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Positive and negative predictive values by the TOC curve

Chong Sun Hong1,a, So Yeon Choia

aDepartment of Statistics, Sungkyunkwan University, Korea
Correspondence to: 1Department of Statistics, Sungkyunkwan University, 25-2 Sungkyunkwan-ro, Jongno-gu, Seoul 03063, Korea. E-mail: cshong@skku.edu
Received October 11, 2019; Revised December 10, 2019; Accepted January 18, 2020.
Abstract
Sensitivity and specificity are popular measures described by the receiver operating characteristic (ROC) curve. There are also two other measures such as the positive predictive value (PPV) and negative predictive value (NPV); however, the PPV and NPV cannot be represented by the ROC curve. Based on the total operating characteristic (TOC) curve suggested by Pontius and Si (International Journal of Geographical Information Science, 97, 570–583, 2014), explanatory methods are proposed to geometrically describe the PPV and NPV by the TOC curve. It is found that the PPV can be regarded as the slope of the right-angled triangle connecting the origin to a certain point on the TOC curve, while 1 − NPV can be represented as the slope of the right-angled triangle connecting a certain point to the top right corner of the TOC curve. When the neutral zone exists, the PPV and 1−NPV can be described as the slopes of two other right-angled triangles of the TOC curve. Therefore, both the PPV and NPV can be estimated using the TOC curve, whether or not the neutral zone is present.
Keywords : accuracy, hypotenuse, performance, sensitivity, specificity, threshold
1. Introduction

Assume that the set C that represents patient states consists of two elements, C = {0, 1}, where elements 0 and 1 represent non-disease / negative and disease / positive states, respectively; and let X be a random variable that measures disease or non-disease. The cumulative distribution function of X, F(x), is assumed to be a linear combination of a distribution function Fd(x) = P(Xx|C = 1) representing a disease state of sample size p, and a distribution function Fn(x) = P(Xx|C = 0) representing a non-disease state of the size q, as follows: F(x) = π1Fd(x) + (1 – π1)Fn(x), where π1 ∈ (0, 1) is regarded as the total probability of disease, and can be estimated as p/(p + q), the ratio of the sizes of the two groups. Fd(x) and Fn(x) are also considered as probability classifiers with Fn(x) ≥ Fd(x) for all x (Metz and Kronman, 1980; Hsieh and Turnbull, 1996; Provost and Fawcett, 2001; Engelmann et al., 2003; Pepe, 2003; Fawcett, 2006).

Classification results can be expressed by a confusion matrix. The true positive (TP) and true negative (TN) in the confusion matrix represent the numbers of correctly classified diseased and non-diseased populations. The true positive rate (TPR, sensitivity) and true negative rate (TNR) have the following relationship: for any threshold (cut-off point) x, $TPR = TP p = 1 - F ^ d ( x ) , TNR = TN q = F ^ n ( x ) .$

However, the false positive (FP) and false negative (FN) are the numbers of populations that predicted disease as non-disease, and predicted non-disease as disease, respectively. The false positive rate (FPR, 1 – specificity) and false negative rate (FNR) have the following relationship: $FPR = FP q = 1 - F ^ n ( x ) , FNR = FN p = F ^ d ( x ) .$

The total number of disease groups, p, is TP + FN in the population, and the number of non-diseased populations, q, is FP+TN. Assume that both p and q are known (Provost and Fawcett, 1997; Fawcett, 2003; Stein, 2005; Hong et al., 2009).

The ‘sensitivity’ and ‘specificity’ are the basic accuracy measures of diagnosis. Whereas the ‘sensitivity’ is the probability that a patient with an actual disease will be positive (Ĉ = 1), ‘specificity’ is the probability of considering a non-diseased person to be negative (Ĉ = 0). Both the ‘sensitivity’ and ‘specificity’ are statistics measured in terms of evaluating the accuracy and performance of the diagnosis that are expressed by the following probability equations for any threshold x on the receiver operating characteristic (ROC) curve (Zhou et al., 2002): $sensitivity = P ( C ^ = 1 | C = 1 ) , specificity = P ( C ^ = 0 | C = 0 ) .$

When the evaluation result of the disease is positive or negative, the probability of the test performance that an actual health condition is disease or non-disease may be required, rather than how accurate the diagnosis is. There exist two other measures: the positive predictive value (PPV) and negative predictive value (NPV) suggested by Raslich et al. (2007) and Zhou et al. (2002). The PPV is the probability that the patient has the disease following a positive test result, and the NPV is the probability that the patient does not have the disease following a negative test result. The PPV tells the clinician what percent of those with a positive finding have the disease; however, the NPV reveals what percent of those with a negative result do not have the disease. For any threshold x, the PPV and NPV can be expressed as (Raslich et al., 2007): $PPV = P ( C = 1 | C ^ = 1 ) , NPV = P ( C = 0 | C ^ = 0 ) .$

The neutral zone was proposed to ensure the retesting of diseases with high diagnostic costs. Both the PPV and NPV were also derived when the neutral zone is present (Daniel and Steven, 2016).

