Artificial neural networks modeling geneenvironment interaction
 Frauke Günther^{1},
 Iris Pigeot^{1} and
 Karin Bammann^{1, 2}Email author
https://doi.org/10.1186/147121561337
© Günther et al.; licensee BioMed Central Ltd. 2012
Received: 14 February 2012
Accepted: 1 April 2012
Published: 14 May 2012
Abstract
Background
Geneenvironment interactions play an important role in the etiological pathway of complex diseases. An appropriate statistical method for handling a wide variety of complex situations involving interactions between variables is still lacking, especially when continuous variables are involved. The aim of this paper is to explore the ability of neural networks to model different structures of geneenvironment interactions. A simulation study is set up to compare neural networks with standard logistic regression models. Eight different structures of geneenvironment interactions are investigated. These structures are characterized by penetrance functions that are based on sigmoid functions or on combinations of linear and nonlinear effects of a continuous environmental factor and a genetic factor with main effect or with a masking effect only.
Results
In our simulation study, neural networks are more successful in modeling geneenvironment interactions than logistic regression models. This outperfomance is especially pronounced when modeling sigmoid penetrance functions, when distinguishing between linear and nonlinear components, and when modeling masking effects of the genetic factor.
Conclusion
Our study shows that neural networks are a promising approach for analyzing geneenvironment interactions. Especially, if no prior knowledge of the correct nature of the relationship between covariables and response variable is present, neural networks provide a valuable alternative to regression methods that are limited to the analysis of linearly separable data.
Keywords
Background
The etiological pathway of any complex disease can be described as an interplay of genetic and nongenetic underlying causes (e.g. [1–3]). Usually, regression based methods are applied in the study of complex diseases (e.g. [4–8]). However, regression methods do not necessarily capture the complexity of the interplay of genetic and nongenetic factors. In particular, regression models require preprocessing of data to reflect any nonlinear relationship. First, continuous variables have to be either categorized or transformed according to their assumed form of relationship to the response. Second, interaction terms have to be explicitly included into the regression models to test for any statistical interaction. Third, if no prior knowledge of the functional form of the doseresponserelationship is present, a variety of regression models has to be explored. With increasing number of variables, finding the best model through trialanderror is no longer feasible due to the large number of possible models.
For modeling complex relationships, especially with little prior knowledge of the exact nature of these relationships, a more flexible statistical tool should be used. One promising alternative is the use of artificial neural networks. Here, variables do not have to be transformed a priori and interactions are modeled implicitly, that is, they do not have to be a priori formulated in the model [9]. We successfully applied neural networks for modeling different twolocus disease models, i.e. different types of genegene interactions as e.g. epistatic models [10].
Since studies using neural networks for modeling continuous covariables have previously shown promising results (see e.g. [11–13]), the aim of this paper is to investigate the usability of neural networks for modeling complex diseases that are determined by a geneenvironment interaction with a continuously measured environmental factor. Based on simulated data in a casecontrol design, we analyze the general modeling ability of neural networks for different structures of geneenvironment interactions. Theoretic risk models are defined representing different types of twoway interactions of one genetic and one environmental factor (e.g. [14]). The predicted risk is compared to the theoretic risk to assess the modeling ability. Additionally, neural networks are trained to a real data set to investigate the practicability of neural networks in a real life situation. All results are compared to those obtained by logistic regression models as reference method. Advantages and disadvantages of using a neural network approach are discussed.
Methods
Simulation study
Casecontrol data sets are generated using a two step design. First, underlying populations are simulated with a controlled prevalence of 10% and an overall sample size of five million observations. These populations carry the information of two marginally independent and randomly drawn factors – one biallelic locus and one continuous environmental factor – and a casecontrol status. The minor allele frequency is 30% to ensure sufficient cell frequencies in the final casecontrol data sets and it is assumed that the HardyWeinberg equilibrium holds. The environmental factor follows a continuous uniform distribution on the interval [0,100]. Depending on the genotype G and the environmental factor U, the casecontrol status is allocated through eight given theoretic risk models as introduced in the next subsection. Considering each theoretic risk model in a high and a low risk scenario, this results in sixteen underlying populations. As the second step, 100 casecontrol data sets are randomly drawn from all underlying populations for each analysis. Thus, for each analysis, mean values over 100 data sets are considered in sixteen situations. Three different sample sizes of 2,000 subjects (1,000 cases + 1,000 controls), 1,000 subjects (500 cases + 500 controls), and 400 subjects (200 cases + 200 controls) are used.
