Comparison of information-theoretic to statistical methods for gene-gene interactions in the presence of genetic heterogeneity

Optimized permutation testing for information theoretic measures of multi-gene interactions

Background

Permutation testing is often considered the “gold standard” for multi-test significance analysis, as it is an exact test requiring few assumptions about the distribution being computed. However, it can be computationally very expensive, particularly in its naive form in which the full analysis pipeline is re-run after permuting the phenotype labels. This can become intractable in multi-locus genome-wide association studies (GWAS), in which the number of potential interactions to be tested is combinatorially large.

Results

In this paper, we develop an approach for permutation testing in multi-locus GWAS, specifically focusing on SNP–SNP-phenotype interactions using multivariable measures that can be computed from frequency count tables, such as those based in Information Theory. We find that the computational bottleneck in this process is the construction of the count tables themselves, and that this step can be eliminated at each iteration of the permutation testing by transforming the count tables directly. This leads to a speed-up by a factor of over 10³ for a typical permutation test compared to the naive approach. Additionally, this approach is insensitive to the number of samples making it suitable for datasets with large number of samples.

Conclusions

The proliferation of large-scale datasets with genotype data for hundreds of thousands of individuals enables new and more powerful approaches for the detection of multi-locus genotype-phenotype interactions. Our approach significantly improves the computational tractability of permutation testing for these studies. Moreover, our approach is insensitive to the large number of samples in these modern datasets. The code for performing these computations and replicating the figures in this paper is freely available at https://github.com/kunert/permute-counts.

Asymptotic Distributions of Empirical Interaction Information

Interaction Information is one of the most promising interaction strength measures with many desirable properties. However, its use for interaction detection was hindered by the fact that apart from the simple case of overall independence, asymptotic distribution of its estimate has not been known. In the paper we provide asymptotic distributions of its empirical versions which are needed for formal testing of interactions. We prove that for three-dimensional nominal vector normalized empirical interaction information converges to the normal law unless the distribution coincides with its Kirkwood approximation. In the opposite case the convergence is to the distribution of weighted centred chi square random variables. This case is of special importance as it roughly corresponds to interaction information being zero and the asymptotic distribution can be used for construction of formal tests for interaction detection. The result generalizes result in Han (Inf Control 46(1):26–45 1980) for the case when all coordinate random variables are independent. The derivation relies on studying structure of covariance matrix of asymptotic distribution and its eigenvalues. For the case of 3 × 3 × 2 contingency table corresponding to study of two interacting Single Nucleotide Polymorphisms (SNPs) for prediction of binary outcome, we provide complete description of the asymptotic law and construct approximate critical regions for testing of interactions when two SNPs are possibly dependent. We show in numerical experiments that the test based on the derived asymptotic distribution is easy to implement and yields actual significance levels consistently closer to the nominal ones than the test based on chi square reference distribution.

Machine learning in pharmacometrics: Opportunities and challenges

The explosive growth in medical devices, imaging and diagnostics, computing, and communication and information technologies in drug development and healthcare has created an ever-expanding data landscape that the pharmacometrics (PMX) research community must now traverse. The tools of machine learning (ML) have emerged as a powerful computational approach in other data-rich disciplines but its effective utilization in the pharmaceutical sciences and PMX modelling is in its infancy. ML-based methods can complement PMX modelling by enabling the information in diverse sources of big data, e.g. population-based public databases and disease-specific clinical registries, to be harnessed because they are capable of efficiently identifying salient variables associated with outcomes and delineating their interdependencies. ML algorithms are computationally efficient, have strong predictive capabilities and can enable learning in the big data setting. ML algorithms can be viewed as providing a computational bridge from big data to complement PMX modelling. This review provides an overview of the strengths and weaknesses of ML approaches vis-à-vis population methods, assesses current research into ML applications in the pharmaceutical sciences and provides perspective for potential opportunities and strategies for the successful integration and utilization of ML in PMX.

Information Theory in Computational Biology: Where We Stand Today

"A Mathematical Theory of Communication" was published in 1948 by Claude Shannon to address the problems in the field of data compression and communication over (noisy) communication channels. Since then, the concepts and ideas developed in Shannon's work have formed the basis of information theory, a cornerstone of statistical learning and inference, and has been playing a key role in disciplines such as physics and thermodynamics, probability and statistics, computational sciences and biological sciences. In this article we review the basic information theory based concepts and describe their key applications in multiple major areas of research in computational biology-gene expression and transcriptomics, alignment-free sequence comparison, sequencing and error correction, genome-wide disease-gene association mapping, metabolic networks and metabolomics, and protein sequence, structure and interaction analysis.