Based on statistical decision theory, the ROC curve is a visual tool that can easily identify the classifier’s performance in binary classification (Egan, 1975; Zweig and Campbell, 1993; Bradley, 1997; Fawcett and Provost, 1997; Pepe, 2000; Hong and Lee, 2018). Since the ROC curve is implemented with (FPR, TPR) = (1 – specificity, sensitivity) in a unit length square, the ‘sensitivity’ and ‘specificity’ can be represented by the ROC curve. In contrast, the PPV and NPV cannot be represented by the ROC curve. Pontius and Si (2014) proposed the total operating characteristic (TOC) curve, which is expressed with the numbers of observations of TP, FN, FP, and TN. The TOC curve is implemented as a curve in a parallelogram with a hypotenuse slope of 45 degrees, belonging to a rectangle, where the lengths of the horizontal and vertical axes are p + q and p, respectively.

In this study, we investigate how the PPV and NPV can be represented by the TOC curve, neither of which can be expressed by the ROC curve. We explain their meanings in terms of geometry, then both the PPV and NPV are examined in relation to the shape of the TOC curve. New explanatory methods are proposed to geometrically describe the PPV and NPV by the TOC curve. In addition, the PPV and NPV are also geometrically described and implemented by the TOC curve, when the neutral zone is present. We discuss these relationships between the descriptions of the PPV and NPV by the TOC curve mathematically.

Section 2 of this paper introduces the TOC curve, explains definitions of the PPV and NPV as well as geometrically discusses the representations by the TOC curve of both the PPV and NPV, so that their representations by the TOC curve could be explored. Section 3 discusses and derives the PPV and NPV with probability equations when the neutral zone is present; expresses the PPV and NPV in this case by the TOC curve, and then geometrically compares this description of the TOC curve with those represented in Section 2. Section 4 generates two random samples representing disease and non-disease states with different sample sizes, then obtains and discusses the values of PPV and NPV with interpretations derived in this work. This section also generates two random samples in the presence of the neutral zone with some kinds of two types of errors, and discusses and explores some results based on the neutral zone. Finally, some conclusions are summarized in Section 5.

2. PPV and NPV by the TOC curve

### 2.1. TOC curve

If two samples are skewed, imbalanced data or very different in size, p and q, the well-known ROC curve has the disadvantage of insufficiently expressing the characteristics of the classification results (Fawcett, 2006; Garcia et al., 2010). Since the ROC curve describes the entire information only in terms of TPR and FPR, it is difficult to explain the frequency of their confusion matrix (Pontius and Si, 2014). To make up for the problems of these ROC curves, Pontius and Si (2014) proposed the TOC curve expressed with cell observations of the confusion matrix.

Figure 1 shows a TOC plot. The unit on both the horizontal and vertical axes of the TOC plot indicates the number of observations. The horizontal axis ranges from 0 to p + q, which is the total number of people in the disease and non-disease groups, respectively. The vertical axis ranges from 0 to p, which is the size of the disease group. The TOC curve is implemented as a curve in a parallelogram with a hypotenuse slope of 1 (45 degrees), belonging to a rectangle whose lengths of horizontal and vertical axes are p + q and p, respectively. The coordinate on the horizontal axis indicates the number of diseased in the diagnosis, TP(x) + FP(x), for a given threshold x, while the vertical axis indicates the number of correctly classified diseases TP(x). That is, the coordinate of TOC curve is given by (TP(x) + FP(x), TP(x)) for a given threshold x.

From any point A whose coordinate is (TP(x) + FP(x), TP(x)) on the TOC curve in Figure 1, the lengths to each side of a parallelogram are represented as TP, FP, TN, and FN of the confusion matrix. That is, the lengths from a point A on the curve to the base and the top sides of the parallelogram are TP and FN, respectively; while the lengths to the left and right sides of the parallelogram are FP and TN, respectively. Therefore, there is the advantage that it is possible to identify and describe information for confusion matrices that cannot be derived from the ROC curve (Pontius and Si, 2014).

