Nonparametric inference of the area under ROC curve under two-phase cluster sampling

ABSTRACT

Nonparametric inference of the area under ROC curve (AUC) has been well developed either in the presence of verification bias or clustering. However, current nonparametric methods are not able to handle cases where both verification bias and clustering are present. Such a case arises when a two-phase study design is applied to a cohort of subjects (verification bias) where each subject might have multiple test results (clustering). In such cases, the inference of AUC must account for both verification bias and intra-cluster correlation. In the present paper, we propose an IPW AUC estimator that corrects for verification bias and derive a variance formula to account for intra-cluster correlations between disease status and test results. Results of a simulation study indicate that the method that assumes independence underestimates the true variance of the IPW AUC estimator in the presence of intra-cluster correlations. The proposed method, on the other hand, provides a consistent variance estimate for the IPW AUC estimator by appropriately accounting for correlations between true disease statuses and between test results.

KEYWORDS:

• Area under a ROC curve
• two-phase studies
• screening tests
• verification bias
• intra-cluster correlation

Disclosure statement

No potential conflict of interest was reported by the author(s).

Appendix A. Proof of Equation (4)(4) $\hat{A}(\gamma )-A=\frac{1}{N}\sum _{i=1}^n\sum _{k=1}^{m_i}{\pi }_i^{-1}{V}_i{\in }_{ik}+o_p(n^{-1/2}),$(4)

First observe that $\hat{A}-A$ can be expressed as

$\widehat{A}-A={\displaystyle \int (\widehat{\overline{F}}(}t)-\overline{F}(t))d(\widehat{\overline{G}}(t)-\overline{G}(t))+{\displaystyle \int \overline{F}}(t)d\left[\widehat{\overline{G}}(t)-\overline{G}(t)\right]+{\displaystyle \int \left[\widehat{\overline{F}}(t)-\overline{F}(t)\right]}d\overline{G}(t)$

$=W_1+W_2+W_3.$

Now, we prove that $W_1=o_p(n^{-1/2})$. To this end, we observe that $W_1$ can be written as

$W_1={\displaystyle \int (\widehat{\overline{F}}(}t)-\overline{F}(t))d\widehat{\overline{G}}(t)-{\displaystyle \int (\widehat{\overline{F}}(}t)-\overline{F}(t))d\overline{G}(t)$

$=\frac{{\displaystyle {\sum }_{j=1}^n{\displaystyle {\sum }_{l=1}^{m_j}{\pi }_j^{-1}{V}_j{D}_{jl}}}\left[\widehat{\overline{F}}(T_{jl})-\overline{F}(T_{jl})-{\displaystyle \int (\widehat{\overline{F}}(}t)-\overline{F}(t))d\overline{G}(t)\right]}{{\displaystyle {\sum }_{j=1}^n{\displaystyle {\sum }_{l=1}^{m_j}{\pi }_j^{-1}{V}_j{D}_{jl}}}}$

Define $H(T_{ik},T_{jl})=\psi (T_{ik},T_{jl})-\bar{F}(T_{jl})+\bar{G}(T_{ik})+\theta -1.$ Then,

$W_1=\frac{\frac{1}{N^2}{\sum }_{i=1}^n{\sum }_{j=1}^n{\sum }_{k=1}^{m_i}{\sum }_{l=1}^{m_j}{\pi }_i^{-1}{\pi }_j^{-1}{V}_i{V}_j(1-D_{ik})D_{jl}H(T_{ik},T_{jl})}{\frac{1}{N}{\sum }_{i=1}^n{\sum }_{k=1}^{m_i}{\pi }_i^{-1}{V}_i(1-D_{ik})\frac{1}{N}{\sum }_{j=1}^n{\sum }_{l=1}^{m_j}{\pi }_j^{-1}{V}_j{D}_{jk}}$

Since $\frac{1}{N}{\sum }_{i=1}^n{\sum }_{k=1}^{m_i}{\pi }_i^{-1}{V}_i(1-D_{ik})\stackrel{p}{⟶}Pr(D=0),\frac{1}{N}{\sum }_{j=1}^n{\pi }_j^{-1}{V}_j{D}_{jl}\stackrel{p}{⟶}Pr(D=1)$, thus it suffices to show

${\hat{W}}_1=\frac{1}{N^2}\sum _{i=1}^n\sum _{j=1}^n\sum _{k=1}^{m_i}\sum _{l=1}^{m_j}{\pi }_i^{-1}{\pi }_j^{-1}{V}_i{V}_j(1-D_{ik})D_{jl}H(T_{ik},T_{jl}).=o_p(n^{-1/2}).$

