78 group and between group variance was carried out.

Writers have commented on the variations in results obtained using 79

different grouping procedures. To check the results of the cluster analysis, the same data were also analysed by means of Veldman's

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hierarchical grouping programme (HGR0UP). The principal

The conversion of distance measures to percentage loss of detail is described by Haggett, Peter, Locational Analysis in Human Geography, Edward Arnold, London, 1966, pp.256-7. The procedure was used by Johnston, R.J., 'Choice in Classification: The Subjectivity of

Objective Methods', Annals of the Association of American Geographers, 1967, 58, pp.575-89.

7 8

Mather noted that, in addition to analysis of variance, a method of testing the validity of groupings - such as discriminant analysis - should be employed, unless there are prior grounds for believing that some structure is present within the data. In the present example, the cluster items were all measures of population change and since there were meaningful relationships between the rates, further testing of the

validity of the groupings was not considered essential. See Mather, P.M., 'Areal Classification in Geomorphology', in Chorley, Richard J. (ed.), Spatial Analysis in Geomorphology, Methuen, London, 1972, pp.313-4. 7 9

For example, Johnston, 'Choice in Classification ...'. 80

Veldman, D.J., Fortran Programming for the Behavioral Sciences, Holt, Rinehart and Winston, New York, 1967. The author is grateful to

Mr L. Neilson, formerly of the Urban Research Unit, ANU for advice on HGR0UP.

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TABLE 2.5 Means and Standard Deviations of Rural Population Change Groups, (a) Mean Rates of Population Change in Rural Areas

Group f 1947-54

Annual growth rate percent

1954-61 1961-66 1966-71 1 7 1.77 3.79 4.32 6.16 2 9 2.51 0.22 1.16 2.52 3 4 0.49 0.17 -2.35 0.84 4 13 1.81 1.83 1.74 0.11 5 67 1.13 0.22 -0.55 -1.69 6 15 0.31 -1.85 -1.16 -1.55 7 6 3.82 4.33 0.80 -4.17 8 6 4.97 1.53 4.93 -1.49 Ungrouped 10 Total 137 (b) Standard Deviations Group f 1947-54 1954-61 1961-66 1966-71 1 7 3.36 1.79 2 .40 2.14 2 9 3.23 0.55 0.93 1.59 3 4 1.19 0.62 0.92 0.82 4 13 1.56 0.45 0.95 1.15 5 67 1.54 0.78 0.64 1.18 6 15 1.28 0.66 0.60 1.57 7 6 5.71 2.23 0.81 1.60 8 6 3.95 1.10 1.06 2.84

Note: f = frequency of cases in each group.

Source: Cluster analysis (CLUSTAN) of revised rural population totals. difference between CLUSTAN and HGROUP is that the former uses centroid replacement, while the latter assigns a new value to one of the original objects and deletes the other member of the pair. Thus HGROUP is less

space conserving than CLUSTAN as the representative value for each group migrates as new members are added. In CLUSTAN the centroid values become more inert as group membership increases.

From the Victorian data, HGROUP yielded twelve groups, the level of error being 9.3. This level was chosen as the most appropriate one by applying Veldman's criterion that a marked increase in the error index denotes the stage below which the groupings are most worthy of study. For eleven and thirteen groupings, the errors were 8.1 and 12.1 respectively. Table 2.6 shows the comparative results of the HGROUP and CLUSTAN analyses.

TABLE 2.6 Victorian Rural Areas - Membership of HGROUP and CLUSTAN Groups. HGROUP 1 2 CLUSTAN 3 4 5 6 7 8 Ungrouped Total Rural Areas 1 6 6 2 1 5 13 3 3 1 26 3 1 1 1 3 4 5 5 5 1 1 6 2 1 3 7 3 4 64 2 73 8 2 2 9 13 1 14 10 2 2 11 1 1 12 1 1 Total Rural Areas 7 9 4 13 67 15 6 6 10 137

Examination of the columns of the table indicates that CLUSTAN groups 1, 3, 4, 5, 6 and 7 were not substantially subdivided by HGROUP. The few areas which were separated from the CLUSTAN groupings were mostly characterised by one or more extreme values. Considering their low frequencies, groups 2 and 8 were more seriously broken up by HGROUP and the presence of

extreme values cannot adequately account for the subdivision. It is likely that some members of these groups occupy a transitional location in four-dimensional space so that under the HGROUP linkage strategy they are partitioned early between different groups, whereas in CLUSTAN they coalesce and retain their separateness at a higher level. Inspection of the rows of Table 2.6 shows that the HGROUP classes 1, 4 and 9 correspond to CLUSTAN groups, while 2 and 7 combine elements of a number of CLUSTAN groups. However, by accepting the HGROUP classes at a lower index level than 9.3, a better approximation to the CLUSTAN groupings is achieved. This suggests that the grouping level indicated by a sudden rise in the error index in HGROUP may not necessarily be the most suitable level to

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adopt for study. The discrepancies between the results of CLUSTAN and HGROUP appear to be minor, the main problem being to determine the grouping

Mather stated that there is no general automatic method of deriving the 'optimum' cut-off point. Mather, 'Areal Classification in Geomorphology', p.313.

level in HGROUP which corresponds to that selected in CLUSTAN. This does not rule out the possibility that other grouping strategies could produce different results, and comparative research in numerical taxonomy remains a field which needs further attention.

Although Figure 2.4 is presented as a summary of the detail mapped in Figure 2.2, three points must be stressed. First, the cluster analysis utilised ungrouped data, and differences such as between 0.1 and -0.1 were less critical than they were in the construction of Figure 2.2. Second, the patterns depicted in the key to Figure 2.4 are only mean trends in the rates of population change, and departures from the mean values are sometimes substantial (Table 2.5). Third, the data used in the cluster analysis were standardised whereas those employed in the other maps (Figure

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