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Bayesian penalized Buckley-James method for high dimensional bivariate censored regression models

Abstract

For high dimensional gene expression data, one important goal is to identify a small number of genes that are associated with progression of the disease or survival of the patients. In this paper, we consider the problem of variable selection for multivariate survival data. We propose an estimation procedure for high dimensional accelerated failure time (AFT) models with bivariate censored data. The method extends the Buckley-James method by minimizing a penalized \(L_2\) loss function with a penalty function induced from a bivariate spike-and-slab prior specification. In the proposed algorithm, censored observations are imputed using the Kaplan-Meier estimator, which avoids a parametric assumption on the error terms. Our empirical studies demonstrate that the proposed method provides better performance compared to the alternative procedures designed for univariate survival data regardless of whether the true events are correlated or not, and conceptualizes a formal way of handling bivariate survival data for AFT models. Findings from the analysis of a myeloma clinical trial using the proposed method are also presented.

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References

  • Ahmed SE, Hossain S, Doksum KA (2012) Lasso and shrinkage estimation in weibull censored regression models. J Stat Plan Inference 142(6):1273–1284

    MathSciNet  MATH  Google Scholar 

  • Barber RF, Candès EJ et al (2015) Controlling the false discovery rate via knockoffs. Annal Stat 43(5):2055–2085

    MathSciNet  MATH  Google Scholar 

  • Barbieri MM, Berger JO (2004) Optimal predictive model selection. Annal Stat 32(3):870–897

    MathSciNet  MATH  Google Scholar 

  • Buckley J, James I (1979) Linear regression with censored data. Biometrika 66(3):429–436

    MATH  Google Scholar 

  • Cai T, Huang J, Tian L (2009) Regularized estimation for the accelerated failure time model. Biometrics 65(2):394–404

    MathSciNet  MATH  Google Scholar 

  • Candès E, Fan Y, Janson L, Lv J (2018) Panning for gold: Model-free knockoffs for high-dimensional controlled variable selection. J R Stat Soc: Ser B (Stat Methodol) 80(3):551–577

    MathSciNet  MATH  Google Scholar 

  • Chang SH (2004) Estimating marginal effects in accelerated failure time models for serial sojourn times among repeated events. Lifetime Data Anal 10(2):175–190

    MathSciNet  MATH  Google Scholar 

  • Chatonnet F, Pignarre A, Sérandour AA, Caron G, Avner S, Robert N, Kassambara A, Laurent A, Bizot M, Agirre X et al (2020) The hydroxymethylome of multiple myeloma identifies fam72d as a 1q21 marker linked to proliferation. Haematologica 105(3):774–783

    Google Scholar 

  • Chiou SH, Kang S, Kim J, Yan J (2014) Marginal semiparametric multivariate accelerated failure time model with generalized estimating equations. Lifetime Data Anal 20(4):599–618

    MathSciNet  MATH  Google Scholar 

  • Cox DR (1972) Regression models and life-tables. J R Stat Soc: Ser B (Methodol) 34(2):187–202

    MathSciNet  MATH  Google Scholar 

  • Duan W, Zhang R, Zhao Y, Shen S, Wei Y, Chen F, Christiani DC (2018) Bayesian variable selection for parametric survival model with applications to cancer omics data. Human Genom 12(1):49

    Google Scholar 

  • George EI, McCulloch RE (1997) Approaches for bayesian variable selection. Stat Sinica 7(2):339–373

    MATH  Google Scholar 

  • Hanagal DD (2006) Bivariate weibull regression model based on censored samples. Stat Papers 47(1):137–147

    MathSciNet  MATH  Google Scholar 

  • Hawley TS, Riz I, Yang W, Wakabayashi Y, DePalma L, Chang YT, Peng W, Zhu J, Hawley RG (2013) Identification of an abcb1 (p-glycoprotein)-positive carfilzomib-resistant myeloma subpopulation by the pluripotent stem cell fluorescent dye cdy1. Am J Hematol 88(4):265–272

