Dependence measure for length-biased survival data using kernel density estimation with a regression procedure

Abstract

In statistical literature, several dependence measures have been extensively established and treated, including Pearson's correlation coefficient, Spearman's ρ and Kendall's τ. In the context of survival analysis with length-biased data, a measure of dependence between survival time and covariates appears to have not received much intention in the literature. The purpose of this paper is to extend Kent's [Information gain and a general measure of correlation. Biometrika. 1983;70(1):163–173.] dependence measure, based on the concept of information gain, to length-biased survival data. Specifically, we develop a new approach to measure the degree of dependence between survival time and several continuous covariates, without censoring, when the relationship is linear. In this regard, kernel density estimation with a regression procedure is proposed. The consistency for all proposed estimators is established. In particular, the performance of the dependence measure for length-biased data is investigated by means of simulations studies.

Keywords:

• Information gain
• correlation
• dependence
• length-biased distribution
• kernel smoothing
• regression

MSCs:

• 62H20
• 62N86

Disclosure statement

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

Appendix

Proof of Corollary 3.2

If ${\mathbf{Z}}^{(2)}$ is absent then $\mathbf{Z}={\mathbf{Z}}^{(1)}.$ So that, from Equation (6(6) $\begin{array}{rl}{\mathrm{\Gamma }}_{\mathtt{\text{PA}}}& =2\{{\int }_{{\mathbb{R}}^{d+1}}\log \left\{f_{LB}(u,{\mathbf{z}}^{(1)}|{\mathbf{z}}^{(2)})\right\}{f}_{LB}(u,\mathbf{z})\mathrm{d}u\mathrm{d}\mathbf{z}\\ & -{\int }_{{\mathbb{R}}^{d+1}}\log \left\{f_{LB}(u,{\mathbf{z}}^{(1)})\right\}{f}_{LB}(u,\mathbf{z})\mathrm{d}u\mathrm{d}\mathbf{z}\}.\end{array}$(6) ), we are in the case where we seek a joint measure of dependence. Equation (6(6) $\begin{array}{rl}{\mathrm{\Gamma }}_{\mathtt{\text{PA}}}& =2\{{\int }_{{\mathbb{R}}^{d+1}}\log \left\{f_{LB}(u,{\mathbf{z}}^{(1)}|{\mathbf{z}}^{(2)})\right\}{f}_{LB}(u,\mathbf{z})\mathrm{d}u\mathrm{d}\mathbf{z}\\ & -{\int }_{{\mathbb{R}}^{d+1}}\log \left\{f_{LB}(u,{\mathbf{z}}^{(1)})\right\}{f}_{LB}(u,\mathbf{z})\mathrm{d}u\mathrm{d}\mathbf{z}\}.\end{array}$(6) ) becomes $\begin{array}{rl}\mathrm{\Gamma }& =2\{{\int }_{{\mathbb{R}}^{p+1}}\log \left\{f_{LB}\left(u|\mathbf{z}\right)f_B\left(\mathbf{z}\right)\right\}{f}_{LB}(u,\mathbf{z})\mathrm{d}u\mathrm{d}\mathbf{z}\\ & -{\int }_{{\mathbb{R}}^{p+1}}\log \left\{f_{LB}\left(u\right)\right\}{f}_{LB}(u,\mathbf{z})\mathrm{d}u\mathrm{d}\mathbf{z}-{\int }_{{\mathbb{R}}^{p+1}}\log \left\{f_{\mathbf{Z}}\left(\mathbf{z}\right)\right\}{f}_{LB}(u,\mathbf{z})\mathrm{d}u\mathrm{d}\mathbf{z}\}\\ & =2\{{\int }_{{\mathbb{R}}^{p+1}}\log \left\{f_{LB}\left(u|\mathbf{z}\right)\right\}{f}_{LB}(u,\mathbf{z})\mathrm{d}u\mathrm{d}\mathbf{z}+{\int }_{{\mathbb{R}}^{p+1}}\log \left\{f_B\left(\mathbf{z}\right)\right\}{f}_{LB}(u,\mathbf{z})\mathrm{d}u\mathrm{d}\mathbf{z}\\ & -{\int }_{{\mathbb{R}}^{p+1}}\log \left\{f_{LB}\left(u\right)\right\}{f}_{LB}(u,\mathbf{z})\mathrm{d}u\mathrm{d}\mathbf{z}-{\int }_{{\mathbb{R}}^{p+1}}\log \left\{f_{\mathbf{Z}}\left(\mathbf{z}\right)\right\}{f}_{LB}(u,\mathbf{z})\mathrm{d}u\mathrm{d}\mathbf{z}\\ & =2\{{\int }_{{\mathbb{R}}^{p+1}}\log \left\{f_{LB}\left(u|\mathbf{z}\right)\right\}{f}_{LB}(u,\mathbf{z})\mathrm{d}u\mathrm{d}\mathbf{z}+{\int }_{{\mathbb{R}}^p}\log \left\{f_B\left(\mathbf{z}\right)\right\}{f}_B\left(\mathbf{z}\right)\mathrm{d}\mathbf{z}\\ & -{\int }_{{\mathbb{R}}^{}}\log \left\{f_{LB}\left(u\right)\right\}{f}_{LB}\left(u\right)\mathrm{d}u-{\int }_{{\mathbb{R}}^p}\log \left\{f_{\mathbf{Z}}\left(\mathbf{z}\right)\right\}{f}_B\left(\mathbf{z}\right)\mathrm{d}\mathbf{z}\},\end{array}$where we used $f_{LB}(u)={\int }_{{\mathbb{R}}^p}{f}_{LB}(u,\mathbf{z})d\mathbf{z}$ and $f_B(\mathbf{z})={\int }_{{\mathbb{R}}^{}}{f}_{LB}(u,\mathbf{z})du$. It follows that, $\mathrm{\Gamma }={\mathrm{\Gamma }}_C+{\mathrm{\Gamma }}_B$, where ${\mathrm{\Gamma }}_C$ is given by (9) and ${\mathrm{\Gamma }}_B=2\{{\int }_{{\mathbb{R}}^p}\log \left\{f_B\left(\mathbf{z}\right)\right\}{f}_B\left(\mathbf{z}\right)\mathrm{d}\mathbf{z}-{\int }_{{\mathbb{R}}^p}\log \left\{f_{\mathbf{Z}}\left(\mathbf{z}\right)\right\}{f}_B\left(\mathbf{z}\right)\mathrm{d}\mathbf{z}\}.$Therefore, ${\rho }_J^2(U,\mathbf{Z})=1-\exp \{-({\mathrm{\Gamma }}_C+{\mathrm{\Gamma }}_B)\}.$

