Estimation of the volume under the ROC surface in presence of nonignorable verification bias

Abstract

The volume under the receiver operating characteristic surface (VUS) is useful for measuring the overall accuracy of a diagnostic test when the possible disease status belongs to one of three ordered categories. In medical studies, the VUS of a new test is typically estimated through a sample of measurements obtained by some suitable sample of patients. However, in many cases, only a subset of such patients has the true disease status assessed by a gold standard test. In this paper, for a continuous-scale diagnostic test, we propose four estimators of the VUS which accommodate for nonignorable missingness of the disease status. The estimators are based on a parametric model which jointly describes both the disease and the verification process. Identifiability of the model is discussed. Consistency and asymptotic normality of the proposed estimators are shown, and variance estimation is discussed. The finite-sample behavior is investigated by means of simulation experiments. An illustration is provided.

Appendices

Appendix 1

Proves

Proof of Theorem 1

We can show that ${\mathbb {E}}\{G_{i\ell r,*}(\mu _0,{\varvec{\xi }}_0)\} = 0$ (see the “Appendix 2”). Then $e_*(\mu _0,{\varvec{\xi }}_0) = 0$, and, by condition (C2) and an application of implicit function theorem, there exists a neighborhood of ${\varvec{\xi }}_0$ in which a continuously differentiable function, $m({\varvec{\xi }})$, is uniquely defined such that $m({\varvec{\xi }}_0) = \mu _0$ and $e_*(m({\varvec{\xi }}),{\varvec{\xi }}) = 0$. Since the maximum likelihood estimator $\hat{{\varvec{\xi }}}$ is consistent, i.e., $\hat{{\varvec{\xi }}} {\mathop {\rightarrow }\limits ^{p}} {\varvec{\xi }}_0$, we have that ${{\tilde{\mu }}}_* = m(\hat{{\varvec{\xi }}}){\mathop {\rightarrow }\limits ^{p}} \mu _0$. On the other hand, $G_*({{\hat{\mu }}}_*, \hat{{\varvec{\xi }}}) = 0$ and condition (C3) implies that $e_*({{\hat{\mu }}}_*,\hat{{\varvec{\xi }}}){\mathop {\rightarrow }\limits ^{p}} 0$. Thus, ${\hat{\mu }}_* {\mathop {\rightarrow }\limits ^{p}} {{\tilde{\mu }}}_*$. $\square$

Proof of Theorem 2

We have

$$\begin{aligned} 0= & {} \sqrt{n}G_{*}({\hat{\mu }}_{*},\hat{{\varvec{\xi }}}) \\ 0= & {} \sqrt{n}G_{*}({\hat{\mu }}_{*},\hat{{\varvec{\xi }}}) + \sqrt{n}e_*({\hat{\mu }}_{*},\hat{{\varvec{\xi }}}) - \sqrt{n}e_*({\hat{\mu }}_{*},\hat{{\varvec{\xi }}}). \end{aligned}$$

Since $e_*(\mu _0,{\varvec{\xi }}_0) = 0$, we get

$$\begin{aligned} 0= & {} \sqrt{n}G_{*}({\hat{\mu }}_{*},\hat{{\varvec{\xi }}}) + \sqrt{n}e_*({\hat{\mu }}_{*},\hat{{\varvec{\xi }}}) - \sqrt{n}e_*({\hat{\mu }}_{*},\hat{{\varvec{\xi }}}) + \sqrt{n}e_*(\mu _0,{\varvec{\xi }}_0) - \sqrt{n}e_*(\mu _0,{\varvec{\xi }}_0) \\= & {} \sqrt{n}\left\{ G_{*}({\hat{\mu }}_{*},\hat{{\varvec{\xi }}}) - e_*({\hat{\mu }}_{*},\hat{{\varvec{\xi }}}) \right\} + \sqrt{n}\left\{ e_*({\hat{\mu }}_{*},\hat{{\varvec{\xi }}}) - e_*(\mu _0,{\varvec{\xi }}_0)\right\} + \sqrt{n}e_*(\mu _0,{\varvec{\xi }}_0) \\&- \, \sqrt{n}G_{*}(\mu _0,{\varvec{\xi }}_0) + \sqrt{n}G_{*}(\mu _0,{\varvec{\xi }}_0) \\= & {} \left[ \sqrt{n}\left\{ G_{*}({\hat{\mu }}_{*},\hat{{\varvec{\xi }}}) - e_*({\hat{\mu }}_{*},\hat{{\varvec{\xi }}}) \right\} - \sqrt{n}\left\{ G_{*}(\mu _0,{\varvec{\xi }}_0) - e_*(\mu _0,{\varvec{\xi }}_0)\right\} \right] \\&+ \, \sqrt{n}\left\{ e_*({\hat{\mu }}_{*},\hat{{\varvec{\xi }}}) - e_*(\mu _0,{\varvec{\xi }}_0)\right\} + \sqrt{n}G_{*}(\mu _0,{\varvec{\xi }}_0). \end{aligned}$$

Condition (C1) implies that the first term in right hand side of the last identity is $o_p(1)$. Using the Taylor expansion, we have

$$\begin{aligned} 0= & {} o_p(1) + \sqrt{n}\left\{ e_*({\hat{\mu }}_{*},\hat{{\varvec{\xi }}}) - e_*(\mu _0,{\varvec{\xi }}_0)\right\} + \sqrt{n}G_{*}(\mu _0,{\varvec{\xi }}_0) \nonumber \\= & {} o_p(1) + \sqrt{n}({\hat{\mu }}_{*} - \mu _0) \frac{\partial e_*(\mu ,{\varvec{\xi }}_0)}{\partial \mu }\Bigg |_{\mu = \mu _0} \nonumber \\&+ \, \sqrt{n}(\hat{{\varvec{\xi }}} - {\varvec{\xi }}_0)\frac{\partial e_*(\mu _0,{\varvec{\xi }})}{\partial {\varvec{\xi }}^\top }\Bigg |_{{\varvec{\xi }} = {\varvec{\xi }}_0} + \sqrt{n}G_{*}(\mu _0,{\varvec{\xi }}_0) . \end{aligned}$$

