Simulated gene expression data for demonstrating the features of roben.
data("GxE_small")
data("GxE_large")GxE_small consists of five components: X, Y, E, clin and coeff. coeff contains the true values of parameters used for generating Y.
GxE_large contains larger datasets: X2, Y2, E2 and clin2
The data model for generating Y
Use subscript \(i\) to denote the \(i\)th subject. Let \((X_{i}, Y_{i}, E_{i}, Clin_{i})\), (\(i=1,\ldots,n\)) be independent and identically distributed random vectors. \(Y_{i}\) is a continuous response variable representing the disease phenotype. \(X_{i}\) is the \(p\)--dimensional vector of G factors. The environmental factors and clinical covariates are denoted as the \(k\)-dimensional vector \(E_{i}\) and the \(q\)-dimensional vector \(Clin_{i}\), respectively. The \(\epsilon\) follows some heavy-tailed distribution. Considering the following model:
$$Y_{i} = \alpha_{0} + \sum_{t=1}^{q}\alpha_{t}Clin_{it} + \sum_{m=1}^{k}\theta_{m}E_{im} + \sum_{j=1}^{p}\gamma_{j}X_{ij} + \sum_{j=1}^{p}\sum_{m=1}^{k}\zeta_{jm}E_{im}X_{ij} +\epsilon_{i},$$ where \(\alpha_{0}\) is the intercept; \(\alpha_{t}\)'s, \(\theta_{m}\)'s, \(\gamma_{j}\)'s and \(\zeta_{jm}\)'s are the regression coefficients for the clinical covariates, environmental factors, genetic factors and G\(\times\)E interactions, respectively.
Define \(\beta_{j}=(\gamma_{j}, \zeta_{j1},\ldots,\zeta_{jk})^\top \equiv (\beta_{j1},\ldots,\beta_{jL})^\top\) and \(U_{ij}=(X_{ij},X_{ij}E_{i1}\ldots,X_{ij}E_{ik})^\top \equiv (U_{ij1},\dots,U_{ijL})^\top\), where \(L=k+1\). The model can be written as $$Y_{i} = \alpha_{0} + \sum_{t=1}^{q}\alpha_{t}Clin_{it} + \sum_{m=1}^{k}\theta_{m}E_{im} + \sum_{j=1}^{p} \big(U_{ij}^\top\beta_{j}\big) +\epsilon_{i},$$ where the coefficient vector \(\beta_{j}\) represents all the main and interaction effects corresponding to the \(j\)th genetic measurement.
The object coeff in GxE_small is a list of four components, corresponding to \(\alpha_{0}\), \(\alpha_{t}\)'s, \(\theta_{m}\)'s and \(\beta_{j}\)'s.
roben
data(GxE_small)
dim(X)
print(coeff)
data(GxE_large)
dim(X)
print(coeff)
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