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Multifactor dimensionality reduction (MDR) was developed as a nonparametric and model-free data mining method for detecting, characterizing, and interpreting epistasis in the absence of significant main effects in genetic and epidemiologic studies of complex traits such as disease susceptibility. The goal of MDR is to change the representation of the data using a constructive induction algorithm to make nonadditive interactions easier to detect using any classification method such as naïve Bayes or logistic regression. Traditionally, MDR constructed variables have been evaluated with a naïve Bayes classifier that is combined with 10-fold cross validation to obtain an estimate of predictive accuracy or generalizability of epistasis models. Traditionally, we have used permutation testing to statistically evaluate the significance of models obtained through MDR. The advantage of permutation testing is that it controls for false-positives due to multiple testing. The disadvantage is that permutation testing is computationally expensive. This is in an important issue that arises in the context of detecting epistasis on a genome-wide scale. The goal of the present study was to develop and evaluate several alternatives to large-scale permutation testing for assessing the statistical significance of MDR models. Using data simulated from 70 different epistasis models, we compared the power and type I error rate of MDR using a 1000-fold permutation test with hypothesis testing using an extreme value distribution (EVD). We find that this new hypothesis testing method provides a reasonable alternative to the computationally expensive 1000-fold permutation test and is 50 times faster. We then demonstrate this new method by applying it to a genetic epidemiology study of bladder cancer susceptibility that was previously analyzed using MDR and assessed using a 1000-fold permutation test.

The workhorse of modern genetic analysis is the parametric linear model. The advantages of the linear modeling framework are many and include a mathematical understanding of the model fitting process and ease of interpretation. However, an important limitation is that linear models make assumptions about the nature of the data being modeled. This assumption may not be realistic for complex biological systems such as disease susceptibility where nonlinearities in the genotype to phenotype mapping relationship that result from epistasis, plastic reaction norms, locus heterogeneity, and phenocopy, for example, are the norm rather than the exception. We have previously developed a flexible modeling approach called symbolic discriminant analysis (SDA) that makes no assumptions about the patterns in the data. Rather, SDA lets the data dictate the size, shape, and complexity of a symbolic discriminant function that could include any set of mathematical functions from a list of candidates supplied by the user. Here, we outline a new five step process for symbolic model discovery that uses genetic programming (GP) for coarse-grained stochastic searching, experimental design for parameter optimization, graphical modeling for generating expert knowledge, and estimation of distribution algorithms for fine-grained stochastic searching. Finally, we introduce function mapping as a new method for interpreting symbolic discriminant functions. We show that function mapping when combined with measures of interaction information facilitates statistical interpretation by providing a graphical approach to decomposing complex models to highlight synergistic, redundant, and independent effects of polymorphisms and their composite functions. We illustrate this five step SDA modeling process with a real case-control dataset.

We introduce a grammar-based hybrid approach to reverse engineering nonlinear ordinary differential equation models from observed time series. This hybrid approach combines a genetic algorithm to search the space of model architectures with a Kalman filter to estimate the model parameters. Domain-specific knowledge is used in a context-free grammar to restrict the search space for the functional form of the target model. We find that the hybrid approach outperforms a pure evolutionary algorithm method, and we observe features in the evolution of the dynamical models that correspond with the emergence of favorable model components. We apply the hybrid method to both artificially generated time series and experimentally observed protein levels from subjects who received the smallpox vaccine. From the observed data, we infer a cytokine protein interaction network for an individual's response to the smallpox vaccine.

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