### 2.2. PPV and NPV by the TOC curve

Both conditional probabilities PPV and NPV of the actual disease or non-disease state given a positive or negative diagnosis result in ( 1.1), can be expressed with frequencies of the confusion matrix as shown in ( 2.1). For any threshold x, $PPV ( x ) = TP ( x ) TP ( x ) + FP ( x ) , NPV ( x ) = TN ( x ) TN ( x ) + FN ( x ) .$

The denominators of PPV(x) and NPV(x) in ( 2.1) are TP(x) + FP(x) and TN(x) + FN(x), respectively, so it is difficult to derive and explain their meanings with the ROC curve, where the vertical and horizontal axes are TPR and FPR, respectively. However, since the total length of the horizontal axis in the TOC curve is p+q, which equals the sum of the denominators of PPV(x) and NPV(x), PPV(x) and NPV(x) could be described by the TOC curve. Based on the coordinate ((TP(x) + FP(x)), TP(x)) of point A on TOC curve in Figure 1, its horizontal and vertical coordinates can be found to be the denominator and numerator of PPV(x), respectively. Hence, PPV(x) can be described by the slope from the origin to point A, considering the right-angled triangle connecting the origin and point A. From now on, 1–NPV(x) may be considered, rather than NPV(x). Then, 1–NPV(x) = FN(x)/(TN(x)+FN(x)). Note that the denominator of 1 – NPV(x) is TN(x) + FN(x), which is the total length of the horizontal axis p+q in the TOC curve minus TP(x)+FP(x), and its numerator is FN(x). Hence, we might regard 1 – NPV(x) as the slope of the right-angled triangle connecting point A and the top right corner of the TOC curve. Therefore, both PPV(x) and 1 – NPV(x) can also be expressed as the slopes of two right-angled triangles as in Result 1:

### Result 1

For any threshold x, both PPV(x) and 1 – NPV(x) could be described as the angles of the two following right-angled triangles on the TOC curve:

• PPV(x) is the slope of the right-angled triangle connecting the origin to point A,

• 1 – NPV(x) is the slope of the right-angled triangle connecting point A to a point (p + q, p),

where the coordinate of A is ((TP(x) + FP(x)), TP(x)). Therefore, $PPV ( x ) = tan θ 1 and 1 - NPV ( x ) = tan θ 2 ,$where θ1 and θ2 are the corresponding angles of two right-angled triangles connecting the origin to point A and point A to the top right corner of the TOC curve, respectively, in

Since both the ROC curve and the TOC curve are concave, both PPV(x) and 1 – NPV(x) are decreasing as the point A moves from the origin to the top right corner of the TOC curve. As the values of X decrease to –∞, point A moves to the top right corner of the TOC curve, so that we may conclude that PPV(x) is decreasing, and NPV(x) is increasing.

The slope from the origin to the rightmost coordinate (p + q, p) of the TOC curve is p/(p + q), which turns out to be the weighted average of PPV(x) and 1 – NPV(x) explained in Result 1. P(C = 1) = PPV(x) × P(Ĉ = 1) + (1 – NPV(x)) × P(Ĉ = 0) with P(C = 1) = p/(p + q); therefore, the following relationship can be obtained: $p p + q = PPV ( x ) × TP ( x ) + FP ( x ) p + q + ( 1 - NPV ( x ) ) × TN ( x ) + FN ( x ) p + q ,$where P(Ĉ = 1) = (TP(x) + FP(x))/(p + q) and P(Ĉ = 0) = (TN(x) + FN(x))/(p + q). Note that each weight indicates the ratio of the lengths of the horizontal axis divided at point A.

### Result 2

For any threshold x, both PPV(x) and NPV(x) have the following properties: $PPV ( x ) ≥ p p + q , NPV ( x ) ≥ q p + q .$

Since PPV(x) is decreasing and NPV(x) is increasing as point A moves from the origin to the top right corner of the TOC curve, the minimum value of PPV(x) can be obtained as p/(p + q), and the minimum value of NPV(x) is q/(p + q), based on Result 2.