Artificial neural networks and logistic regression models are fitted to the data, i.e. separately to all 100 casecontrol data sets for each situation. A multilayer perceptron (MLP, see e.g. [15]) is chosen as neural network. It is briefly described in the Appendix. For neural networks, the genotype information is coded codominant, i.e. the genotype takes possible values 0, 1, and 2 representing the number of mutated alleles. The environmental factor is included in the analyses as continuous variable. For all data sets, six different network topologies, from zero up to five hidden neurons, are trained to avoid an overfitting of the data. For training purposes, the data set is always used as a whole. Each training process is replicated five times each with randomly initialized starting weights drawn from a standard normal distribution to enhance the chance that the training process stops within a global instead of a local minimum. The best trained neural network for each data set, i.e. the best network topology and the best repetition, is selected based on the Bayesian Information Criterion (BIC, [16]), which takes the number of parameters into account and penalizes additional parameters. Thus in each situation, 100 best neural networks predict the underlying risk model and the mean prediction can be used to evaluate the model fit (see below).
For comparison purposes, logistic regression models are fitted to the same data sets. The genotype is coded codominant counting the number of risk alleles and using two dichotomous design variables, one representing the heterozygous and one representing the homozygous mutated genotype. Five different models are used: the null model, three main effect models – containing only one or both main effects – and the full model – containing both main effects and one or two interaction terms depending on the genotype coding. For both coding approaches, the best model is selected based on BIC.
where g = 0,1,2 denotes the genotype and f(g,u^{ ′ }) refers to the theoretic risk model of the casecontrol data set and ${\widehat{f}}^{\left(k\right)}(g,{u}^{\prime})$ to the prediction of the kth casecontrol data set. The smaller $\sum _{g{u}^{\prime}}{E}_{g{u}^{\prime}}$ is, the better the mean model fit of neural networks or logistic regression models is since the estimated risk model and the theoretic risk model coincide for $\sum _{g{u}^{\prime}}{E}_{g{u}^{\prime}}=0$. To take variation into account, pointwise prediction intervals are calculated as empirical 95% intervals. In particular, for all u^{ ′ }= 0, 0.1, 0.2,…,100 and g = 0,1,2 a prediction interval is determined as the interval $\left[\widehat{f}{(g,{u}^{\prime})}_{\left(3\right)};\phantom{\rule{0.3em}{0ex}}\widehat{f}{(g,{u}^{\prime})}_{\left(98\right)}\right]$, where $\widehat{f}{(g,{u}^{\prime})}_{\left(3\right)}$ and $\widehat{f}{(g,{u}^{\prime})}_{\left(98\right)}$ denote the 3rd ordered and the 98th ordered prediction, respectively.
Data generation and all analyses are done using R [17]. The package for training the MLP was implemented by our group and is published on CRAN [18].
Theoretic risk models
Two different types of theoretic risk models for geneenvironment interactions are used, namely the models introduced by Amato et al. [14] and models mainly representing a masking effect of the involved locus as defined below. For all risk models, the kind of functional relationship between the penetrance and the environmental factor depends on the genotype information, i.e. the curve shape is in general different depending on the three genotypes. The relationship is defined on a population level, i.e. the penetrance function F :{0,1,2} × [0,100] → [0,1] with F(g u) = P(Y = 1G = g U = u), where Y ∈ {0,1} denotes the casecontrol status, G ∈ {0,1,2} the genotype, and U ∈ [0,100] the environmental factor, only holds in the corresponding underlying population and has to be converted to f(g u) if a casecontrol data set is analyzed [10].