    Google Scholar 

  • He W, Lawless JF (2005) Bivariate location-scale models for regression analysis, with applications to lifetime data. J R Stat Soc: Ser B (Stat Methodol) 67(1):63–78

    MathSciNet  MATH  Google Scholar 

  • Hornsteiner U, Hamerle A (1996) A combined gee/buckley-james method for estimating an accelerated failure time model of multivariate failure times. Discussion Paper 47, Ludwig-Maximillians Universitat, Munchen. Also available from http://stat.unimuenchen.de/sfb386/publikation.html

  • Hu J, Chai H (2013) Adjusted regularized estimation in the accelerated failure time model with high dimensional covariates. J Multiv Anal 122:96–114

    MathSciNet  MATH  Google Scholar 

  • Huang J, Ma S (2010) Variable selection in the accelerated failure time model via the bridge method. Lifetime Data Anal 16(2):176–195

    MathSciNet  MATH  Google Scholar 

  • Huang J, Ma S, Xie H (2006) Regularized estimation in the accelerated failure time model with high-dimensional covariates. Biometrics 62(3):813–820

    MathSciNet  MATH  Google Scholar 

  • Huang J, Ma S, Xie H (2007) Least absolute deviations estimation for the accelerated failure time model. Stat Sinica 17(4):1533–1548

    MathSciNet  MATH  Google Scholar 

  • Huang J, Ma S, Xie H, Zhang CH (2009) A group bridge approach for variable selection. Biometrika 96(2):339–355

    MathSciNet  MATH  Google Scholar 

  • Huang L, Kopciuk K, Lu X (2020) Adaptive group bridge selection in the semiparametric accelerated failure time model. J Multiv Anal 175:104562

    MathSciNet  MATH  Google Scholar 

  • Huang Y (2002) Censored regression with the multistate accelerated sojourn times model. J R Stat Soc: Ser B (Stat Methodol) 64(1):17–29

    MathSciNet  MATH  Google Scholar 

  • Jin Z, Lin D, Wei L, Ying Z (2003) Rank-based inference for the accelerated failure time model. Biometrika 90(2):341–353

    MathSciNet  MATH  Google Scholar 

  • Jin Z, Lin D, Ying Z (2006) On least-squares regression with censored data. Biometrika 93(1):147–161

    MathSciNet  MATH  Google Scholar 

  • Jin Z, Lin D, Ying Z (2006) Rank regression analysis of multivariate failure time data based on marginal linear models. Scandinavian J Stat 33(1):1–23

    MathSciNet  MATH  Google Scholar 

  • Johnson BA et al (2009) On lasso for censored data. Electron J Stat 3:485–506

    MathSciNet  MATH  Google Scholar 

  • Kalbfleisch JD, Prentice RL (2011) The statistical analysis of failure time data. Wiley, New Jersey

    MATH  Google Scholar 

  • Khan MHR, Shaw JEH (2016) Variable selection for survival data with a class of adaptive elastic net techniques. Stat Comput 26(3):725–741

    MathSciNet  MATH  Google Scholar 

  • Khan MHR, Shaw JEH (2019) Variable selection for accelerated lifetime models with synthesized estimation techniques. Stat Methods Med Res 28(3):937–952

    MathSciNet  Google Scholar 

  • Khan MHR, Bhadra A, Howlader T (2019) Stability selection for lasso, ridge and elastic net implemented with aft models. Stat Appl Genet Mol Biol 18(5):742

    MathSciNet  MATH  Google Scholar 

  • Konrath S, Fahrmeir L, Kneib T (2015) Bayesian accelerated failure time models based on penalized mixtures of gaussians: regularization and variable selection. AStA Adv Stat Anal 99(3):259–280

    MathSciNet  MATH  Google Scholar 

  • Koul H, Vv Susarla, Van Ryzin J et al (1981) Regression analysis with randomly right-censored data. Annal Stat 9(6):1276–1288