Proof of Theorem 3.2

Under Assumptions 3.3 (c), we have by Lemma 3.1 (23) ${\hat{\nu }}_{\hat{\mathit{\beta }}}=(\frac{1}{n}\sum _{i=1}^n\exp \left\{{\hat{\mathit{\beta }}}^{\mathrm{\prime }}{\mathbf{Z}}_i^{\ast }\right\}){\mathcal{M}}_{\mathbf{S}}\left({\mathbf{H}}^{\mathrm{\prime }}\hat{\mathit{\beta }}\right).$(23) Under some regularity conditions [21], Assumptions 3.3 (a) holds. Further, $\mathbf{H}\stackrel{}{\to }\mathbf{0}$ as $n\to \mathrm{\infty }.$ Using Slutsky's theorem and Assumptions 3.3 (b), one has (24) ${\mathcal{M}}_{\mathbf{S}}\left({\mathbf{H}}^{\mathrm{\prime }}\hat{\mathit{\beta }}\right)\stackrel{a.s.}{\to }{\mathcal{M}}_{\mathbf{S}}\left(0\right)=1\text{as}n\to \mathrm{\infty }\cdot$(24) Let $\epsilon >0.$ From Assumptions 3.3 (a), $\mathrm{\exists }{\delta }_1$, $\mathrm{\forall }n\ge {\delta }_1$, $\mathit{\beta }-\epsilon \le \hat{\mathit{\beta }}\le \mathit{\beta }+\epsilon ,\text{a.s.}$ which implies $\hat{\mathit{\beta }}=\mathit{\beta }+o(\epsilon )$. Notice that Assumptions 3 (c) and the strong law of large numbers lead to ${\mathcal{M}}_n\left(\mathit{\beta }\right)=\frac{1}{n}\sum _{i=1}^n\exp \left\{{\mathit{\beta }}^{\mathrm{\prime }}{\mathbf{Z}}_i^{\ast }\right\}\stackrel{a.s.}{\to }\mathcal{M}\left(\mathit{\beta }\right)={\mathrm{E}}_{f_{\mathbf{Z}}}[\exp \left\{{\mathit{\beta }}^{\mathrm{\prime }}\mathbf{Z}\right\}]\text{as}n\to \mathrm{\infty }$because ${\nu }_{\mathit{\beta }}<\mathrm{\infty }$. It follows that, $\mathrm{\exists }{\delta }_2$, $\mathrm{\forall }n\ge {\delta }_2$, ${\mathcal{M}}_n(\mathit{\beta })=\mathcal{M}(\mathit{\beta })+o(\epsilon )\text{a.s.}$ and we can write $\mathrm{\forall }n\ge {\delta }_2$, ${\mathcal{M}}_n(\hat{\mathit{\beta }})={\mathcal{M}}_n(\mathit{\beta }+o(\epsilon )).$ It follows that $\mathrm{\forall }n\ge \text{max}({\delta }_1,{\delta }_2)$, ${\mathcal{M}}_n(\hat{\mathit{\beta }})=\mathcal{M}(\mathit{\beta })+o(\epsilon ),\text{a.s.}$ which implies that (25) $\frac{1}{n}\sum _{i=1}^n\exp \left\{{\hat{\mathit{\beta }}}^{\mathrm{\prime }}{\mathbf{Z}}_i^{\ast }\right\}\stackrel{a.s.}{\to }{\text{E}}_{f_{\mathbf{Z}}}[\exp \left\{{\mathit{\beta }}^{\mathrm{\prime }}\mathbf{Z}\right\}]\text{as}n\to \mathrm{\infty }.$(25) According to (23(23) ${\hat{\nu }}_{\hat{\mathit{\beta }}}=(\frac{1}{n}\sum _{i=1}^n\exp \left\{{\hat{\mathit{\beta }}}^{\mathrm{\prime }}{\mathbf{Z}}_i^{\ast }\right\}){\mathcal{M}}_{\mathbf{S}}\left({\mathbf{H}}^{\mathrm{\prime }}\hat{\mathit{\beta }}\right).$(23) ), (24(24) ${\mathcal{M}}_{\mathbf{S}}\left({\mathbf{H}}^{\mathrm{\prime }}\hat{\mathit{\beta }}\right)\stackrel{a.s.}{\to }{\mathcal{M}}_{\mathbf{S}}\left(0\right)=1\text{as}n\to \mathrm{\infty }\cdot$(24) ) and (25(25) $\frac{1}{n}\sum _{i=1}^n\exp \left\{{\hat{\mathit{\beta }}}^{\mathrm{\prime }}{\mathbf{Z}}_i^{\ast }\right\}\stackrel{a.s.}{\to }{\text{E}}_{f_{\mathbf{Z}}}[\exp \left\{{\mathit{\beta }}^{\mathrm{\prime }}\mathbf{Z}\right\}]\text{as}n\to \mathrm{\infty }.$(25) ), one concludes by Slutsky's theorem that ${\hat{\nu }}_{\hat{\mathit{\beta }}}\stackrel{a.s.}{\to }{\nu }_{\mathit{\beta }}$ as $n\to \mathrm{\infty }$. Hence, ${\hat{\nu }}_{\hat{\mathit{\beta }}}$ is a consistent estimator of ${\nu }_{\mathit{\beta }}.$

