Comparison of subset selection methods in linear regression in the context of health-related quality of life and substance abuse in Russia

The problem of selecting a set of explanatory variables in regression modeling is well known and described in epidemiology and other disciplines [1‐4]. The goal of the analysis, i.e. prediction, explanation, data mining, etc. determines the point of balance in the bias—variance tradeoff, where larger models generally reduce bias at the cost of increasing variance. In epidemiology, exploratory or hypothesis generating analysis aims to identify important correlates (or predictors) of the outcome in terms of clinical and statistical significance, and it normally involves subset selection techniques [5].

Automatic variable selection methods, including forward selection, backward elimination, and stepwise selection (hereafter, ‘stepwise methods’) [6, 7] were developed in 1960s and gained popularity in epidemiology and other fields for a number of reasons, including computational simplicity, relative ease of interpretation, and their implementation in major statistical software packages [3]. Stepwise methods became standard in epidemiology and remain so [8] despite a body of statistical and epidemiologic literature accumulated since early 1970s that provides theoretical and simulation evidence of their deficiencies [9‐16].

Briefly, the main pitfalls of stepwise methods include: (a) the standard errors of the model coefficients are biased downward, and so are the p-values; (b) the absolute values of coefficients are biased upward; (c) the number of variables in the full model affects the number of noise variables in the final model; and (d) reliance on the single best model, while ignoring model uncertainty in producing the estimates [3, 9, 10].

A recent review determined that the most widely used variable selection methods in leading epidemiology journals were selection of variables based on prior knowledge and stepwise algorithms [8]. Stepwise algorithms still remain most widely used variable selection methods in epidemiology outside of genetics. A number of alternative variable selection methods have been proposed during the last couple of decades for a range of disciplines including epidemiology [17‐19]. Model averaging methods allow parameter estimation that accounts for model uncertainty by averaging over all (or selected) models considered, and weighting each model by its likelihood [5]. Bayesian model averaging (BMA) incorporates prior knowledge about the covariates into the estimation procedure [20]. Penalized regression methods, including the lasso [21], the elastic net [22] and their extensions simultaneously select regression variables and estimate regression coefficients conditional on the selected penalization parameter(s). Although none of these methods is ideal, they enrich the epidemiologists’ methodological toolkit, and can offer better solutions in many situations.

Analysis of the study data is particularly suitable for testing different subset selection methods, since the dataset includes a large number of potentially correlated variables, many of which are expected to have a decent explanatory power for the outcome of interest. While no general conclusions about the performance of different approaches can be drawn based on a single case study, our analysis provides useful insights into the problem of subset selection by applying various methods to the real world dataset. In this case study we explore properties of subset selection methods and demonstrate how they influence final model composition and the size of confidence intervals of regression coefficients, emphasizing the importance of careful choice of a proper method. Finally, for this case study we recommend BMA and adaptive elastic net as preferred methods. We argue that in situations of high uncertainty about the final model composition, the conservative approach would be to employ various robust methods and assign the degree of confidence to variables depending on how many methods selected a particular variable in the final subset.

The characteristics of study subjects along with regression coefficients and 95 % CIs for bivariate and full multivariate regressions are presented in Table 1. The dependent variable of interest—a EuroQoL 5D VAS measure of the health related quality of life—has a mean value of 63.97 (standard deviation = 15.34), and a median value of 65.00 (range is 5–100). Summary information on the distribution of dependent variable is presented in the Additional file 4. Correlation matrix of regression covariates is presented in the Additional file 5.

The regression coefficients along with their 95 % CIs estimated asymptotically and with bootstrap for the models selected using BE and FS techniques with AIC, BIC and the LRT (p = 0.05) are presented in the Additional file 3. BE and FS algorithms resulted in very similar models, while models differed substantially depending on the inclusion criterion used. The number of variables retained in the model with AIC is 29 (of which 14 are significant at 0.05 level) for BE method, and 27 (16 are significant) for FS method. Stepwise regression with BIC resulted in models that included 13 and 11 variables for BE and FS correspondingly (all selected variables are significant). When LRT (p = 0.05) was used the algorithms retained 18 and 19 variables in the final model for BE and FS correspondingly (with 18 being significant in both cases). Figure 1 presents the results of the model stability evaluation using bootstrap. In all cases except FS: LRT (p = 0.05) the highest inclusion frequency among non-selected variables was bigger than the lowest inclusion frequency among selected variables, and in case of FS: LRT (p = 0.05) these frequencies were equal. When AIC or LRT (p = 0.05) were used as model selection criteria, the differences in mentioned inclusion probabilities were relatively small ranging from 0 to 0.07. In the case of BIC, however, the highest inclusion frequency among non-selected variables was substantially bigger than the lowest inclusion frequency among selected variables, being 0.23 and 0.32 for BE and FS correspondingly. The percentage of variables with inclusion frequency over 0.9 (of the number of variables in the final model) ranged from 15 % (BE: BIC) to 31 % (BE: AIC) (Fig. 1).