Risk models by Amato et al
The four models are defined as follows:

the genetic model: α_{1} ≤ α_{2} ≤ α_{3} and β_{1} = β_{2} = β_{3} = 0,

the environmental model: α_{1} = α_{2} = α_{3} and β_{1} = β_{2} = β_{3} ≠ 0,

the additive model: α_{1} ≤ α_{2} ≤ α_{3} and β_{1} = β_{2} = β_{3} ≠ 0,

the interaction model: α_{1} = α_{2} = α_{3} and β_{1} ≤ β_{2} ≤ β_{3}.
Used values for α _{ g } , β _{ g } (g = 0,1,2), c , and z
Risk model  Risk scenario  Constant values α_{g}, β_{g} ( g = 0,1,2)  Constant values c, z  

Genetic model  High risk  ${\alpha}_{0}=\frac{2}{3}\xb7{\alpha}_{1},\phantom{\rule{0.3em}{0ex}}{\alpha}_{1}=2.5,\phantom{\rule{0.3em}{0ex}}{\alpha}_{2}=\frac{4}{3}\xb7{\alpha}_{1}$  z = 0.886  
β_{0} = β_{1} = β_{2} = 0  
Low risk  ${\alpha}_{0}=\frac{2}{3}\xb7{\alpha}_{1},\phantom{\rule{0.3em}{0ex}}{\alpha}_{1}=1.25,\phantom{\rule{0.3em}{0ex}}{\alpha}_{2}=\frac{4}{3}\xb7{\alpha}_{1}$  z = 0.390  
β_{0} = β_{1} = β_{2} = 0  
Environmental model  High risk  α_{0} = α_{1} = α_{2} = 7.5,  z = 0.200  
β_{0} = β_{1} = β_{2} = −0.15,  
Low risk  α_{0} = α_{1} = α_{2} = 3.75,  z = 0.200  
Risk models by Amato et al. [14]  β_{0} = β_{1} = β_{2} = −0.075,  
Additive model  High risk  ${\alpha}_{0}=\frac{2}{3}\xb7{\alpha}_{1},\phantom{\rule{0.3em}{0ex}}{\alpha}_{1}=7.5,\phantom{\rule{0.3em}{0ex}}{\alpha}_{2}=\frac{4}{3}\xb7{\alpha}_{1}$,  z = 0.177  
β_{0} = β_{1} = β_{2} = −0.15,  
Low risk  ${\alpha}_{0}=\frac{2}{3}\xb7{\alpha}_{1},\phantom{\rule{0.3em}{0ex}}{\alpha}_{1}=3.75,\phantom{\rule{0.3em}{0ex}}{\alpha}_{2}=\frac{4}{3}\xb7{\alpha}_{1}$,  z = 0.178  
β_{0} = β_{1} = β_{2} = −0.075,  
Interaction model  High risk  α_{0} = α_{1} = α_{2} = 7.5,  z = 0.171  
β_{0} = 2 · β_{1}, β_{1} = −0.15, β_{2} = 0.5·β_{1},  
Low risk  α_{0} = α_{1} = α_{2} = 3.75,  z = 0.169  
β_{0} = 2 · β_{1}, β_{1} = −0.075, β_{2} = 0.5 · β_{1},  
Model 1  High risk (r = 0.150)  c = 0.05, z = 0.254  
Low risk (r = 0.075)  
Risk model representing a masking effect of the genetic factor  Model 2  High risk (r = 0.150)  c = 0.05, z = 0.286  
Low risk (r = 0.075)  
Model 3  High risk (r = 0.150)  c = 0.075, z = 0.631  
Low risk (r = 0.075)  
Model 4  High risk (r = 0.150)  c = 0.075, z = 0.964  
Low risk (r = 0.075) 
Risk models representing a masking effect of the genetic factor
 1.The structure of the first risk model is given by the following penetrance function F :{0,1,2} × [0,100] → [0,1]$F(g,u)=\left\{\begin{array}{cc}\frac{zc}{1+\text{exp}(r(u50\left)\right)}+c& \text{if}g=0\\ c& \text{if}g=1.\\ c& \text{if}g=2\end{array}\right.$
 2.The second risk model is defined by$F(g,u)=\left\{\begin{array}{cc}\frac{z}{1+\text{exp}(r(u50\left)\right)}& \text{if}g=0\\ c& \text{if}g=1\phantom{\rule{1em}{0ex}}.\\ 2c& \text{if}g=2\end{array}\right.$
 3.In the third risk model, the penetrance function is given by$F(g,u)=\left\{\begin{array}{cc}c& \text{if}g=0\\ c& \text{if}g=1\phantom{\rule{1em}{0ex}}.\\ \frac{zc}{1+\text{exp}(r(u50\left)\right)}+c& \text{if}g=2\end{array}\right.$