Proof of Theorem 3.4

From Equation (21(21) ${\hat{\mathrm{\Gamma }}}_{\mathtt{\text{PA}}}={\hat{\mathrm{\Gamma }}}_{\mathtt{\text{PA}},1}+{\hat{\mathrm{\Gamma }}}_{\mathtt{\text{PA}},2},$(21) ), we have ${\hat{\mathrm{\Gamma }}}_{\mathtt{\text{PA}}}={\hat{\mathrm{\Gamma }}}_{\mathtt{\text{PA}},1}+{\hat{\mathrm{\Gamma }}}_{\mathtt{\text{PA}},2}.$ Firstly, we may write ${\hat{\mathrm{\Gamma }}}_{\mathtt{\text{PA}},1}=2\{V_1+W_{1,1}-W_{1,0}\}$, where

• $V_1=\frac{1}{n}\sum _{j=1}^n\log \{f_{LB}(U_j|{\mathbf{Z}}_j)\}-\frac{1}{n}\sum _{j=1}^n\log \{f_{LB}(U_j|{\mathbf{Z}}_j^{(1)})\};$

• $W_{1,1}=\frac{1}{n}\sum _{j=1}^n\log \{{\hat{f}}_{LB}(U_j|{\mathbf{Z}}_j)\}-\frac{1}{n}\sum _{j=1}^n\log \{f_{LB}(U_j|{\mathbf{Z}}_j)\};$

• $W_{1,0}=\frac{1}{n}\sum _{j=1}^n\log \{{\hat{f}}_{LB}(U_j|{\mathbf{Z}}_j^{(1)})\}-\frac{1}{n}\sum _{j=1}^n\log \{f_{LB}(U_j|{\mathbf{Z}}_j^{(1)})\}.$