    MathSciNet  MATH  Google Scholar 

  • Lee KE, Mallick BK (2004) Bayesian methods for variable selection in survival models with application to dna microarray data. Sankhyā: Ind J Stat 66(4):756–778

    MathSciNet  MATH  Google Scholar 

  • Lee KH, Chakraborty S, Sun J (2017) Variable selection for high-dimensional genomic data with censored outcomes using group lasso prior. Comput Stat Data Anal 112:1–13

    MathSciNet  MATH  Google Scholar 

  • Li H, Yin G (2009) Generalized method of moments estimation for linear regression with clustered failure time data. Biometrika 96(2):293–306

    MathSciNet  MATH  Google Scholar 

  • Li Y, Dicker L, Zhao SD (2014) The dantzig selector for censored linear regression models. Stat Sinica 24(1):251

    MathSciNet  MATH  Google Scholar 

  • Lu W (2007) Tests of independence for censored bivariate failure time data. Lifetime Data Anal 13(1):75–90

    MathSciNet  MATH  Google Scholar 

  • Miller RG (1976) Least squares regression with censored data. Biometrika 63(3):449–464

    MathSciNet  MATH  Google Scholar 

  • Mitchell TJ, Beauchamp JJ (1988) Bayesian variable selection in linear regression. J Am Stat Assoc 83(404):1023–1032

    MathSciNet  MATH  Google Scholar 

  • Narisetty NN, He X et al (2014) Bayesian variable selection with shrinking and diffusing priors. Annal Stat 42(2):789–817

    MathSciNet  MATH  Google Scholar 

  • Noll JE, Vandyke K, Hewett DR, Mrozik KM, Bala RJ, Williams SA, Kok CH, Zannettino AC (2015) Pttg1 expression is associated with hyperproliferative disease and poor prognosis in multiple myeloma. J Hematol Oncol 8(1):106

    Google Scholar 

  • Pan W, Kooperberg C (1999) Linear regression for bivariate censored data via multiple imputation. Stat Med 18(22):3111–3121

    Google Scholar 

  • Pan W, Louis TA (2000) A linear mixed-effects model for multivariate censored data. Biometrics 56(1):160–166

    MATH  Google Scholar 

  • Park T, Casella G (2008) The bayesian lasso. J Am Stat Assoc 103(482):681–686

    MathSciNet  MATH  Google Scholar 

  • Ročková V, George EI (2014) Emvs: the em approach to bayesian variable selection. J Am Stat Assoc 109(506):828–846

    MathSciNet  MATH  Google Scholar 

  • Sabourin JA, Valdar W, Nobel AB (2015) A permutation approach for selecting the penalty parameter in penalized model selection. Biometrics 71(4):1185–1194

    MathSciNet  MATH  Google Scholar 

  • Schneider H, Weissfeld L (1986) Estimation in linear models with censored data. Biometrika 73(3):741–745

    MathSciNet  MATH  Google Scholar 

  • Sha N, Tadesse MG, Vannucci M (2006) Bayesian variable selection for the analysis of microarray data with censored outcomes. Bioinformatics 22(18):2262–2268

    Google Scholar 

  • Shaughnessy J (2005) Amplification and overexpression of cks1b at chromosome band 1q21 is associated with reduced levels of p27 kip1 and an aggressive clinical course in multiple myeloma. Hematology 10:117–126

    Google Scholar 

  • Shaughnessy JD Jr, Zhan F, Burington BE, Huang Y, Colla S, Hanamura I, Stewart JP, Kordsmeier B, Randolph C, Williams DR et al (2007) A validated gene expression model of high-risk multiple myeloma is defined by deregulated expression of genes mapping to chromosome 1. Blood 109(6):2276–2284

    Google Scholar 

  • Shi L, Campbell G, Jones W, Campagne F, Wen Z, Walker S, Su Z, Chu T, Goodsaid F, Pusztai L et al (2010) The maqc-ii project: a comprehensive study of common practices for the development and validation of microarray-based predictive models. Nature Biotechnol 28:827–838

