Technical Note Reduction of artefact in scatter plots of spherocylindrical data W. F. Harris Optometric Science Research Group, Department of Optometry, University of Johannesburg, PO Box 524, Auckland Park, 2006 South Africa Abstract Round-off of spherocylindrical powers, to multiples of 0.25 D (for example) in the case of sphere and cylinder, and 1 or 5° in the case of axis, represents a type of distortion of the data. The result can be artefacts in graphical representations, which can mislead the researcher. Lines and clusters can appear, some caused by moire ´ effects, which have no deeper significance. Furthermore artefacts can obscure meaningful information in the data including bimodality and other forms of departure from normality. A process called unrounding is described which largely eliminates these artefacts; each rounded power is replaced by a power chosen randomly from the powers that make up what is called the error cell of the rounded power. Keywords: artefact, dioptric power matrix, dioptric power space, graphical representation, round- off, spherocylindrical power, statistics Scatter plots in three dimensions are a useful adjunct to quantitative analyses of refractions and other sphero- cylindrical powers (Harris, 1991, 2001a; Harris and Malan, 1992a,b; Elliott et al., 1997; Raasch, 1997; Thibos et al., 1997; Naeser and Hjortdal, 2001; Rubin and Harris, 2001). For example they can reveal struc- tures in the data including skewness, multimodality, outliers and other forms of departure from normality. Occasionally, however, they exhibit spurious structures that are a consequence of round-off (Malan, 1994; Rubin and Harris, 1995; Blackie and Harris, 1997; Cronje´ and Harris, 1999; Gillan, 2000; van Gool, 2000; also see references cited in Harris, 2001b). The resear- cher may be misled into reading structures into the data that have no basis in reality apart from the round-off. Furthermore meaningful structures in the data may be obscured. The purpose of this note is to describe a simple technique, which more or less removes artefact that arises out of round-off. Consider 1.21 )0.60 · 33 for example. We shall refer to it as a hypothetical power. Following common practice we round it to the nearest multiple of 0.25 D for sphere and cylinder and 5° for axis. We obtain the rounded power 1.25 )0.50 · 35. There are an infinite number of hypothetical powers that round to the same rounded power, 1.25 )0.50 · 35. We call this set of hypothetical powers the error cell (Harris and Abelman, 2001; Harris, 2001b, 2002) for the rounded power 1.25 )0.50 · 35. Rounding as we have just done represents a bias, a subjective distortion of the data. It is subjective because is depends on the arbitrary choice of units and the rounding routine. An alternative choice of units or rounding routine would result in a different rounded power. By rounding powers in a set of powers in the same way one is introducing a consistent bias the cumulative result of which may be spurious patterns in the data. Examples based on made-up spherocylindrical data are shown in Figure 1 (See the caption to the figure for more details). In an attempt to overcome the problem we might try to ÔunroundÕ the data. If presented with 1.25 )0.50 · 35, for example, we might suppose that the hypothetical power was 1.26 )0.49 · 36. We shall distinguish the latter as the unrounded power. Of course, beyond the constraints implied by the rounding routine, such a power is no more than a guess. In the absence of any additional information Received: 1 July 2004 Revised form: 12 August 2004 Accepted: 23 August 2004 Correspondence and reprint request to: W. F. Harris. Fax: +27-11-489-2091. E-mail address: wfh@na.rau.ac.za Ophthal. Physiol. Opt. 2005 25: 13–17 ª 2005 The College of Optometrists 13