3. PPV and NPV with the neutral zone by the TOC curve

### 3.1. PPV and NPV with the neutral zone

Sometimes, one of the rates FPR and FNR can be viewed as more important to control. However, it is not possible to choose a point on the ROC curve that achieves the targeted values for both FPR and FNR. In addition, difficult classification problems incur the risk of high FPR and/or high FNR when the group features are not very different between the two groups. In these situations, the most desirable solution is to identify features that have higher discrimination ability. Hence, Daniel and Steven (2016) proposed neutral zone classifiers that add a soft classification outcome, “neutral” in order to handle ambiguous cases and allow the control of both FPR and FNR. Consider the following classifier: $C ^ = { 1 , disease , N , neutral zone , 0 , non-disease .$

Two constants x0 and x1 with x0 < x1 are found by exerting explicit controls of FPR and FNR. Specifically, x0 and x1 are the solutions of the following two equations: for two types of errors, α and β, $FPR ( x ) = 1 - F ^ n ( x ) ≤ P ( X ≥ x 1 | C = 0 ) = P ( C ^ = 1 | C = 0 ) = α , FNR ( x ) = F ^ d ( x ) ≤ P ( X ≤ x 0 | C = 1 ) = P ( C ^ = 0 | C = 1 ) = β .$

The solutions for the two constants are x0 = d−1(β) and x1 = n−1(1 – α) for fixed values of α and β. A neutral zone exists if and only if d−1(β) < n−1(1 – α) (see Daniel and Steven (2016) for more detail).

Both NZR0 and NZR1 are denoted as probabilities for the neutral zone, $NZR 0 = P ( C ^ = N | C = 0 ) = F ^ n ( x 1 ) - F ^ n ( x 0 ) = TNR ( x 1 ) - TNR ( x 0 ) , NZR 1 = P ( C ^ = N | C = 1 ) = F ^ d ( x 1 ) - F ^ d ( x 0 ) = TPR ( x 0 ) - TPR ( x 1 ) .$

The NZ0 and NZ1 were defined as NZ0 = q × NZR0 and NZ1 = p × NZR1, respectively, where areas of NZR0 and NZR1 were demonstrated for two probability density functions, and two straight lines were implemented around the ROC curve.

Daniel and Steven (2016, p.2350) defined both PPV = P(C = 1|Ĉ = 1) and NPV = P(C = 0|Ĉ = 0) in the presence of the neutral zone. Here, we could derive the following PPV and NPV using ( 3.1): $PPV ( x 0 , x 1 ) = TP ( x 0 ) - NZ 1 TP ( x 0 ) - NZ 1 + FP ( x 1 ) , NPV ( x 0 , x 1 ) = TN ( x 1 ) - NZ 0 TN ( x 1 ) - NZ 0 + FN ( x 0 ) .$

Note that both the PPV and NPV in ( 3.2) depend on x0, as well as x1.

### 3.2. PPV and NPV with the neutral zone by the TOC curve

Suppose that A0 and A1 in Figure 2 are the points on the TOC curve corresponding to two arbitrary thresholds x0 and x1 (x0 < x1), respectively. Their coordinates are then assumed to be A0 = (TP(x0) + FP(x0), TP(x0)) and A1 = (TP(x1) + FP(x1), TP(x1)). Considering two coordinates of A0 and A1, the horizontal length between A0 and A1 can be found to be NZ0 + NZ1. That is, (TP(x0) + FP(x0)) – (TP(x1) + FP(x1)) = NZ0 + NZ1.

It is also found that subtracting NZ1 from the vertical height of A0 is equal to the vertical height of A1, so that TP(x0) – NZ1 = TP(x1) can be obtained. The lengths from the two points A0 and A1 to the right side of the parallelogram in Figure 2 are also TN(x0) and TN(x1), respectively. Since the slope of the parallelogram hypotenuse is 1, it can be confirmed that TN(x1) – NZ0 = TN(x0). This relationship can be seen in Figure 1, and the following ( 3.3) can be derived using ( 2.1) and ( 3.1): $TP ( x 0 ) - NZ 1 = TP ( x 1 ) , TN ( x 1 ) - NZ 0 = TN ( x 0 ) .$

The PPV and NPV in the presence of the neutral zone could be also expressed as in Result 3 by using ( 3.3).

### Result 3

For any two thresholds x0 and x1, both PPV and 1 – NPV in the presence of the neutral zone are represented as: $PPV ( x 1 ) = TP ( x 1 ) TP ( x 1 ) + FP ( x 1 ) , 1 - NPV ( x 0 ) = FN ( x 0 ) TN ( x 0 ) + FN ( x 0 ) .$

Note that whereas both the PPV and NPV in ( 3.2) depend on x0 and x1 together, the PPV and NPV in Result 3 turn out to be the functions of x1 and x0 only, respectively.

