A simplified method to approximate a ROC curve with a Bézier curve to calculate likelihood ratios of quantitative test results

In order to calculate likeli hood ratios (LR) values for quantitative test results, a distribution-independent algorithm based on Bézier curves is proposed. Receiver operating characteristic (ROC) analysis provides the LR as the slope of the tangent to the ROC curve at the point corresponding to the test result. • Here, we make use of cubic Bézier curves defined by Bernstein polynomials of degree 3. • A simplified method to adjust a Bézier curve to a ROC curve is presented • The crucial advantage of this procedure is that Bézier curves are constructed by tangents to the ROC curve, whose slopes immediately provide the LR of a specific point on the curve.


Background: Bayes' theorem
In medical laboratory diagnostics, the measured quantity in a test gives the physician a measure for estimating the significance of a test result for a particular diagnosis. However, it is left to the physician to assess a quantitative result as to its influence on judging the relevance for a particular disease. In principle, the physician starts with a suspicion for a particular disease. In order to confirm or refute the preliminary suspicion, the physician orders a specific laboratory test and a high result gives a stronger confirmation than a low result.
Formally, in Bayes' view: pretest odds multiplied by the likelihood ratio (LR) of the quantitative test result give the posttest odds.

Method
Receiver operating characteristic (ROC) analysis provides the LR as the slope of the tangent to the ROC curve at the point corresponding to the test result. A distribution-independent method has recently been described to calculate these slopes by adjusting Bézier curves to the ROC that are defined by tangents to a curve [1] (Fierz, 2018). The mathematical basis of Bézier curves are Bernstein polynomials of degree n defined by For the purpose here, we make use of cubic Bézier curves defined by Here, a simplified method to adjust a Bézier curve to a ROC curve is presented. The procedure is based on a simplified method for the shape control of cubic Bézier curves [2] . In principle, the shape of the curve is defined by the two endpoints (P 0 and P 3 ) with the corresponding tangents to them and a shape control point B(t) = S (u) on the curve ( Fig. 2 ).

Procedure
Step1: Definition of the variable t of the Bézier curve The variable t of the Bernstein polynomial has to be introduced and is defined here by t xy = ( x + y )/2 of the empirical ROC points.
Step 2: Selection of the endpoints of the Bézier curve ( Fig. 2 ) The endpoints of the Bézier curve P 0 = (x 0 , y 0 ) and P 3 = (x 3 , y 3 ) have to be chosen. Subsequently, the range of t has to be adjusted to the range from 0 to 1 as follows: t xy = ((x-x 0 )/(x 3 -x 0 ) + (y-y 0 )/y 3 -y 0 ))/2 Step 3: Selection of the shape point of the Bézier curve ( Fig. 2 ) A shape point S(u) has to be chosen. It is recommended here that from the empirical ROC points the one which is closest to the upper left corner is used. This corresponds to the maximal Youden index ( J = Se + Sp-1). The t value at that point is denoted u.
Step 4: Selection of tangent vectors at the endpoints of the Bézier curve ( Fig. 2 ) Apart from the endpoints P 0 and P 3 , and the shape point S the shape of the Bézier curve is defined by the two tangents at the endpoints: T 0 = (g 0 , h 0 ) and T 3 = (g 3 , h 3 ) are the tangents vectors at P 0 and P 3 respectively. To start with it is recommended to set Fig. 1. Principle of constructing cubic Bézier curves. First, the lines between the control points P0, P1, P2, and P3 are divided by the ratio t leading to T1, T2, and T3. Second, the lines between T1, T2, and T3 are again divided by the ratio t leading to T4, and T5. Third, the line between T4, and T5 is again divided by the ratio t leading to B(t) on the Bézier curve. The line between T4, and T5 is the tangent to B(t). Pierre Bézier (1910-1999) was a French engineer who developed a method of producing computer-driven curves to be used in the design of automobiles at Renault, which came to be known as Bézier curves.

Fig. 2.
Shape definition of the Bézier curve. Apart from the endpoints P 0 and P 3 , and the shape point S the shape of the Bézier curve is defined by the two tangents at the endpoints. Fig. 3. Calculation of P 1 and P 2 . P 1 and P 2 of the Bézier curve can be calculated according to (Yang & Huang, 1993).
T 0 = (0, 1) (vertical line) and T 3 = (1, 0) (horizontal line) as in an ideal ROC curve. Later, g 0 and h 3 can be adjusted to reach an optimal fit of the Bézier curve to the empirical ROC points.
Step 5: Calculation of P1 and P2 of the Bézier curve ( Fig. 3 ) P 1 and P 2 of the Bézier curve can be calculated according to (Yang & Huang, 1993) with two shape parameters α and β and the following relations: where the notation | | denotes the determinant operator Step 6: Calculation of the tangent slopes ( Fig. 4 ) With P0,1,2,3 being established in this way, the slopes of the tangents, i.e. the LR(t), can be calculated for all t with

Validation
To demonstrate the procedure, data from a prosteate-specific antigen (PSA) study are used [3] (Thompson et al., 2005) with the ROC analysis: Gleason Grade > 7 ( n = 250) vs Gleason Grade < 7 or No Cancer ( n = 5325):  "Conclusion (by the authors): There is no cutpoint of PSA with simultaneous high sensitivity and high specificity for monitoring healthy men for prostate cancer, but rather a continuum of prostate cancer risk at all values of PSA." Step 7: Application to PSA test results ( Figs. 5 and 6 ) In either way, preferring the method that gives the best fit, the diagnostic LR can be calculated from all quantitative test results.