Detailed outputs of lasso, adaptive lasso, and adaptive elastic net regressions are presented in the Additional file 6, and include regularization paths, graphs of CV results, estimates of regression coefficients along with 95 % CIs, and the bootstrap inclusion frequencies for variables. For lasso the highest inclusion frequency among non-selected variables was bigger than the lowest inclusion frequency among selected variables (both for λ_min and λ_1se). In adaptive lasso and adaptive elastic net, however, the situation was opposite—inclusion frequencies of selected model variables were bigger than those of non-selected in all cases except adaptive elastic net λ_1se, where the corresponding frequencies were equal (Fig. 2). Adaptive lasso demonstrated better stability in terms of the difference between the lowest selected and the highest non-selected variables inclusion frequency. On the other hand, lasso and adaptive elastic net demonstrated better performance in terms of percentage of model variables with inclusion frequency over 0.9 (Fig. 2). The number of significant variables was generally smaller in all penalized regression methods compared to stepwise methods (Fig. 4).

Figure 3 presents posterior inclusion probabilities for each variable from the BMA analysis. Subsets selected based on aggregate information and 100 best models were very similar (the latter subset included one additional variable). The model size was 12 and 13 correspondingly. In aggregate information model 95 % credible intervals of 4 out of 12 variables did not include zero, and in 100 best subset model it was 8 out of 13 (Fig. 4). BMA posterior inclusion probabilities for model variables, along with regression coefficients and 95 % credible intervals, as well as graphs presenting information regarding the sampling process and posterior distribution of model size are presented in the Additional file 7.

Figure 4 presents the summary and comparison of the variable selection methods used in this analysis. It shows the inclusion and significance in the final subset for every variable for every method used in the analysis. It provides summary of the model size and number of significant variables by method, as well as subset inclusion summary by variable. Out of 48 variables analyzed, 8 were included into the final model by all variable selection methods. 7 were not selected by either of methods (besides bivariate and full multivariate regression).

Correlates of the worse health-related quality of life selected most often included: older age, lower income level, younger age of the drug use initiation, using non-sterile injecting equipment during the last 4 weeks, having higher mental health problems score, being HIV positive and not getting regular HIV care (compared to being HIV negative or unaware), being Hepatitis C positive and never being offered treatment (compared to being Hepatitis C negative or unaware), and having high internalized drug use stigma.

Table 2 provides the summary of performance of the analyzed methods in terms of model stability, whether model uncertainty is properly incorporated in the statistical inference procedure, as well as computational efficiency of the algorithms.

In this paper we present a case study that compares the outputs of conventional (stepwise) and alternative (penalized regression, BMA) subset selection methods in linear regression. This is a valuable real-world example, common in epidemiology, of the challenges of subset selection when (i) the number of candidate covariates is large, (ii) most of candidate variables have good explanatory power for the outcome of interest, and (iii) multicollinearity is present. Previous theoretical work and simulation studies have shown that conventional stepwise algorithms perform particularly poorly when thus challenged [2, 9], and are outperformed by BMA [11, 48] and penalized regression methods [49, 50].

Some statisticians, however, argue that disadvantages of stepwise methods are over-emphasized [51]. Studies that report agreement between stepwise and alternative variable selection methods [52] may mislead the reader to the conclusion of their similar performance. While it is not unlikely to observe agreement in subset selection methods in the situation of relatively few potentially weakly correlated variables, the deficiencies of stepwise methods become more apparent when number of candidate variables increase and issues of multicollinearity come into play. As such, the salient illustration of the importance of careful choice of the subset selection method is the huge variability in the model size and composition derived with different methods in our analysis (Fig. 4). The final model size varies from 41 (lasso, λ_min) to 11 (adaptive lasso, λ_1se and FS: BIC), and the percentage of significant variables among those selected to enter final model varies from 100 % (BE: BIC, FS: BIC, and BE: LRT) to 27 % (lasso, λ_min).

