Feature ranking based on synergy networks to identify prognostic markers in DPT-1
© Ahmadi Adl et al.; licensee Springer. 2013
Received: 1 June 2013
Accepted: 6 September 2013
Published: 19 September 2013
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© Ahmadi Adl et al.; licensee Springer. 2013
Received: 1 June 2013
Accepted: 6 September 2013
Published: 19 September 2013
Interaction among different risk factors plays an important role in the development and progress of complex disease, such as diabetes. However, traditional epidemiological methods often focus on analyzing individual or a few ‘essential’ risk factors, hopefully to obtain some insights into the etiology of complex disease. In this paper, we propose a systematic framework for risk factor analysis based on a synergy network, which enables better identification of potential risk factors that may serve as prognostic markers for complex disease. A spectral approximate algorithm is derived to solve this network optimization problem, which leads to a new network-based feature ranking method that improves the traditional feature ranking by taking into account the pairwise synergistic interactions among risk factors in addition to their individual predictive power. We first evaluate the performance of our method based on simulated datasets, and then, we use our method to study immunologic and metabolic indices based on the Diabetes Prevention Trial-Type 1 (DPT-1) study that may provide prognostic and diagnostic information regarding the development of type 1 diabetes. The performance comparison based on both simulated and DPT-1 datasets demonstrates that our network-based ranking method provides prognostic markers with higher predictive power than traditional analysis based on individual factors.
Type 1 diabetes (T1D) is an autoimmune disorder and one of the common pediatric diseases with a diverse pathogenesis, clinical phenotype, and outcome. Despite the emergence of T1D as a global issue with a steady increase in incidence worldwide over the past decade, the etiology of T1D is still not fully understood. Recent studies, including the Diabetes Prevention Trial-Type 1 (DPT-1), have suggested that this complex disease has multiple risk factors, including genetic predisposition, diet, viruses, and geography in addition to autoimmunity[1, 4–7]. The previous epidemiology studies mostly focus on studying hypotheses regarding individual risk factors, which have obtained important initial understanding, including the predisposing roles from genetic markers such as human leukocyte antigens. However, traditional hypothesis-driven approaches focusing on ‘essential’ factors may not be sufficient for fully understanding T1D. With large-scale perspective studies such as DPT-1, we believe that data-driven investigation considering all candidate factors with their interactions can serve as a critical complement for previous hypothesis-driven research.
Data-driven methods have been proven to be useful in both identifying probable mechanisms involved in disease and providing accurate biomarkers for early prediction[8, 9]. However, as shown in genome-wide association studies (GWAS), single marker analysis is not sufficient for genetic studies of complex diseases[10, 11]. In order to better explain the missing heritability of complex disease through analyzing high-dimensional genotype data, several methods have been proposed to take into account the interactive effect among single-nucleotide polymorphisms as well as multiple genes in GWAS and other -omic data analysis[12–14]. In this work, we propose a network-based mathematical model for systematically analyzing candidate risk factors for disease. We consider that the individual effect and interactions from potential risk factors are all manifested as statistical associations with the disease outcome. Based on this, we construct a synergy network which integrates both the individual and synergistic interactive effects of factors in one single graph structure. We then propose a novel algorithm based on this synergy network to identify biomarkers for early prediction of disease. Specifically, we verify the effectiveness of our method using simulated case-control datasets. With such validated results, we apply our method to identify biomarkers for prognosis of T1D from measured immunologic and metabolic indices in DPT-1. The performance of the identified markers is then compared to the performance of traditional forward feature selection which only considers the individual statistical association with outcome. Our comprehensive results show that our network-based method identifies better biomarkers with better predictive performance.
Feature selection approaches are commonly used to identify biomarkers by finding a subset of biomedical measurements with high predictive power with respect to disease outcome[15–17]. As it is computationally very expensive to exhaustively search for the best subset of variables, these methods mostly rely on heuristic approaches. Filtering variables based on their individual effect on disease outcome has been a common practice in biomedical research. Heuristic approaches based on filtering have been successful in identifying biomarkers with strong individual effects. However, they may miss variables with weak individual effects but having synergistic interactive effects that produce high predictive accuracy[15, 17]. To avoid missing these critical variables with high synergistic effects on outcome, we propose a new approach which takes into account both individual and synergistic interactive effects. In our approach, we first construct a synergy network based on the individual and synergistic effects of all the observed variables. Then, we solve the problem of finding the best subnetwork by an efficient graph spectral algorithm which leads to a novel feature ranking that improves the traditional ranking by taking into account the interaction among variables. Finally, we use this feature ranking together with traditional forward feature selection to achieve the final set of biomarkers.