 4.For the fourth risk model, the penetrance function is determined as follows:$F(g,u)=\left\{\begin{array}{cc}\frac{1}{2}c& \text{if}g=0\\ c& \text{if}g=1\phantom{\rule{1em}{0ex}}.\\ \frac{z2c}{1+exp(r(u50\left)\right)}+2c& \text{if}g=2\end{array}\right.$
Real data application
To study the performance of a neural network in a real life situation, we applied this approach to a crosssectional study dealing with a lifestyle induced complex disease. This application should serve as an example for the general practicability of our approach without describing the study from a subject point of view. The common effect of an SNP and a continuous environmental factor on a binary outcome is investigated while controlling for the effect of one binary confounder. The data set includes 138 cases and 1599 controls. As in the simulation study, neural networks with up to five hidden neurons are trained each five times with randomly initialized weights drawn from of a standard normal distribution and the best neural network is chosen based on BIC. The analysis is done once using the whole data set and once stratified by the confounding factor. For the stratified analysis, 95% bootstrap percentile intervals are calculated using 100 bootstrap replications [19].
Results
Risk models by Amato et al.
Differences between theoretic and estimated penetrance functions (models by Amato et al. [[14]])
High risk scenario  Low risk scenario  

Neural network  Logistic regression  Logistic regression (DV)  Neural network  Logistic regression  Logistic regression (DV)  
n =1000 + 1000  n=1000 + 1000  
Genetic model  40.79  31.31  48.15  48.22  40.85  83.62  
$\sum _{g{u}^{\prime}}{E}_{g{u}^{\prime}}$  Environmental model  46.14  277.11  277.11  52.45  171.61  171.36 
Additive model  45.13  256.52  260.10  47.99  163.19  189.92  
Interaction model  119.77  345.77  247.93  132.47  225.61  194.37  
n =500 + 500  n = 500 + 500  
Genetic model  59.28  47.14  68.22  64.27  92.02  159.80  
$\sum _{g{u}^{\prime}}{E}_{g{u}^{\prime}}$  Environmental model  60.57  277.51  277.15  90.76  174.37  174.16 
Additive model  56.10  268.11  297.62  80.66  190.25  242.34  
Interaction model  138.91  344.50  268.75  153.56  233.16  210.98  
n = 200 + 200  n = 200 + 200  
Genetic model  101.95  85.67  152.25  97.23  167.48  207.66  
$\sum _{g{u}^{\prime}}{E}_{g{u}^{\prime}}$  Environmental model  96.32  278.40  278.93  163.16  177.14  175.27 
Additive model  96.16  329.55  374.17  177.24  246.06  292.39  
Interaction model  168.90  349.88  316.01  207.81  256.22  291.88 
If the sample size decreases, the modeling ability becomes worse for neural networks as well as for logistic regression models (see Table 2). However, neural networks still show the best model fit if the environmental factor has an effect. The prediction intervals include the true underlying risk model in all but two situations (interaction model, n = 500 + 500, low risk scenario and interaction model, n = 200 + 200, high risk scenario, data not shown).
