Appendix D: An uncalibrated indicator of folding accuracy

One striking, and so far uncommented, feature of the results of the Sect. 4 (Roscoe 1999A) analysis is the very large increase in the degree of $(R_{{\rm min}},R_{{\rm opt}})$ correlation that arises from the hole-cutting process: specifically, we see that the t-ratio statistic for the coefficient of $R_{{\rm min}}$ in the model $R_{{\rm opt}}\,=\,b_0\,+\,b_1\,R_{{\rm min}}$ increases from 14 to 24 whilst the coefficient of determination increases from 19.2% to 38.6%. The significance of these numerical changes is given a startling visual impact by the "before and after'' diagrams of Figs. 2 and 3 in that paper.

In a qualitative sense, this result is to be expected for the following reasons: given that, after any folding process, absolute errors will inevitably exist in the estimated position of the dynamical centre of any given galaxy, then there will be absolute errors of similar magnitudes in the pre-hole cutting and post-hole cutting estimates of $R_{{\rm min}}$. Since the pre-hole cutting estimate of $R_{{\rm min}}$ is less than the post-hole cutting estimate of $R_{{\rm min}}$, it follows that the corresponding percentage errors in the pre-hole cutting estimates of $R_{{\rm min}}$ will be greater than those in the post-hole cutting estimates of $R_{{\rm min}}$. Consequently, given that $(R_{{\rm min}},R_{{\rm opt}})$ correlations exist in the first place, these correlations must automatically increase through the hole-cutting process.

The real significance of the foregoing results is therefore not that the $(R_{{\rm min}},R_{{\rm opt}})$ correlations increase at all through the hole-cutting process, but is rather that the magnitude of this increase acts potentially as an uncalibrated statistical measure for the magnitude of the original absolute errors in the estimated positions of the dynamical centres taken over the whole sample.

Thus, we might expect that relatively large changes in $(R_{{\rm min}},R_{{\rm opt}})$correlations through the hole-cutting process will indicate the presence of relatively large errors in the estimates of $O_{{\rm dyn}}$ over the set of folding solutions, and vice-versa. These considerations play a significant part in the developments from Sect. 8 onwards.