(18)

It is straightforward to show that

$$\begin{aligned} \frac{\partial e_*(\mu ,{\varvec{\xi }}_0)}{\partial \mu }\Bigg |_{\mu = \mu _0} = - \mathrm {Pr}(D_1 = 1) \mathrm {Pr}(D_2 = 1) \mathrm {Pr}(D_3 = 1) = - \theta _1 \theta _2 \theta _3. \end{aligned}$$

By standard results on the limit distribution of U-statistics (van der Vaart 2000, Theorem 12.3, Chap. 12),

$$\begin{aligned} \sqrt{n}U_{n,*}(\mu _0,{\varvec{\xi }}_0)= & {} \sqrt{n}\left\{ G_{*}(\mu _0,{\varvec{\xi }}_0) - e_*(\mu _0,{\varvec{\xi }}_0)\right\} \\= & {} \sqrt{n}G_{*}(\mu _0,{\varvec{\xi }}_0) {\mathop {\rightarrow }\limits ^{p}} \sqrt{n}{\tilde{G}}_{*}(\mu _0,{\varvec{\xi }}_0), \end{aligned}$$

where $\sqrt{n}{\tilde{G}}_{*}(\mu ,{\varvec{\xi }})$ is the projection of $U_{n,*}$ onto the set of all statistics of the form

$$\begin{aligned} \sqrt{n}{\tilde{G}}_{n,*}(\mu ,{\varvec{\xi }})= & {} \frac{1}{2\sqrt{n}}\sum _{i=1}^{n} {\mathbb {E}}\bigg \{ G_{i\ell r,*}(\mu ,{\varvec{\xi }}) + G_{ir \ell ,*}(\mu ,{\varvec{\xi }}) + G_{\ell ir,*}(\mu ,{\varvec{\xi }}) \\&+ \, G_{\ell r i,*}(\mu ,{\varvec{\xi }}) + G_{r i\ell ,*}(\mu ,{\varvec{\xi }}) + G_{r \ell i,*}(\mu ,{\varvec{\xi }}) \big |O_i \bigg \} \end{aligned}$$

for $\ell \ne i$ and $r \ne \ell , r \ne i$. For the maximum likelihood estimator $\hat{{\varvec{\xi }}}$, we can write

$$\begin{aligned} \sqrt{n}\left( \hat{{\varvec{\xi }}} - {\varvec{\xi }}_0\right)= & {} \frac{1}{\sqrt{n}}\left[ -\frac{\partial {\mathbb {E}}\left\{ {\mathcal {S}}_i({\varvec{\xi }})\right\} }{\partial {\varvec{\xi }}^\top }\Bigg |_{{\varvec{\xi }} = {\varvec{\xi }}_0}\right] ^{-1}\sum _{i=1}^{n}{\mathcal {S}}_i({\varvec{\xi }}_0) + o_p(1) \\= & {} \frac{1}{\sqrt{n}}{\mathcal {I}}({\varvec{\xi }})^{-1} \sum _{i=1}^{n}{\mathcal {S}}_i({\varvec{\xi }}_0) + o_p(1). \end{aligned}$$

Hence, from (18),

$$\begin{aligned}&\theta _1 \theta _2 \theta _3 \sqrt{n}({\hat{\mu }}_{*} - \mu _0) \nonumber \\&\quad = o_p(1) + \frac{1}{\sqrt{n}} \frac{\partial e_*(\mu _0,{\varvec{\xi }})}{\partial {\varvec{\xi }}^\top }\Bigg |_{{\varvec{\xi }} = {\varvec{\xi }}_0} {\mathcal {I}}({\varvec{\xi }})^{-1} \sum _{i=1}^{n}{\mathcal {S}}_i({\varvec{\xi }}_0) \nonumber \\&\qquad + \, \frac{1}{2\sqrt{n}}\sum _{i=1}^{n} {\mathbb {E}}\bigg \{ G_{i\ell r,*}(\mu _0,{\varvec{\xi }}_0) + G_{ir \ell ,*}(\mu _0,{\varvec{\xi }}_0) + G_{\ell ir,*}(\mu _0,{\varvec{\xi }}_0) \nonumber \\&\qquad + \, G_{\ell r i,*}(\mu _0,{\varvec{\xi }}_0) + G_{r i\ell ,*}(\mu _0,{\varvec{\xi }}_0) + G_{r \ell i,*}(\mu _0,{\varvec{\xi }}_0) \big |O_i \bigg \}\nonumber \\&\quad = o_p(1) + \frac{1}{\sqrt{n}}\sum _{i=1}^{n} \Bigg [ \frac{\partial e_*(\mu _0,{\varvec{\xi }})}{\partial {\varvec{\xi }}^\top }\Bigg |_{{\varvec{\xi }} = {\varvec{\xi }}_0} {\mathcal {I}}({\varvec{\xi }})^{-1} {\mathcal {S}}_i({\varvec{\xi }}_0) \nonumber \\&\qquad + \, \frac{1}{2} {\mathbb {E}}\bigg \{ G_{i\ell r,*}(\mu _0,{\varvec{\xi }}_0) + G_{ir \ell ,*}(\mu _0,{\varvec{\xi }}_0) + G_{\ell ir,*}(\mu _0,{\varvec{\xi }}_0) \nonumber \\&\qquad + \, G_{\ell r i,*}(\mu _0,{\varvec{\xi }}_0) + G_{r i\ell ,*}(\mu _0,{\varvec{\xi }}_0) + G_{r \ell i,*}(\mu _0,{\varvec{\xi }}_0) \big |O_i \bigg \} \Bigg ] \nonumber \\&\quad = o_p(1) + \frac{1}{\sqrt{n}}\sum _{i=1}^{n}Q_{i,*}(\mu _0,{\varvec{\xi }}_0) = o_p(1) + \frac{1}{\sqrt{n}} Q_*(\mu _0,{\varvec{\xi }}_0). \end{aligned}$$

