A Clarification of Slope and Scale

Abstract

Improvements in the quantification and visual analysis of data, plotted across non-standardized graphs, are possible with the equations introduced in this paper. Equation 1 (an expression of graphic scale variability) forms part of the foundation for Equation 2 (an expansion on the traditional calculation of the tangent inverse of a line’s algebraic slope). These equations provide clarification regarding aspects of “slope” and graphic scaling that have previously confused mathematicians. The apparent lack of correspondence between geometric slope (the angle of inclination) and algebraic slope (the m in y = mx + b) on “non-homogeneous” graphs (graphs where the scale values/distances on the y-axis are not the same as on the x-axis) is identified and directly resolved. This is important because nearly all behavior analytic graphs are “non-homogeneous” and problems with consistent visual inspection of such graphs have yet to be fully resolved. This paper shows how the precise geometric slope for any trend line on any non-homogeneous graph can quickly be determined—potentially improving the quantification and visual analysis of treatment effects in terms of the amount/magnitude of change in slope/variability. The equations herein may also be used to mathematically control for variability inherent in a graph’s idiosyncratic construction, and thus facilitate valid comparison of data plotted on various non-standard graphs constructed with very different axes scales—both within and across single case design research studies. The implications for future research and the potential for improving effect size measures and meta-analyses in single-subject research are discussed.

A proposal for the assessment of replication of effects in single‐case experimental designs

“…A great variety in these two ratios has been found in time‐series line graphs, leading to potential distortions when performing visual analysis (Kubina et al, 2017; Ledford, Barton, Severini, Zimmerman, & Pokorski, 2019; Peltier, McKenna, et al, 2022; Peltier et al, 2021; Peltier, Muharib, et al, 2022). Third, the confounding between slope and scale (Kinney, 2022) is also not likely for modified Brinley plots, given that they are square. Finally, the modified Brinley plot is efficient in that it makes possible representing the results (e.g., within‐phase means and mean differences) for several comparisons within participants and across participants on the same plot, as was illustrated previously in the text.…”

A Proposal for the Assessment of Replication of Effects in Single-Case Experimental Designs

“…A great variety in these two ratios has been found in time-series line graphs, leading to potential distortions when performing visual analysis (Kubina et al, 2017;Ledford, Barton, Severini, Zimmerman, & Pokorski, 2019;Peltier, McKenna, et al, 2022;Peltier et al, 2021;Peltier, Muharib, et al, 2022). Third, the confounding between slope and scale (Kinney, 2022) is also not likely for modified Brinley plots, given that they are square. Finally, the modified Brinley plot is efficient in that it makes possible representing the results (e.g., within-phase means and mean differences) for several comparisons within participants and across participants on the same plot, as was illustrated previously in the text.…”

Evaluating Behavior Modification for Tantrums in Early Childhood: Approaches for Enhanced Management and Intervention

“…4) The techniques used in behavior modification can be explained and regulated rationally, allowing for objective prediction and evaluation of treatment outcomes. 5) The time required to implement these procedures is important (Kinney, 2022).…”