Now, we have

$E\left[{\pi }_i^{-1}{\pi }_{i{ }^{\mathrm{\prime }}}^{-1}{\pi }_j^{-2}{V}_i{V}_{i{ }^{\mathrm{\prime }}}{V}_j(1-D_{ik})(1-D_{i{ }^{\mathrm{\prime }}k{ }^{\mathrm{\prime }}})D_{jl}H(T_{ik},T_{jl})H(T_{i{ }^{\mathrm{\prime }}k{ }^{\mathrm{\prime }}},T_{jl})\right]=0,i\ne i{ }^{\mathrm{\prime }},$

$E\left[{\pi }_i^{-2}{\pi }_j^{-1}{\pi }_{j{ }^{\mathrm{\prime }}}^{-1}{V}_i{V}_j{V}_{j{ }^{\mathrm{\prime }}}(1-D_{ik})D_{jl}{D}_{j{ }^{\mathrm{\prime }}l{ }^{\mathrm{\prime }}}H(T_{ik},T_{jl})H(T_{ik},T_{j{ }^{\mathrm{\prime }}l{ }^{\mathrm{\prime }}})\right]=0,j\ne j{ }^{\mathrm{\prime }}.$

Therefore,

$E{\hat{W}}_1^2=\frac{1}{N^4}\sum _{i,j=1}^n\sum _{i{ }^{\mathrm{\prime }},j{ }^{\mathrm{\prime }}=1}^n\sum _{k=1}^{m_i}\sum _{l=1}^{m_j}\sum _{k{ }^{\mathrm{\prime }}=1}^{m_{i{ }^{\mathrm{\prime }}}}\sum _{l{ }^{\mathrm{\prime }}=1}^{m_{j{ }^{\mathrm{\prime }}}}E[{\pi }_i^{-1}{\pi }_j^{-1}{V}_i{V}_j(1-D_{ik})D_{jl}H(T_{ik},T_{jl})$

${\pi }_{i{ }^{\mathrm{\prime }}}^{-1}{\pi }_{j{ }^{\mathrm{\prime }}}^{-1}{V}_{i{ }^{\mathrm{\prime }}}{V}_{j{ }^{\mathrm{\prime }}}(1-D_{i{ }^{\mathrm{\prime }}k{ }^{\mathrm{\prime }}})D_{j{ }^{\mathrm{\prime }}l{ }^{\mathrm{\prime }}}H(T_{i{ }^{\mathrm{\prime }}k{ }^{\mathrm{\prime }}},T_{j{ }^{\mathrm{\prime }}l{ }^{\mathrm{\prime }}})]$

$=\frac{1}{N^4}\sum _{i,j=1}^n\sum _{k=1}^{m_i}\sum _{l=1}^{m_j}\sum _{k{ }^{\mathrm{\prime }}=1}^{m_{i{ }^{\mathrm{\prime }}}}\sum _{l{ }^{\mathrm{\prime }}=1}^{m_{j{ }^{\mathrm{\prime }}}}E[{\pi }_i^{-2}{\pi }_j^{-2}{V}_i{V}_j(1-D_{ik})D_{jl}H(T_{ik},T_{jl})H(T_{ik{ }^{\mathrm{\prime }}},T_{jl{ }^{\mathrm{\prime }}})]$

$\le \frac{4}{N^4}\sum _{i,j=1}^n\sum _{k=1}^{m_i}\sum _{l=1}^{m_j}\sum _{k{ }^{\mathrm{\prime }}=1}^{m_{i{ }^{\mathrm{\prime }}}}\sum _{l{ }^{\mathrm{\prime }}=1}^{m_{j{ }^{\mathrm{\prime }}}}{\pi }_i^{-2}{\pi }_j^{-2}$

$\le \frac{4}{N^4}\sum _{i,j=1}^n\frac{m_{max}^4}{{\pi }_{min}^4}=o_p(n^{-1/2}),$

where $m_{max}$ is the maximum cluster size and ${\pi }_{min}$ is the minimum sampling probability. This shows that $W_1=o_p(n^{-1/2})$. Finally, note that

$W_2=\frac{{\sum }_{i=1}^n{\sum }_{k=1}^{m_i}{\pi }_i^{-1}{V}_i{D}_{ik}\left[\bar{F}(T_{ik})-\theta \right]}{{\sum }_{i=1}^n{\sum }_{k=1}^{m_i}{\pi }_i^{-1}{V}_i{D}_{ik}}.$

$W_3=\frac{{\sum }_{i=1}^n{\sum }_{k=1}^{m_i}{\pi }_i^{-1}{V}_i(1-D_{ij})\left[1-\bar{G}(T_{ik})-\theta \right]}{{\sum }_{i=1}^n{\sum }_{k=1}^{m_i}{\pi }_i^{-1}{V}_i(1-D_{ik})},$

which completes the proof of Equation (4)(4) $\hat{A}(\gamma )-A=\frac{1}{N}\sum _{i=1}^n\sum _{k=1}^{m_i}{\pi }_i^{-1}{V}_i{\in }_{ik}+o_p(n^{-1/2}),$(4) .

Additional information

Funding

The author(s) reported that there is no funding associated with the work featured in this article.