(19)

Note that the observed data $O_i$ are i.i.d, then $Q_{i,*}(\mu _0,{\varvec{\xi }}_0)$ are also i.i.d. In addition, we easily show that

$$\begin{aligned} 0= & {} {\mathbb {E}}\Bigg [{\mathbb {E}}\bigg \{ G_{i\ell r,*}(\mu _0,{\varvec{\xi }}_0) + G_{ir \ell ,*}(\mu _0, {\varvec{\xi }}_0) + G_{\ell ir,*}(\mu _0,{\varvec{\xi }}_0) + G_{\ell r i,*}(\mu _0, {\varvec{\xi }}_0) \\&+ \, G_{r i\ell ,*}(\mu _0, {\varvec{\xi }}_0) + G_{r \ell i,*}(\mu _0, {\varvec{\xi }}_0) \big |O_i \bigg \} \Bigg ]. \end{aligned}$$

Therefore, ${\mathbb {E}}\{Q_{i,*} (\mu _0,{\varvec{\xi }}_0)\} = 0$, and $\frac{1}{\sqrt{n}} Q_* (\mu _0,{\varvec{\xi }}_0) {\mathop {\rightarrow }\limits ^{d}} {\mathcal {N}}(0, {\mathbb {V}}\mathrm {ar}\left\{ Q_{i,*} (\mu _0,{\varvec{\xi }}_0)\right\} )$ by the Central Limit Theorem. It follows that

$$\begin{aligned} \sqrt{n}\left( {\hat{\mu }}_{*} - \mu _0 \right) {\mathop {\rightarrow }\limits ^{d}} {\mathcal {N}}\left( 0, \varLambda _*\right) , \end{aligned}$$

where

$$\begin{aligned} \varLambda _* = \frac{{\mathbb {V}}\mathrm {ar}\left\{ Q_{i,*} (\mu _0,{\varvec{\xi }}_0)\right\} }{\theta _1^2\theta _2^2\theta _3^2}. \end{aligned}$$

(20)

$\square$

Variance estimation

Under condition (C3), a consistent estimator of $\varLambda _*$ can be obtained as

$$\begin{aligned} {\hat{\varLambda }}_* = \frac{{\mathbb {V}}\mathrm {ar}\left\{ {\hat{Q}}_{i,*} ({\hat{\mu }}_{*},\hat{{\varvec{\xi }}})\right\} }{{\hat{\theta }}_{1,*}^2 {\hat{\theta }}_{2,*}^2 {\hat{\theta }}_{3,*}^2} = \frac{\frac{1}{n - 1} \sum \limits _{i=1}^{n}{\hat{Q}}_{i,*}^2({\hat{\mu }}_{*},\hat{{\varvec{\xi }}})}{{\hat{\theta }}_{1,*}^2 {\hat{\theta }}_{2,*}^2 {\hat{\theta }}_{3,*}^2}, \end{aligned}$$

(21)

where ${\hat{\theta }}_{k,*}$ are the estimates of the disease probabilities, $\theta _{k}$ for $k = 1,2,3$. Specifically, ${\hat{\theta }}_{k,\mathrm {FI}} = \frac{1}{n}\sum \nolimits _{i=1}^{n} {\hat{\rho }}_{ki}$, ${\hat{\theta }}_{k,\mathrm {MSI}} = \frac{1}{n}\sum \nolimits _{i=1}^{n} {\tilde{D}}_{ki,\mathrm {MSI}}$, ${\hat{\theta }}_{k,\mathrm {PDR}} = \frac{1}{n}\sum \nolimits _{i=1}^{n} {\tilde{D}}_{ki,\mathrm {PDR}}$ and ${\hat{\theta }}_{k,\mathrm {IPW}} = \sum \nolimits _{i=1}^{n} V_i D_{ki}{\hat{\pi }}_i^{-1} \bigg /\sum \nolimits _{i=1}^{n} V_i{\hat{\pi }}_i^{-1}$. According to (19), we have that

