The predictive receiver operating characteristic curve for the joint assessment of the positive and negative predictive values

Binary test outcomes typically result from dichotomizing a continuous test variable, observable or latent. The effect of the threshold for test positivity on test sensitivity and specificity has been studied extensively in receiver operating characteristic (ROC) analysis. However, considerably less attention has been given to the study of the effect of the positivity threshold on the predictive value of a test. In this paper we present methods for the joint study of the positive (PPV) and negative predictive values (NPV) of diagnostic tests. We define the predictive receiver operating characteristic (PROC) curve that consists of all possible pairs of PPV and NPV as the threshold for test positivity varies. Unlike the simple trade-off between sensitivity and specificity exhibited in the ROC curve, the PROC curve displays what is often a complex interplay between PPV and NPV as the positivity threshold changes. We study the monotonicity and other geometric properties of the PROC curve and propose summary measures for the predictive performance of tests. We also formulate and discuss regression models for the estimation of the effects of covariates.

Definition 1. (The Hazard Rate Order) Let X and Y be two random variables with absolutely continuous distribution functions F and G, respectively, such that Then X is said to be smaller than Y in the hazard rate order. The hazard rate order is defined on the ratio of the PDF and the survival function. This ratio f(t)/(1 − F (t)) is the first derivative of log(1 − F (t)).
Therefore, the hazard rate order implies that the slope of the log of one survival function is uniformly smaller than that of the other.
If one function does not have a uniformly smaller slope than the other, and if the two functions have the same starting points (which are zero here), then for the function initially having a lower speed therefore falling behind at the beginning may pass over the other function because of a higher speed later. In other words, the two functions are likely to cross at some point. Figure 1 demonstrates the failure of the hazard rate order. The solid curve represents the log of the survival function from N(0,1), and the dashed curve is from N(1,2). Note that the two curves cross at some point around minus one.
To summarize, one interpretation of the hazard rate order is that the log of the survival function of one random variable declines uniformly faster than that of the other. A common example that the hazard rate order fails, is that the log of the survival functions of two random variables cross. However, the non-crossing of the log of the two survival functions does not imply that the hazard rate order is satisfied.

Proofs of Proposition 3.1 and Proposition 3.2
In this appendix, we will prove Proposition 3.1 and Proposition 3.2. For convenience of notation, let We begin with studying the geometric properties of h + a,b and h − a,b . The following lemma describes the properties, and will be used to prove Proposition 3.1 and Proposition 3.2.
Proof. In the following, we only prove statement (i). Statement (ii) can be obtained by similar arguments.
Proof of Proposition 3.1. Since h + a,b is strictly decreasing, and since h − a,b is strictly increasing, we have These two inequalities complete the proof.

Proof of Proposition 3.2.
In the following, we will only prove Proposition 3.2 for b > 1. The statements for b < 1 can be obtained using similar arguments.
(a) Since b > 1, we have (c) Since h + a,b is increasing, and since Combine results from (a), (b) and (c), we conclude that there exists a unique Therefore, the positive predictive value is strictly decreasing on t ∈ (−∞, c * P P V ) and is strictly increasing on t ∈ (c * P P V , ∞). Similarly, we can prove that the negative predictive value is strictly increasing on t ∈ (−∞, c * N P V ) and is strictly decreasing on t ∈ (c * N P V , ∞), where c * N P V > a/ (1 − b). Therefore, we also conclude that c * P P V ≤ c * N P V .

Existence of pseudo-likelihood estimates and the variance-covariance matrix
(i) The existence of pseudo-likelihood estimates depends on the identifiability of the model. For the predictive model presented in Section 4, the identifiability is guaranteed whenever the number of ordinal categories K ≥ 3.
(ii) Examine if the matrix which shows that the matrix J is of the form V k V T k c k , where V k is the vector of ∂h k /∂Ψ and c k is a constant larger than zero. Hence, it is proved that J is positive definite.