Models representing a masking effect of the genetic factor
Differences between theoretic and estimated penetrance functions (models representing a masking effect of the genetic factor)
High risk scenario  Low risk scenario  

Neural network  Logistic regression  Logistic regression (DV)  Neural network  Logistic regression  Logistic regression (DV)  
n = 1000 + 1000  n = 1000 + 1000  
Model 1  38.63  211.62  105.83  41.07  195.15  87.57  
$\sum _{g{u}^{\prime}}{E}_{g{u}^{\prime}}$  Model 2  117.94  359.10  155.40  101.92  323.89  114.71 
Model 3  40.67  253.01  85.51  43.15  258.19  65.87  
Model 4  103.37  228.10  85.16  103.63  227.50  59.74  
n = 500 + 500  n = 500 + 500  
Model 1  54.58  219.39  136.26  70.40  207.97  140.74  
$\sum _{g{u}^{\prime}}{E}_{g{u}^{\prime}}$  Model 2  144.35  363.36  176.74  183.28  327.58  143.06 
Model 3  60.98  261.86  110.93  66.25  278.61  114.68  
Model 4  143.62  235.44  102.13  115.59  237.14  81.13  
n = 200 + 200  n = 200 + 200  
Model 1  126.56  252.88  251.70  192.47  244.17  225.63  
$\sum _{g{u}^{\prime}}{E}_{g{u}^{\prime}}$  Model 2  262.92  371.69  230.25  297.68  348.46  215.70 
Model 3  139.27  324.55  215.12  141.28  328.64  191.61  
Model 4  189.69  287.39  169.86  164.13  280.21  149.95 
With decreasing sample sizes, the model fit again becomes worse and the variance increases (data not shown). If the sample size is 500 + 500 subjects, neural networks again have the best model fit for the first three risk models in the high risk scenario. In the low risk scenario, this is only true for the first and the third risk model. A sample size of just 200 + 200 subjects leads to a considerably worse model fit of neural networks. In this situation, logistic regression models with design variables coding the genotype have the best model fit for the second and fourth risk model in both risk scenarios. Neural networks still have the best model fit if the gene has a masking effect only.
Real data application
Discussion
In this paper, we studied the ability of neural networks and logistic regression models to capture different types of geneenvironment interactions. Neural networks were able to predict the theoretic risk models in all sixteen investigated situations such that the prediction intervals contained the true underlying risk models in most situations and were thus superior to logistic regression models. Logistic regression models without design variables completely failed to model the constant effects. Employing design variables led to a considerably better model fit only when average values over the 100 data sets were considered. Single predictions for one data set often had a misleading form and did not distinguish between linear and nonlinear components especially for the first two risk models. Nevertheless for risk model 4, logistic regression models using design variables provided the best model fit compared with neural networks as could be seen by the mean absolute differences although the prediction interval did not include the whole true risk model. However, the reasoning behind this fact is still unknown. The real data set application showed the general usability of neural networks in real life situations. Neural networks discovered different risk slopes for each genotype, which also became obvious from the corresponding bootstrap confidence intervals.
Neural networks do not use interaction terms. In our application, they mainly needed one or two hidden neurons if the environmental factor had an effect (risk models by [14]) and they needed one hidden neuron if the locus only had a masking effect and two hidden neurons if the locus had an own main effect (risk models representing a masking effect of the genetic factor). For logistic regression, the correct main effect models were mainly selected for the genetic and the environmental model as best models based on BIC and full models were selected for the additive and interaction model. Thus, the latter two risk models cannot be distinguished from each other based on the covariables included. Logistic regression models mainly needed an interaction term to model the underlying risk models representing a masking effect of the genetic factor regardless of whether the genotype was coded codominant or using design variables (data not shown).
Logistic regression models belong to the class of generalized linear models and as such are limited in their modeling capacity to linearly separable data. On the contrary, neural networks can adapt to any piecewise continuous function. Since linear and nonlinear relationships can be modeled simultaneously, neural networks are a promising tool if little is known about the exact relationship between covariables and a response variable or especially, if a nonlinear relationship is assumed.
Thus, our results suggest that neural networks can be a valuable approach already in the situation of 500 cases and 500 controls. However, there are two main drawbacks of neural networks. First, the computing time needed to train them is very high. In our application, the analyses for one situation (100 replications, six network topologies each) sometimes took more than 30 hours. Second, neural networks are still considered as blackbox approach since both network topology and trained weights have no direct interpretation. Thus, there is no established way for model selection and parameter testing. One possibility to estimate the effect of a covariable is provided by the concept of generalized weights [20]. The aim of this paper was to investigate the general modeling ability of neural networks as a first step. Further research should to be devoted to the missing interpretability of trained neural networks.