We also conclude that both the PPV and NPV in the presence of the neutral zone could be explained as the slopes of two right-angled triangles. Based on the coordinate on the TOC curve, TP(x1) + FP(x1) and TP(x1) correspond to the denominator and numerator of PPV(x1) in Result 3, respectively. Note that TP(x0) – NZ1 + FP(x1) and TP(x0) – NZ1, which are the denominator and numerator of PPV(x0, x1) in ( 3.2), are identical to TP(x1) + FP(x1) and TP(x1) in PPV(x1) in Result 3. Hence, PPV(x1) can be explained as the slope of the right-angled triangle connecting the origin and point A1.

With analogous arguments in Section 2, the denominator of 1 – NPV(x0, x1), TN(x1) – NZ0 + FN(x0), is the entire horizontal length, p + q, minus the sum of both TP(x0) – NZ1 + FP(x1) and NZ0 + NZ1, where TP(x0) – NZ1 + FP(x1) is the horizontal length from the origin to point A1, and NZ0 + NZ1 is the horizontal distance of the two points A0 and A1. Alternatively, the denominator of 1 – NPV(x0), TN(x0) + FN(x0), in Result 3 is also the entire horizontal length minus TP(x1) + FP(x1). That is, the denominator of 1 – NPV(x0) means the horizontal distance from point A0 to the far right of the horizontal axis of the TOC curve. The numerator of 1 – NPV(x0), FN(x0), indicates the vertical distance from point A0 to the top of the TOC curve. That is FN(x0) = p–TP(x0). Hence, 1–NPV(x0) can be explained by the slope of the right-angled triangle connecting point A0 to the top right corner of the TOC curve.

### Result 4

For any two thresholds x1 and x0, PPV(x1) and 1 – NPV(x0) could be described as the angles of the two following right-angled triangles on the TOC curve:

• PPV(x1) is the slope of the right-angled triangle connecting the origin to point A1,

• 1 – NPV(x0) is the slope of the right-angled triangle connecting point A0 to the point (p + q, p),

where A0 = (TP(x0) + FP(x0), TP(x0)) and A1 = (TP(x1) + FP(x1), TP(x1)). Therefore, for any two thresholds x1 and x0, both PPV(x1) and 1 – NPV(x0) have the following relationship on the TOC curve. $PPV ( x 1 ) = tan θ 1 and 1 - NPV ( x 0 ) = tan θ 2 ,$where θ1 and θ2 are the corresponding angles of the two right-angled triangles connecting the origin to point A1 and point A0 to the top right corner of the TOC curve, respectively, in

Note that the coordinates of A0 and A1 are (TP(x0)+FP(x0), TP(x0)) and (TP(x1)+FP(x1), TP(x1)), which are also dependent on β and α, respectively. Therefore, it can be concluded that whereas PPV(x) and NPV(x) in Section 2 depend on the x value, both PPV(x1) and 1 – NPV(x0) in Section 3 depend on x1 and x0, which also depend on the fixed values of β and α, respectively.

For x0xx1, where x1 = n−1(1 – α) and x0 = d−1(β) for fixed values of α and β, we found that α ≥ 1 – Fn(x) and βFd(x). Since PPV(x1) ≤ PPV(x) and NPV(x0) ≥ NPV(x), p/(p + q) ≤ PPV(x) ≤ PPV(x1) and q/(p + q) ≤ NPV(x) ≤ NPV(x0) for x0xx1 using ( 2.3). Moreover, Daniel and Steven (2016, p. 2350) derived that α ≤ (1 – Fd(x1))(1 – Fn(x))/(1 – Fd(x)) and βFn(x0)Fd(x)/Fn(x). With these inequalities, as well as α ≥ 1 – Fn(x) and βFd(x), we might obtain the following specific inequalities: $F d ( x ) ≤ F d ( x 0 ) ≤ F n ( x 0 ) F n ( x ) × F d ( x ) , 1 - F n ( x ) ≤ 1 - F n ( x 1 ) ≤ 1 - F d ( x 1 ) 1 - F d ( x ) × ( 1 - F n ( x ) ) .$

4. Illustrative examples and simulation

Let the distribution function representing a disease state be a normal distribution function with mean μ and unit variance, Xd ~ N(μ, 1), of sample size p; and the other distribution function representing a non-disease state, be the standard normal distribution function, Xn ~ N(0, 1), of size q. In order to generate two sample data, set μ = (1.0, 2.0), p = (100, 200), and q = (200, 100). Hence the following four cases are considered: $Case 1 : F d ( x ) = Φ ( x ; 1 , 1 ) , p = 100 , F n ( x ) = Φ ( x ; 0 , 1 ) , q = 200 , Case 2 : F d ( x ) = Φ ( x ; 2 , 1 ) , p = 100 , F n ( x ) = Φ ( x ; 0 , 1 ) , q = 200 , Case 3 : F d ( x ) = Φ ( x ; 1 , 1 ) , p = 200 , F n ( x ) = Φ ( x ; 0 , 1 ) , q = 100 , Case 4 : F d ( x ) = Φ ( x ; 2 , 1 ) , p = 200 , F n ( x ) = Φ ( x ; 0 , 1 ) , q = 100.$

For each case, the coordinate of point A in the TOC curve can be obtained corresponding to the random variable X. For values of X near the optimal threshold μ/2, Tables 1 (a)4 (a) contain the coordinates of A, as well as the values of PPV(x) and NPV(x).

