Sensitivity of AJUS to tolerance parameter

Last week I introduced Galtung’s AJUS system and its implementation in R. As noted, the tolerance parameter is not a trivial matter, but we can use the provided function ajusCheck to explore how different tolerance parameters affect the outcome. Basically we throw a number of tolerance parameters at the ajus function, and see what happens.

In this example, the default values give type “S” (i.e. multi-peaked). Next, we visually inspect the distribution using ajusPlot:


So maybe the last increase is not significant, so we can try a different tolerance value (e.g. 0.2). We can continue this, or just try them all (e.g. between 0.1 and 2):

library(agrmt) # install from R-Forge
# Data:
V <- c(0,0,1,2,1.5,1.6)
ajus(V) # using default tolerance = 0.1
ajusPlot(V) # visual inspection
ajus(V, tolerance=0.2) # exploring
ajusCheck(V, t=seq(0.1,2,0.1)) # let's try them all

[1] 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9
[20] 2.0

[1] “S” “A” “A” “A” “J” “J” “J” “J” “J” “J” “J” “J” “J” “J” “J” “F” “F” “F” “F”
[20] “F”

Which is it? Usually the substantive meaning of our numbers is the guide but sometimes we deal with constructs that are more difficult to assess. Using ajusCheck we can easily check whether we should spend hours thinking about whether 0.2 or 0.3 is more appropriate – or whether we get the same result.

Galtung’s AJUS System

Galtung (1967) introduced the AJUS system as a way to classify distributions according to shape. This is a means to reduce complexity. The underlying idea is to classify distributions by ignoring small differences that are not important. The system was originally developed for eye-balling, but having it done by a computer makes the classification more systematic.

All distributions are classified as being one of AJUS, and I have added a new type “F” to complement the ones identified by Galtung.

  • A: unimodal distribution, peak in the middle
  • J: unimodal, peak at either end
  • U: bimodal, peak at both ends
  • S: bimodal or multi-modal, multiple peaks
  • F: flat, no peak; this type is new

The skew is given as -1 for a negative skew, 0 for absence of skew, or +1 for a positive skew. The skew is important for J-type distributions: it distinguishes monotonous increase from monotonous decrease.

I have implemented the AJUS system in my R package agrmt. By setting the tolerance, we can determine what size of differences we consider small enough to be ignored. The default tolerance is 0.1, equivalent to 10% if using 0 to 1. AJUS implemented in R sets a systematic threshold, something we do not do when eye-balling differences.

The tolerance parameter is not a trivial choice, but a test is included in the R package to directly test sensitivity to the tolerance parameter (ajusCheck).

Here are some examples (using the experimental ajusPlot function and tolerance = 10):
Differences smaller than the tolerance set (10) are ignored.

Reference: Galtung, J. 1969. Theory and Methods of Social Research. Oslo: Universitetsforlaget.