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Section 2.3, entitled The General MDS Algorithm, made reference to a num- ber of adjustable parameters that are involved in most Multidimensional Scaling procedures. While most of these parameters have been elaborated on in previous sections, this Section will summarise the decisions a researcher will need to make when performing MDS procedures. This will be done with particular emphasis on the choices that must be made before a first attempt at MDS. Each of these parameters is expected to affect the final coordinate configuration in some way, albeit some tend to have a greater influence than others.

Firstly, before any form of ordination can be attempted, it is naturally important that the multidimensional data be decided upon. Referring to Section 2.8, it can be seen that there are a number of choices that must be made with regards to the proximity matrix used in the MDS process. These choices are namely: deciding whether to form the proximity matrix directly (manual counting) or indirectly (proximity calculations); having the prox- imities as similarities or dissimilarities; and finally accepting a symmetric

form of the matrix or allowing asymmetry amongst the proximities. In the event that an asymmetric proximity matrix is accepted, it should be noted that this will double the number of points in the configuration and increase the number of points in the Shepard Diagram to 2n₂
. For convenience, it will be assumed that all proximity matrices are symmetric throughout the remainder of the dissertation.

With the input data decided upon, the crucial decision of which form of Multidimensional Scaling must be made. In many cases, the form of the proximities will have an influence on the type of MDS to be used. If the proximities place importance on the extent of the dissimilarities, then a metric MDS is appropriate, while a non metric MDS should be used if only rank order of the proximities are relevant. Within each of the categories there are further choices of the specific MDS, all of which will be covered in Chapter 3. The researcher may want to assess the results of a number of the various methods for comparison.

The output of the ordination then needs to be defined in terms of p, the number of dimensions in which the points will be mapped. The concept of the number of dimensions on which to derive the MDS configuration is discussed in Section 2.5 and the Scree Plot in Section 2.6. The researcher may have a preexisting idea of the most appropriate number of dimensions for the specific data, and if not, the Scree plot will be useful in determin- ing what p should be. In practice, a common method is to first perform MDS in two dimensions and then make an assessment of whether additional dimensions will be appropriate or necessary. Some care must be put into the choice of p as the ordination is sensitive to it. If too few dimensions are used, multiple axes of variation will be portrayed by a single ordination dimension, whereas too many dimensions will cause a single axis of varia- tion to be portrayed over many dimensions of ordination (Holland, 2008). In order to explain this, the example of mapping the distance between the tops of buildings is revisited. Three axes of variation for the scenario could be seen as the axis of latitude, the axis of longitude and the axis of altitude. In the event that a one dimensional ordination is performed, the three axes of variation will be portrayed on a single dimension. If alternatively, the ordination is done in more than three dimensions, the three axes of varia-

tion are portrayed by too many dimensions. In both cases the nature of the differences between the building tops is nearly impossible to determine by observing the MDS configuration. This example makes very literal use of the word ‘axes’: realistically the axes of variation may require somewhat more thought in their interpretation, such as aggressiveness when exploring data on breeds of dog. Details on how to interpret these axes of variation will be provided in Section 2.10. A rule of thumb suggested by some specialists of MDS is that a p dimensional ordination requires at least 4p variables in the raw data (Wickelmaier, 2003). This is clearly just a guide line however, as a data set consisting of only six or seven variables should still be able to have p = 2 without any problem. This is despite the rule specifying that there should be at least eight variables for this ordination output.

The starting configuration should then be provided. With the exception of Classical Scaling, every form of MDS requires an initial configuration of n points in the p dimensional space. The starting configuration is a parameter that tends to have a substantial influence on the final configuration of the result. The default starting configuration is often accepted to be the result of a Classical Scaling on the data, however it may be of interest to instead use a completely random set of coordinates or perhaps the resulting configuration obtained by some other method of ordination. The starting configuration is usually something that should be experimented with, as it should be noted how susceptible a set of data is to different starting positions.

The final parameters that need to be set are those that correspond to how the progress of the MDS process is being monitored. The form of stress analysing the configuration of each iteration must be defined and the tolerance of differences between stress of iterations needs to be set. Sec- tion 2.4 provides a few examples of the variation of stress formulas, namely: STRESS-1, STRESS-2 and Normalised Raw Stress. With regards to the tolerance: the higher the value the less the number of iterations and there- fore, the possibility of a less reliable result. Alternatively a very small level of tolerance will imply more iterations and the probability of a more reli- able configuration, however the procedure becomes more time consuming and more computationally intensive. The most appealing trade-off between the two extremes is a tolerance that is small enough to bring about an ad-

equately converged result where if the value were to be lowered the change in the result would be negligible. In practice a tolerance of say 0.001 tends to be more than adequate.