    Google Scholar 

  • Stute W, Wang JL (1993) The strong law under random censorship. Annal Stat 36:1591–1607

    MathSciNet  MATH  Google Scholar 

  • Tanner MA, Wong WH (1987) The calculation of posterior distributions by data augmentation. J Am Stat Assoc 82(398):528–540

    MathSciNet  MATH  Google Scholar 

  • Tibshirani R (1997) The lasso method for variable selection in the cox model. Stat Med 16(4):385–395

    Google Scholar 

  • Tsiatis AA (1990) Estimating regression parameters using linear rank tests for censored data. Annal Stat 90:354–372

    MathSciNet  MATH  Google Scholar 

  • Uno H, Cai T, Pencina MJ, D’Agostino RB, Wei L (2011) On the c-statistics for evaluating overall adequacy of risk prediction procedures with censored survival data. Stat Med 30(10):1105–1117

    MathSciNet  Google Scholar 

  • Van Erp S, Oberski DL, Mulder J (2019) Shrinkage priors for bayesian penalized regression. J Math Psychol 89:31–50

    MathSciNet  MATH  Google Scholar 

  • Visser M (1996) Nonparametric estimation of the bivariate survival function with an application to vertically transmitted aids. Biometrika 83(3):507–518

    MATH  Google Scholar 

  • Wang S, Nan B, Zhu J, Beer DG (2008) Doubly penalized buckley-james method for survival data with high-dimensional covariates. Biometrics 64(1):132–140

    MathSciNet  MATH  Google Scholar 

  • Wang X, Song L (2011) Adaptive lasso variable selection for the accelerated failure models. Commun Stat-Theory Methods 40(24):4372–4386

    MathSciNet  MATH  Google Scholar 

  • Wang YG, Fu L (2011) Rank regression for accelerated failure time model with clustered and censored data. Comput Stat Data Anal 55(7):2334–2343

    MathSciNet  MATH  Google Scholar 

  • Wei LJ (1992) The accelerated failure time model: a useful alternative to the cox regression model in survival analysis. Stat Med 11(14–15):1871–1879

    Google Scholar 

  • Wei LJ, Ying Z, Lin D (1990) Linear regression analysis of censored survival data based on rank tests. Biometrika 77(4):845–851

    MathSciNet  Google Scholar 

  • Xu J, Leng C, Ying Z (2010) Rank-based variable selection with censored data. Stat Comput 20(2):165–176

    MathSciNet  Google Scholar 

  • Yi GY, He W (2006) Methods for bivariate survival data with mismeasured covariates under an accelerated failure time model. Commun Stat-Theory Methods 35(8):1539–1554

    MathSciNet  MATH  Google Scholar 

  • Yin G, Cai J (2005) Quantile regression models with multivariate failure time data. Biometrics 61(1):151–161

    MathSciNet  MATH  Google Scholar 

  • Zhan F, Huang Y, Colla S, Stewart JP, Hanamura I, Gupta S, Epstein J, Yaccoby S, Sawyer J, Burington B et al (2006) The molecular classification of multiple myeloma. Blood 108(6):2020–2028

    Google Scholar 

  • Zhu LP, Li L, Li R, Zhu LX (2011) Model-free feature screening for ultrahigh-dimensional data. J Am Stat Assoc 106(496):1464–1475

    MathSciNet  MATH  Google Scholar 

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Appendices

Multicollinearity design

Let sample size n be 100 and dimension p be 100. Let the first 10 variables be independently generated from standard normal distribution. Then for \(j = 11,\cdots ,20\), consider

$$\begin{aligned} {\varvec{X}}_j = {\varvec{X}}_{j-10} + \tau , \end{aligned}$$

where \(\tau \) is a random error from a standard normal distribution. The rest of the variables are further generated from multivariate normal distribution with mean zero and covariance matrix with elements \(\varvec{\varSigma }_{ij} = 0.5^{|i-j|}\). Following Sect. 4.1, generate \({\varvec{T}}_{\cdot 1}\) and \({\varvec{T}}_{\cdot 2}\) and corresponding censoring times and censoring indicators. Furthermore, we assume the relevant variables as the following