$$\begin{aligned}&{\hat{Q}}_{i,*} ({\hat{\mu }}_{*},\hat{{\varvec{\xi }}}) \\&\quad = \left\{ \frac{1}{(n-1)(n-2)} \sum _{i=1}^{n}\sum _{{\mathop {\ell \ne i}\limits ^{\ell =i}}}^{n}\sum _{{\mathop {r \ne \ell , r \ne i}\limits ^{r = 1}}}^{n} \frac{\partial G_{i\ell r,*}({\hat{\mu }}_{*},{\varvec{\xi }})}{\partial {\varvec{\xi }}^\top }\bigg |_{{\varvec{\xi }} = \hat{{\varvec{\xi }}}} \right\} \\&\qquad \times \, \left\{ -\sum _{i=1}^{n}\frac{\partial {\mathcal {S}}_i({\varvec{\xi }})}{\partial {\varvec{\xi }}^\top }\bigg |_{{\varvec{\xi }} = \hat{{\varvec{\xi }}}}\right\} ^{-1} {\mathcal {S}}_i(\hat{{\varvec{\xi }}}) \\&\qquad + \, \frac{1}{2(n-1)(n-2)} \sum _{{\mathop {\ell \ne i}\limits ^{\ell =1}}}^{n} \sum _{{\mathop {r \ne i, r \ne \ell }\limits ^{r = 1}}}^{n}\bigg \{ G_{i\ell r,*}({\hat{\mu }}_{*},\hat{{\varvec{\xi }}}) + G_{ir \ell ,*}({\hat{\mu }}_{*},\hat{{\varvec{\xi }}}) + G_{\ell ir,*}({\hat{\mu }}_{*},\hat{{\varvec{\xi }}}) \\&\qquad + \, G_{\ell r i,*}({\hat{\mu }}_{*},\hat{{\varvec{\xi }}}) + G_{r i\ell ,*}({\hat{\mu }}_{*},\hat{{\varvec{\xi }}}) + G_{r \ell i,*}({\hat{\mu }}_{*},\hat{{\varvec{\xi }}})\bigg \}. \end{aligned}$$

In addition, for fixed i, we also have that

$$\begin{aligned} \sum _{{\mathop {\ell \ne i}\limits ^{\ell = 1}}}^{n} \sum _{{\mathop {r \ne i, r \ne \ell }\limits ^{r = 1}}}^{n} \left\{ G_{i\ell r,*}({\hat{\mu }}_{*},\hat{{\varvec{\xi }}}) + G_{ikr,*}({\hat{\mu }}_{*},\hat{{\varvec{\xi }}})\right\}= & {} 2\sum _{{\mathop {\ell \ne i}\limits ^{\ell = 1}}}^{n} \sum _{{\mathop {r \ne i, r \ne \ell }\limits ^{r = 1}}}^{n}G_{i\ell r,*}({\hat{\mu }}_{*},\hat{{\varvec{\xi }}}), \\ \sum _{{\mathop {\ell \ne i}\limits ^{\ell = 1}}}^{n} \sum _{{\mathop {r \ne i, r \ne \ell }\limits ^{r = 1}}}^{n} \left\{ G_{\ell ir,*}({\hat{\mu }}_{*},\hat{{\varvec{\xi }}}) + G_{r i\ell ,*}({\hat{\mu }}_{*},\hat{{\varvec{\xi }}})\right\}= & {} 2\sum _{{\mathop {\ell \ne i}\limits ^{\ell = 1}}}^{n} \sum _{{\mathop {r \ne i, r \ne \ell }\limits ^{r = 1}}}^{n}G_{\ell ir,*}({\hat{\mu }}_{*},\hat{{\varvec{\xi }}}), \\ \sum _{{\mathop {\ell \ne i}\limits ^{\ell = 1}}}^{n} \sum _{{\mathop {r \ne i, r \ne \ell }\limits ^{r = 1}}}^{n} \left\{ G_{\ell r i,*}({\hat{\mu }}_{*},\hat{{\varvec{\xi }}}) + G_{r \ell i,*}({\hat{\mu }}_{*},\hat{{\varvec{\xi }}})\right\}= & {} 2\sum _{{\mathop {\ell \ne i}\limits ^{\ell =1}}}^{n} \sum _{{\mathop {r \ne i, r \ne \ell }\limits ^{r = 1}}}^{n}G_{r \ell i,*}({\hat{\mu }}_{*},\hat{{\varvec{\xi }}}). \end{aligned}$$

Therefore,

$$\begin{aligned}&{\hat{Q}}_{i,*} ({\hat{\mu }}_{*},\hat{{\varvec{\xi }}}) \nonumber \\&\quad = \left\{ \frac{1}{(n-1)(n-2)}\sum _{i = 1}^{n}\sum _{{\mathop {\ell \ne i}\limits ^{\ell = i}}}^{n}\sum _{{\mathop {r \ne \ell , r \ne i}\limits ^{r = 1}}}^{n} \frac{\partial G_{i\ell r,*}({\hat{\mu }}_{*},{\varvec{\xi }})}{\partial {\varvec{\xi }}^\top }\bigg |_{{\varvec{\xi }} = \hat{{\varvec{\xi }}}}\right\} \nonumber \\&\qquad \times \, \left\{ -\sum _{i=1}^{n}\frac{\partial {\mathcal {S}}_i({\varvec{\xi }})}{\partial {\varvec{\xi }}^\top }\bigg |_{{\varvec{\xi }} = \hat{{\varvec{\xi }}}}\right\} ^{-1} {\mathcal {S}}_i(\hat{{\varvec{\xi }}}) \nonumber \\&\qquad + \, \frac{1}{(n-1)(n-2)} \sum _{{\mathop {\ell \ne i}\limits ^{\ell = 1}}}^{n} \sum _{{\mathop {r \ne i, r \ne \ell }\limits ^{r = 1}}}^{n}\bigg \{ G_{i\ell r,*}({\hat{\mu }}_{*},\hat{{\varvec{\xi }}}) + G_{\ell ir,*}({\hat{\mu }}_{*},\hat{{\varvec{\xi }}}) + G_{r \ell i,*}({\hat{\mu }}_{*},\hat{{\varvec{\xi }}})\bigg \}.\nonumber \\ \end{aligned}$$

(22)