Differences between theoretic and estimated penetrance functions (sensitivity analysis: low minor allele frequency)
High risk scenario  Low risk scenario  

Neural network  Logistic regression  Logistic regression (DV)  Neural network  Logistic regression  Logistic regression (DV)  
n = 1000 + 1000  n = 1000 + 1000  
Genetic model  80.29  80.39  303.07^{∗}  87.65  209.74  249.96  
$\sum _{g{u}^{\prime}}{E}_{g{u}^{\prime}}$  Environmental model  79.60  278.32  277.18  78.18  170.94  170.94 
Additive model  74.67  369.57  443.10  92.18  303.98  348.50  
Interaction model  180.02  415.60  541.02^{∗}  191.77  327.44  481.62^{∗}  
Model 1  113.62  244.87  375.43^{∗}  179.23  226.03  355.59^{∗}  
$\sum _{g{u}^{\prime}}{E}_{g{u}^{\prime}}$  Model 2  232.75  389.70  472.47^{∗}  318.57  346.57  460.08^{∗} 
Model 3  253.00  230.12  232.20  256.38  253.67  254.80  
Model 4  133.91  126.27  97.92  138.28  132.11  93.04 
Conclusions
To the best of our knowledge, neural networks have not been used for modeling geneenvironment interactions so far. In other contexts, MLPs were clearly superior to logistic regression models [21, 22]. Previously, we have successfully employed neural networks for the analysis of genegene interactions in simulation studies [10]. This paper shows that the advantages of neural networks are even more pronounced when modeling geneenvironment interactions with continuous environmental factors.
In practice, neural networks can be applied in casecontrol studies to investigate the common effect of two genetic factors or one genetic and one environmental factor. Since the functional form of the model has not to be specified in neural networks, it has neither to be known whether the two involved factors indeed have an effect on the disease nor whether an interaction between both factors is present. The prediction of a neural network generates insight in the kind of relationship between covariables and disease, for example, whether the underlying relationship is nonlinear or whether there are different relationships per genotype. Thus, although there is still need for further research regarding the interpretability of neural networks, neural networks are already a valuable statistical tool especially for exploratory analyses and/or when little is known about the functional relationship of risk factors and investigated disease.
Appendix
Artificial neural networks
The general idea of a multilayer perceptron (MLP) is to approximate functional relationships between covariables and response variable(s). It consists of neurons and synapses that are organized as a weighted directed graph. The neurons are arranged in layers and subsequent layers are usually fully connected by synapses. Each synapse is attached by a weight indicating the effect of this synapse. A positive weight indicates an amplifying, a negative weight a repressing effect. Neural networks have to be trained using a learning algorithm to adjust the synaptic weights according to given data. The learning algorithm minimizes the deviation of predicted output and given response variable measured by an error function.
where w_{0}, w_{ j }, and w_{ ij }, i = 0,…,n, j = 1,…,m, denote the weights including intercepts, x = (x_{0}x_{1},…,x_{ n })^{ T } the vector of all covariables including a constant neuron x_{0} and σ the activation function that maps the output of each neuron to a given range. MLPs are a direct extension of generalized linear models (GLM, [24]) and an MLP without hidden layer is algebraically equivalent to a generalized linear model with σ as inverse link function. In this case, trained weights and estimated regression coefficients coincide.
To train neural networks according to the casecontrol data sets, resilient backpropagation [25] as learning algorithm with cross entropy as error function and logistic function as activation function is used.
Author’s contributions
FG planned and carried out the simulation study and drafted the manuscript. IP drafted the manuscript. KB planned the simulation study and drafted the manuscript. All authors read and approved the final manuscript.
Declarations
Acknowledgements
We gratefully acknowledge the financial support for this research by the grant PI 345/31 from the German Research Foundation (DFG).
We would like to thank two anonymous reviewers for their valuable remarks.
Authors’ Affiliations
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