  • no sharing: \(\left\{ j: \varvec{\beta }_{j1} \ne 0\right\} = \left\{ 1,\cdots ,10\right\} , \left\{ j: \varvec{\beta }_{j2} \ne 0\right\} = \left\{ 21,\cdots ,30\right\} \)

  • all sharing: \(\left\{ j: \varvec{\beta }_{jk} \ne 0\right\} = \left\{ 1,\cdots ,10\right\} \)

  • some sharing: \(\left\{ j: \varvec{\beta }_{j1} \ne 0\right\} = \left\{ 1,\cdots ,10\right\} , \left\{ j: \varvec{\beta }_{j2} \ne 0\right\} = \left\{ 1,\cdots ,5,\cdots ,\right. \left. 21,\cdots ,25\right\} \)

All of the true relevant variables are generated independently from \(\textsf {N}(3,0.5)\). We repeat all simulation setups for 200 times and fix the true coefficient values for all simulation runs.

Table 11 False positives and false negatives reported for multicollinearity design
Table 12 Sensitivity, specificity, and MCC scores reported for multicollinearity design

The results of the multicollinearity design can be found in Tables 9 and 10. For this simulation design, the univariate AEnet failed to give any results due to the issue with singular matrix computation, therefore we only report results from the other four competing methods. We see that all of the methods tend to recognize the ten irrelevant variables as signals. For no-sharing and all-sharing cases, the proposed method is able to give the smallest number of false positives while being able to recognize almost all of the relevant variables, giving almost zero false negatives. For some-sharing cases, we observe more obvious trade-off between false positives and false negatives for using \(\lambda _{min}\) and \(\lambda _{1se}\) while the proposed method selects the variables more strictly, returning with lower false positive scores and higher false negative scores. However, in terms of MCC score as an overall measure, the proposed method is able to achieve the highest MCC scores for all setups, demonstrating that the proposed method is able to outperform existing methods and to handle complicated data examples.

Dense design

Let \(n = 100\) and \(p = 100\). Following Sect. 4.1, we generate design matrix \({\varvec{X}}\) from multivariate normal distribution with mean zero and covariance matrix with elements \(\varvec{\varSigma }_{ij} = 0.5^{|i-j|}\). Then we generate \({\varvec{T}}_{\cdot 1}\) and \({\varvec{T}}_{\cdot 2}\) and corresponding censoring times and censoring indicators in a similar manner. In this simulation design, we assume that for each column of the true coefficient matrix, there are 20 relevant variables. That is, for some-sharing setups, we will have in total 45 relevant variables. All of the true relevant variables are generated independently from \(\textsf {N}(3,0.5)\). We repeat all simulation setups for 200 times and fix the true coefficient values for all simulation runs.

Table 13 False positives and false negatives reported for dense design
Table 14 Sensitivity, specificity, and MCC scores reported for dense design

The results of the dense design can be found in Tables 11 and  12. We see that the proposed method gives consistent performance to have the best MCC scores among all competing methods. For no-sharing and all-sharing setups, the proposed method is able to give the best combination of false positives and false negatives, achieving highest sensitivity and specificity scores. For some-sharing setups, when \(c \ne 1\), the proposed method is more strict in selecting signals which results in missing almost half of the relevant variables. However the proposed method is still able to correctly identify more relevant variables and noise variables compared with other competing methods, achieving the highest MCC scores.

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Yin, W., Zhao, S.D. & Liang, F. Bayesian penalized Buckley-James method for high dimensional bivariate censored regression models. Lifetime Data Anal 28, 282–318 (2022). https://doi.org/10.1007/s10985-022-09549-5

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Keywords

  • Buckley-James estimator
  • Bayesian penalization
  • Multivariate survival data
  • Variable selection