Robust sparse accelerated failure time model for survival analysis

1.Introduction

Accurate estimation for the cancer patients’ survival time with high dimension and low sample size gene expression dataset is a significant challenge in survival analysis. The efficient method which can identify the relevant genes associated with tumours may be helpful for cancer research and treatment. In the last two decades, the Cox proportional hazards (Cox) model with the regularization approach has been widely used for the patient risk classification and relevant biomarkers identification [1, 2, 3]. However the Cox proportional hazards model may not be suitable if the data does not meet the proportional hazards assumption. Meanwhile, the patient’s survival time estimationhas become a very important requirement in clinical treatment. Hence the accelerated failure time (AFT) model has already become one succedaneum of the Cox model in cancer survival analysis. Nevertheless the small sample size limits the performance of AFT model construction, such as the censored data in the cancer clinical data cannot be directly used in the model training. To increase the number of available data in the AFT model, some different imputation methods were proposed in cancer survival analysis. The most widely used one is Buckley-James estimation method [4, 5, 6], it estimates the censored data using the Kaplan-Meier approach; the other one is called ranking based estimator, it estimated the survival time from computing the score function of the partial likelihood [7, 8]. In our proposed RS-AFT model, we used Kaplan-Meier approach [9] to deal with the censored data.

The traditional ordinary least squares (OLS) approach has been used to construct the prediction model for a long time. However the OLS approach is sensitive to noise in the data, which significantly reduces its robustness in the practical application. Meanwhile the OLS estimation cannot achieve an unbiased solution under some certain conditions, and its estimated variance is quite large [10]. To improve the performance of the OLS estimation, the robust regression and the regularization methods were proposed. The least absolute deviation (LAD) is the kind of the robust regression method to confront the noise. The regularization approaches are widely used for variable selection in high dimensional data analysis. To overcome the shortcomings in the OLS method, Li et al. [11] proposed a RLAD method that combines the robust regression and regularization approach together. After that the LAD-lasso [12] and LAD-Adaptive lasso [13] were implemented. However compared to the L1 type regularization, the Lq(0 <q< 1) type regularization can obtain more sparse result, and it has some attractive properties, such as unbiasedness, oracle properties and consistency of variable selection [14, 15]. Therefore, Chang et al. [10] proposed LAD-L q regularization, which outperforms some existing methods based on the OLS with L1 type regularization approaches in variable selection.

Considering the high dimensional and low sample size data in cancer survival analysis, many different kinds of regularization methods were used as the penalty function to combine with the regression loss function, such as lasso [16], elastic net [17], the smoothly clipped absolute deviation (SCAD) [18], L1/2 regularization [19]. These methods help the model predict the objective function value and select the feature genes related to the disease; however we found it was a difficult work to get a balance between the prediction accuracy and sparsity. Usually high prediction accuracy means large numbers of the selected genes; it means people have to waste much time for researching some unrelated genes. We considered the LAD-Lq regularization, which have the advantages of LAD and Lq (0 <q< 1), was a good choice to instead of these old regularization methods. Hence we proposed a robust sparse AFT model with LAD-Lq regularization approach (RS-AFT), we thought the new model can generate good performances in survival time estimation, and it has a powerful ability to find the cancer related genes because of its sparsity.

2.Method

Supposing the dataset included n samples, (yi,δi,xi)i=1n represents the single patient’s sample, where yi is the observed survival time pf the patient, δi=0 represents the sample is the censored data and if δi=1 means the sample is the completed data, xi=(xi⁢1,…,xi⁢p)indicate the p dimensional covariates.

The AFT model can be written as a linear regression model: h⁢(yi)=xiT⁢β+εi,i=1,…,n where h(.) is the log transformation or some other monotone functions, εi is the independent random error with a normal distribution function, and β=(β1,β2,…,βp) is the regression coefficient vector of p variables.

For estimating the censored time, we used the Kaplan-Meier weights estimation method because of its simple and fast [6]. The estimated value h⁢(yi) of the censored time yi can be written as:

where Δ⁢S^⁢(t(r)) is the step of at time t⁢(r) [20].

As we know, the least squares approach method is widely used to find β:

To overcome the shortcomings of least squares approach, especially for data X with high noise, the least absolute deviation (LAD) was adopted:

In fact, not all genes in the microarray dataset may be associated with the patient’s survival time, which means some coefficients β may be zero in the true model. A good method should select bio-mark genes consistently and efficiently. Some regularization methods have been widely used to find the true disease related genes. The different penalty function regularized AFT model using LAD approach will be written as:

The AFT model with the LAD-lasso regularization approach is:

Trying to get more sparse solutions, we proposed the robust sparse AFT model with the LAD-Lq approach (RS-AFT):

3.Algorithm

Solving RS-AFT model is a non-convex optimization problem. We designed the weighted iterative algorithm to solve it. The regularization part |β|q in the RS-AFT model can be replaced by the first-order Taylor expansion:

The minimization problem of the RS-AFT model will be shown:

In the literature [21], the BIC method was used to select the optimal regularization parameter λ. The likelihood function of the posterior probability by BIC is given by:

Minimizing l⁢(β):

|β𝐿𝐴𝐷|q (β𝐿𝐴𝐷 is obtained by the least absolute deviation of the AFT model) can be seen as the estimator of |β|q,λ can be written as:

Since the variable selection consistency of the L(0<q⩽1)q method has been proved in [1], we simply set q= 1 in the weighted iterative algorithm for turning the parameter λ. The detail procedure of the weighted iterative algorithm for the RS-AFT model is given:

4.Results

5.Conclusion

For biological analysis of the results, 15 top-ranked genes selected by the different AFT methods in Lung cancer dataset were shown in Table 7. Compared with the other AFT models based on the least squares approaches with different regularization methods, the RS-AFT model selected some unique genes, such as SMAD4, ENPP2, LLGL1. SMAD4 belongs to the member of Smad family which is one kind of signal transduction proteins. The Smad family proteins play a key role in transmitting the TGF-beta signals from the cell-surface receptor to cell nucleus, mutation or deletion of SMAD4, which has been proved to lead to the pancreatic cancer [27]. We think it may be strongly associated with the lung cancer. ENPP2 is also known as ATX, this gene can stimulate the motility of tumour cells. The expression of ENPP2 has been found to be up regulated in some different kinds of cancers [28]. The protein encoded by the gene LLGL1 was said to be very similar to the tumour suppressor of drosophila which is a highly relevant gene to cancer [29]. What is more, some relevant genes selected by other AFT models with lasso, elastic net and SCAD, were also found by the RS-AFT, for example, TRA2A, WWP1, DOC2A and HUWE1. They are significantly associated to the lung cancer which has been discussed [30].

We also obtained the similar experimental results from the analysis of the other three real datasets. The biological analysis showed that the RS-AFT model not only can find the relevant genes which were selected by AFT models with other regularization methods, but also can find some unique genes, which were not selected by other AFT models but also significantly associated to disease. Hence, we can say the RS-AFT model may find the disease related genes accurately and efficiently.

The experiment results show that the RS-AFT model outperforms some existing survival estimation approaches. It can effectively select the bio-mark genes and estimate the patients’ survival time accurately in high dimensional and low sample size biological datasets. With the less mark genes and accurate survival time prediction, this method will be a more practical tool for cancer research and treatment.

In the data experiments we found that large number of the censored data great effect the accuracy of the RS-AFT model. The more censored data, the more difficulty we get in the experiments. Hence in the future work, we will try to combine the RS-AFT model with some machine learning methods, such as some semi supervised methods, we thought they may have strong ability to against with the censored data, the more completed data will improve the accuracy of our RS-AFT model obviously.