To construct the synergy network, we need to measure the individual predictive power of all variables together with their pairwise synergistic power. One natural way to measure both individual and synergistic powers is to use a logistic regression model. In order to measure the individual power of variable v i , we can learn the following logistic model log(g/(1−g))=α0+α1v i in which g is the probability p(y=1|v i ), where y denotes the disease outcome of interest. After fitting this model to the given data, the magnitude of the coefficient α1 measures the individual power of v i . To make sure that the measurements for different variables are with the same unit and comparable to each other, we use − log(p i ) as the individual power of variable v i , in which p i is the coefficient p-value for α1 and measures the statistical significance of the individual power of v i . Similarly, in order to measure the synergistic predictive power between two variables v i and v j , we fit the following logistic model log(g/(1−g))=α0+α1v i +α2v j +β v i v j (where g=p(y=1|v i ,v j )) to data and consider − log(p ij ) as the synergistic power of variables v i and v j , in which p ij is the coefficient p-value of β. With that, we construct the synergy network which can be represented by a graph G(V,E). In this synergy network, V is the set of nodes corresponding to all the variables, and each v i ∈V has the node weight f(v i ) equal to − log(p i ); E is the set of edges (v i ,v j ) with the edge weight s(v i ,v j ) equal to − log(p ij ).
where C denotes potential subnetworks and 0≤λ≤1 is a weighting coefficient between individual and synergistic effects. As both f(v i ) and s(v i ,v j ) are nonnegative, the previous optimization problem has the degenerated solution to include all the risk factors in C. To overcome this problem, we further impose another constraint to restrict the size of selected subnetworks to have |C|≤K. This formulation is in fact the problem of finding a maximum weighted clique (MWCP) which is a generalization of the classical maximum clique problem (MCP). As MCP is nondeterministically polynomial (NP)-hard, it can be easily shown that MWCP is NP-hard as well. Thus, our biomarker identification problem formulated in Equation 1 is also an NP-hard problem. Several approaches have been previously proposed to find the exact optimal solution of the problem by employing branch-and-bound techniques, but it is probable that exhaustive search over all possible subnetworks is needed. In this paper, we propose a fast approximate algorithm for MWCP which also provides a ranked list of features based on both their individual and synergistic effects.
By straightforward algebraic manipulations, we can show that the potential solution x∗ has to satisfy M x∗=α x∗. Therefore, the relaxed solution x∗ to the MWCP is an eigenvector of the matrix M. Furthermore, we want the objective function x∗TM x∗=α x∗Tx∗=α K to have the maximum value with x∗, which means that we want α to be as large as possible. Hence, the solution x∗ will be the eigenvector of M with the largest corresponding eigenvalue. Also given the relaxed solution x∗, for any K, the approximate solution to the original integer programming optimization problem is to take top K nodes with the largest corresponding magnitudes in x∗. This also shows that the candidate risk factors with larger magnitudes in x∗ are more desirable to be selected in the final subset of risk factors as potential prognostic biomarkers. Thus, we can use the absolute values in x∗ as a score to rank the risk factors. We note that K can be an arbitrary number without loss of generality, which will not affect our final ranking as the x∗ only depends on the matrix M. As one can see, the proposed method combines both individual power and synergistic power among all candidate risk factors into one single score that can be used to rank them.
In order to select a subset of risk factors based on any ranking, a common approach is to use forward feature selection. We replace the ranking step of the forward feature selection, which is only based on individual power, by our network-based spectral ranking which takes into account the interaction among factors as well. In forward feature selection, we sequentially add potential risk factors from the top of the ranked list to the current set of selected factors only if it improves the classification performance; otherwise, we move to the next factor in the ranked list. This procedure is repeated until we reach the end of the ranked list.
We evaluate the performance of our network-based biomarker identification based on both simulated datasets and datasets obtained from the DPT-1 study and compare it with the individual-based biomarker identification, which only considers individual effects. In order to properly estimate and compare the performance of biomarker identification methods, we perform an ‘embedded’ cross-validation procedure.