The most widely used subset selection method—stepwise regression with LRT (p = 0.05) returns a model of size 18 (BE) and 19 (FS), where 18 covariates are significant (in both cases), illustrating the failure to incorporate model uncertainty into the statistical inference procedure. Stepwise BIC selects models of size 13 (BE) and 11 (FS), since BIC generally favors smaller models by imposing a larger model size penalty. While stepwise BIC provides a desired parsimony (with large sample size), the selected models are highly unstable (Fig. 1), and suffer from the same problem of underestimated standard errors. Compared to BIC, stepwise regression with AIC performs better in terms of model selection stability (Fig. 1). The improved stability is likely a result of a less severe model size penalty, which indirectly alleviates the problem of multicollinearity. The stepwise AIC, however, provides little improvement compared to LRT as it still fails to account for model uncertainty and treats a selected model as if it was pre-specified, thus returning biased standard errors [53].

Penalized regression methods provide flexible model selection tools. By specifying the form of the penalty term (L1, L2, or their combination), and by selecting the strength of the penalty (the value of regularization parameter λ), the researcher can control the model size and address the problem of multicollinearity. There are theoretical challenges in estimation of standard errors in lasso and its extensions, however use of the bootstrap with fixed λ allows estimating the confidence intervals for coefficients that are not shrunk to zero conditional on the selected value of λ [21]. The advantages of lasso are mostly notable in situations when the number of covariates is close to or bigger than number of observations. This explains its wide use in genetic studies. Several drawbacks of lasso have been noted, including unsatisfactory performance in the presence of multicollinearity [54]. Elastic net regression improves the performance of lasso in this respect [22], and assigning weights equal to 0.5 to L1 and L2 penalty tends to select or exclude groups of correlated variables together [55]. The use of a single unique regularization parameter can lead to selection of irrelevant variables and over-shrink large coefficients of important correlates. Adaptive lasso [41] reduces estimation bias and improves stability by putting individual penalty on every regression variable. Adaptive elastic net [42] demonstrated good performance in dealing with multicollinearity, estimation bias and model selection stability.

In our analysis all penalized regression methods select models of relatively large size (between 33 and 41) for λ_min (Fig. 4). This is explained by the fact that most of the covariates have some explanatory power, and the sample size is large compared to the number of covariates; thus the cross-validation with MSE loss favors bigger models. When we drew a random sample of 80 observations from the dataset, we obtained a model size of 9, 5 and 9 for lasso, adaptive lasso and adaptive elastic net, respectively, for λ_min. Given such property, the use of λ_1se is a more sensible option in our example. It provides the desired parsimony, while the percentage of deviance explained reduces moderately (from 0.40 for λ_min to 0.32 for λ_1se in all three methods). The model size obtained through penalized regression methods corresponding to λ_1se (14 for lasso, 11 for adaptive lasso, and 12 for adaptive elastic net) is comparable to that of BIC stepwise (13 and 11 variables). All three penalized regression methods, however, demonstrate a substantial improvement in stability compared to stepwise BIC (Figs. 1 and 2, Table 2). Further, unlike in stepwise regression, the tuning of λ with CV and the estimation of standard errors using the bootstrap offer an improvement in addressing the issue of model uncertainty. Thus the number of significant correlates in penalized regression methods with λ_1se is substantially smaller than in stepwise BIC (Fig. 4, Table 2).

BMA offers an improvement compared to conventional methods by directly incorporating model uncertainty into the process of model selection, and estimation of regression coefficients and their standard deviations (Table 2). Model ranking by the marginal likelihood provides information regarding the model uncertainty, and posterior inclusion probabilities of covariates offer an intuitive and convenient aid to subset selection. BMA is primarily a method for estimation of regression coefficients and their standard deviations. While making proper inferences is itself highly valuable, the BMA algorithm includes estimation of likelihoods of different models, and can therefore return the highest likelihood model (often called best subset). An alternative approach, which we used in this paper, suggests using the median probability model as a subset selection method [47]. It performs subset selection based on posterior inclusion probabilities of covariates, setting a threshold at the level of 0.5.

