Sure independence screening

Suppose we have a genomic dataset (y _i , x _i ), where y _i is the response and x _i  = (x _i1, x _i2, …, x _ip ) is the vector of p covariates, for i = 1, 2, …, n. In real applications, Y could be measurements of some phenotypes or quantitative traits, such as weights, drug response, etc. X could be some high-dimensional omics-measurements, such as gene expression, CpG methylations, etc. In a typical genomic setting, p could be far larger than n. To deal with high dimensionality, effective variable selection techniques are required.

The sure independence screening (SIS) method introduced by Fan and Lv [15] is a two stage approach. First, it selects significant predictors by sorting the corresponding marginal likelihood (correlation in linear model), thus fast reducing the ultra-high dimensionality to a relatively large scale d (e.g., o(n)). Subsequent to SIS, a more sophisticated lower dimensional model selection technique such as SCAD [14], the Dantzig selector [26], LASSO [12], or adaptive LASSO [13] could be applied to perform the final variable selection and parameter estimation. Apparently, SIS could dramatically speed up variable selection when the p is extremely large. Fan and Lv proved SIS enjoys the sure screening property and model selection consistency under certain conditions.

Screening with prior knowledge integration

We noted that the idea of SIS is based on marginal correlation to first select important variables. Based on this idea, we proposed an novel approach, screening with prior knowledge integration (SKI), to select variables in the first stage. The basic procedure of SKI is drawn in Fig. 1. The idea of the SKI is to rank the variables not only based on marginal correlation but to also incorporate external information. The rationale here is that the variables supported by both marginal correlation and external information are more likely to be important features, and thus should be included in the second stage for parameter estimation with larger probability.

The initial ranking represents the importance of each variable known ahead of the ongoing study. For example, if the goal of this study is to predict drug response based on gene expressions, other genetic measurements such as copy number variants (CNV) might be available. We could first rank each CNV by its marginal correlation with drug response obtained by univariate linear regression and then we map CNV ranks back onto the genes to get an initial rank of genes. More commonly, we could rank genes based on their importance scores obtained by expert domain knowledge or literature searching.

Typically, we require that each variable has an initial rank. For those variables with no information, an average rank can be assigned. For instance, among 100 predictors, 10 of them are found associated with response from existing knowledge. We could assign ranks (ranged from 1 to 10) to these 10 predictors based on their association strength and 55 for the rest. Alternatively, if we don’t know the association strength, we could set the ranks of 10 predictors as the average of 1 to 10, which is 5.

Estimation of α

As mentioned above, the selection of α could control the relative strength of influence imposed by prior knowledge, which is essential for the success of the proposed methods. Unfortunately, there is no pleasant way for tuning this parameter. LASSO or elastic-net [27], uses cross-validation strategy to select α with lowest internal prediction errors. However, the problem we face here is a ultra-high dimensional problem, where the number of covariates p is already much larger than sample size n. Cross validation will require us to further spit the sample into training and testing, which can make the ultra-high dimensionality issue worse. To alleviate these concerns, we develop the following alternative strategy.

Here loglike _sat refers the log-likelihood or the saturated model (i.e., a model with a free parameter per observation). And loglike _null refers to the intercept model. We compare the dev. ratio across different α’s, and select the α yields largest dev. ratio as the final α.

The rationale of this method is that if one set of variables is more biologically meaningful than the other, the better it could fit a ridge regression model. We do notice that the number of variables selected d will affect the performance of SKI in terms of estimation of α. In the most extreme case, if only one variable is selected (d = 1), then the estimated α will always be zero. But our experiences suggest the number of variables selected won’t affect the results significantly if this number is not too small. Although some methods have been proposed to tune this parameter [29], how to determine the number of variables is out of the scope for this study.

Extension: iterative SKI

Fan and Lv demonstrated that when too many predictors are involved, the basic sure screening methods might miss some important variables due to collinearity issues. In their paper they developed an iterative version of SIS to use fully the joint information of the covariates rather than marginal information. Briefly, in the first step, a subset of k ₁ variables is selected using an SIS-based method. Next, a n -vector of residuals are obtained from regressing the response Y over k ₁ variables are treated as new responses and the same method is applied to the remaining p − k ₁ variables. The process is repeated until desired number (e.g., d) of variables is selected or (predefined) maximum iteration is reached.

