A LASSO Method to Identify Protein Signature Predicting Post-transplant Renal Graft Survival

Abstract

Identifying novel biomarkers to predict renal graft survival is important in post-transplant clinical practice. Serum creatinine, currently the most popular surrogate biomarker, offers limited information on the underlying allograft profiles. It is known to perform unsatisfactorily to predict renal function. In this paper, we apply a LASSO machine-learning algorithm in the Cox proportional hazards model to identify promising proteins that are associated with the hazard of allograft loss after renal transplantation, motivated by a clinical pilot study that collected 47 patients receiving renal transplants at the University of Michigan Hospital. We assess the association of 17 proteins previously identified by Cibrik et al. (PROTEOMICS Clin Appl 7(11–12): 839–849, 2013) with allograft rejection in our regularized Cox regression analysis, where the LASSO variable selection method is applied to select important proteins that predict the hazard of allograft loss. We also develop a post-selection inference to further investigate the statistical significance of the proteins on the hazard of allograft loss, and conclude that two proteins KIM-1 and VEGF-R2 are important protein markers for risk prediction.

Appendix: Derivation of the Asymptotic Distribution of the Bias-Corrected Estimator \(\hat{\varvec{\beta }}_\mathrm{bc}\)

Denote the partial score function by $\varvec{s}_n(\varvec{\beta }) = \frac{1}{n}\sum _{i=1}^n\delta _i\left[ \varvec{X}_i - \frac{\sum _{j=1}^nI(\mathcal {T}_j \ge \mathcal {T}_i)\exp (\varvec{X}_j^\mathrm{T}\varvec{\beta })\varvec{X}_j}{\sum _{j=1}^nI(\mathcal {T}_j \ge \mathcal {T}_i)\exp (\varvec{X}_j^\mathrm{T}\varvec{\beta })}\right]$. Denote $\hat{\varvec{\beta }} = \hat{\varvec{\beta }}(\lambda _n)$ given in (2), and let $\varvec{\beta }^0$ be the true value of $\varvec{\beta }$. It is known that the regularized estimator of $\varvec{\beta }$ in (2) satisfies KKT condition (3), namely,

$$\begin{aligned} \lambda _n \hat{\varvec{\kappa }} = \frac{1}{n}\sum _{i=1}^n\delta _i\left[ \varvec{X}_i - \frac{\sum _{j=1}^nI(\mathcal {T}_j \ge \mathcal {T}_i)\exp (\varvec{X}_j^\mathrm{T}\hat{\varvec{\beta }})\varvec{X}_j}{\sum _{j=1}^nI(\mathcal {T}_j \ge \mathcal {T}_i) \exp (\varvec{X}_j^\mathrm{T}\hat{\varvec{\beta }})}\right] . \end{aligned}$$

(6)

Adding $s_n(\varvec{\beta }^0)$ on both sides of (3), we obtain

$$\begin{aligned}&\frac{1}{n}\sum _{i=1}^n\delta _i\left[ \frac{S_{n2}(\mathcal {T}_i)}{S_{n0}(\mathcal {T}_i)} - \frac{S^{\otimes 2}_{n1}(\mathcal {T}_i)}{S^{2}_{n0}(\mathcal {T}_i)}\right] (\hat{\varvec{\beta }} - \varvec{\beta }^0) \\&\qquad +\, \lambda _n\hat{\varvec{\kappa }} + O_{ p}(\Vert \hat{\varvec{\beta }} - \varvec{\beta }^0\Vert _2^2) = s_n\big (\varvec{\beta }^0\big ), \end{aligned}$$

where $S_{nk}(y; \varvec{\beta }) = \sum _{j=1}^nI(\mathcal {T}_j \ge y)\exp (\varvec{X}_j^\mathrm{T}\varvec{\beta })\varvec{X}_j^{\otimes k}$ and $S_{nk}(y) = S_{nk}(y; \varvec{\beta }^0)$. Then the KKT conditions can be rewritten as follows:

$$\begin{aligned} \hat{\varvec{\beta }} - \varvec{\beta }^0 + \left\{ I_n\big (\varvec{\beta }^0\big )\right\} ^{-1}\lambda _n\hat{\varvec{\kappa }} = \left\{ I_n\big (\varvec{\beta }^0\big )\right\} ^{-1}s_n\big (\varvec{\beta }^0\big ) + O_{p}(\Vert \hat{\varvec{\beta }} - \varvec{\beta }^0\Vert _2^2), \end{aligned}$$

(7)

where the observed Fisher information matrix $I_n(\varvec{\beta }) =\frac{1}{n}\sum _{i=1}^n\delta _i\left[ \frac{S_{n2}(\mathcal {T}_i; \varvec{\beta })}{S_{n0}(\mathcal {T}_i; \varvec{\beta })} - \frac{S^{\otimes 2}_{n1}(\mathcal {T}_i; \varvec{\beta })}{S^{ 2}_{n0}(\tilde{T}_i; \varvec{\beta })}\right]$, which is assumed to be non-singular. For any vector $\varvec{a}$, denote $\varvec{a}^{\otimes 0} = 1, \varvec{a}^{\otimes 1} = \varvec{a}$, and $\varvec{a}^{\otimes 2} = \varvec{a}\varvec{a}^\mathrm{T}$.

Since the second term in (6) is asymptotically negligible, the following bias-corrected estimator is given by

$$\begin{aligned} \hat{\varvec{\beta }}_\mathrm{bc}= & {} \hat{\varvec{\beta }} + \left\{ I_n\big (\varvec{\beta }^0\big )\right\} ^{-1}\lambda _n\hat{\varvec{\kappa }}= \hat{\varvec{\beta }} + \left\{ I_n\big (\varvec{\beta }^0\big )\right\} ^{-1}\varvec{s}_n(\hat{\varvec{\beta }}), \end{aligned}$$

(8)

where the second equality holds due to Eq. (6). Replacing with consistent estimates $I_n(\hat{\varvec{\beta }})$ and $s_n(\hat{\varvec{\beta }})$ for $I_n(\varvec{\beta }^0)$ and $\varvec{s}_n(\varvec{\beta }^0)$, respectively, we immediately obtain an asymptotic pivot. Denote the Fisher information matrix by $I(\varvec{\beta }) = E\left\{ I_n(\varvec{\beta })\right\}$. By the central limit theorem, the asymptotic distribution of ${\hat{\varvec{\beta }}}_\mathrm{bc}$ follows from expression (8) under the same regularity conditions required for the consistency of the LASSO estimator.