## 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 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```

```\$tolerance [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```

\$type
[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.