The quantity $\sum \nolimits _{i=1}^{n} \frac{\partial {\mathcal {S}}_i({\varvec{\xi }})}{\partial {\varvec{\xi }}^\top }\bigg |_{{\varvec{\xi }} = \hat{{\varvec{\xi }}}}$ could be obtained as the Hessian matrix of the log-likelihood function at $\hat{{\varvec{\xi }}}$. In order to compute $\frac{\partial G_{i\ell r,*}({\hat{\mu }}_{*},{\varvec{\xi }})}{\partial {\varvec{\xi }}^\top }\bigg |_{{\varvec{\xi }} = \hat{{\varvec{\xi }}}}$, we have to get the derivatives $\frac{\partial }{\partial {\varvec{\xi }}^\top } \rho _{ki}({\varvec{\tau }}_{0\rho _k})$, $\frac{\partial }{\partial {\varvec{\xi }}^\top } \rho _{k(0)i}({\varvec{\xi }})$, $\frac{\partial }{\partial {\varvec{\xi }}^\top } \pi ^{-1}_{i}({\varvec{\lambda }}, {\varvec{\tau }}_\pi )$, $\frac{\partial }{\partial {\varvec{\xi }}^\top } \pi _{10i}({\varvec{\lambda }}, {\varvec{\tau }}_\pi )$, $\frac{\partial }{\partial {\varvec{\xi }}^\top } \pi _{01i}({\varvec{\lambda }}, {\varvec{\tau }}_\pi )$ and $\frac{\partial }{\partial {\varvec{\xi }}^\top } \pi _{00i}({\varvec{\lambda }}, {\varvec{\tau }}_\pi )$.

In Sect. 2.3, we obtain

$$\begin{aligned} \begin{array}{ll} \dfrac{\partial }{\partial \lambda _1} \pi _{10i}({\varvec{\lambda }}, {\varvec{\tau }}_\pi ) = \pi _{10i}(1 - \pi _{10i}); &{} \quad \dfrac{\partial }{\partial \lambda _2} \pi _{10i}({\varvec{\lambda }}, {\varvec{\tau }}_\pi ) = 0; \\ \dfrac{\partial }{\partial \lambda _1} \pi _{01i}({\varvec{\lambda }}, {\varvec{\tau }}_\pi ) = 0; &{} \quad \dfrac{\partial }{\partial \lambda _2} \pi _{01i}({\varvec{\lambda }}, {\varvec{\tau }}_\pi ) = \pi _{01i}(1 - \pi _{01i}); \\ \dfrac{\partial }{\partial \lambda _1} \pi _{00i}({\varvec{\lambda }}, {\varvec{\tau }}_\pi ) = 0; &{} \quad \dfrac{\partial }{\partial \lambda _2} \pi _{00i}({\varvec{\lambda }}, {\varvec{\tau }}_\pi ) = 0. \end{array} \end{aligned}$$

and

$$\begin{aligned} \frac{\partial }{\partial {\varvec{\tau }}_\pi ^\top }\pi _{d_1 d_2 i} = {\varvec{U}}_i (1 - \pi _{d_1 d_2 i})\pi _{d_1 d_2 i}, \end{aligned}$$

where $(d_1, d_2)$ belongs to the set $\{(1,0), (0,1), (0,0)\}$. Also, we have

$$\begin{aligned} \begin{array}{ll} \dfrac{\partial }{\partial {\varvec{\tau }}^\top _{\rho _1}} \rho _{1i}(\tau _\rho ) = {\varvec{U}}_i\rho _{1i}(1 - \rho _{1i}); &{}\quad \dfrac{\partial }{\partial {\varvec{\tau }}^\top _{\rho _2}} \rho _{1i}({\varvec{\tau }}_\rho ) = - {\varvec{U}}_i\rho _{1i}\rho _{2i}; \\ \dfrac{\partial }{\partial {\varvec{\tau }}^\top _{\rho _2}} \rho _{2i}({\varvec{\tau }}_\rho ) = {\varvec{U}}_i\rho _{2i}(1 - \rho _{2i}); &{}\quad \dfrac{\partial }{\partial {\varvec{\tau }}^\top _{\rho _1}} \rho _{2i}({\varvec{\tau }}_\rho ) = - {\varvec{U}}_i \rho _{1i}\rho _{2i}. \end{array} \end{aligned}$$

Moreover,

$$\begin{aligned} \frac{\partial }{\partial \lambda _s} \pi ^{-1}_i({\varvec{\lambda }}, {\varvec{\tau }}_\pi ) = -D_{si}\frac{1 - \pi _i}{\pi _i}; \qquad \frac{\partial }{\partial {\varvec{\tau }}_\pi ^\top }\pi ^{-1}_{i}({\varvec{\lambda }}, {\varvec{\tau }}_\pi ) = -{\varvec{U}}_{i}\frac{1 - \pi _i}{\pi _i}, \end{aligned}$$

with $s = 1, 2$. Then, recall that

$$\begin{aligned} \rho _{1(0)i}= & {} \frac{(1 - \pi _{10i})\rho _{1i}}{(1 - \pi _{10i})\rho _{1i} + (1 - \pi _{01i})\rho _{2i} + (1 - \pi _{00i})\rho _{3i}}, \\ \rho _{2(0)i}= & {} \frac{(1 - \pi _{01i})\rho _{2i}}{(1 - \pi _{10i})\rho _{1i} + (1 - \pi _{01i})\rho _{2i} + (1 - \pi _{00i})\rho _{3i}}, \\ \rho _{3(0)i}= & {} \frac{(1 - \pi _{00i})\rho _{3i}}{(1 - \pi _{10i})\rho _{1i} + (1 - \pi _{01i})\rho _{2i} + (1 - \pi _{00i})\rho _{3i}}. \end{aligned}$$