To explore the presence of the neutral zone explained in Section 3, set α = (0.05, 0.10) and β = (0.10, 0.05). Then find two thresholds x1 and x0. Their corresponding coordinates of points A1 and A0 in the TOC curve; in addition, the values of PPV(x1) and NPV(x0), are also found and summarized in Tables 1 (b)4 (b). Figures 3 and 4 show four TOC curves corresponding to the two cases including points A1 and A0, respectively. The values of PPV(x), NPV(x), PPV(x1), and NPV(x0) for each case are now examined and discussed.

### 4.1. Case 1 (μ = 1.0, p = 100, q = 200)

For the first case, area under the curve (AUC) = 0.76025, so that the discriminate power is adequate. Based on Table 1 (a), it can be found that as the value of X decreases, the coordinate of point A moves from the origin to point (p+q, p); and as point A moves to (p+q, p), the value of PPV(x) is decreasing, while that of NPV(x) is increasing.

Equation (2.2) is also confirmed for X = 0.5, which can be regarded as an optimal threshold (the middle between two means of two distribution functions): $100 300 = 0.52842 × 131 300 + ( 1 - 0.81759 ) × 169 300 .$

From Table 1 (b), when α increases from 0.05 to 0.10, the coordinate of A1 moves to (p + q, p), and the value of PPV(x1) is decreasing. However, when β decreases from 0.10 to 0.05, the coordinate of A0 moves to (p + q, p), and the value of NPV(x0) is increasing. As α increases and β decreases, the distance between points A1 and A0 reduces. This means that as the discriminate power is increasing, the two points A1 and A0 get closer.

Figures 3 (a) and (b) demonstrate the TOC curve, points A1 and A0 for the two kinds α and β. Figures 3(a) and (b) show that as α increases and β decreases, the coordinates of A1 and A0 move from the origin to point (p + q, p). Points A1 and A0 also become closer to one another.

### 4.2. Case 2 (μ = 2.0, p = 100, q = 200)

Case 2 has a different mean value of the distribution function for disease state, compared with Case 1. For Case 2, AUC = 0.9214, which has better discriminate power than Case 1, so that points A1 and A0 approach each other closer than for Case 1. Based on Tables 2 (a) and (b), similar results are obtained, except that the values of both PPV(x) and NPV(x) are larger than those in Case 1. It can also be found that equation (2.2) is also satisfied in this case, for example, when X = 1.0, which is an optimal threshold, the equality holds: $200 300 = 0.72614 × 116 300 + ( 1 - 0.91384 ) × 184 300 .$

### 4.3. Case 3 (μ = 1.0, p = 200, q = 100)

Case 3 has the same mean value and different sample sizes of p and q, compared with Case 1. For Case 3, it is obtained that AUC = 0.76025, which has the same discriminate power as Case 1. However, the height of the TOC curve is different. Since the TOC curve in Case 3 is taller than in Case 1, the values of PPV(x) are much larger than those in Case 1, whereas the values of NPV(x) are smaller those in Case 1.

### 4.4. Case 4 (μ = 2.0, p = 200, q = 100)

Case 4 has different sample sizes of p and q compared with Case 2, and has a larger mean than Case 3. Based on Tables 4 (a) and (b) for Case 4, AUC = 0.9214, which has the same discriminate power as Case 2, so that both points A1 and A0 approach each other closer than for Cases 1 and 3. Also, the values of both PPV(x) and NPV(x) can be found to be larger than those in Case 3 (Figure 4).

5. Real data example

A credit evaluation data was collected by a Korean domestic K bank in June 1918. This sample contains two random variables: one means the risk score (RS) rank which is a risk evaluation variable with 20 grades; the other variable divides each sample into three stages which are the classifications of credit risk. The first stage was not used in this work since it is almost a non-default state and has too large sample. Hence, we take the second and third stages of size 6,951 and 1,191, respectively. The second and third stages are regarded as the non-default and default groups, respectively. Note that the total sample size we are concerned with is 8,142. The higher the RS rank, the more likely that bankruptcy will be judged.

