Many complex human diseases are likely the consequence of the joint

Many complex human diseases are likely the consequence of the joint actions of genetic and environmental factors. categories. There is a dearth of statistical methods for detecting gene by time-varying environmental exposure interactions. Here we propose a powerful functional logistic regression (FLR) approach to model the time-varying effect of longitudinal environmental OSI-906 exposure and its conversation with genetic factors OSI-906 on disease risk. Capitalizing on the powerful functional data analysis framework our proposed FLR model is usually capable of accommodating longitudinal exposures measured at irregular time points and contaminated by measurement errors commonly encountered in observational studies. We use considerable OSI-906 simulations to show that this proposed method can control the Type I error and is more powerful than alternative ad hoc methods. We demonstrate the power of this new method using data from a case-control study of pancreatic malignancy to identify the windows of vulnerability of lifetime body mass index on the risk of pancreatic malignancy as well as genes which may change this association. including cases and controls (= + denote the binary disease status of individual = 1 .. denote the covariate vector including for example sex age and leading theory components capturing populace substructure. Given a SNP to be tested for GxE conversation let denote the genotype of the SNP in subject is the time of the individual = 1 … is the individual takes values in the time interval = [is usually the individuals that is usually = and = = ··· = for all those = 1 … way for example into 5-12 months intervals [Sanchez et al. 2011 Measurements in the same interval are then averaged leading to a single environmental exposure value for each individual. We re-write to denote the individual = 1 … = 0 against the alternative is not 0. This can be carried out by calculating a p-value for each and taking the minimum of the p-values denoted as minP. We then compare minP with the Bonferroni correction significance threshold 0.05/longitudinal exposures = 0 against the alternative is not 0. This seems to be the same as OSI-906 that in Model (1); however the key difference is usually that and in Equation (1) are estimated for each environmental exposure measurement separately whereas those in Equation (2) are estimated jointly for all those measurements. We can employ a is usually large [Pan et al. 2011 Another caveat of Model (2) is usually that correlated longitudinal exposures may lead to unstable numerical solutions due to multicollinearity. AGIF New method: functional logistic regression and FPCA The FDA including the FPCA and functional linear/generalized linear models has emerged as a powerful approach to modeling noisy and irregularly measured longitudinal data in association with a scalar response variable e.g. disease outcome [Li et al. 2010 Müller 2005 Here we propose to model the longitudinal environmental exposure in the FDA framework. First we decompose the longitudinal exposure trajectory into a few uncorrelated components using the FPCA taking into account possible measurement errors and then model gene by longitudinal exposure conversation using the functional logistic regression (FLR) model. FPCA We model the individuals’ exposure trajectories as impartial realizations from a square integrable stochastic process ∈ with imply function ∈ = [Thorem [Leng and Muller 2006 we have eigendecomposition OSI-906 and are eigenfunctions and eigenvalues ordered by size = = and 0 normally. By the decomposition [Yao et al. 2005 a random curve = (is the functional principle component (FPC) score for the subject. In addition satisfies steps the similarity between the deviation of individual curve decomposition. We further presume that we observe the = 1 … time points for = 1 … from the entire observed data = 1 .. and = 1 … = 1 .. and = 1 … ∈ = [? ≠ are obtained by applying spectral decomposition to the smoothed covariance surface = (and to be jointly normal and predicting the random effects based on its conditional expectation: ∈ can be well approximated by the function space spanned by the leading eigenfunctions leading to a truncated version of the representation can be based on either the portion of variance explained (FVE) or some model selection criteria such as modfied AIC and BIC [Li et al. 2010 Yao et al. 2005 As exhibited OSI-906 in Equation (5) the infinite-dimensional trajectory FPC scores for each individual. The PACE method is usually implemented in the Matlab.