After some algebra, we get

$$\begin{aligned} \frac{\partial }{\partial \lambda _1} \rho _{1(0)i}({\varvec{\xi }})= & {} \frac{1}{z^2}\left[ -\pi _{10i}(1 - \pi _{10i})\rho _{1i}\left\{ (1 - \pi _{01i})\rho _{2i} + (1 - \pi _{00i})\rho _{3i} \right\} \right] , \\ \frac{\partial }{\partial \lambda _2} \rho _{1(0)i}({\varvec{\xi }})= & {} \frac{1}{z^2} \rho _{1i}\rho _{2i}\pi _{01i}(1 - \pi _{01i}) (1 - \pi _{10i}), \\ \frac{\partial }{\partial {\varvec{\tau }}_\pi ^\top } \rho _{1(0)i}({\varvec{\xi }})= & {} -\frac{{\varvec{U}}_i}{z^2} \rho _{1i}(1 - \pi _{10i}) \bigg \{ \rho _{2i}(1 - \pi _{01i})(\pi _{10i} - \pi _{01i}) \\&+ \, \rho _{3i}(1 - \pi _{00i})(\pi _{10i} - \pi _{00i})\bigg \}, \\ \frac{\partial }{\partial {\varvec{\tau }}_{\rho _1}^\top } \rho _{1(0)i}({\varvec{\xi }})= & {} \frac{{\varvec{U}}_i}{z^2} \rho _{1i} (1 - \pi _{10i}) \left\{ \rho _{2i}(1 - \pi _{01i}) + \rho _{3i}(1 - \pi _{00i}) \right\} , \\ \frac{\partial }{\partial {\varvec{\tau }}_{\rho _2}^\top } \rho _{1(0)i}({\varvec{\xi }})= & {} -\frac{{\varvec{U}}_i}{z^2} \rho _{1i}\rho _{2i} (1 - \pi _{10i}) (1 - \pi _{01i}). \end{aligned}$$

Finally, we set $z = (1 - \pi _{10i})\rho _{1i} + (1 - \pi _{01i})\rho _{2i} + (1 - \pi _{00i})\rho _{3i}$, and get

$$\begin{aligned} \frac{\partial }{\partial \lambda _1} \rho _{2(0)i}({\varvec{\xi }})= & {} \frac{1}{z^2} \rho _{1i}\rho _{2i}\pi _{10i}(1 - \pi _{10i}) (1 - \pi _{01i}), \\ \frac{\partial }{\partial \lambda _2} \rho _{2(0)i}({\varvec{\xi }})= & {} \frac{1}{z^2} \left[ -\pi _{01i}(1 - \pi _{01i})\rho _{2i}\left\{ (1 - \pi _{10i})\rho _{1i} + (1 - \pi _{00i})\rho _{3i} \right\} \right] , \\ \frac{\partial }{\partial {\varvec{\tau }}_\pi ^\top } \rho _{2(0)i}({\varvec{\xi }})= & {} -\frac{{\varvec{U}}_i}{z^2} \rho _{2i}(1 - \pi _{01i}) \bigg \{ \rho _{1i}(1 - \pi _{10i})(\pi _{01i} - \pi _{10i}) \\&+ \, \rho _{3i}(1 - \pi _{00i})(\pi _{01i} - \pi _{00i})\bigg \}, \\ \frac{\partial }{\partial {\varvec{\tau }}_{\rho _1}^\top } \rho _{2(0)i}({\varvec{\xi }})= & {} -\frac{{\varvec{U}}_i}{z^2} \rho _{1i}\rho _{2i} (1 - \pi _{10i}) (1 - \pi _{01i}), \\ \frac{\partial }{\partial {\varvec{\tau }}_{\rho _2}^\top } \rho _{2(0)i}({\varvec{\xi }})= & {} \frac{{\varvec{U}}_i}{z^2} \rho _{2i} (1 - \pi _{01i}) \left\{ \rho _{1i}(1 - \pi _{10i}) + \rho _{3i}(1 - \pi _{00i}) \right\} . \end{aligned}$$

The derivative $\dfrac{\partial }{\partial {\varvec{\xi }}^\top } \rho _{3(0)i}({\varvec{\xi }})$ can be computed by using the fact that $\rho _{3(0)i} = 1 - \rho _{1(0)i} - \rho _{2(0)i}$.

Appendix 2

Here, we show that the estimating functions $G_{i\ell r,*}$ are unbiased under the working disease and verification models. Recall that ${\varvec{\xi }} = ({\varvec{\lambda }}^\top , {\varvec{\tau }}^\top _\pi , {\varvec{\tau }}^\top _\rho )^\top$.

• FI estimator

  We have

  $$\begin{aligned} {\mathbb {E}}\left\{ G_{i\ell r,\mathrm {FI}}(\mu _0, {\varvec{\xi }}_0)\right\}= & {} {\mathbb {E}}\left\{ \rho _{1i}({\varvec{\tau }}_{0\rho }) \rho _{2\ell }({\varvec{\tau }}_{0\rho }) \rho _{3r}({\varvec{\tau }}_{0\rho }) (I_{i\ell r} - \mu ) \right\} \\= & {} {\mathbb {E}}\left\{ \rho _{1i}\rho _{2\ell }\rho _{3r}(I_{i\ell r} - \mu _0) \right\} . \end{aligned}$$

  Hence, ${\mathbb {E}}\left\{ G_{i\ell r,\mathrm {FI}}(\mu _0, {\varvec{\xi }}_0)\right\} = 0$ from (13).