BMA is not free of problems, of which the most important are specification of priors and dealing with computationally intractable number of candidate models [20]. In our analysis we chose conservative non-informative uniform priors that did not take a full advantage of the power of Bayesian approach. In situations when an investigator has more information about the model covariates before performing the data analysis, specification of informative priors is desirable, however the choice should be justified. With the advancement of technology, BMA is now implemented in many statistical software packages allowing for its wider use [35, 56]. Software packages that implement BMA normally include several versions of the model space search algorithms, of which MCMC is the most widely used. Computational efficiency can be an issue for the implementation of BMA, because the MCMC chain has to converge, and convergence can be hard to reach, especially in the big data context (Table 2). Moreover, with MCMC one can never be fully confident that the algorithm has successfully searched the entire model space. One of the common ways to address this problem is to run the chain with different starting models and compare the results. Obtaining similar results increases the confidence that an entire model space was properly searched, but doesn’t offer a guarantee.

In our analysis we present two ways of posterior inclusion probability estimation: based on aggregate information from all models based on MCMC frequencies, and based on exact marginal likelihoods of 100 best models. These two methods result in very similar selected subsets that differ by one (of 48) variable. Since it is desirable to incorporate full model uncertainly, which is already limited by non-complete enumeration of models, it is advised to use posterior inclusion probabilities based on aggregate information. In our example this approach, consistent with expectation, returns a parsimonious model consisting of 12 variables, and the most conservative 95 % credible intervals of all methods, where only 4 variables are significant.

Several limitations of our analysis should be mentioned. In the absence of a gold standard for subset selection, and not knowing the data generating process, such as in cases of simulation studies, we had to rely on indirect measures of methods performance, such as investigation of model stability and assessment of how well the model uncertainty is addressed by different methods. This is an inherent limitation of using real-life data for analyses such as ours compared to simulation studies. However, this is also one of the strengths of this paper, since it offers an example of the real-life behavior of different methods that were extensively evaluated in simulation studies. All of the methods analyzed in this paper and the results of our analysis are only applicable to the linear regression. While the methods themselves can be extended to logistic regression, proportional hazard model, etc., the results we presented cannot be directly extrapolated to other types of regression models. Moreover, we assumed the form of the model (i.e. OLS regression), and only considered the uncertainty coming from its composition, while a broader model selection problem also includes considerations of the uncertainty regarding the functional form of the model. We have performed our case study using one dataset that represents a typical example of data used in epidemiological research among PWIDs. In our analysis we aimed to focus on analytical approaches and present very detailed outputs of the methods and their comparison in order for this example to serve as both the demonstration of approach to methods comparison, and the presentation of actual findings from such comparison. Extending similar analysis to multiple datasets might be one of the potential future research directions. From the practical perspective, we hope that our example would motivate investigators to employ similar strategies in applied data analysis.

Our analysis demonstrates that it is beneficial to apply different subset selection methods, and explore where their outputs do and do not agree (Fig. 4). This is especially useful in exploratory analysis, situations of high uncertainty about the correct model, and if one is interested in finding a set of the strongest correlates or predictors of the outcome, and wants to improve the credibility of findings. When interpreting the results, it is important to differentiate between the features of statistically sound methods that stem from varying tradeoff between bias and variance [5], and the deficiencies in selection, estimation, and inference inherent to methods that violate the principles of statistical theory [10]. This approach should not be confused, however, with using multiple methods, selecting one that gives the most ‘desired’ result, and presenting it as if it was the only one method deployed.

In our analysis all correlates that are selected most often using different methods make intuitive sense, and the findings are generally in line with other studies conducted in similar populations [57‐60].

Our analysis emphasizes the importance of understanding of the properties of various subset selection methods, and a careful choice of method (or a combination thereof) that would correspond to the goals of analysis. As we have shown, different subset selection methods return models with very different sizes and estimated coefficients. It is, therefore, vital that the researcher defend the reasons behind choosing a particular technique based on assumptions, theoretical considerations, research questions, and intended use of the results.

Based on performance of different methods in this case study and previous theoretical work, we discourage the use of stepwise algorithms, and recommend BMA that accounts for the full model uncertainty, and adaptive elastic net (with λ_1se when N is large) in cases such as ours. We also encourage researchers to explore model uncertainty and stability as part of their analyses, and report these details in epidemiological papers.