We extend this idea to SKI and developed an iterative version of SKI (iSKI). The similar procedure was used. In the first step, the rank of each variable is obtained as weighted geometric mean of knowledge-based rank and the sorting marginal correlation between responses and predictors. For the rest of the steps, the rank is weighted geometric mean of the knowledge-based rank and the sorting marginal correlation between residuals and predictors.

Simulations

We adopted a similar simulation in Ma 2012 [30]. In total n _x  = 200 samples (X, Y _x ) were simulated, with gene number p = 10, 000. 200 clusters were simulated independently, and 50 genes in each cluster were simulated from a multivariate normal distribution with μ = 0, σ ² = 1 and AR(1) correlation structure ρ = 0. 6. (i.e., cor(i, j) = ρ ^{|i − j|}). This is to mimic a real gene expression studies with taking pathway structure into account. In each cluster, the coefficients β’s of first ten genes were simulated from a uniform distribution with minimum 0.5 and maximum 1. All other β’s were set to be zeros. This is consistent with the parsimonious assumption that only few genes and pathways were associated with phenotypes or diseases. Continuous responses were generated from linear regression models with σ _x ²  = 1 (or 3).

Another n _z  = 200 samples (Z, Y _z ) with gene number p = 10, 000 were simulated to mimic an external gene expression study, where our prior knowledge was drawn from. Gene expressions and responses were simulated from the same structure as described above. But the non-zero coefficients β were simulated to have 0, 50, and 100% overlap with non-zero β in the internal settings. This is to mimic the situation that the prior knowledge completely disagrees, partially agrees and exactly agrees with our true experiment settings.

Real application: drug response analysis

We next applied the SKI procedure to a drug response study and compared it to the results obtained with the SIS procedure. Selumetinib (AZD6224) is a drug used to treat various types of cancer such as non-small cell lung cancer (NSCLC). It is a potent, highly selective MEK1 inhibitor. Unfortunately, despite intensive studies, the genetic mechanism for Selumetinib resistant remains controversial [31–34]. We applied the SKI procedure to identify the potential biomarkers of response to Selumetinib. We downloaded the drug response data (i.e., Active Area) from the Cancer Cell Line Encyclopedia (CCLE) project [35] together with its baseline omics measurement, which includes gene expression, mutation data, and copy numbers. In total there were 489 cell lines and 41,872 genomic features measured. For a single feature, we assign a specific gene annotation on it. We then searched the Drug2Gene database [25] to acquire prior knowledge of association between selumetinib and genes. Drug2Gene is an integrative knowledge base reporting relations between genes/proteins and drugs/compounds including bioactivity data where available. The data has been collected from 23 public databases and integrated to provide a 'one-stop shop’ for identifying tool compounds for genes or finding all known targets of a drug. In total, 383 genes were identified to have associations with selumetinib. We gave an initial rank to 41,872 genomic features based on whether its annotated genes have a known association with selumetinib. For 1105 features with annotated genes having association with selumetinib, we set their ranks as 553, and for others, we set the ranks as 21,489.

The SKI and SIS procedure were used for variable selection, respectively. The top 100 features were selected and SCAD was used to fit the final model. In other studies, external information (e.g., biological relevance) are used to judge whether the variables identified are accurate. Since here we already used this knowledge in SKI, it is unfair to judge the results by this criteria. So we used leave-out-out cross validation (LOOCV) to compare the prediction squared error of these two methods.

The average of α estimated in SKI was 0.382, indicating that the prior known associated genes are very informative in variable selection. In Fig. 2, we showed the LOOCV prediction square error of two methods. In general SKI methods outperforms SIS in terms of small prediction error. The median (mean) prediction square errors are 0.324 (0.828) and 0.158 (0.397) for SIS and SKI, respectively. By integrating prior known information, SKI selects the variables more accurately.