025000000 0.217

3 0.3 100 100 310.8760 320.0196 -9.143593 -33.63576 32.95577465 0.050000000 0.589

4 0.4 100 100 322.5014 344.0439 -21.542475 -40.43463 0.03938696 0.010000000 0.054

5 0.5 100 100 331.4413 360.3548 -28.913498 -44.78068 -9.11259108 0.007142857 0.006 *

6 0.6 100 100 344.8502 374.7056 -29.855369 -46.88886 -9.69559705 0.006250000 0.005 *

7 0.7 100 100 363.6210 388.0228 -24.401872 -47.41493 -4.13498039 0.008333333 0.016

8 0.8 100 100 385.8985 406.3956 -20.497097 -47.09522 2.23935390 0.012500000 0.080

9 0.9 100 100 419.4520 444.7892 -25.337206 -55.84177 11.49107833 0.016666667 0.174

To summarize, we see that we have significant differences between both groups: the red group has significantly more faster RTs, but in their central tendency longer RTs.

Recommendations for comparing groups:

- Always plot the densities of both distributions.
- Make a visual scan: Where do the groups differ? Is the central tendency a reasonable summary of the distributions and of the difference between both distributions?
- If you are interested in the central tendency, think about the test for trimmed means, as in most cases this describes the central tendency better than the arithmetic mean.
- If you are interested in comparing quantiles in the
*tails*of the distribution, use the qcomhd function.

### References

Wilcox, R. R., Erceg-Hurn, D. M, Clark, F., & Carlson, M. (in press). Comparing two independent groups via the lower and upper quantiles. *Journal of Statistical Computation and Simulation.* doi:10.1080/00949655.2012.754026.