Validation

4.4.6.4.1 POINT BISERIAL CORRELATION

An interesting result was noted in Point Biserial Correlation validation. While r- values varied, and the strongest of these was produced by the 4-cluster solution, all of the Point Biserial Correlations for different cluster solutions were significant at p < 0.001 for both the original and replication analyses. This suggested a lack of sensitivity to different cluster structures or a very similar clustering arrangement between cases for all clusters (and therefore similar Point Biserial Correlation results). For example, the change from the 5-cluster to the 4-cluster solution was only two ‘outliers’ clustering together which might be expected to affect the correlation less than if two large groups were clustered together. The very small change in r-value between these solutions indicates that this is the case. However a small change in r- values (0.04) also existed between the 6-cluster and 5-cluster solutions. Given two large groups clustered between the 6-cluster and 5-cluster solutions, it might be expected to show a large change in r-value for the Point Biserial Correlation as the cluster solutions were considerably different. As such, the similarity between Point Biserial Correlation values for these two solutions was not due to a small change in clustering. Rather, it indicated that the measure was not sensitive to a large cluster

change at a late stage of clustering (i.e. in the last five cluster solutions moving from 62 clusters to 2 clusters). Table 4.29 repeats the Point Biserial Correlation results for original and replication cluster analyses.

Table 4.29: Point Biserial Correlation results for cluster analyses.

r-value for p=0.001 11 10 9 8 7 6 5 4 3 2

N=62 0.41 0.47 0.52 0.52 0.54 0.58 0.58 0.621 0.622 0.61 0.58

Subset1 (N=41) 0.48 0.56 0.56 0.59 0.66 0.65 0.63

Subset 2 (N=41) 0.48 0.55 0.56 0.57 0.59 0.57

Subset 3 (N=41) 0.48 0.55 0.55 0.61 0.59

This is an important point for this cluster analysis. Point Biserial Correlation would have validated cluster solutions from the 10-cluster solution to the 2-cluster solution based on setting significance at p < 0.001 or effect size large (r > 0.50) for the

original data. Also, all subset cluster solutions would have been validated. This being the case, a poor choice of cluster solution would not have been detected at the

validation stage by Point Biserial Correlation (although a good analysis, as has been performed in this study, should not arrive at a poor solution at the validation stage). This highlights the importance of appropriate selection of the number of clusters in a cluster analysis. It also emphasises the need to use more than one technique to validate the solution as recommended by Milligan (1996).

4.4.6.4.2 REPLICATION

Replication subsets indicated that the larger groups (Front Foot and Reverse) were stable. All cases were reclassified into the same cluster as in the original analysis. That is, all Reverse group golfers were clustered together in each subset while all

Front Foot golfers were clustered together in each subset. Also, the patterns of weight transfer for the Front Foot and Reverse groups were evident in each subset, in spite of reduced numbers in each group. If a pattern is unstable, then a reduced N forming the group means might be expected to alter the pattern. This was not the case in this analysis for either the Front Foot or the Reverse group.

Replication subsets indicated that neither of the two small groups was stable. The Extreme Back Foot Reverse group appeared in two of three subset analyses while the Midstance backswing Front Foot group appeared in only one subset analysis. This supported other evidence that these groups were not valid; the small size of the cluster, the late clustering (7-cluster and 5-cluster solutions) of the two cases and the fact that they were composed of high handicap or social golfers.

While the results for the small groups can be interpreted as being unstable (and based on the criteria set they should be), it needs to be considered in light of what happens to small groups in replication analysis. While a useful validation method, replication disadvantages small clusters. For example, using the Extreme Back Foot Reverse group (N = 2), there was an 89% chance that at least one of the two golfers would be chosen in a sample of two thirds of the original sample [2/3 chance that golfer 1 will be selected + 2/3 chance that golfer 2 will be selected – 4/9 (chance that both are chosen) = 8/9 = 89%]. However there is only 44% chance that both will be chosen. With only N = 2 some instability of the mean can be expected, even for valid groups. Hence the removal of one of the two cases can alter the group mean considerably – enough to move the golfer to another cluster and hence not form the original cluster. While this is the strength of the reclassification procedure as more disparate groups

are less likely to reclassify, the effect will be more pronounced on smaller groups compared with larger groups and so a limitation of the method exists. While a better method of replication might be bootstrapping to generate a large number of datasets (N = 1000 compared with only three in this study), this is not realistic due to the amount of work required. Also, as cluster analysis is heuristic in nature (Milligan, 1996), researcher input is required at different stages of the analysis, limiting automation. It should be noted that if the small cluster was represented in 89% of bootstrap subsets, it would still be outside the likely significance levels that might be set (e.g. appeared in 95% of analyses, p < 0.05).

Small groups (i.e. N = 1 or 2) might be outliers or might represent valid clusters that represent only a small percentage of the population or that have been under-sampled in the study. As such, a negative replication result should not on its own discount the cluster. Rather, other aspects, such as theoretical assessment of the groups, are also required to assess the validity of small clusters. While the small cluster cannot be considered robust for that particular study if it does not appear in replication analysis, it does still exist. As such, discounting a small cluster needs to be supported by strong theoretical arguments. It was for this reason that replication, other validity tests, clustering issues and theoretical assessment were all considered in making the decision that the small clusters in this study were technical errors rather than a valid and unique technique/styles (see section 4.4.4).

4.4.6.5 Leave-one-out reclassification

Both large groups passed ‘leave-one-out’ reclassification, with 100% success. All Front Foot golfers were reclassified into the Front Foot group and all Reverse golfers were reclassified into the Reverse group. This supported the validity of the two large clusters.

Both the Midstance Backswing Front Foot group and the Extreme Back Foot Reverse group failed ‘leave-one-out’ reclassification. This provided support for these clusters being invalid, or outliers. However, similar limitations exist for the ‘leave-one-out’ reclassification method as for the replication method; large groups tend to be

advantaged and small groups tend to be disadvantaged. The removal of one case from a small group is more likely to change the group means more considerably than the removal of one case from a large group. This being the case, it is more likely that small groups will fail the leave-one-out classification procedure. As for replication, the validation results for leave-one-out classification need to be considered along with other indicators as well as a strong theoretical basis for including or discounting the cluster as a valid ad useful technique.