Transferring entropy to the realm of GxG interactions

Genome-wide association studies are moving to genome-wide interaction studies, as the genetic background of many diseases appears to be more complex than previously supposed. Thus, many statistical approaches have been proposed to detect gene-gene (GxG) interactions, among them numerous information theory-based methods, inspired by the concept of entropy. These are suggested as particularly powerful and, because of their nonlinearity, as better able to capture nonlinear relationships between genetic variants and/or variables. However, the introduced entropy-based estimators differ to a surprising extent in their construction and even with respect to the basic definition of interactions. Also, not every entropy-based measure for interaction is accompanied by a proper statistical test. To shed light on this, a systematic review of the literature is presented answering the following questions: (1) How are GxG interactions defined within the framework of information theory? (2) Which entropy-based test statistics are available? (3) Which underlying distribution do the test statistics follow? (4) What are the given strengths and limitations of these test statistics?

Key Considerations and Methods in the Study of Gene-Environment Interactions

With increased involvement of genetic data in most epidemiological investigations, gene-environment (G × E) interactions now stand as a topic, which must be meticulously assessed and thoroughly understood. The level, mode, and outcomes of interactions between environmental factors and genetic traits have the capacity to modulate disease risk. These must, therefore, be carefully evaluated as they have the potential to offer novel insights on the "missing heritability problem", reaching beyond our current limitations. First, we review a definition of G × E interactions. We then explore how concepts such as the early manifestation of the genetic components of a disease, the heterogeneity of complex traits, the clear definition of epidemiological strata, and the effect of varying physiological conditions can affect our capacity to detect (or miss) G × E interactions. Lastly, we discuss the shortfalls of regression models to study G × E interactions and how other methods such as the ReliefF algorithm, pattern recognition methods, or the LASSO (Least Absolute Shrinkage and Selection Operator) method can enable us to more adequately model G × E interactions. Overall, we present the elements to consider and a path to follow when studying genetic determinants of disease in order to uncover potential G × E interactions.

CINOEDV: a co-information based method for detecting and visualizing n-order epistatic interactions

BACKGROUND

Detecting and visualizing nonlinear interaction effects of single nucleotide polymorphisms (SNPs) or epistatic interactions are important topics in bioinformatics since they play an important role in unraveling the mystery of "missing heritability". However, related studies are almost limited to pairwise epistatic interactions due to their methodological and computational challenges.

RESULTS

We develop CINOEDV (Co-Information based N-Order Epistasis Detector and Visualizer) for the detection and visualization of epistatic interactions of their orders from 1 to n (n ≥ 2). CINOEDV is composed of two stages, namely, detecting stage and visualizing stage. In detecting stage, co-information based measures are employed to quantify association effects of n-order SNP combinations to the phenotype, and two types of search strategies are introduced to identify n-order epistatic interactions: an exhaustive search and a particle swarm optimization based search. In visualizing stage, all detected n-order epistatic interactions are used to construct a hypergraph, where a real vertex represents the main effect of a SNP and a virtual vertex denotes the interaction effect of an n-order epistatic interaction. By deeply analyzing the constructed hypergraph, some hidden clues for better understanding the underlying genetic architecture of complex diseases could be revealed.

CONCLUSIONS

Experiments of CINOEDV and its comparison with existing state-of-the-art methods are performed on both simulation data sets and a real data set of age-related macular degeneration. Results demonstrate that CINOEDV is promising in detecting and visualizing n-order epistatic interactions. CINOEDV is implemented in R and is freely available from R CRAN: http://cran.r-project.org and https://sourceforge.net/projects/cinoedv/files/ .