• MSI estimator

  Consider ${\mathbb {E}}\left\{ D_{ki,\mathrm {MSI}}({\varvec{\xi }}_0)|T_i, {\varvec{A}}_i\right\}$. We have

  $$\begin{aligned}&{\mathbb {E}}\left\{ D_{ki,\mathrm {MSI}}({\varvec{\xi }}_0)|T_i, {\varvec{A}}_i\right\} \\&\quad = {\mathbb {E}}\left\{ V_i D_{ki} + (1 - V_i)\rho _{k(0)i}({\varvec{\xi }}_0)|T_i, {\varvec{A}}_i\right\} \\&\quad = {\mathbb {E}}\left[ {\mathbb {E}}\left\{ V_i D_{ki} + (1 - V_i)\rho _{k(0)i}({\varvec{\xi }}_0)|T_i, {\varvec{A}}_i, V_i \right\} | T_i, {\varvec{A}}_i \right] \\&\quad = \mathrm {Pr}(V_i = 1|T_i, {\varvec{A}}_i){\mathbb {E}}\left( D_{ki}|V_i = 1, T_i, {\varvec{A}}_i\right) \\&\qquad + \, \mathrm {Pr}(V_i = 0|T_i, {\varvec{A}}_i){\mathbb {E}}\left( \rho _{k(0)i}({\varvec{\xi }}_0)|V_i = 0, T_i, {\varvec{A}}_i \right) \\&\quad = \mathrm {Pr}(V_i = 1|T_i, {\varvec{A}}_i)\mathrm {Pr}(D_{ki} = 1|V_i = 1, T_i, {\varvec{A}}_i) \\&\qquad + \, \mathrm {Pr}(V_i = 0|T_i, {\varvec{A}}_i)\mathrm {Pr}(D_{ki} = 1|V_i = 0, T_i, {\varvec{A}}_i) \\&\quad = \mathrm {Pr}(D_{ki} = 1|T_i, {\varvec{A}}_i) = \rho _{ki}. \end{aligned}$$

  Therefore,

  $$\begin{aligned}&{\mathbb {E}}\left\{ G_{i\ell r, \mathrm {MSI}}(\mu _0,{\varvec{\xi }}_0) \right\} \\&\quad = {\mathbb {E}}\left\{ D_{1i,\mathrm {MSI}}({\varvec{\xi }}_0) D_{2\ell , \mathrm {MSI}}({\varvec{\xi }}_0) D_{3r, \mathrm {MSI}}({\varvec{\xi }}_0) \left( I_{i\ell r} - \mu _0 \right) \right\} \\&\quad = {\mathbb {E}}\Big [ \left( I_{i\ell r} - \mu _0 \right) {\mathbb {E}}\left\{ D_{1i,\mathrm {MSI}}({\varvec{\xi }}_0) | T_i, {\varvec{A}}_i \right\} {\mathbb {E}}\left\{ D_{2\ell ,\mathrm {MSI}}({\varvec{\xi }}_0) | T_\ell , {\varvec{A}}_\ell \right\} \\&\qquad \times \, {\mathbb {E}}\left\{ D_{3r,\mathrm {MSI}}({\varvec{\xi }}_0) | T_r, {\varvec{A}}_r \right\} \Big ] \\&\quad = {\mathbb {E}}\left\{ \rho _{1i}\rho _{2\ell }\rho _{3r}(I_{i\ell r} - \mu _0) \right\} . \end{aligned}$$

• IPW estimator

  In this case,

  $$\begin{aligned} {\mathbb {E}}\left( \frac{V_i D_{ki}}{\pi _i({\varvec{\xi }}_0)} \bigg |T_i, {\varvec{A}}_i \right)= & {} \frac{{\mathbb {E}}\left( V_i D_{ki}|T_i, {\varvec{A}}_i\right) }{\pi _i({\varvec{\xi }}_0)} \\= & {} \frac{{\mathbb {E}}\left\{ D_{ki} {\mathbb {E}}\left( V_i |D_{1i}, D_{2i}, T_i, {\varvec{A}}_i\right) \big | T_i, {\varvec{A}}_i\right\} }{\pi _i({\varvec{\xi }}_0)} \\= & {} \frac{{\mathbb {E}}\left( \pi _i D_{ki}|T_i, {\varvec{A}}_i\right) }{\pi _i} = \rho _{ki}. \end{aligned}$$

  Thus,

  $$\begin{aligned}&{\mathbb {E}}\left\{ G_{i\ell r, \mathrm {IPW}}(\mu _0, {\varvec{\xi }}_0)\right\} \\&\quad = {\mathbb {E}}\left\{ \frac{V_i V_\ell V_r D_{1i} D_{2\ell } D_{3r}}{\pi _i({\varvec{\xi }}_0) \pi _\ell ({\varvec{\xi }}_0) \pi _k({\varvec{\xi }}_0)} \left( I_{i\ell r} - \mu _0\right) \right\} \\&\quad = {\mathbb {E}}\Bigg \{ \left( I_{i\ell r} - \mu _0\right) {\mathbb {E}}\left( \frac{V_i D_{1i}}{\pi _i({\varvec{\xi }}_0)} \bigg | T_i, {\varvec{A}}_i\right) {\mathbb {E}}\left( \frac{V_\ell D_{2\ell }}{\pi _\ell ({\varvec{\xi }}_0)} \bigg |T_\ell , {\varvec{A}}_\ell \right) \\&\qquad \times \, {\mathbb {E}}\left( \frac{V_r D_{3r}}{\pi _r({\varvec{\xi }}_0)} \bigg | T_r, {\varvec{A}}_r\right) \Bigg \} \\&\quad = {\mathbb {E}}\left\{ \rho _{1i} \rho _{2\ell } \rho _{3r}(I_{i\ell r} - \mu _0) \right\} . \end{aligned}$$