As explained earlier, our feature selection approach includes two steps: First, we construct a synergy network based on the given dataset and rank the candidate risk factors using our spectral algorithm. Second, we use the ranked list of factors obtained in the first step to perform a forward feature selection. To make sure that we do not overestimate the performance of our biomarker identification approach, we perform the following embedded cross-validation procedure: Similar to the regular ten-fold cross validation, we first randomly divide the dataset into ten folds, within which one fold is used as the testing set to test the performance and the remaining nine folds are used as the training set to select biomarkers and learn the classifier. In order to select biomarkers based on the training set, we first use all the data points in the training set to construct a synergy network and perform our spectral algorithm to obtain the ranked list. Then, using the ranked list, we perform a forward feature selection method to select the best performing set of biomarkers. In the forward feature selection method, we sequentially add candidate factors to the current feature set (starting with an empty set), if it improves the classification performance; otherwise, we move to the next factor in the ranked list. To evaluate the performance of a set of potential risk factors during forward feature selection, we use another standard ten-fold cross validation in which we further divide the training set into ten folds, nine of which are used to train the classifier and the remaining is used to test the performance. After performing the forward feature selection and identifying the biomarkers, we learn a classifier based on the training dataset using those selected features and compute the performance based on the testing set. During our performance evaluation procedure, we adopt the MATLAB implementation of quadratic discriminant analysis as the classifier to make sure that the pairwise interaction among risk factors is taken into account by the classifier. To measure the performance of any classifier in our performance evaluation procedure, in addition to the accuracy, we also compute the area under the ROC curve (AUC) which is a more reliable measure of prediction performance in our experiments. When we use accuracy as the performance measure during forward feature selection, the identified biomarkers are optimized to provide better accuracy. We also take AUC as the performance measure for forward feature selection so that the biomarkers are optimized to provide better AUC. This two sets of biomarkers are not necessarily the same, especially with unbalanced datasets, as they are supposed to optimize for different criteria. Thus, for each dataset, we have two sets of results: one based on accuracy and one based on AUC.
In this logistic model, the magnitude of each individual coefficient α i determines the individual effect of the corresponding variable v i on outcome y, and the magnitude of the interaction coefficient β ij determines the amount of synergistic effect of two variables v i and v j on the outcome. To obtain the previously described case-control data, we simulate 30 random features with each variable v i following a mixture-of-Gaussian distribution with equally weighted (mixture parameters equal to 0.5) Gaussian distributions with the same variance of 1.0 and the means equal to −1.0 and 1.0, respectively. For 435 interaction coefficients β ij , we randomly set 425 of them to zero, and the values of the other ten are drawn from the standard normal distribution (mean 0.0 and variance 1.0). We also set all the individual coefficients α i to zero which means that there is no feature with significant individual effect. To simulate the outcome y, we first compute the probability p(y=1|v) based on the previous logistic model (Equation 5). Then, we generate the value for y from a Bernoulli distribution with the success parameter equal to p(y=1|v). We have generated 20 of such case-control datasets with 200 data samples in each set for the performance evaluation of our method. In order to make sure that our performance comparison results are independent of how we set the values of these coefficients, each of these 20 datasets is simulated with different random values for coefficients β ij .
Finally, in our simulation model, we always have p(y=1)=p(y=0), which is due to the symmetry of the logistic function, symmetry of distribution of all features, and symmetry of distribution of coefficients around zero. As a result, the datasets simulated from the model are balanced, i.e., they have almost the same number of case and control samples. Because of this, the accuracy and AUC performance measures are very similar for all of our simulated datasets which might not be the case for unbalanced datasets.
