A medical researche at the University of Warwick has found the 2.500 year-old Pythagoras theorem could be the most effective way to identify the point at which a patient´s health begins to improve.

 
In a paper published in PLOS ONE, Dr. Rob Froud from Warwick Medical School at the University of Warwick worked with his colleague Gary Abel from the University of Cambridge. They made the discovery after looking at data from ROC (Receiver Operating Characteristic) curves.

 
Dr. Froud said:” It all comes down to choosing a point on a curve to determine when recovery has occurred. For many chronic conditions, epidemiologists agree that the correct point to choose is that which is closest to the top-left corner o the plot containing the curve. As we stopped to think about it, it struck us as obvious that the way to choose this point was by using Pythagorus Theorem. We set about exploring the implications of this and how it might change conclusions in research. We conducted several experiments using real trial data and it seems using Pythagoras´theorem makes a material difference. It helps to identify the point at which a patient has improved with more consistency and accuracy than other methods commonly used”.

 
Receiver Operator Characteristic (ROC) curves are being used to identify Minimally Impotant Change (MIC) thresholds on scales that mesaure a change in health status. In quasi-continous patient reported outcome measures, such as those that measure changes in chronic diseases with variable clinical tajectories, sensitivity and specificity are often valued equally.

 
ROC curves have may uses, all of which fall into the theme of comparing a continous (or quasi-continous) measure with a dichotomous one.
In statistics, a receiver operating characteristic curve, i.e. ROC curve, is a graphical plot that illustrates the diagnostic ability o a binary classifier system as its discrimination threshold is varied.

 
The ROC curve was first developed by electrical engineers and radar engineers during World War II for detecting enemy objects in battlefields and was son introduced to psycology to account for perceptual detection of stimuli. ROC analysis since then has been used in medicine, radiology, biometrics, forecasting of natural hazards, meteorology, model performance assessment, and other areas for many decades and is increasingly used in machine learning and data mining research.

 
The ROC is also known as a relative operating characteristic curve, because it is a comparison of two operating characteristics (TPR and FPR) as the criterion changes.
The ROC curve is created by plotting the true positive rate (TPR) against the false positive rate (FPR) at various threshold settings. The true-positive rate is also kown as sensitivity, recall or probability of detection in machine learning. The false-positive rate is also known as the fall-out or probability of false alarm and can be calculated as (1-specificity). In general, if the probability distributions for both detection an false alarm are known, the ROC curve can be generated by plotting the cumulative distribution function (area under the probability distribution from to the discrimination threshold) of the detection probability in the y-axis versus the cumulative distribution function of the false-alarm probability on the x-axis.

 
The diagonal divides the ROC space. Points above the diagonal represent good classification results (better than random), points below the line represent poor results (worse than random). Note that the output of a consistently poor predictor could simply be inverted to obtain a good predictor.

 
Pythagorean theorem and health

 
Since the fourth century AD, Pythagoras has commoly been given credit for discovering the Pythagorean theorem, a theorem in geometry that states that in a right-angled triangle the area of square on the hypotense ( the side opposite the right angle) is equal to the sum of the areas of the squares o the other two sides.

 
By plotting a ROC curve we display the sensitivity and specificity of the change in a continous measure for detecting a dichotomous judgement about change.
It should be obvious that to find the point closest to the top left hand corner, one can calcúlate the distance of each point to the corner and select the one where that distance is smallest. To calcúlate that distance you can apply Pythagoras´theorem. So if (1-specificity) and (1-sensitivity) form two sides of a right angled triangle, the distance to the corner is the hypotenuse.

 
Choise of MIC (Minimally Important Change) estimator is important. Different methods for choosing cut-points can lead to materially different MIC threscholds and thus affect results of responder analyses and trial conclusions. An stimator base don the smallest sum of squares of 1-sensitivity and 1-specificity is preferable when sensitivity and specificity are valued equally. Unlike other methods currently in use, the cut-point chosen by teh sum squares method always and efficiently chooses the cut-point closest to the top-left corner of ROC space, regardeless of the shape of the ROC curves.

Bibliography:
• “Detector Performance Analysis Using ROC Curves- MATLAB& Simulink Example”
http://www.mathworks.com. Retrieved 2016

• Ton J. Cleophas; AH. Zwinderman; Toine F. Cleophas; Eugene P. Cleophas, “Statistics Applied to Clinical Trials” 2006

• Froud ,R,; Abiel, G, “Using ROC curves to choose minimally important change thresholds when sensitivity and spcecificity are valued equally: the forgotten lesson of Pythagoras. Theoretical considerations and an example application of change in health status” PLOS One 2014

• Wikipedia

Relation links:

• Cambridge; Curves ROC and Pythagoras
https://www.cchsr.iph.cam.ac.uk/2174

• Froud ,R,; Abiel, G, “Using ROC curves to choose minimally important change thresholds when sensitivity and spcecificity are valued equally: the forgotten lesson of Pythagoras. Theoretical considerations and an example application of change in health status” PLOS One 2014
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4256421/#!po=55.2632