Based on this real data in Table 5, Figure 5 (a) represents the ROC curve. The AUC for this ROC curve is 0.9622. It can be obtained that sensitivity = 0.8371 and specificity = 1 – 0.0499 = 0.9500, corresponding to an optimal threshold at RS rank=16. These values can also be obtained from Table 6 (a): sensitivity = 997/1191 = 0.8371 and specificity = 6604/6951 = 0.9500. All of these values are close to 1; therefore, we might conclude that the values of these statistics that assess the accuracy and performance of the diagnosis are very good.

Figure 5 (b) expresses the TOC curve since the PPV and NPV cannot be explained on the ROC curve. Figure 5 (b) shows that the values of PPV and NPV outside A0 and A1 are almost 1, because the PPV is the slope from origin to point A1 and the NPV is the slope from point A0 to the top right corner of the TOC curve. We can obtain the value of PPV and NPV from Table 6 (b): PPV = 997/(997+347) = 0.74182 and NPV = 2590/(2590+10) = 0.99615. The points A1 and A0 correspond to the RS ranks of 16 and 11, respectively.

The neutral zone can be set from A1 to A0 on the TOC curve, and the corresponding RS ranks are x1 = 16 and x0 = 11. Two types of errors α and β are also obtained, which are α = 0.0500, and β = 0.0084. Therefore, based on the TOC curve in Figure 5, it can be concluded that the value of PPV (the probability that a sample determined as default is actually default) and NPV (the probability that a sample determined as non-default is actually non-default) are 1 almost everywhere with small error probabilities.

6. Conclusion

The ‘sensitivity’ and ‘specificity’ are well known measures for evaluating the accuracy and performance of diagnosis. When the evaluation result of the disease is positive or negative, the probability of the test performance that an actual health condition is disease or non-disease may be more important than how accurate the diagnosis is. To satisfy this interest, there are two other measures such as the PPV and NPV proposed by Zhou et al. (2002) and Raslich et al. (2007): the PPV is the probability that the patient has the disease following a positive test result, while the NPV is the probability that the patient does not have the disease following a negative test result.

The ‘sensitivity’ and ‘specificity’ can be described on the ROC curve; however, PPV and NPV cannot be represented by the ROC curve. In this paper, some explanatory methods are proposed to describe the PPV and NPV by the TOC curve, which can be geometrically explained by the TOC curve. The coordinate of a certain point corresponding to any threshold x on the TOC curve is ((TP(x) + FP(x)), TP(x)). If a right-angled triangle is considered connecting the origin to a certain point, PPV(x) can be regarded as the slope of the right-angled triangle. The denominator of 1–NPV(x) is TN(x) + FN(x) and its numerator is FN(x); therefore, 1 – NPV(x) can be represented as the slope of the right-angled triangle connecting a certain point to the top right corner of the TOC curve.

Therefore, it may be concluded that PPV(x) = tan θ1 and 1 – NPV(x) = tan θ2, where θ1 and θ2 are the corresponding angles of two right-angled triangles connecting the origin to a certain point, and from the certain point to the top right corner of the TOC curve, respectively.

The neutral zone was suggested to ensure the retesting of diseases with high diagnostic costs. Daniel and Steven (2016) derived definitions of both the PPV and NPV when the neutral zone is present. When the neutral zone is present, the PPV and NPV are also geometrically described and implemented on the TOC curve in this study. The two constants x0 and x1 are defined with x0 < x1 as x0 = d−1(β) and x1 = n−1(1 – α) for fixed values of α and β. Then these PPV(x1) and NPV(x0), in the presence of the neutral zone, are found to be functions of x1 and x0, respectively. It may also be concluded that both PPV(x1) and NPV(x0), in the presence of the neutral zone, can be explained as the slopes of two right-angled triangles. When the neutral zone exists, PPV(x1) can be described as the slope of the right-angled triangle connecting the origin to a certain point that is a function of x1 on the TOC curve, while 1–NPV(x0) can be represented as the slope of the right-angled triangle connecting a certain point that is a function of x0 to the top right corner of the TOC curve. Therefore, it may be concluded that PPV(x1) = tan θ1 and 1 – NPV(x0) = tan θ2, where θ1 and θ2 are the corresponding angles of two right-angled triangles connecting the origin to a certain point depending on x1, and from the certain point depending on x0 to the top right corner of the TOC curve, respectively; therefore, both the PPV and NPV can be estimated by using the TOC curve, whether or not the neutral zone exists.