DPT-1 was a study designed to determine if T1D can be prevented or delayed by preclinical intervention of insulin supplement. It focuses on first- and second-degree nondiabetic relatives of patients with T1D before the age of 45, since they have more than tenfold risk of developing T1D compared to the general population. DPT-1 screened 103,391 subjects altogether and categorized them into four risk groups based on genetic susceptibility, age, the presence of autoantibodies (including islet cell autoantibodies (ICA), insulin autoantibodies (IAA), glutamic acid decarboxylase (GAD), insulinoma-associated protein 2 (ICA512)), and the change of metabolic markers during oral glucose tolerance test (OGTT) and IV glucose tolerance test (IVGTT). The 3,483 subjects positive for ICA were staged to quantify the projected 5-year risk of diabetes. Our analysis focuses on the study for the ‘high risk’ and ‘intermediate risk’ groups[7–9], which contain 339 and 372 subjects, respectively. The subjects of each group were randomly divided into two roughly equal subgroups: one received parenteral or oral insulin supplement, while the other was assigned to the placebo arm of the study. In this paper, we focus on the subjects of the placebo group. We consider the placebo subgroups of both high-risk and intermediate-risk groups as a dataset for our data-driven analysis (analysis based on the treated group is provided in Additional file1). The dataset contains the following 19 features from baseline characteristics in DPT-1, focusing on immunologic and metabolic markers. We have taken the available titer values for different autoantibodies, including ICA, IAA, GAD, ICA512, and micro-insulin autoantibodies. For metabolic indices, we have fasting glucose, glycated hemoglobin (HbA1c), fasting insulin, and first-phase insulin response (FPIR) from IVGTTs. Homeostasis model assessment of insulin resistance (HOMA-IR) and FPIR-to-HOMA-IR ratio are also computed as in. From OGTTs, in addition to 2-h glucose and fasting glucose, we have collected blood samples for C-peptide measurements in the fasting state and then 30, 60, 90, and 120 min after oral glucose, from which we have computed peak C-peptide as the maximum point of all measurements and AUC C-peptide using the trapezoid rule. Furthermore, as age and body mass index (BMI) have been conjectured to be important confounding factors, we also include them in our set of features. We are interested in identifying the most predictive group of features as biomarkers from the above described candidates to predict the outcome which is the development of T1D at the end of the DPT-1 study. The dataset contains 356 subjects within which 133 subjects developed T1D at the end of the study.
Accuracy and AUC performance of network-based ranking and individual-based ranking based on the DPT-1 dataset
Final sets of biomarkers and their corresponding accuracy and AUC performances for the DPT-1 dataset
2-h glucose, IAA, ICA512, peakC-peptide, AUC C-peptide
2-h glucose, IAA, fasting glucose (IVGTT), ICA512, peak C-peptide, AUC C-peptide, FPIR-to-HOMA-IR ratio
2-h glucose, AUC C-peptide, BMI, FPIR-to-HOMA-IR ratio, fasting insulin (IVGTT), HOMAIR, HbA1c, IAA, ICA512, peak C-peptide
age, 2-h glucose, IAA, ICA512, peak C-peptide, AUC C-peptide
2-h glucose, IAA, FPIR, fasting glucose (IVGTT), ICA512, peak C-peptide, AUCC-peptide, FPIR-to-HOMA-IR ratio
2-h glucose, age, FPIR-to-HOMA-IR ratio, fasting glucose (IVGTT), IAA, peak C-peptide, weight
Note that, as mentioned previously, the features selected during the forward feature selection step of our biomarker identification method might vary when we optimize different performance measures. As a result, the final set of biomarkers when we use accuracy in our performance evaluation is different from the final set of biomarkers when we use AUC. The final set of biomarkers using both accuracy and AUC is reported; however, based on the fact that AUC measurement is more reliable than accuracy for unbalanced datasets, we believe that the final set of biomarkers obtained by AUC is more reliable.
We have proposed a new feature ranking method that significantly improves the traditional feature ranking by considering the synergistic interaction among potential risk factors. The comprehensive results based on simulated datasets and the dataset from DPT-1 have shown that our network-based feature ranking can help identify more predictive biomarkers than traditional individual-based feature ranking. The set of final biomarkers identified for T1D may help find more predictive models for T1D which may provide early prediction of disease for timely treatment. Furthermore, the improvement obtained by our network-based data-driven method suggests that a more comprehensive systematic data-driven analysis of biomedical variables will be helpful for the better understanding of T1D etiology.
Area under ROC curve
Body mass index
Diabetes Prevention Trial-Type 1 (DPT-1) study
First-phase insulin response
Glutamic acid decarboxylase
Genome-wide association studies
Homeostasis model assessment of insulin resistance
Islet cell autoantibodies
Insulinoma-associated protein 2
IV glucose tolerance test
Maximum clique problem
Maximum weighted clique problem
Oral glucose tolerance test
Receiver operating characteristic
Type 1 diabetes.
The project was supported in part by Award R21DK092845 from the National Institute Of Diabetes and Digestive and Kidney Diseases, National Institutes of Health. The data from the DPT-1 reported here were supplied by the NIDDK Central Repositories. This manuscript was not prepared in collaboration with Investigators of the DPT-1 study and does not necessarily reflect the opinions or views of the DPT-1 study, the NIDDK Central Repositories, or the NIDDK.
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.