• PDR estimator

  $$\begin{aligned}&{\mathbb {E}}\left\{ D_{ki, \mathrm {PDR}}({\varvec{\xi }}_0)|T_i, {\varvec{A}}_i\right\} \\&\quad = {\mathbb {E}}\left[ {\mathbb {E}}\left\{ \frac{V_i D_{ki}}{\pi _i({\varvec{\xi }}_0)} - \rho _{k(0)i}({\varvec{\xi }}_0)\left( \frac{V_i}{\pi _i({\varvec{\xi }}_0)} - 1\right) \bigg | D_{1i}, D_{2i}, T_i, {\varvec{A}}_i\right\} \bigg | T_i, {\varvec{A}}_i\right] \\&\quad = {\mathbb {E}}\Bigg \{D_{ki} {\mathbb {E}}\left( \frac{V_i}{\pi _i({\varvec{\xi }}_0)} \bigg | D_{1i}, D_{2i}, T_i, {\varvec{A}}_i\right) \\&\qquad - \, \rho _{k(0)i}({\varvec{\xi }}_0) {\mathbb {E}}\left( \frac{V_i}{\pi _i({\varvec{\xi }}_0)} - 1 \bigg | D_{1i}, D_{2i}, T_i, {\varvec{A}}_i\right) \bigg | T_i, {\varvec{A}}_i \Bigg \} \\&\quad = {\mathbb {E}}(D_{ki} | T_i, {\varvec{A}}_i) = \rho _{ki}. \end{aligned}$$

  Hence,

  $$\begin{aligned}&{\mathbb {E}}\left\{ G_{i\ell r, \mathrm {PDR}}(\mu _0,{\varvec{\xi }}_0)\right\} \\&\quad = {\mathbb {E}}\left\{ D_{1i,\mathrm {PDR}}({\varvec{\xi }}_0) D_{2\ell , \mathrm {PDR}}({\varvec{\xi }}_0) D_{3r, \mathrm {PDR}}({\varvec{\xi }}_0) \left( I_{i\ell r} - \mu _0 \right) \right\} \\&\quad = {\mathbb {E}}\Big [ \left( I_{i\ell r} - \mu _0 \right) {\mathbb {E}}\left\{ D_{1i,\mathrm {PDR}}({\varvec{\xi }}_0) | T_i, {\varvec{A}}_i \right\} {\mathbb {E}}\left\{ D_{2\ell ,\mathrm {PDR}}({\varvec{\xi }}_0) | T_\ell , {\varvec{A}}_\ell \right\} \\&\qquad \times \, {\mathbb {E}}\left\{ D_{3r,\mathrm {PDR}}({\varvec{\xi }}_0) | T_r, {\varvec{A}}_r \right\} \Big ] \\&\quad = {\mathbb {E}}\left\{ \rho _{1i}\rho _{2\ell }\rho _{3r}(I_{i\ell r} - \mu _0) \right\} . \end{aligned}$$

Table 4 Monte Carlo means (MCmean) for the maximum likelihood estimators of the elements of nuisance parameters $\varvec{\lambda }$, $\varvec{\tau }_\pi$, $\varvec{\tau }_{\rho _1}$ and $\varvec{\tau }_{\rho _2}$, when the true missing data mechanism is MAR

Table 5 Monte Carlo means (MCmean), relative bias (Bias), Monte Carlo standard deviations (MCsd) and estimated standard deviations (Esd) for the proposed VUS estimators and the SPE estimator, when the true missing data mechanism is MAR

Table 6 Monte Carlo means (MCmean) for the maximum likelihood estimators of the elements of nuisance parameters $\varvec{\lambda }$, $\varvec{\tau }_\pi$, $\varvec{\tau }_{\rho _1}$ and $\varvec{\tau }_{\rho _2}$, when the estimated models are misspecified

Table 7 Monte Carlo means (MCmean), relative bias (Bias), Monte Carlo standard deviations (MCsd) and estimated standard deviations (Esd) for the proposed VUS estimators and the SPE estimator, when the estimated models are misspecified

Appendix 3

Here, we present results of an additional simulation study, that covers the cases of: (i) missing at random (MAR) assumption for the missigness of the disease status; (ii) model misspecification in the estimation process.

In the study, the diagnostic test T, covariate A and the disease status ${\mathcal {D}}$ are generated as in scenario I of Sect. 4 of the paper. Moreover, the verification status V is:

• (i) generated as in scenario I with $h(T,A;{\varvec{\tau }}_\pi )=-1 + T - 1.2A$ and $\lambda _1 = \lambda _2 = 0$, i.e., under MAR assumption (verification rate roughly equal to 0.57);

• (ii) generated as in scenario I, but models for the verification and disease processes used in the fitting procedure are misspecified, because the estimated verification model uses as predictors $T^{1/3}$ and $\log |A|$ instead of T and A, respectively, and the estimated disease model uses $A^{1/3}$ instead of A.

In both (i) and (ii), the true VUS is 0.791. We consider three different values of sample size, i.e., 250, 500 and 1500. The number of replications in each simulation experiment is set to 1000.

Simulation results are given in Tables 4 and 5, for the case (i), and in Tables 6 and 7, for the case (ii). As expected, in case (i) results show some bias of the proposed VUS estimators when compared to the SPE estimator which is properly used here. However, the bias decreases when the sample size increases. In case (ii), all estimators appear to be biased, even when the sample size is large. Moreover, although in the considered case the bias seems to stay on acceptable levels, we expect that, given the nature of the estimators, it could be even dramatically high with other kinds of misspecification.