Figures
Fig. 1.

PPV and NPV by the TOC curve. PPV = positive predictive value; NPV = negative predictive value; TOC = total operating characteristic.

Fig. 2.

PPV and NPV with neutral zone by the TOC curve. PPV = positive predictive value; NPV = negative predictive value; TOC = total operating characteristic.

Fig. 3.

Total operating characteristic curves for Case 1.

Fig. 4.

Total operating characteristic curves for Case 4.

Fig. 5.

ROC curve and TOC curve for real data. ROC = receiver operating characteristic; TOC = total operating characteristic.

TABLES

### Table 1

Results of PPV and NPV for Case 1

(a) Results corresponding to A
x A PPV(x) NPV(x)
0.70 (110, 62) 0.56080 0.79871
0.60 (120, 66) 0.54440 0.80815
0.50 (131, 69) 0.52842 0.81759
0.40 (141, 73) 0.51293 0.82698
0.30 (152, 76) 0.49798 0.83627
(b) Results with α and β
α β x1 A1 PPV(x1) x0 A0 NPV(x0)
0.05 0.10 1.6449 (36, 26) 0.7219 −0.2816 (212, 90) 0.8861
0.10 0.05 1.2816 (59, 39) 0.6605 −0.6448 (243, 95) 0.9121

PPV = positive predictive value; NPV = negative predictive value.

### Table 2

Results of PPV and NPV for Case 2

(a) Results corresponding to A
x A PPV(x) NPV(x)
1.2 (102, 19) 0.77399 0.89309
1.1 (109, 82) 0.75045 0.90377
1.0 (116, 84) 0.72614 0.91384
0.9 (123, 86) 0.70131 0.92325
0.8 (131, 88) 0.67622 0.93197
(b) Results with α and β
α β x1 A1 PPV(x1) x0 A0 NPV(x0)
0.05 0.10 1.6449 (74, 64) 0.8646 0.7285 (137, 90) 0.9386
0.10 0.05 1.2816 (96, 76) 0.7925 0.3552 (167, 95) 0.9623

PPV = positive predictive value; NPV = negative predictive value.

### Table 3

Results of PPV and NPV for Case 3

(a) Results corresponding to A
x A PPV(x) NPV(x)
0.70 (148, 124) 0.83627 0.49798
0.60 (159, 131) 0.82698 0.51293
0.50 (169, 138) 0.81759 0.52842
0.40 (180, 145) 0.80815 0.54440
0.30 (190, 152) 0.79871 0.56080
(b) Results with α and β
α β x1 A1 PPV(x1) x0 A0 NPV(x0)
0.05 0.10 1.6449 (57, 52) 0.9121 −0.2816 (241, 180) 0.6605
0.10 0.05 1.2816 (88, 78) 0.8861 −0.6448 (264, 190) 0.7218

PPV = positive predictive value; NPV = negative predictive value.

### Table 4

Results of PPV and NPV for Case 4

(a) Results corresponding to A
x A PPV(x) NPV(x)
1.2 (169, 158) 0.93197 0.67622
1.1 (177, 163) 0.92325 0.70131
1 (184, 168) 0.91384 0.72614
0.9 (191, 173) 0.90377 0.75045
0.8 (198, 177) 0.89309 0.77399
(b) Results with α and β
α β x1 A1 PPV(x1) x0 A0 NPV(x0)
0.05 0.10 1.6449 (133, 128) 0.9626 0.7185 (204, 180) 0.7925
0.10 0.05 1.2816 (163, 153) 0.9386 0.3552 (226, 190) 0.8646

PPV = positive predictive value; NPV = negative predictive value.

### Table 5

Real data with risk score rank and stages

Risk score rank Second stage Third stage Sum
1 4 0 4
2 1 0 1
3 17 0 17
4 55 0 55
5 56 0 56
6 175 0 175
7 189 0 189
8 279 0 279
9 518 0 518
10 421 0 421
11 875 10 885
12 761 7 768
13 631 34 665
14 1102 0 1102
15 555 92 647
16 965 51 1,016
17 347 64 411
18 0 306 306
19 0 111 111
20 0 516 516

sum 6,951 1,191 8,142

### Table 6

Confusion matrix in optimal threshold

(a) At RS rank = 16
Real non-default Real default
Non-default 6604 194
Default 347 997
(b) At RS rank = 16 and RS rank = 11
Real non-default Real default
Non-default 2590 10
Neutral zone 4014 184
Default 347 997

RS = risk score.

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