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bioRxiv · 10.1101/2025.08.18.670747

Graphical examples used to show why caution is required if using the coefficient of determination (R2) to interpret data for medical case reports.

Hurr, T. J.
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Abstract

A patient with a medical condition can have medical tests or symptoms scored that generate numerical results before a treatment, during a treatment or after a treatment, usually over several days, to determine if any benefits have occurred. The changes in the numerical measurements or scores over time can be readily plotted using computer software to show an equation for the line of best fit for either linear or log equations, together with the coefficient of determination (R2). Despite the ease of generating this type of graphical representations caution is required in interpreting the R2 value with reference to medical case reports. To understand why this is so, at a basic level, four scenarios using hypothetical patient scores were used to generate scatter plots showing the equation for the line of best fit and R2 values with comparison to the average and standard deviation (SD) values. The graphical examples are used to supplement the more complex mathematical and statistical explanations and choice for effect measures that are available. It was found R2 values for log equations for the line of best fit did not follow a trend with increasing treatment days. For linear equations, higher R2 value may not necessarily correspond to a lower standard deviation (SD) value for the averaged scores. The R2 value can be influenced by the day on which the scores were recorded, despite the equivalence of the average scores and SD values. R2 values may not indicate the strength of a treatment benefit or the magnitude of scatter between data sets. Score averaging can increase R2 values, while average values remain the same but with the SD value decreasing. The graphical examples shown provide an explanation why line graphs may be the simplest and best option for reporting, particularly non-linear numerical data, in case reports. Graphical Abstract O_FIG O_LINKSMALLFIG WIDTH=154 HEIGHT=200 SRC="FIGDIR/small/670747v2_ufig1.gif" ALT="Figure 1"> View larger version (34K): org.highwire.dtl.DTLVardef@1ad866org.highwire.dtl.DTLVardef@75282forg.highwire.dtl.DTLVardef@1a1592forg.highwire.dtl.DTLVardef@1e64166_HPS_FORMAT_FIGEXP M_FIG Graphical examples of the line of best fit and R2 values from hypothetical patient scores are compared with average (Av.) and standard deviation (SD) values A. From the line of best fit, Patient 1 has a higher R2 value than Patient 2 even though the average score has a higher SD value. B. Patient 1 records scores on days 6 and 7 and Patient 2 records the same scores on days 9 and 10, yet Patient 1 has a higher R2 value for the line of best fit despite the scores average and SD values being the same. C. For Patients 1 and 2, the R2 values for the line of best fit are the same, despite the score averages and SD values being different and show R2 values do not predict a treatment benefit or allow a comparison of the magnitude of a benefit between data sets. D. Averaging daily scores removes scatter, increasing R2 values however the average scores remain the same, but the SD value ({+/-} 0.51) was reduced despite an identical slope and intercept for the line of best fit. C_FIG

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Hurr, T. J.. 2025-08-22. Graphical examples used to show why caution is required if using the coefficient of determination (R2) to interpret data for medical case reports.. https://doi.org/10.1101/2025.08.18.670747

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