Organizaciones como Instrumentos de Dominación-Explotación

An MDS procedure that is performed where p is either one, two or three will produce a configuration that may be graphically portrayed, albeit that a three dimensional configuration is likely to need slightly more advanced graphical software. These visualised configurations will entail n points, where n is the number of objects in the data set. Two other outputs from an MDS procedure that are vital in interpretation are the value of stress and the Shepard Diagram. The resulting stress value should be scrutinised before any real analysis of the configuration is to take place. Only results

with stress values low enough need even be considered for further analysis. The reader is referred to Section 2.4.6 for guidelines on stress interpretation. The Shepard Diagram however, should be assessed in conjunction with the configuration, as any observation made on a pairing illustrated by the config- uration must be cross checked with the corresponding point on the Shepard Diagram. Therefore, a major claim based on the supposed relationship be- tween two points is likely to be invalid if the Shepard Diagram reveals it has been inaccurately portrayed.

Figure 2.7 shows a simple non-metric MDS result, using Sammon Map- ping, on the Skulls data. The corresponding STRESS-1 value for this partic- ular result is roughly 0.18, which is considered slightly unsatisfactory. Under normal circumstances the high stress value would warrant further steps be taken, in terms of adjusting parameters of the procedure, however since this has purely demonstrative intent, the current solution will be retained. The figure shows the resulting configuration of the procedure (Figure 2.7(a)) and the corresponding Shepard Diagram (Figure 2.7(b)). Inspection of the configuration reveals only basic observations of the relative relationship be- tween individual points, where points close together are seen to be similar and those further apart are dissimilar. The Shepard Diagram also displays a fairly generic output with the majority of the points lying close to the diagonal, showing accurate fit. The diagram does however also show a num- ber of points lying above the majority trend and other lying below it. This suggests that a number of points have been either under or over stated com- pared to their true distance. However, considering the configuration resulted in a high stress value, this observation is unsurprising.

When analysing the configuration there are two major concepts that must be remembered, both of which have been discussed already but de- serve reiteration. The first of these is that the orientation of the points is completely arbitrary: and following this, the axes of the plotting area are meaningless. The researcher is thus able to rotate, reflect and zoom in and out of the configuration without losing any information, provided that the distances between the points remain constant on a relative scale. The second concept is that most distances between points are usually distorted with con- sideration that longer distances are less distorted. One general explanation

(a) MDS Configuration

(b) Shepard Plot

is provided for this in Section 2.4.6, where it was shown that the Normalised Raw Stress of a longer distance, with an inaccuracy of 1 unit, has a smaller stress value than a shorter distance with the same inaccuracy. In the spe- cific example shown the stress value of the scenario with longer lengths was almost half that of the scenario with shorter lengths. Another valid reason exists, however, when MDS is performed by an MDS computer program. Since all stress calculations are based on a sum of squared difference com- ponent, it follows that bigger discrepancies have the higher influence on stress. Since longer distances have the greater potential to be inaccurate by greater amounts, most MDS algorithms and softwares account for this and focus on achieving higher accuracy on longer distances. Shorter distances can be inaccurate by the same proportion and have much less influence on the stress value. They therefore require less attention by the algorithms.

Ease of interpretation of a configuration may often rely on how the re- searcher has set up the procedure and categorised their data. In the event that the objects in the data set fall under specific categories, it is sug- gested that each category be assigned a different colour. For example, if the subjects of the data set are people, one might consider having the points representing males being one colour and those representing females another. Alternatively, if gender is a non important factor in the study, age groups could be considered important and each group given a different colour, etc. The benefit of colour coding categories is that any notable differences be- tween categories should be picked up from the MDS configuration straight away by the researcher. More subtle differences may take slightly longer to identify and interpret, however the presence of colours will make this visual component more manageable. A more in-depth discussion on the dif- ferent forms of colour coding can be found in Section 4.2.2.7, of which the ‘RColorBrewer’ package is the subject. Figure 2.8 demonstrates the same configuration as before but with the inclusion of category differentiation. The skulls identified to be from males have been coded blue while those from females have been coded pink. Immediately it is noticed that the dif- ferent genders tend to different sides of the configuration; an observation that may not be as clear without the addition of colours.

Figure 2.8: Skull Data With Coloured Categories

the data. Clusters can be highly informative in a sense of suggesting which groups of subjects have exhibited strong relationships to one another. Some- times certain clusters are to be expected and thus serve useful in confirming the similarities, and in other cases the presence of a cluster may provide new information and suggest relationships that were not previously suspected. One vital concept that researchers must be vigilant of when interpreting clusters is that only the presence of clusters may be commented on and not the internal relationships of points within clusters. This follows from the fact that shorter distances tend to be more inaccurate and tight clusters are made up primarily of short distances. Internal analysis of clusters should be carried out by extracting sub matrices from the data which will allow fur- ther MDS procedures to be performed on the cluster individually. This will provide a more accurate representation on the intra-cluster relationships.

The Sammon Mapping configuration shown in Figures 2.7(a) and 2.8 does not demonstrate any clear clusters, however a Classical Scaling output on the same data does suggest some cluster like features may exist. Figure 2.9 shows the Classical Scaling result with a possible cluster identified.

Figure 2.9: Skull Data: Cluster

In the event that p > 3 and consequently the final configuration is in a space that cannot be visualised, the researcher is usually required to perform analyses based on the numerical coordinates. An option however does exist to analyse separate components of the configuration in a visual manner, and attempt to make conclusions by combining the different visuals. So if p = 4, there is the option to plot dimension 1 vs. dimension 2, dimension 3 vs. di- mension 4, etc. In order to perform this analysis in a comprehensive manner with two dimensional mappings a total of p₂
mappings exist. Similarly, if this were to be done using three dimensions, p₃
possible mappings exist.