Simulated gene expression data for demonstrating the features of BayesQVGEL.
The data object consists of seven components: y, e, C, g, w ,k and coeff. coeff contains the true values of parameters used for generating Y.
The data and model setting
Consider a longitudinal study on \(n\) subjects with \(k\) repeated measurement for each subject. Let \(Y_{ij}\) be the measurement for the \(i\)th subject at each time point \(j\)(\(1\leq i \leq n, 1\leq j \leq k\)) .We use a \(m\)-dimensional vector \(G_{ij}\) to denote the genetics factors, where \(G_{ij} = (G_{ij1},...,G_{ijm})^{T}\). Also, we use \(p\)-dimensional vector \(E_{ij}\) to denote the environment factors, where \(E_{ij} = (E_{ij1},...,E_{ijp})^{T}\). \(X_{ij} = (1, T_{ij})^{T}\), where \(T_{ij}^{T}\) is a vector of time effects . \(Z_{ij}\) is a \(h \times 1\) covariate associated with random effects and \(\alpha_{i}\) is a \(h\times 1\) vector of random effects. At the beginning, the interaction effects is modeled as the product of genomics features and environment factors with 4 different levels. After representing the environment factors as three dummy variables, the identification of the gene by environment interaction needs to be performed as group level. Combing the genetics factors, environment factors and their interactions that associated with the longitudinal phenotype, we have the following mixed-effects model: $$Y_{ij} = X_{ij}^{T}\gamma_{0}+E_{ij}^{T}\gamma_{1}+G_{ij}^{T}\gamma_{2}+(G_{ij}\bigotimes E_{ij})^{T}\gamma_{3}+Z_{ij}^{T}\alpha_{i}+\epsilon_{ij}.$$ where \(\gamma_{1}\),\(\gamma_{2}\),\(\gamma_{3}\) are \(p\),\(m\) and \(mp\) dimensional vectors that represent the coefficients of the environment effects, the genetics effects and interactions effects, respectively. Accommodating the Kronecker product of the \(m\) - dimensional vector \(G_{ij}\) and the \(p\)-dimensional vector \(E_{ij}\), the interactions between genetics and environment factors can be expressed as a \(mp\)-dimensional vector, denoted as the following form: $$G_{ij}\bigotimes E_{ij} = [E_{ij1}E_{ij1},E_{ij2}E_{ij2},...,E_{ij1}E_{ijp},E_{ij2}E_{ij1},...,E_{ijm}E_{ijp}]^{T}.$$ When \(Z_{ij}^{T} = (1,j)\) and \(\alpha_{i} = (\alpha_{i1},\alpha_{i2})^{T}\), the model becomes random intercept and slope model. When \(Z_{ij}^{T} = 1\) and \(\alpha_{i} = \alpha_{i1}\), the model becomes random intercept model.
BayesQVGEL
data(data)
dim(y)
dim(g)
dim(e)
dim(w)
print(k)
print(C)
print(coeff)
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