The application of information theory for the research of aging and aging-related diseases

This article reviews the application of information-theoretical analysis, employing measures of entropy and mutual information, for the study of aging and aging-related diseases. The research of aging and aging-related diseases is particularly suitable for the application of information theory methods, as aging processes and related diseases are multi-parametric, with continuous parameters coexisting alongside discrete parameters, and with the relations between the parameters being as a rule non-linear. Information theory provides unique analytical capabilities for the solution of such problems, with unique advantages over common linear biostatistics. Among the age-related diseases, information theory has been used in the study of neurodegenerative diseases (particularly using EEG time series for diagnosis and prediction), cancer (particularly for establishing individual and combined cancer biomarkers), diabetes (mainly utilizing mutual information to characterize the diseased and aging states), and heart disease (mainly for the analysis of heart rate variability). Few works have employed information theory for the analysis of general aging processes and frailty, as underlying determinants and possible early preclinical diagnostic measures for aging-related diseases. Generally, the use of information-theoretical analysis permits not only establishing the (non-linear) correlations between diagnostic or therapeutic parameters of interest, but may also provide a theoretical insight into the nature of aging and related diseases by establishing the measures of variability, adaptation, regulation or homeostasis, within a system of interest. It may be hoped that the increased use of such measures in research may considerably increase diagnostic and therapeutic capabilities and the fundamental theoretical mathematical understanding of aging and disease.

SYMPHONY, an information-theoretic method for gene-gene and gene-environment interaction analysis of disease syndromes

We develop an information-theoretic method for gene-gene (GGI) and gene-environmental interactions (GEI) analysis of syndromes, defined as a phenotype vector comprising multiple quantitative traits (QTs). The K-way interaction information (KWII), an information-theoretic metric, was derived for multivariate normal distributed phenotype vectors. The utility of the method was challenged with three simulated data sets, the Genetic Association Workshop-15 (GAW15) rheumatoid arthritis data set, a high-density lipoprotein (HDL) and atherosclerosis data set from a mouse QT locus study, and the 1000 Genomes data. The dependence of the KWII on effect size, minor allele frequency, linkage disequilibrium, population stratification/admixture, as well as the power and computational time requirements of the novel method was systematically assessed in simulation studies. In these studies, phenotype vectors containing two and three constituent multivariate normally distributed QTs were used and the KWII was found to be effective at detecting GEI associated with the phenotype. High KWII values were observed for variables and variable combinations associated with the syndrome phenotype compared with uninformative variables not associated with the phenotype. The KWII values for the phenotype-associated combinations increased monotonically with increasing effect size values. The KWII also exhibited utility in simulations with non-linear dependence between the constituent QTs. Analysis of the HDL and atherosclerosis data set indicated that the simultaneous analysis of both phenotypes identified interactions not detected in the analysis of the individual traits. The information-theoretic approach may be useful for non-parametric analysis of GGI and GEI of complex syndromes.

Genomics in the post-GWAS era

The field of genomics has entered a new era in which the ability to identify genetic variants that impact complex human traits and disease in an unbiased fashion using genome-wide approaches is widely accessible. To date, the workhorse of these efforts has been the genome-wide association study (GWAS), which has quickly moved from novel to routine, and has provided key insights into aspects of the underlying allelic architecture of complex traits. The main lesson learned from the early GWAS efforts is that though many disease-associated variants are often discovered, most have only a minor effect on disease, and in total explain only a small amount of the apparent heritability. Here we provide a brief overview of the genetic variation classes that may harbor the heritability missing from GWAS, and touch on approaches that will be leveraged in the coming years as genomics-and by extension medicine-becomes increasingly personalized.

Modeling of environmental and genetic interactions with AMBROSIA, an information-theoretic model synthesis method

To develop a model synthesis method for parsimoniously modeling gene-environmental interactions (GEI) associated with clinical outcomes and phenotypes. The AMBROSIA model synthesis approach utilizes the k-way interaction information (KWII), an information-theoretic metric capable of identifying variable combinations associated with GEI. For model synthesis, AMBROSIA considers relevance of combinations to the phenotype, it precludes entry of combinations with redundant information, and penalizes for unjustifiable complexity; each step is KWII based. The performance and power of AMBROSIA were evaluated with simulations and Genetic Association Workshop 15 (GAW15) data sets of rheumatoid arthritis (RA). AMBROSIA identified parsimonious models in data sets containing multiple interactions with linkage disequilibrium present. For the GAW15 data set containing 9187 single-nucleotide polymorphisms, the parsimonious AMBROSIA model identified nine RA-associated combinations with power >90%. AMBROSIA was compared with multifactor dimensionality reduction across several diverse models and had satisfactory power. Software source code is available from http://www.cse.buffalo.edu/DBGROUP/bioinformatics/resources.html. AMBROSIA is a promising method for GEI model synthesis.