The law of large numbers and Slutsky's theorem, lead to conclude that as $n\to \mathrm{\infty }$ $V_1\stackrel{P}{\to }\mathrm{E}[\log \{f_{LB}(U_{}|{\mathbf{Z}}_{})\}]-\mathrm{E}[\log \{f_{LB}(U_{}|{\mathbf{Z}}_{}^{(1)})\}].$ Moreover, by Theorem 1, $W_{1,1}\stackrel{P}{\to }0$ and $W_{1,0}\stackrel{P}{\to }0$ as $n\to \mathrm{\infty }$. So that, (26) ${\hat{\mathrm{\Gamma }}}_{\mathtt{\text{PA}},1}\stackrel{P}{\to }2\{\mathrm{E}[\log \left\{f_{LB}\left(U_{}|{\mathbf{Z}}_{}\right)\right\}]-\mathrm{E}[\log \left\{f_{LB}\left(U_{}|{\mathbf{Z}}_{}^{(1)}\right)\right\}]\}={\mathrm{\Gamma }}_{\mathtt{\text{PA}},1}\text{as}n\to \mathrm{\infty }.$(26) Secondly, we may write ${\hat{\mathrm{\Gamma }}}_{\mathtt{\text{PA}},2}=2\{V_2+W_{2,1}-W_{2,0}\}$, where

• $V_2=\frac{1}{n}\sum _{j=1}^n\log \{f_B({\mathbf{Z}}_j^{(1)}|{\mathbf{Z}}_j^{(2)})\}-\frac{1}{n}\sum _{j=1}^n\log \{f_B({\mathbf{Z}}_j^{(1)})\};$

• $W_{2,1}=\frac{1}{n}\sum _{j=1}^n\log \{{\hat{f}}_B({\mathbf{Z}}_j^{(1)}|{\mathbf{Z}}_j^{(2)})\}-\frac{1}{n}\sum _{j=1}^n\log \{f_B({\mathbf{Z}}_j^{(1)}|{\mathbf{Z}}_j^{(2)})\};$

• $W_{2,0}=\frac{1}{n}\sum _{j=1}^n\log \{{\hat{f}}_B({\mathbf{Z}}_j^{(1)})\}-\frac{1}{n}\sum _{j=1}^n\log \{f_B({\mathbf{Z}}_j^{(1)})\}.$

Applying the law of large numbers and Slutsky's theorem, we conclude that as $n\to \mathrm{\infty }$, $V_2\stackrel{P}{\to }\mathrm{E}[\log \{f_B({{\mathbf{Z}}^{(1)}}_{}|{\mathbf{Z}}_{}^{(2)})\}]-\mathrm{E}[\log \{f_B({\mathbf{Z}}_{}^{(1)})\}].$ In addition, according to Theorem 3.3, $W_{2,1}\stackrel{P}{\to }0$ and $W_{2,0}\stackrel{P}{\to }0$ as $n\to \mathrm{\infty }.$ So that, (27) ${\hat{\mathrm{\Gamma }}}_{\mathtt{\text{PA}},2}\stackrel{P}{\to }2\mathrm{E}[\log \left\{f_B\left({{\mathbf{Z}}^{(1)}}_{}|{\mathbf{Z}}_{}^{(2)}\right)\right\}]-\mathrm{E}[\log \left\{f_B\left({\mathbf{Z}}_{}^{(1)}\right)\right\}]={\mathrm{\Gamma }}_{\mathtt{\text{PA}},2}\text{as}n\to \mathrm{\infty }.$(27) Therefore, from Equations (26(26) ${\hat{\mathrm{\Gamma }}}_{\mathtt{\text{PA}},1}\stackrel{P}{\to }2\{\mathrm{E}[\log \left\{f_{LB}\left(U_{}|{\mathbf{Z}}_{}\right)\right\}]-\mathrm{E}[\log \left\{f_{LB}\left(U_{}|{\mathbf{Z}}_{}^{(1)}\right)\right\}]\}={\mathrm{\Gamma }}_{\mathtt{\text{PA}},1}\text{as}n\to \mathrm{\infty }.$(26) ) and (27(27) ${\hat{\mathrm{\Gamma }}}_{\mathtt{\text{PA}},2}\stackrel{P}{\to }2\mathrm{E}[\log \left\{f_B\left({{\mathbf{Z}}^{(1)}}_{}|{\mathbf{Z}}_{}^{(2)}\right)\right\}]-\mathrm{E}[\log \left\{f_B\left({\mathbf{Z}}_{}^{(1)}\right)\right\}]={\mathrm{\Gamma }}_{\mathtt{\text{PA}},2}\text{as}n\to \mathrm{\infty }.$(27) ), one obtains ${\hat{\mathrm{\Gamma }}}_{\mathtt{\text{PA}}}={\hat{\mathrm{\Gamma }}}_{\mathtt{\text{PA}},1}+{\hat{\mathrm{\Gamma }}}_{\mathtt{\text{PA}},2}\stackrel{P}{\to }{\mathrm{\Gamma }}_{\mathtt{\text{PA}}}={\mathrm{\Gamma }}_{\mathtt{\text{PA}},1}+{\mathrm{\Gamma }}_{\mathtt{\text{PA}},2}\text{as}\to \mathrm{\infty }.$

Additional information

Funding

This work was supported by Natural Sciences and Engineering Research Council of Canada [Mayer Alvo/OGP0009068,Mhamed Mesfioui/ 06536-2018].